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The Higher Integrability of Commutators of Calderón...
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Research Article The Higher Integrability of Commutators of Calderón-Zygmund Singular Integral Operators on Differential Forms Jinling Niu and Yuming Xing Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China Correspondence should be addressed to Yuming Xing; [email protected] Received 18 August 2018; Accepted 30 September 2018; Published 6 November 2018 Academic Editor: Gestur ´ Olafsson Copyright © 2018 Jinling Niu and Yuming Xing. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this article, the higher integrability of commutators of Calder´ on-Zygmund singular integral operators on differential forms is derived. Also, the higher order Poincar´ e-type inequalities for the commutators acting on the solutions of Dirac-harmonic equations are obtained. Finally, some applications of the main results are demonstrated by examples. 1. Introduction Differential forms as the key tools are widely used in many fields including quasiconformal analysis, nonlinear elasticity, and differential geometry, due to their advantage of being coordinate system independent; see [1–5]. e integrability of various operators and the upper bound estimates for the norms of operators are very important and core topics while studying the -theory of differential forms and investigating the qualitative and quantitative properties of the solutions of partial differential equations. In last few decades, a lot of related research has been done and many results on estimates for the -norms of various operators applied to the differential form in terms of the -norms of have been obtained; see [6–12]. In this paper, we define the commutators of Calder´ on-Zygmund singular integral operators [, Ω ] on differential forms and give the strong type estimates for the commutators, which allows one to estimate ‖[, Ω ]‖ in terms of the norm ‖‖ with > 1. Meanwhile, we make a contribution to the estimates of the commutators ‖[, Ω ]‖ in terms of the norm ‖‖ , where >, that is the higher integrability of commutators of Calder´ on-Zygmund singular integral operators on differential forms. en, we will establish the higher order Poincar´ e-type inequalities for the commutators applied to the solutions of Dirac- harmonic equations. e higher integrability and higher order inequalities in this paper can be used to study the regu- larity properties of the related operators. More results on the problem of higher order estimates and their applications in potential theory, quantum mechanics, and partial differential equations can be found in [13–17]. is paper is organised as follows. Section 2 contains, in addition to definitions and other preliminary material, the main lemmas. In Section 3, eorems 12 and 13 show the local higher integrability of commutators of Calder´ on- Zygmund singular integral operators on differential forms. Based on these local results, the global higher integrability is presented in eorems 14 and 15 by the well-known covering lemma. Especially, when the differential forms satisfy the Dirac-harmonic equations (in [18]), we establish the higher order Poincar´ e-type inequalities for the commutators in Section 4. e local higher order Poincar´ e-type inequalities are given in eorems 16 and 17 and the global higher order Poincar´ e-type inequalities are obtained in eorems 18 and 19. Finally, we demonstrate some applications of the main results by examples in Section 5. ese results obtained in this paper will provide a further insight into the -theory and regularity theory of the related operators and differential forms. 2. Preliminary Before specifying the main results precisely, we introduce some notation. Let ⊂ R be a bounded domain, ≥2, and be the balls with the same center, and () = () for any real number >1. We denote by () the Hindawi Journal of Function Spaces Volume 2018, Article ID 1949747, 9 pages https://doi.org/10.1155/2018/1949747
Transcript of The Higher Integrability of Commutators of Calderón...
Research ArticleThe Higher Integrability of Commutators of Calderoacuten-ZygmundSingular Integral Operators on Differential Forms
Jinling Niu and Yuming Xing
Department of Mathematics Harbin Institute of Technology Harbin 150001 China
Correspondence should be addressed to Yuming Xing xyuminghiteducn
Received 18 August 2018 Accepted 30 September 2018 Published 6 November 2018
Academic Editor Gestur Olafsson
Copyright copy 2018 JinlingNiu and Yuming XingThis is an open access article distributed under the Creative Commons AttributionLicensewhichpermits unrestricteduse distribution and reproduction in anymedium provided the original work is properly cited
In this article the higher integrability of commutators of Calderon-Zygmund singular integral operators on differential forms isderived Also the higher order Poincare-type inequalities for the commutators acting on the solutions of Dirac-harmonic equationsare obtained Finally some applications of the main results are demonstrated by examples
1 Introduction
Differential forms as the key tools are widely used in manyfields including quasiconformal analysis nonlinear elasticityand differential geometry due to their advantage of beingcoordinate system independent see [1ndash5] The integrabilityof various operators and the upper bound estimates for thenorms of operators are very important and core topics whilestudying the 119871119901-theory of differential forms and investigatingthe qualitative and quantitative properties of the solutionsof partial differential equations In last few decades a lotof related research has been done and many results onestimates for the 119871119901-norms of various operators applied to thedifferential form 119906 in terms of the 119871119901-norms of 119906 have beenobtained see [6ndash12] In this paper we define the commutatorsof Calderon-Zygmund singular integral operators [119887 119879Ω] ondifferential forms and give the strong type estimates forthe commutators which allows one to estimate [119887 119879Ω]119906119901in terms of the norm 119906119901 with 119901 gt 1 Meanwhile wemake a contribution to the estimates of the commutators[119887 119879Ω]119906119904 in terms of the norm 119906119901 where 119904 gt 119901 that is thehigher integrability of commutators of Calderon-Zygmundsingular integral operators on differential forms Then wewill establish the higher order Poincare-type inequalitiesfor the commutators applied to the solutions of Dirac-harmonic equations The higher integrability and higherorder inequalities in this paper can be used to study the regu-larity properties of the related operators More results on the
problem of higher order estimates and their applications inpotential theory quantum mechanics and partial differentialequations can be found in [13ndash17]
This paper is organised as follows Section 2 containsin addition to definitions and other preliminary materialthe main lemmas In Section 3 Theorems 12 and 13 showthe local higher integrability of commutators of Calderon-Zygmund singular integral operators on differential formsBased on these local results the global higher integrability ispresented inTheorems 14 and 15 by the well-known coveringlemma Especially when the differential forms satisfy theDirac-harmonic equations (in [18]) we establish the higherorder Poincare-type inequalities for the commutators inSection 4 The local higher order Poincare-type inequalitiesare given in Theorems 16 and 17 and the global higher orderPoincare-type inequalities are obtained in Theorems 18 and19 Finally we demonstrate some applications of the mainresults by examples in Section 5 These results obtained inthis paper will provide a further insight into the 119871119901-theoryand regularity theory of the related operators and differentialforms
2 Preliminary
Before specifying the main results precisely we introducesome notation Let 119872 sub R119899 be a bounded domain 119899 ge 2119861 and 120590119861 be the balls with the same center and 119889119894119886119898(120590119861) =120590119889119894119886119898(119861) for any real number 120590 gt 1 We denote by 120583(119872) the
HindawiJournal of Function SpacesVolume 2018 Article ID 1949747 9 pageshttpsdoiorg10115520181949747
2 Journal of Function Spaces
119899-dimensional Lebesgue measure of a set119872 sube R119899 The set ofall 119897-forms denoted by Λ119897 = Λ119897(R119899) is a 119897-vector spanned byexterior products 119890119868 = 1198901198941 and 1198901198942 sdot sdot sdot and 119890119894119897 for all ordered 119897-tuples119868 = (1198941 1198942 119894119897) 1 le 1198941 lt sdot sdot sdot lt 119894119897 le 119899 The 119897-form 119906(119909) =sum119868 119906119868(119909)119889119909119868 is called a differential 119897-form if its coefficients119906119868 are differentiable functions We shall denote by1198631015840(119872Λ119897)the differential 119897-forms space on119872 and denote by 119871119901(119872Λ119897)the space of differential 119897-forms on 119872 satisfying int
119872|119906119868|119901 ltinfin and with norm 119906119901119872 = (int119872(sum119868 |119906119868(119909)|2)1199012119889119909)1119901 In
particular we know that a 0-form is a function A differential119897-form 119906 isin 1198631015840(119872Λ119897) is called a closed form if 119889119906 = 0in 119872 Similarly a differential (119897 + 1)-form V isin 1198631015840(119872Λ119897+1)is called a coclosed form if 119889⋆V = 0 From the Poincarelemma 119889119889119906 = 0 we know that 119889119906 is a closed form Theoperator ⋆ Λ119897(R119899) 997888rarr Λ119899minus119897(R119899) is the Hodge-star operatorwhich is an isometric isomorphism and the linear operator119889 1198631015840(119872Λ119897) 997888rarr 1198631015840(119872Λ119897+1) 0 le 119897 le 119899 minus 1 is calledthe exterior differential The Hodge codifferential operator119889⋆ 1198631015840(119872Λ119897+1) 997888rarr 1198631015840(119872Λ119897) the formal adjoint of 119889 isdefined by 119889⋆ = (minus1)119899119897+1⋆119889⋆ see [19] for more introductionThe following Dirac-harmonic equation for differential formswas initially introduced by S Ding and B Liu in [18]
119889⋆119860 (119909 119863119906) = 0 (1)
for almost every 119909 isin 119872 where 119863 = 119889 + 119889⋆ is the Diracoperator 119872 sub R119899 is a domain and 119860 119872 times Λ119897(R119899) 997888rarrΛ119897(R119899) satisfies the following conditions
1003816100381610038161003816119860 (119909 120585)1003816100381610038161003816 le 119886 10038161003816100381610038161205851003816100381610038161003816119901minus1 ⟨119860 (119909 120585) 120585⟩ ge 10038161003816100381610038161205851003816100381610038161003816119901 (2)
for almost every 119909 isin 119872 and all 120585 isin Λ119897(R119899) Here 119886 gt 0 is aconstant and 1 lt 119901 lt infin is a fixed exponent associated with(1)
We should point out that the Dirac-harmonic equationis a kind of general equation which includes many existingharmonic equations as special cases such as the 119860-harmonicequation see [18 20 21] for more information
The Calderon-Zygmund singular integral operator 119879Ω ondifferential forms is defined by
119879Ω119906 (119909) = sum119868
(intR119899
Ω(119909 minus 119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899 119906119868 (119910) 119889119910)119889119909119868 (3)
whereΩ(119909) is defined on 119878119899minus1 has mean 0 and is sufficientlysmooth
If 119887 isin 119861119872119874(R119899) the commutator of Calderon-Zygmundsingular integral operator on differential forms is of the form
[119887 119879Ω] 119906 (119909) = 119887 (119909) 119879Ω119906 (119909) minus 119879Ω (119887119906) (119909)= sum119868
(intR119899(119887 (119909) minus 119887 (119910)) Ω (119909 minus 119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899 119906119868 (119910) 119889119910)119889119909119868
(4)
When taking 119906(119909) as a 0-form the commutator in (4)reduces to the corresponding operator on function space asfollows
[119887 119879Ω] 119891 (119909) = intR119899(119887 (119909) minus 119887 (119910)) Ω (119909 minus 119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899 119891 (119910) 119889119910 (5)
For the degenerated operator and the related applicationsin partial differential equations see [22ndash24]
In order to prove our conclusions we need several lem-mas The following 119871119901-boundedness result for commutator[119887 119879Ω] on function spaces was proved in [25]
Lemma 1 Let 119908 be a weight satisfying 119860119901 condition for allcubes 119876 if there is a constant 119862 such that
( 1|119876| int119876119908 (119909) 119889119909)( 1|119876| int119876119908 (119909)1minus1199011015840 119889119909)119901minus1 le 119862
lt infin(6)
where 1 lt 119901 lt infin and 1119901 + 11199011015840 = 1 119879Ω is any Calderon-Zygmund singular integral operator Then given any function119887 isin 119861119872119874(R119899) [119887 119879Ω] satisfies the following inequality
(intR119899
1003816100381610038161003816[119887 119879Ω] 119891 (119909)1003816100381610038161003816119901 119908 (119909) 119889119909)1119901
le 119862 119887119861119872119874 (intR119899
1003816100381610038161003816119891 (119909)1003816100381610038161003816119901 119908 (119909) 119889119909)1119901 (7)
The following lemma was given by S Ding and B Liu in[18]
Lemma2 Let119906 isin 1198631015840(119872Λ119897) be a solution of119863119894119903119886119888-harmonicequation (1) in119872 120590 gt 1 and 0 lt 119904 119905 lt infin are constants Thenthere exists a constant 119862 independent of 119906 such that
119906119904119861 le 119862120583 (119861)(119905minus119904)119904119905 119906119905120590119861 (8)
for all cubes or balls 119861 with 120590119861 sub 119872
In [26] T Iwaniec and A Lutoborski gave the followingthree lemmas which will be used repeatedly in this paper
Lemma 3 Let 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 1 lt119901 lt infin be a differential form and 119879 119871119901(119872Λ119897) 997888rarr1198821119901(119872Λ119897minus1) be the homotopy operator Then we have thefollowing decomposition
119906 = 119889 (119879119906) + 119879 (119889119906) (9)
and the inequality
119879119906119901119872 le 119862 (119899 119901) 120583 (119872)119889119894119886119898 (119872) 119906119901119872 (10)
holds for any bounded domain 119872 sub R119899 where 119862(119899 119901) is aconstant independent of 119906
From [26] the 119897-form 119906119872 isin 119871119901(119872Λ119897) is defined by 119906119872 =(1120583(119872))int119872119906(119910)119889119910 if 119897 = 0 and 119906119872 = 119889(119879119906) if 119897 = 1 1198991 le 119901 le infin
Journal of Function Spaces 3
Lemma 4 Let 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 Then 119906119872 isin119871119901(119872Λ119897) and10038171003817100381710038171199061198721003817100381710038171003817119901119872 le 119862 (119899 119901) 120583 (119872) 119906119901119872 (11)
where 119862(119899 119901) is a constant independent of 119906 and 1 lt 119901 lt infin
Lemma 5 Let 119906 isin 1198631015840(119872Λ119897) and 119889119906 isin 119871119901(119872Λ119897+1) Then119906 minus 119906119872 is in 119871119899119901(119899minus119901)(119872Λ119897) and(int119872
1003816100381610038161003816119906 minus 1199061198721003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 119862119901 (119899) (int119872|119889119906|119901 119889119909)1119901
(12)
where 119897 = 1 2 119899 1 lt 119901 lt 119899 and 119862119901(119899) is a constantindependent of 119906
The following lemma appears in [27]
Lemma 6 Let 120595 defined on [0 +infin) be a strictly increasingconvex function 120595(0) = 0 and119872 sub R119899 be a domain Assumethat 119906(119909) isin 1198631015840(119872Λ119897) satisfies 120595(119896(|119906| + |119906119872|)) isin 1198711(119872 120583)for any real number 119896 gt 0 and 120583(119909 isin 119872 |120583 minus 120583119872| gt 0) gt 0where 120583 is a Radon measure defined by 119889120583(119909) = 120596(119909)119889119909 witha weight 120596(119909) then for any 119886 gt 0 we obtain
int119872120595(12 1003816100381610038161003816119906 minus 1199061198721003816100381610038161003816) 119889120583 le 1198621 int119872120595 (119886 |119906|) 119889120583le 1198622 int
119872120595 (2119886 1003816100381610038161003816119906 minus 1199061198721003816100381610038161003816) 119889120583
(13)
where 1198621 gt 0 and 1198622 gt 0 are constantsChoose 120595(119905) = 119905119901 119901 gt 1 and119908(119909) = 1 and let119872 be a ball119861 in Lemma 6 we find that the norms 119906119901119861 and 119906 minus 119906119861119901119861
are comparable that is
1003817100381710038171003817119906 minus 1199061198611003817100381710038171003817119901119861 le 1198621 119906119901119861 le 1198622 1003817100381710038171003817119906 minus 1199061198611003817100381710038171003817119901119861 (14)
for any ball 119861 with |119909 isin 119861 |119906 minus 119906119861| gt 0| gt 0The covering lemma below belongs to [19]
Lemma 7 Each domain119872 has a modified Whitney cover ofcubesV = 119876119894 such that
cup119894119876119894 = 119872sum119876119894isinV
120594radic54119876119894 le 119873120594Ω (15)
and some 119873 gt 1 and if 119876119894 cap 119876119895 = 0 then there exists acube 119877 (this cube need not be a member ofV) in119876119894 cap 119876119895 suchthat 119876119894 cup 119876119895 sub 119873119877 Moreover if119872 is 120575-John then there is adistinguished cube 1198760 isinV which can be connected with everycube 119876 isin V by a chain of cubes 1198760 1198761 119876119896 = 119876 from V
and such that 119876 sub 120588119876119894 119894 = 0 1 2 119896 for some 120588 = 120588(119899 120575)
3 Higher Integrability
In this section we show the higher integrability of commu-tators of Calderon-Zygmund singular integral operators ondifferential forms We first concentrate on the local higherintegrability of the commutators We need the followinglemma appearing in [28]
Lemma 8 Let
119860 (119891) (119909) = lim120598997888rarr0int|119909minus119910|gt120598
119896 (119909 minus 119910) 119891 (119910) 119889119910 (16)
where 119909 119910 are points in n-dimensional Euclidean spaceR119899 and119896(119909) is a homogeneous function of degree minus119899 with mean valuezero on |119909| = 1 and let 119861(119891) = 119887(119909)119891(119909) If 119896 and 119887 aresufficiently smooth and 119887 is bounded then
1003817100381710038171003817100381710038171003817100381710038171003817120597120597119909119895 (119860119861 minus 119861119860) 119891
1003817100381710038171003817100381710038171003817100381710038171003817119901 le 11988810038171003817100381710038171198911003817100381710038171003817119901 (17)
for 1 lt 119901 lt infin where 119888 is a constant independent of 119891Lemma 9 Let 119906 isin 119871119901(R119899 Λ119897) 119897 = 1 2 119899 1 lt 119901 lt infin119879Ω be the Calderon-Zygmund singular integral operator ondifferential forms and 119887 be sufficiently smooth and boundedThen there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901R119899 le 119862 119906119901R119899 (18)
Proof For any differential 119897-form 119906(119909) by the definition ofcommutator of Calderon-Zygmund singular integral opera-tor on differential forms we have
[119887 119879Ω] (119906) (119909) = 119887 (119909) 119879Ω119906 (119909) minus 119879Ω (119887119906) (119909)= sum119868
(intR119899
Ω(119909 minus 119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899 (119887 (119909) minus 119887 (119910)) 119906119868 (119910) 119889119910)119889119909119868(19)
Let 119870(119909 minus 119910) = Ω(119909 minus 119910)|119909 minus 119910|119899 and then according to thedefinition of the exterior differential operator we obtain
119889 [119887 119879Ω] (119906) (119909)= sum119868
119899sum119896=1
120597 intR119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896 119889119909119896
and 119889119909119868(20)
Using the elementary inequality | sum119873119894=1 119905119894|119904 le 119873119904minus1sum119873119894=1 |119905119894|119904for constants 119873 119904 gt 0 we deduce1003817100381710038171003817119889 [119887 119879Ω] 119906 (119909)1003817100381710038171003817119901119901R119899le int
R119899(sum119868
119899sum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
10038161003816100381610038161003816100381610038161003816100381610038162)1199012
119889119909
4 Journal of Function Spaces
le 1198621 intR119899sum119868
119899sum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003816100381610038161003816100381610038161003816100381610038161003816119901 119889119909
= 1198621sum119868
119899sum119896=1
intR119899
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003816100381610038161003816100381610038161003816100381610038161003816119901 119889119909
= 1198621sum119868
119899sum119896=1
1003817100381710038171003817100381710038171003817100381710038171003817120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003817100381710038171003817100381710038171003817100381710038171003817119901
119901R119899
(21)
By Lemma 8 we obtain
1003817100381710038171003817100381710038171003817100381710038171003817120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003817100381710038171003817100381710038171003817100381710038171003817119901R119899le 1198622 1003817100381710038171003817119906119868 (119909)1003817100381710038171003817119901R119899
(22)
Substituting (22) into (21) and applying the fundamentalinequality 119886119904 + 119887119904 le (119886 + 119887)119904 119904 ge 1 yields that1003817100381710038171003817119889 [119887 119879Ω] 119906 (119909)1003817100381710038171003817119901119901R119899 le 1198623sum
119868
1003817100381710038171003817119906119868 (119909)1003817100381710038171003817119901119901R119899= 1198623sum
119868
intR119899
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816119901 119889119909= 1198623 int
R119899sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816119901 119889119909le 1198624 int
R119899(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816)119901 119889119909
(23)
Using the inequality | sum119873119894=1 119905119894|119904 le 119873119904minus1sum119873119894=1 |119905119894|119904 again weeasily have
(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816)2 le 1198625sum
119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162 (24)
that is
(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816) le (1198625sum119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162)12 (25)
Substituting (25) into (23) gives
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901119901R119899 le 1198626 intR119899(sum119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162)1199012 119889119909
= 1198626 intR119899|119906 (119909)|119901 119889119909 = 1198626 119906119901119901R119899
(26)
This completes the proof of Lemma 9
In Lemma 9 let 119906 = 119906 in bounded domain119872 and 119906 = 0inR119899 119872 we can easily obtain the following lemma
Lemma 10 Let 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt infin119879Ω be the Calderon-Zygmund singular integral operator on
differential forms and 119887 be sufficiently smooth and boundedThen there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901119872 le 119862 119906119901119872 (27)
where119872 sub R119899 is any bounded domain
Using Lemma 1 and the analogous method developed inLemma 9 we have the following estimate for [119887 119879Ω]Lemma 11 Let 119906 isin 119871119901(119872Λ119897) 119897 = 0 1 119899 1 lt 119901 lt infinand 119879Ω be a Calderon-Zygmund singular integral operator ondifferential forms Then given any function 119887 isin 119861119872119874(R119899)[119887 119879Ω] satisfies the strong (119901 119901) inequality
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901119872 le 119862 119887119861119872119874 119906119901119872 (28)
for any bounded domain 119872 sub R119899 where 119862 is a constantindependent of 119906
We now present the local higher integrability of commu-tators of Calderon-Zygmund singular integral operators ondifferential forms
Theorem 12 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt 119899then [119887 119879Ω]119906 isin 119871119904119897119900119888(119872Λ119897) for any 0 lt 119904 lt 119899119901(119899 minus 119901)Moreover there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 119862120583 (119861)1119904+1119899minus1119901 119906119901120590119861 (29)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof For any ball 119861 with 120590119861 sub 119872 for some 120590 gt 1 in thecase that the measure 120583119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0 = 0 We have obviously that [119887 119879Ω]119906 = ([119887 119879Ω]119906)119861 almosteverywhere in 119861 which implies [119887 119879]119906 is a closed form and asolution of the Dirac-harmonic equation (1) in119872 ApplyingLemmas 4 and 2 to [119887 119879Ω]119906 with any 119904 119901 gt 0 we have1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 = 1003817100381710038171003817([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861
le 1198621120583 (119861) 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861le 1198622120583 (119861)1+1119904minus1119901 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901120590119861
(30)
for all balls B with 120590119861 sub 119872 for some 120590 gt 1 We note that 119887 isbounded thus 119887 isin 119861119872119874(R119899) By Lemma 11 it follows that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 1198622120583 (119861)1+1119904minus1119901 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901120590119861le 1198623120583 (119861)1+1119904minus1119901 119887119861119872119874 119906119901120590119861le 1198624120583 (119861)1+1119904minus1119901 119906119901120590119861
(31)
Journal of Function Spaces 5
On the other hand if the measure 120583119909 isin 119861 |[119887 119879Ω]119906 minus([119887 119879Ω]119906)119861| gt 0 gt 0 Applying Lemma 5 to [119887 119879Ω]119906 we get(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198625 (119899 119901) (int119861
1003816100381610038161003816119889 [119887 119879Ω] 1199061003816100381610038161003816119901 119889119909)1119901 (32)
Then by Lemma 10 it follows that
(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198626 (119899 119901) (int119861|119906|119901 119889119909)1119901
(33)
Noticing that the measure |119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0| gt 0 we could use Lemma 6 Taking 120595(119905) = 119905119899119901(119899minus119901) inLemma 6 we have
(int119861|V|119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198627 (int119861
1003816100381610038161003816V minus V1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901(34)
for any differential form V and any ball B with |119909 isin 119861 |V minusV119861| gt 0| gt 0 Replacing V by [119887 119879Ω]119906 in (34) we obtain
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901 le 1198628 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
(35)
Combining (33) and (35) yields that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198629 (int119861|119906|119901 119889119909)1119901
(36)
In view of the monotonic property of the 119871119901-space with 0 lt119904 lt 119899119901(119899 minus 119901) we have( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)
1119904
le ( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
(37)
Substituting (36) and (37) gives
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904
le 11986210120583 (119861)1119904minus1119901+1119899 (int119861|119906|119901 119889119909)1119901
(38)
Therefore inequality (29) follows in both cases which indi-cates that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879]119906 isin 119871119904119897119900119888(119872Λ119897) Wehave completed the proof of Theorem 12
Moreover inequality (38) can be written as the followingintegral average inequality
( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904
le 11986211120583 (119861)1119899 ( 1120583 (119861) int120590119861 |119906|119901 119889119909)1119901
(39)
Clearly with 119901 close to 119899 the integral exponent 119904 on thelefthand side could bemuch larger than the integral exponent119901 on the right hand side since the condition 0 lt 119904 lt 119899119901(119899 minus119901) Hence the higher integrability of operator [119887 119879Ω] for thecase that 1 lt 119901 lt 119899 is obtained
Next we consider the higher integrability of [119887 119879Ω] forthe case 119901 ge 119899Theorem 13 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904119897119900119888(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 119862120583 (119861)1119904+1119899minus1119901 119906119901120590119861 (40)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof It is immediate by the same method developed in theproof of Theorem 12 that (40) holds for 120583119909 isin 119861 |[119887 119879Ω]119906 minus([119887 119879Ω]119906)119861| gt 0 = 0 Let us consider the remain case that120583119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt 0 gt 0 Select 120572 =max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Then it is easy to checkthat
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (41)
since 119899 minus 119901 le 0 Notice that 1 lt 119902 = 120572119899119901(119899 + 120572119901) lt 119899Using Lemma 5 to [119887 119879Ω]119906 and combining Lemma 10 and themonotonic property of the 119871119901-space we have(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198621 (int119861
1003816100381610038161003816119889 [119887 119879Ω] 1199061003816100381610038161003816119902 119889119909)1119902
le 1198622 (int119861|119906|119902 119889119909)1119902
le 1198623120583 (119861)1119902minus1119901 (int119861|119906|119901 119889119909)1119901
(42)
Notice that the measure |119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0| gt 0 Therefore we can select 120595(119905) = 119905119899119902(119899minus119902) in Lemma 6Then for any differential form 119908 we have
(int119861|119908|119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198624 (int119861
1003816100381610038161003816119908 minus 1199081198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 (43)
6 Journal of Function Spaces
Replacing 119908 by [119887 119879Ω]119906 in (43) we get
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 le 1198625 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
(44)
Taking into account the fact that 119899119902(119899 minus 119902) = 120572119901 gt 119904 we seefrom the monotonic property of the 119871119901-space (42) and (44)that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904 le 1198626120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902le 1198627120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198628120583 (119861)1119902minus1119901+1119904minus1120572119901 (int119861|119906|119901 119889119909)1119901
= 1198628120583 (119861)1119899+1119904minus1119901 (int119861|119906|119901 119889119909)1119901
(45)
which means that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879Ω]119906 isin119871119904119897119900119888(119872Λ119897)We have completed the proof ofTheorem 13
Now we are ready to assert the global higher integrabilityof the commutator of Calderon-Zygmund singular integraloperator on differential forms
Theorem 14 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 0 lt 119904 lt 119899119901(119899 minus 119901) Moreoverthere exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (46)
for any bounded domain119872 sub R119899Proof Notice that 1119904+1119899minus1119901 gt 0 since 0 lt 119904 lt 119899119901(119899minus119901)By Lemma 7 andTheorem 12 we have
[119887 119879] 119906119904119872 le sum119861isin][119887 119879] 119906119904119861
le sum119861isin](1198621120583 (119861)1119904+1119899minus1119901 119906119901120590119861)
le sum119861isin](1198621120583 (119872)1119904+1119899minus1119901 119906119901120590119861)
le 1198622120583 (119872)1119904+1119899minus1119901119873119906119901119872le 1198623 119906119901119872
(47)
which finishes the proof of Theorem 14
Using the similar method as we did in Theorem 14 andcombining with Theorem 13 we can deduce the followingglobal result for the case 119901 ge 119899Theorem 15 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant C independent of u such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (48)
for any bounded domain119872 sub R1198994 Higher Order Poincareacute-Type Inequalities
In this section we shall state the higher order Poincare-typeinequalities for commutator of Calderon-Zygmund singularintegral operator acting on the solutions of the Dirac-harmonic equations
Theorem 16 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (49)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof For any 119906 isin 119871119901119897119900119888(119872Λ119897) using the 119871119901-decompositionformula (9) to [119887 119879Ω]119906 we obtain
[119887 119879Ω] 119906 = 119889119879 ([119887 119879Ω] 119906) + 119879119889 ([119887 119879Ω] 119906) (50)
Noticing that 119889119879([119887119879Ω]119906) = ([119887 119879Ω]119906)119861 for any differentialform 119906 by (50) (10) and Lemma 10 we get1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901(119899minus119901)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198621 (119899 119901) 120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198622 (119899 119901) 120583 (119861)1+1119899 119906119899119901(119899minus119901)119861
(51)
Since 119906 is a solution of the Dirac-harmonic equation (1) byLemma 2 we have
119906119899119901(119899minus119901)119861 le 120583 (119861)minus1119899 119906119901120590119861 (52)
where120590 gt 1 is a constant Substituting (52) into (51) gives that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861le 1198623 (119899 119901) 120583 (119861) 119906119901120590119861 (53)
Journal of Function Spaces 7
Moreover for any 0 lt 119904 lt 119899119901(119899minus119901) applying the monotonicproperties of 119871119901-space it follows that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119901+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861 (54)
Combined with (53) the above inequality becomes
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198624 (119899 119901) 120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (55)
which finishes the proof of Theorem 16
Next we will prove the higher order Poincare-typeinequality still holds for the case that 119901 ge 119899Theorem 17 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of the Dirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then for any119904 gt 0 there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (56)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof Select 120572 = max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Thenwe easily obtain
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (57)
that is 119902 lt 119901 since 119899 minus 119901 le 0 We can also find 1 lt 119902 =120572119899119901(119899 + 120572119901) lt 119899 Using the same technique as in the proofof Theorem 16 we have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902(119899minus119902)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198621120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198622120583 (119861)1+1119899 119906119899119902(119899minus119902)119861le 1198623120583 (119861)1+1119899 120583 (119861)1119902minus1119899minus1119901 119906119901120590119861le 1198624120583 (119861)1+1119902minus1119901 119906119901120590119861
(58)
Noticing that 119899119902(119899 minus 119902) = 120572119901 gt 119904 and combining themonotonic property of the 119871119901-space we obtain1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119902+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902119899minus119902119861 (59)
Combining (58) and (59) we finally have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198625120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (60)
The proof of Theorem 17 is completed
Based onTheorems 16 and 17 we can obtain the followingglobal higher order Poincare-type inequalities for the com-mutator [119887 119879Ω] using the analogous method developed inTheorem 14
Theorem 18 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (61)
for any bounded domain119872 sub R119899Theorem 19 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then forany 119904 gt 0 there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (62)
for any bounded domain119872 sub R1198995 Applications
In this section we demonstrate some applications of ourmainresults established in the previous sections
Example 20 Let 119903 gt 0 and 119896 gt 0 be any constants and1198721 =(1199091 1199092 1199093) 11990921 + 11990922 + 11990923 le 1199032 sub R3 Consider the 1-form119906 (1199091 1199092 1199093) = 1199091119896 + 11990921 + 11990922 + 11990923 1198891199091
+ 1199092119896 + 11990921 + 11990922 + 11990923 1198891199092+ 1199093119896 + 11990921 + 11990922 + 11990923 1198891199093
(63)
defined in119872 such that1198721 sub 119872 It is easy to check that 119889119906 =0 Hence 119906 is a solution of the Dirac-harmonic equation (1)
8 Journal of Function Spaces
for any operators 119860 satisfying (2) Also it can be calculatedthat
|119906| = (( 1199091119896 + 11990921 + 11990922 + 11990923)2 + ( 1199092119896 + 11990921 + 11990922 + 11990923)
2
+ ( 1199093119896 + 11990921 + 11990922 + 11990923)2)12
= ( 11990921 + 11990922 + 11990923(119896 + 11990921 + 11990922 + 11990923)2)12 lt 1
(64)
for any 119909 = (1199091 1199092 1199093) isin 1198721 Thus
1199061199011198721 = (int1198721|119906|119901 119889119909)1119901
= (int1198721
100381610038161003816100381610038161003816100381610038161003816100381611990921 + 11990922 + 119909231003816100381610038161003816119896 + 11990921 + 11990922 + 1199092310038161003816100381610038162
10038161003816100381610038161003816100381610038161003816100381610038161199012 119889119909)
1119901
le (int1198721
1119889119909)1119901 = (41205873 1199033)1119901
(65)
Applying Theorem 14 for 1 lt 119901 lt 3 we have [119887 119879Ω]119906 isin119871119904(119872Λ119897) for any 0 lt 119904 lt 3119901(3 minus 119901) Analogously we have[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0when119901 ge 3 byTheorem 15In the meantime we have the higher estimate for [119887 119879Ω]119906that is
1003817100381710038171003817[119887 119879Ω] 11990610038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (66)
In addition we can evaluate the following integrals byTheorem 18 andTheorem 19 that
10038171003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721100381710038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (67)
We should notice that the above example can be extendedto the case of R119899 as follows
Example 21 We can check that the 1-form defined inR119899
119906 (1199091 119909119899) = 119899sum119894=1
119909119894119896 + 11990921 + sdot sdot sdot + 1199092119899 119889119909119894 119896 gt 0 (68)
is a solution of the Dirac-harmonic equation (1) for anyoperators 119860 satisfying (2) Hence Theorems 14 15 18 and 19are also applicable to 119906(1199091 119909119899)Remark It is worth pointing out that the global resultsobtained in this paper can be extended to larger classes ofdomains such as 119871119901-averaging domains and 119871120593(120583)-averagingdomains see [1 27] Also the techniques developed inthis paper provide an effective method to study the higherintegrability of bilinear commutators of singular integrals ondifferential forms which are defined in [25] We leave thestatements and proofs to the interested readers
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The work was supported by the National Natural ScienceFoundation of China (Grant no 11601105)
References
[1] R P Agarwal S Ding andCNolder Inequalities forDifferentialForms Springer New York NY USA 2009
[2] Y Xing and C Wu ldquoGlobal weighted inequalities for operatorsand harmonic forms on manifoldsrdquo Journal of MathematicalAnalysis and Applications vol 294 no 1 pp 294ndash309 2004
[3] J B Perot and C J Zusi ldquoDifferential forms for scientists andengineersrdquo Journal of Computational Physics vol 257 no partB pp 1373ndash1393 2014
[4] C Scott ldquoL119901-theory of differential forms onmanifolds theory ofdifferential forms on manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 6 pp 2075ndash2096 1995
[5] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004
[6] S Ding and B Liu ldquoA singular integral of the compositeoperatorrdquo Applied Mathematics Letters vol 22 no 8 pp 1271ndash1275 2009
[7] S Ding and B Liu ldquoGlobal estimates for singular integrals ofthe composite operatorrdquo Illinois Journal of Mathematics vol 53no 4 pp 1173ndash1185 2009
[8] S Ding and B Liu ldquoGlobal integrability of the Jacobian of acomposite mappingrdquo Journal of Inequalities and ApplicationsArt ID 89134 9 pages 2006
[9] H Bi and S Ding ldquoSome strong (pq)-type inequalities forthe homotopy operatorrdquo Computers and Mathematics withApplications An International Journal vol 62 no 4 pp 1780ndash1789 2011
[10] H Bi ldquoWeighted inequalities for potential operators on differ-ential formsrdquo Journal of Inequalities and Applications Art ID713625 13 pages 2010
[11] Y Ling and B Gejun ldquoSome local Poincare inequalities for thecomposition of the sharp maximal operator and the Greenrsquosoperatorrdquo Computers and Mathematics with Applications vol63 no 3 pp 720ndash727 2012
[12] S Ding ldquoIntegral estimates for the Laplace-Beltrami andGreenrsquos operators applied to differential forms on manifoldsrdquoJournal for Analysis and its Applications vol 22 no 4 pp 939ndash957 2003
[13] G Lu and R L Wheeden ldquoHigh order representation formulasand embedding theorems on stratified groups and generaliza-tionsrdquo Studia Mathematica vol 142 no 2 pp 101ndash133 2000
[14] G Lu ldquoPolynomials higher order Sobolev extension theo-rems and interpolation inequalities on weighted Folland-STEin
Journal of Function Spaces 9
spaces on stratified groupsrdquo Acta Mathematica Sinica vol 16no 3 pp 405ndash444 2000
[15] E W Stredulinsky ldquoHigher integrability from reverse Holderinequalitiesrdquo Indiana University Mathematics Journal vol 29no 3 pp 407ndash413 1980
[16] F W Gehring ldquoThe L119901-integrability of the partial derivativesof a quasiconformal mappingrdquo Acta Mathematica vol 130 pp265ndash277 1973
[17] W S Cohn G Lu and S Lu ldquoHigher order Poincare inequal-ities associated with linear operators on stratified groups andapplicationsrdquoMathematische Zeitschrift vol 244 no 2 pp 309ndash335 2003
[18] S Ding and B Liu ldquoDirac-harmonic equations for differentialformsrdquo Nonlinear Analysis Theory Methods and ApplicationsAn International Multidisciplinary Journal vol 122 pp 43ndash572015
[19] C A Nolder ldquoHardy-Littlewood theorems for 119860-harmonictensorsrdquo Illinois Journal of Mathematics vol 43 no 4 pp 613ndash632 1999
[20] A Carbonaro A McIntosh and A J Morris ldquoLocal Hardyspaces of differential forms on Riemannian manifoldsrdquo TheJournal of Geometric Analysis vol 23 no 1 pp 106ndash169 2013
[21] K Kodaira ldquoHarmonic fields in Riemannian manifolds (gener-alized potential theory)rdquo Annals of Mathematics Second Seriesvol 50 pp 587ndash665 1949
[22] R R Coifman R Rochberg and G Weiss ldquoFactorizationtheorems for Hardy spaces in several variablesrdquo Annals ofMathematics vol 103 no 3 pp 611ndash635 1976
[23] L Chaffee and D Cruz-Uribe ldquoNecessary conditions for theboundedness of linear and bilinear commutators on Banachfunction spacesrdquo Mathematical Inequalities and Applicationsvol 21 no 1 pp 1ndash16 2018
[24] C Segovia and J L Torrea ldquoWeighted inequalities for com-mutators of fractional and singular integralsrdquo PublicacionsMatematiques vol 35 no 1 pp 209ndash235 1991
[25] C Perez ldquoSharp estimates for commutators of singular integralsvia iterations of the Hardy-Littlewood maximal functionrdquoJournal of Fourier Analysis and Applications vol 3 no 6 pp743ndash756 1997
[26] T Iwaniec and A Lutoborski ldquoIntegral estimates for nullLagrangiansrdquo Archive for Rational Mechanics and Analysis vol125 no 1 pp 25ndash79 1993
[27] S Ding ldquoL120593(120583)-averaging domains and the quasi-hyperbolicmetricrdquo Computers and Mathematics with Applications AnInternational Journal vol 47 no 10-11 pp 1611ndash1618 2004
[28] A P Calderon ldquoCommutators of singular integral operatorsrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 53 pp 1092ndash1099 1965
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2 Journal of Function Spaces
119899-dimensional Lebesgue measure of a set119872 sube R119899 The set ofall 119897-forms denoted by Λ119897 = Λ119897(R119899) is a 119897-vector spanned byexterior products 119890119868 = 1198901198941 and 1198901198942 sdot sdot sdot and 119890119894119897 for all ordered 119897-tuples119868 = (1198941 1198942 119894119897) 1 le 1198941 lt sdot sdot sdot lt 119894119897 le 119899 The 119897-form 119906(119909) =sum119868 119906119868(119909)119889119909119868 is called a differential 119897-form if its coefficients119906119868 are differentiable functions We shall denote by1198631015840(119872Λ119897)the differential 119897-forms space on119872 and denote by 119871119901(119872Λ119897)the space of differential 119897-forms on 119872 satisfying int
119872|119906119868|119901 ltinfin and with norm 119906119901119872 = (int119872(sum119868 |119906119868(119909)|2)1199012119889119909)1119901 In
particular we know that a 0-form is a function A differential119897-form 119906 isin 1198631015840(119872Λ119897) is called a closed form if 119889119906 = 0in 119872 Similarly a differential (119897 + 1)-form V isin 1198631015840(119872Λ119897+1)is called a coclosed form if 119889⋆V = 0 From the Poincarelemma 119889119889119906 = 0 we know that 119889119906 is a closed form Theoperator ⋆ Λ119897(R119899) 997888rarr Λ119899minus119897(R119899) is the Hodge-star operatorwhich is an isometric isomorphism and the linear operator119889 1198631015840(119872Λ119897) 997888rarr 1198631015840(119872Λ119897+1) 0 le 119897 le 119899 minus 1 is calledthe exterior differential The Hodge codifferential operator119889⋆ 1198631015840(119872Λ119897+1) 997888rarr 1198631015840(119872Λ119897) the formal adjoint of 119889 isdefined by 119889⋆ = (minus1)119899119897+1⋆119889⋆ see [19] for more introductionThe following Dirac-harmonic equation for differential formswas initially introduced by S Ding and B Liu in [18]
119889⋆119860 (119909 119863119906) = 0 (1)
for almost every 119909 isin 119872 where 119863 = 119889 + 119889⋆ is the Diracoperator 119872 sub R119899 is a domain and 119860 119872 times Λ119897(R119899) 997888rarrΛ119897(R119899) satisfies the following conditions
1003816100381610038161003816119860 (119909 120585)1003816100381610038161003816 le 119886 10038161003816100381610038161205851003816100381610038161003816119901minus1 ⟨119860 (119909 120585) 120585⟩ ge 10038161003816100381610038161205851003816100381610038161003816119901 (2)
for almost every 119909 isin 119872 and all 120585 isin Λ119897(R119899) Here 119886 gt 0 is aconstant and 1 lt 119901 lt infin is a fixed exponent associated with(1)
We should point out that the Dirac-harmonic equationis a kind of general equation which includes many existingharmonic equations as special cases such as the 119860-harmonicequation see [18 20 21] for more information
The Calderon-Zygmund singular integral operator 119879Ω ondifferential forms is defined by
119879Ω119906 (119909) = sum119868
(intR119899
Ω(119909 minus 119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899 119906119868 (119910) 119889119910)119889119909119868 (3)
whereΩ(119909) is defined on 119878119899minus1 has mean 0 and is sufficientlysmooth
If 119887 isin 119861119872119874(R119899) the commutator of Calderon-Zygmundsingular integral operator on differential forms is of the form
[119887 119879Ω] 119906 (119909) = 119887 (119909) 119879Ω119906 (119909) minus 119879Ω (119887119906) (119909)= sum119868
(intR119899(119887 (119909) minus 119887 (119910)) Ω (119909 minus 119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899 119906119868 (119910) 119889119910)119889119909119868
(4)
When taking 119906(119909) as a 0-form the commutator in (4)reduces to the corresponding operator on function space asfollows
[119887 119879Ω] 119891 (119909) = intR119899(119887 (119909) minus 119887 (119910)) Ω (119909 minus 119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899 119891 (119910) 119889119910 (5)
For the degenerated operator and the related applicationsin partial differential equations see [22ndash24]
In order to prove our conclusions we need several lem-mas The following 119871119901-boundedness result for commutator[119887 119879Ω] on function spaces was proved in [25]
Lemma 1 Let 119908 be a weight satisfying 119860119901 condition for allcubes 119876 if there is a constant 119862 such that
( 1|119876| int119876119908 (119909) 119889119909)( 1|119876| int119876119908 (119909)1minus1199011015840 119889119909)119901minus1 le 119862
lt infin(6)
where 1 lt 119901 lt infin and 1119901 + 11199011015840 = 1 119879Ω is any Calderon-Zygmund singular integral operator Then given any function119887 isin 119861119872119874(R119899) [119887 119879Ω] satisfies the following inequality
(intR119899
1003816100381610038161003816[119887 119879Ω] 119891 (119909)1003816100381610038161003816119901 119908 (119909) 119889119909)1119901
le 119862 119887119861119872119874 (intR119899
1003816100381610038161003816119891 (119909)1003816100381610038161003816119901 119908 (119909) 119889119909)1119901 (7)
The following lemma was given by S Ding and B Liu in[18]
Lemma2 Let119906 isin 1198631015840(119872Λ119897) be a solution of119863119894119903119886119888-harmonicequation (1) in119872 120590 gt 1 and 0 lt 119904 119905 lt infin are constants Thenthere exists a constant 119862 independent of 119906 such that
119906119904119861 le 119862120583 (119861)(119905minus119904)119904119905 119906119905120590119861 (8)
for all cubes or balls 119861 with 120590119861 sub 119872
In [26] T Iwaniec and A Lutoborski gave the followingthree lemmas which will be used repeatedly in this paper
Lemma 3 Let 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 1 lt119901 lt infin be a differential form and 119879 119871119901(119872Λ119897) 997888rarr1198821119901(119872Λ119897minus1) be the homotopy operator Then we have thefollowing decomposition
119906 = 119889 (119879119906) + 119879 (119889119906) (9)
and the inequality
119879119906119901119872 le 119862 (119899 119901) 120583 (119872)119889119894119886119898 (119872) 119906119901119872 (10)
holds for any bounded domain 119872 sub R119899 where 119862(119899 119901) is aconstant independent of 119906
From [26] the 119897-form 119906119872 isin 119871119901(119872Λ119897) is defined by 119906119872 =(1120583(119872))int119872119906(119910)119889119910 if 119897 = 0 and 119906119872 = 119889(119879119906) if 119897 = 1 1198991 le 119901 le infin
Journal of Function Spaces 3
Lemma 4 Let 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 Then 119906119872 isin119871119901(119872Λ119897) and10038171003817100381710038171199061198721003817100381710038171003817119901119872 le 119862 (119899 119901) 120583 (119872) 119906119901119872 (11)
where 119862(119899 119901) is a constant independent of 119906 and 1 lt 119901 lt infin
Lemma 5 Let 119906 isin 1198631015840(119872Λ119897) and 119889119906 isin 119871119901(119872Λ119897+1) Then119906 minus 119906119872 is in 119871119899119901(119899minus119901)(119872Λ119897) and(int119872
1003816100381610038161003816119906 minus 1199061198721003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 119862119901 (119899) (int119872|119889119906|119901 119889119909)1119901
(12)
where 119897 = 1 2 119899 1 lt 119901 lt 119899 and 119862119901(119899) is a constantindependent of 119906
The following lemma appears in [27]
Lemma 6 Let 120595 defined on [0 +infin) be a strictly increasingconvex function 120595(0) = 0 and119872 sub R119899 be a domain Assumethat 119906(119909) isin 1198631015840(119872Λ119897) satisfies 120595(119896(|119906| + |119906119872|)) isin 1198711(119872 120583)for any real number 119896 gt 0 and 120583(119909 isin 119872 |120583 minus 120583119872| gt 0) gt 0where 120583 is a Radon measure defined by 119889120583(119909) = 120596(119909)119889119909 witha weight 120596(119909) then for any 119886 gt 0 we obtain
int119872120595(12 1003816100381610038161003816119906 minus 1199061198721003816100381610038161003816) 119889120583 le 1198621 int119872120595 (119886 |119906|) 119889120583le 1198622 int
119872120595 (2119886 1003816100381610038161003816119906 minus 1199061198721003816100381610038161003816) 119889120583
(13)
where 1198621 gt 0 and 1198622 gt 0 are constantsChoose 120595(119905) = 119905119901 119901 gt 1 and119908(119909) = 1 and let119872 be a ball119861 in Lemma 6 we find that the norms 119906119901119861 and 119906 minus 119906119861119901119861
are comparable that is
1003817100381710038171003817119906 minus 1199061198611003817100381710038171003817119901119861 le 1198621 119906119901119861 le 1198622 1003817100381710038171003817119906 minus 1199061198611003817100381710038171003817119901119861 (14)
for any ball 119861 with |119909 isin 119861 |119906 minus 119906119861| gt 0| gt 0The covering lemma below belongs to [19]
Lemma 7 Each domain119872 has a modified Whitney cover ofcubesV = 119876119894 such that
cup119894119876119894 = 119872sum119876119894isinV
120594radic54119876119894 le 119873120594Ω (15)
and some 119873 gt 1 and if 119876119894 cap 119876119895 = 0 then there exists acube 119877 (this cube need not be a member ofV) in119876119894 cap 119876119895 suchthat 119876119894 cup 119876119895 sub 119873119877 Moreover if119872 is 120575-John then there is adistinguished cube 1198760 isinV which can be connected with everycube 119876 isin V by a chain of cubes 1198760 1198761 119876119896 = 119876 from V
and such that 119876 sub 120588119876119894 119894 = 0 1 2 119896 for some 120588 = 120588(119899 120575)
3 Higher Integrability
In this section we show the higher integrability of commu-tators of Calderon-Zygmund singular integral operators ondifferential forms We first concentrate on the local higherintegrability of the commutators We need the followinglemma appearing in [28]
Lemma 8 Let
119860 (119891) (119909) = lim120598997888rarr0int|119909minus119910|gt120598
119896 (119909 minus 119910) 119891 (119910) 119889119910 (16)
where 119909 119910 are points in n-dimensional Euclidean spaceR119899 and119896(119909) is a homogeneous function of degree minus119899 with mean valuezero on |119909| = 1 and let 119861(119891) = 119887(119909)119891(119909) If 119896 and 119887 aresufficiently smooth and 119887 is bounded then
1003817100381710038171003817100381710038171003817100381710038171003817120597120597119909119895 (119860119861 minus 119861119860) 119891
1003817100381710038171003817100381710038171003817100381710038171003817119901 le 11988810038171003817100381710038171198911003817100381710038171003817119901 (17)
for 1 lt 119901 lt infin where 119888 is a constant independent of 119891Lemma 9 Let 119906 isin 119871119901(R119899 Λ119897) 119897 = 1 2 119899 1 lt 119901 lt infin119879Ω be the Calderon-Zygmund singular integral operator ondifferential forms and 119887 be sufficiently smooth and boundedThen there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901R119899 le 119862 119906119901R119899 (18)
Proof For any differential 119897-form 119906(119909) by the definition ofcommutator of Calderon-Zygmund singular integral opera-tor on differential forms we have
[119887 119879Ω] (119906) (119909) = 119887 (119909) 119879Ω119906 (119909) minus 119879Ω (119887119906) (119909)= sum119868
(intR119899
Ω(119909 minus 119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899 (119887 (119909) minus 119887 (119910)) 119906119868 (119910) 119889119910)119889119909119868(19)
Let 119870(119909 minus 119910) = Ω(119909 minus 119910)|119909 minus 119910|119899 and then according to thedefinition of the exterior differential operator we obtain
119889 [119887 119879Ω] (119906) (119909)= sum119868
119899sum119896=1
120597 intR119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896 119889119909119896
and 119889119909119868(20)
Using the elementary inequality | sum119873119894=1 119905119894|119904 le 119873119904minus1sum119873119894=1 |119905119894|119904for constants 119873 119904 gt 0 we deduce1003817100381710038171003817119889 [119887 119879Ω] 119906 (119909)1003817100381710038171003817119901119901R119899le int
R119899(sum119868
119899sum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
10038161003816100381610038161003816100381610038161003816100381610038162)1199012
119889119909
4 Journal of Function Spaces
le 1198621 intR119899sum119868
119899sum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003816100381610038161003816100381610038161003816100381610038161003816119901 119889119909
= 1198621sum119868
119899sum119896=1
intR119899
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003816100381610038161003816100381610038161003816100381610038161003816119901 119889119909
= 1198621sum119868
119899sum119896=1
1003817100381710038171003817100381710038171003817100381710038171003817120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003817100381710038171003817100381710038171003817100381710038171003817119901
119901R119899
(21)
By Lemma 8 we obtain
1003817100381710038171003817100381710038171003817100381710038171003817120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003817100381710038171003817100381710038171003817100381710038171003817119901R119899le 1198622 1003817100381710038171003817119906119868 (119909)1003817100381710038171003817119901R119899
(22)
Substituting (22) into (21) and applying the fundamentalinequality 119886119904 + 119887119904 le (119886 + 119887)119904 119904 ge 1 yields that1003817100381710038171003817119889 [119887 119879Ω] 119906 (119909)1003817100381710038171003817119901119901R119899 le 1198623sum
119868
1003817100381710038171003817119906119868 (119909)1003817100381710038171003817119901119901R119899= 1198623sum
119868
intR119899
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816119901 119889119909= 1198623 int
R119899sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816119901 119889119909le 1198624 int
R119899(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816)119901 119889119909
(23)
Using the inequality | sum119873119894=1 119905119894|119904 le 119873119904minus1sum119873119894=1 |119905119894|119904 again weeasily have
(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816)2 le 1198625sum
119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162 (24)
that is
(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816) le (1198625sum119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162)12 (25)
Substituting (25) into (23) gives
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901119901R119899 le 1198626 intR119899(sum119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162)1199012 119889119909
= 1198626 intR119899|119906 (119909)|119901 119889119909 = 1198626 119906119901119901R119899
(26)
This completes the proof of Lemma 9
In Lemma 9 let 119906 = 119906 in bounded domain119872 and 119906 = 0inR119899 119872 we can easily obtain the following lemma
Lemma 10 Let 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt infin119879Ω be the Calderon-Zygmund singular integral operator on
differential forms and 119887 be sufficiently smooth and boundedThen there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901119872 le 119862 119906119901119872 (27)
where119872 sub R119899 is any bounded domain
Using Lemma 1 and the analogous method developed inLemma 9 we have the following estimate for [119887 119879Ω]Lemma 11 Let 119906 isin 119871119901(119872Λ119897) 119897 = 0 1 119899 1 lt 119901 lt infinand 119879Ω be a Calderon-Zygmund singular integral operator ondifferential forms Then given any function 119887 isin 119861119872119874(R119899)[119887 119879Ω] satisfies the strong (119901 119901) inequality
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901119872 le 119862 119887119861119872119874 119906119901119872 (28)
for any bounded domain 119872 sub R119899 where 119862 is a constantindependent of 119906
We now present the local higher integrability of commu-tators of Calderon-Zygmund singular integral operators ondifferential forms
Theorem 12 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt 119899then [119887 119879Ω]119906 isin 119871119904119897119900119888(119872Λ119897) for any 0 lt 119904 lt 119899119901(119899 minus 119901)Moreover there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 119862120583 (119861)1119904+1119899minus1119901 119906119901120590119861 (29)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof For any ball 119861 with 120590119861 sub 119872 for some 120590 gt 1 in thecase that the measure 120583119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0 = 0 We have obviously that [119887 119879Ω]119906 = ([119887 119879Ω]119906)119861 almosteverywhere in 119861 which implies [119887 119879]119906 is a closed form and asolution of the Dirac-harmonic equation (1) in119872 ApplyingLemmas 4 and 2 to [119887 119879Ω]119906 with any 119904 119901 gt 0 we have1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 = 1003817100381710038171003817([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861
le 1198621120583 (119861) 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861le 1198622120583 (119861)1+1119904minus1119901 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901120590119861
(30)
for all balls B with 120590119861 sub 119872 for some 120590 gt 1 We note that 119887 isbounded thus 119887 isin 119861119872119874(R119899) By Lemma 11 it follows that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 1198622120583 (119861)1+1119904minus1119901 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901120590119861le 1198623120583 (119861)1+1119904minus1119901 119887119861119872119874 119906119901120590119861le 1198624120583 (119861)1+1119904minus1119901 119906119901120590119861
(31)
Journal of Function Spaces 5
On the other hand if the measure 120583119909 isin 119861 |[119887 119879Ω]119906 minus([119887 119879Ω]119906)119861| gt 0 gt 0 Applying Lemma 5 to [119887 119879Ω]119906 we get(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198625 (119899 119901) (int119861
1003816100381610038161003816119889 [119887 119879Ω] 1199061003816100381610038161003816119901 119889119909)1119901 (32)
Then by Lemma 10 it follows that
(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198626 (119899 119901) (int119861|119906|119901 119889119909)1119901
(33)
Noticing that the measure |119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0| gt 0 we could use Lemma 6 Taking 120595(119905) = 119905119899119901(119899minus119901) inLemma 6 we have
(int119861|V|119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198627 (int119861
1003816100381610038161003816V minus V1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901(34)
for any differential form V and any ball B with |119909 isin 119861 |V minusV119861| gt 0| gt 0 Replacing V by [119887 119879Ω]119906 in (34) we obtain
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901 le 1198628 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
(35)
Combining (33) and (35) yields that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198629 (int119861|119906|119901 119889119909)1119901
(36)
In view of the monotonic property of the 119871119901-space with 0 lt119904 lt 119899119901(119899 minus 119901) we have( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)
1119904
le ( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
(37)
Substituting (36) and (37) gives
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904
le 11986210120583 (119861)1119904minus1119901+1119899 (int119861|119906|119901 119889119909)1119901
(38)
Therefore inequality (29) follows in both cases which indi-cates that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879]119906 isin 119871119904119897119900119888(119872Λ119897) Wehave completed the proof of Theorem 12
Moreover inequality (38) can be written as the followingintegral average inequality
( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904
le 11986211120583 (119861)1119899 ( 1120583 (119861) int120590119861 |119906|119901 119889119909)1119901
(39)
Clearly with 119901 close to 119899 the integral exponent 119904 on thelefthand side could bemuch larger than the integral exponent119901 on the right hand side since the condition 0 lt 119904 lt 119899119901(119899 minus119901) Hence the higher integrability of operator [119887 119879Ω] for thecase that 1 lt 119901 lt 119899 is obtained
Next we consider the higher integrability of [119887 119879Ω] forthe case 119901 ge 119899Theorem 13 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904119897119900119888(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 119862120583 (119861)1119904+1119899minus1119901 119906119901120590119861 (40)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof It is immediate by the same method developed in theproof of Theorem 12 that (40) holds for 120583119909 isin 119861 |[119887 119879Ω]119906 minus([119887 119879Ω]119906)119861| gt 0 = 0 Let us consider the remain case that120583119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt 0 gt 0 Select 120572 =max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Then it is easy to checkthat
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (41)
since 119899 minus 119901 le 0 Notice that 1 lt 119902 = 120572119899119901(119899 + 120572119901) lt 119899Using Lemma 5 to [119887 119879Ω]119906 and combining Lemma 10 and themonotonic property of the 119871119901-space we have(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198621 (int119861
1003816100381610038161003816119889 [119887 119879Ω] 1199061003816100381610038161003816119902 119889119909)1119902
le 1198622 (int119861|119906|119902 119889119909)1119902
le 1198623120583 (119861)1119902minus1119901 (int119861|119906|119901 119889119909)1119901
(42)
Notice that the measure |119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0| gt 0 Therefore we can select 120595(119905) = 119905119899119902(119899minus119902) in Lemma 6Then for any differential form 119908 we have
(int119861|119908|119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198624 (int119861
1003816100381610038161003816119908 minus 1199081198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 (43)
6 Journal of Function Spaces
Replacing 119908 by [119887 119879Ω]119906 in (43) we get
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 le 1198625 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
(44)
Taking into account the fact that 119899119902(119899 minus 119902) = 120572119901 gt 119904 we seefrom the monotonic property of the 119871119901-space (42) and (44)that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904 le 1198626120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902le 1198627120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198628120583 (119861)1119902minus1119901+1119904minus1120572119901 (int119861|119906|119901 119889119909)1119901
= 1198628120583 (119861)1119899+1119904minus1119901 (int119861|119906|119901 119889119909)1119901
(45)
which means that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879Ω]119906 isin119871119904119897119900119888(119872Λ119897)We have completed the proof ofTheorem 13
Now we are ready to assert the global higher integrabilityof the commutator of Calderon-Zygmund singular integraloperator on differential forms
Theorem 14 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 0 lt 119904 lt 119899119901(119899 minus 119901) Moreoverthere exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (46)
for any bounded domain119872 sub R119899Proof Notice that 1119904+1119899minus1119901 gt 0 since 0 lt 119904 lt 119899119901(119899minus119901)By Lemma 7 andTheorem 12 we have
[119887 119879] 119906119904119872 le sum119861isin][119887 119879] 119906119904119861
le sum119861isin](1198621120583 (119861)1119904+1119899minus1119901 119906119901120590119861)
le sum119861isin](1198621120583 (119872)1119904+1119899minus1119901 119906119901120590119861)
le 1198622120583 (119872)1119904+1119899minus1119901119873119906119901119872le 1198623 119906119901119872
(47)
which finishes the proof of Theorem 14
Using the similar method as we did in Theorem 14 andcombining with Theorem 13 we can deduce the followingglobal result for the case 119901 ge 119899Theorem 15 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant C independent of u such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (48)
for any bounded domain119872 sub R1198994 Higher Order Poincareacute-Type Inequalities
In this section we shall state the higher order Poincare-typeinequalities for commutator of Calderon-Zygmund singularintegral operator acting on the solutions of the Dirac-harmonic equations
Theorem 16 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (49)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof For any 119906 isin 119871119901119897119900119888(119872Λ119897) using the 119871119901-decompositionformula (9) to [119887 119879Ω]119906 we obtain
[119887 119879Ω] 119906 = 119889119879 ([119887 119879Ω] 119906) + 119879119889 ([119887 119879Ω] 119906) (50)
Noticing that 119889119879([119887119879Ω]119906) = ([119887 119879Ω]119906)119861 for any differentialform 119906 by (50) (10) and Lemma 10 we get1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901(119899minus119901)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198621 (119899 119901) 120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198622 (119899 119901) 120583 (119861)1+1119899 119906119899119901(119899minus119901)119861
(51)
Since 119906 is a solution of the Dirac-harmonic equation (1) byLemma 2 we have
119906119899119901(119899minus119901)119861 le 120583 (119861)minus1119899 119906119901120590119861 (52)
where120590 gt 1 is a constant Substituting (52) into (51) gives that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861le 1198623 (119899 119901) 120583 (119861) 119906119901120590119861 (53)
Journal of Function Spaces 7
Moreover for any 0 lt 119904 lt 119899119901(119899minus119901) applying the monotonicproperties of 119871119901-space it follows that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119901+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861 (54)
Combined with (53) the above inequality becomes
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198624 (119899 119901) 120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (55)
which finishes the proof of Theorem 16
Next we will prove the higher order Poincare-typeinequality still holds for the case that 119901 ge 119899Theorem 17 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of the Dirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then for any119904 gt 0 there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (56)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof Select 120572 = max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Thenwe easily obtain
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (57)
that is 119902 lt 119901 since 119899 minus 119901 le 0 We can also find 1 lt 119902 =120572119899119901(119899 + 120572119901) lt 119899 Using the same technique as in the proofof Theorem 16 we have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902(119899minus119902)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198621120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198622120583 (119861)1+1119899 119906119899119902(119899minus119902)119861le 1198623120583 (119861)1+1119899 120583 (119861)1119902minus1119899minus1119901 119906119901120590119861le 1198624120583 (119861)1+1119902minus1119901 119906119901120590119861
(58)
Noticing that 119899119902(119899 minus 119902) = 120572119901 gt 119904 and combining themonotonic property of the 119871119901-space we obtain1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119902+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902119899minus119902119861 (59)
Combining (58) and (59) we finally have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198625120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (60)
The proof of Theorem 17 is completed
Based onTheorems 16 and 17 we can obtain the followingglobal higher order Poincare-type inequalities for the com-mutator [119887 119879Ω] using the analogous method developed inTheorem 14
Theorem 18 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (61)
for any bounded domain119872 sub R119899Theorem 19 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then forany 119904 gt 0 there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (62)
for any bounded domain119872 sub R1198995 Applications
In this section we demonstrate some applications of ourmainresults established in the previous sections
Example 20 Let 119903 gt 0 and 119896 gt 0 be any constants and1198721 =(1199091 1199092 1199093) 11990921 + 11990922 + 11990923 le 1199032 sub R3 Consider the 1-form119906 (1199091 1199092 1199093) = 1199091119896 + 11990921 + 11990922 + 11990923 1198891199091
+ 1199092119896 + 11990921 + 11990922 + 11990923 1198891199092+ 1199093119896 + 11990921 + 11990922 + 11990923 1198891199093
(63)
defined in119872 such that1198721 sub 119872 It is easy to check that 119889119906 =0 Hence 119906 is a solution of the Dirac-harmonic equation (1)
8 Journal of Function Spaces
for any operators 119860 satisfying (2) Also it can be calculatedthat
|119906| = (( 1199091119896 + 11990921 + 11990922 + 11990923)2 + ( 1199092119896 + 11990921 + 11990922 + 11990923)
2
+ ( 1199093119896 + 11990921 + 11990922 + 11990923)2)12
= ( 11990921 + 11990922 + 11990923(119896 + 11990921 + 11990922 + 11990923)2)12 lt 1
(64)
for any 119909 = (1199091 1199092 1199093) isin 1198721 Thus
1199061199011198721 = (int1198721|119906|119901 119889119909)1119901
= (int1198721
100381610038161003816100381610038161003816100381610038161003816100381611990921 + 11990922 + 119909231003816100381610038161003816119896 + 11990921 + 11990922 + 1199092310038161003816100381610038162
10038161003816100381610038161003816100381610038161003816100381610038161199012 119889119909)
1119901
le (int1198721
1119889119909)1119901 = (41205873 1199033)1119901
(65)
Applying Theorem 14 for 1 lt 119901 lt 3 we have [119887 119879Ω]119906 isin119871119904(119872Λ119897) for any 0 lt 119904 lt 3119901(3 minus 119901) Analogously we have[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0when119901 ge 3 byTheorem 15In the meantime we have the higher estimate for [119887 119879Ω]119906that is
1003817100381710038171003817[119887 119879Ω] 11990610038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (66)
In addition we can evaluate the following integrals byTheorem 18 andTheorem 19 that
10038171003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721100381710038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (67)
We should notice that the above example can be extendedto the case of R119899 as follows
Example 21 We can check that the 1-form defined inR119899
119906 (1199091 119909119899) = 119899sum119894=1
119909119894119896 + 11990921 + sdot sdot sdot + 1199092119899 119889119909119894 119896 gt 0 (68)
is a solution of the Dirac-harmonic equation (1) for anyoperators 119860 satisfying (2) Hence Theorems 14 15 18 and 19are also applicable to 119906(1199091 119909119899)Remark It is worth pointing out that the global resultsobtained in this paper can be extended to larger classes ofdomains such as 119871119901-averaging domains and 119871120593(120583)-averagingdomains see [1 27] Also the techniques developed inthis paper provide an effective method to study the higherintegrability of bilinear commutators of singular integrals ondifferential forms which are defined in [25] We leave thestatements and proofs to the interested readers
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The work was supported by the National Natural ScienceFoundation of China (Grant no 11601105)
References
[1] R P Agarwal S Ding andCNolder Inequalities forDifferentialForms Springer New York NY USA 2009
[2] Y Xing and C Wu ldquoGlobal weighted inequalities for operatorsand harmonic forms on manifoldsrdquo Journal of MathematicalAnalysis and Applications vol 294 no 1 pp 294ndash309 2004
[3] J B Perot and C J Zusi ldquoDifferential forms for scientists andengineersrdquo Journal of Computational Physics vol 257 no partB pp 1373ndash1393 2014
[4] C Scott ldquoL119901-theory of differential forms onmanifolds theory ofdifferential forms on manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 6 pp 2075ndash2096 1995
[5] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004
[6] S Ding and B Liu ldquoA singular integral of the compositeoperatorrdquo Applied Mathematics Letters vol 22 no 8 pp 1271ndash1275 2009
[7] S Ding and B Liu ldquoGlobal estimates for singular integrals ofthe composite operatorrdquo Illinois Journal of Mathematics vol 53no 4 pp 1173ndash1185 2009
[8] S Ding and B Liu ldquoGlobal integrability of the Jacobian of acomposite mappingrdquo Journal of Inequalities and ApplicationsArt ID 89134 9 pages 2006
[9] H Bi and S Ding ldquoSome strong (pq)-type inequalities forthe homotopy operatorrdquo Computers and Mathematics withApplications An International Journal vol 62 no 4 pp 1780ndash1789 2011
[10] H Bi ldquoWeighted inequalities for potential operators on differ-ential formsrdquo Journal of Inequalities and Applications Art ID713625 13 pages 2010
[11] Y Ling and B Gejun ldquoSome local Poincare inequalities for thecomposition of the sharp maximal operator and the Greenrsquosoperatorrdquo Computers and Mathematics with Applications vol63 no 3 pp 720ndash727 2012
[12] S Ding ldquoIntegral estimates for the Laplace-Beltrami andGreenrsquos operators applied to differential forms on manifoldsrdquoJournal for Analysis and its Applications vol 22 no 4 pp 939ndash957 2003
[13] G Lu and R L Wheeden ldquoHigh order representation formulasand embedding theorems on stratified groups and generaliza-tionsrdquo Studia Mathematica vol 142 no 2 pp 101ndash133 2000
[14] G Lu ldquoPolynomials higher order Sobolev extension theo-rems and interpolation inequalities on weighted Folland-STEin
Journal of Function Spaces 9
spaces on stratified groupsrdquo Acta Mathematica Sinica vol 16no 3 pp 405ndash444 2000
[15] E W Stredulinsky ldquoHigher integrability from reverse Holderinequalitiesrdquo Indiana University Mathematics Journal vol 29no 3 pp 407ndash413 1980
[16] F W Gehring ldquoThe L119901-integrability of the partial derivativesof a quasiconformal mappingrdquo Acta Mathematica vol 130 pp265ndash277 1973
[17] W S Cohn G Lu and S Lu ldquoHigher order Poincare inequal-ities associated with linear operators on stratified groups andapplicationsrdquoMathematische Zeitschrift vol 244 no 2 pp 309ndash335 2003
[18] S Ding and B Liu ldquoDirac-harmonic equations for differentialformsrdquo Nonlinear Analysis Theory Methods and ApplicationsAn International Multidisciplinary Journal vol 122 pp 43ndash572015
[19] C A Nolder ldquoHardy-Littlewood theorems for 119860-harmonictensorsrdquo Illinois Journal of Mathematics vol 43 no 4 pp 613ndash632 1999
[20] A Carbonaro A McIntosh and A J Morris ldquoLocal Hardyspaces of differential forms on Riemannian manifoldsrdquo TheJournal of Geometric Analysis vol 23 no 1 pp 106ndash169 2013
[21] K Kodaira ldquoHarmonic fields in Riemannian manifolds (gener-alized potential theory)rdquo Annals of Mathematics Second Seriesvol 50 pp 587ndash665 1949
[22] R R Coifman R Rochberg and G Weiss ldquoFactorizationtheorems for Hardy spaces in several variablesrdquo Annals ofMathematics vol 103 no 3 pp 611ndash635 1976
[23] L Chaffee and D Cruz-Uribe ldquoNecessary conditions for theboundedness of linear and bilinear commutators on Banachfunction spacesrdquo Mathematical Inequalities and Applicationsvol 21 no 1 pp 1ndash16 2018
[24] C Segovia and J L Torrea ldquoWeighted inequalities for com-mutators of fractional and singular integralsrdquo PublicacionsMatematiques vol 35 no 1 pp 209ndash235 1991
[25] C Perez ldquoSharp estimates for commutators of singular integralsvia iterations of the Hardy-Littlewood maximal functionrdquoJournal of Fourier Analysis and Applications vol 3 no 6 pp743ndash756 1997
[26] T Iwaniec and A Lutoborski ldquoIntegral estimates for nullLagrangiansrdquo Archive for Rational Mechanics and Analysis vol125 no 1 pp 25ndash79 1993
[27] S Ding ldquoL120593(120583)-averaging domains and the quasi-hyperbolicmetricrdquo Computers and Mathematics with Applications AnInternational Journal vol 47 no 10-11 pp 1611ndash1618 2004
[28] A P Calderon ldquoCommutators of singular integral operatorsrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 53 pp 1092ndash1099 1965
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Journal of Function Spaces 3
Lemma 4 Let 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 Then 119906119872 isin119871119901(119872Λ119897) and10038171003817100381710038171199061198721003817100381710038171003817119901119872 le 119862 (119899 119901) 120583 (119872) 119906119901119872 (11)
where 119862(119899 119901) is a constant independent of 119906 and 1 lt 119901 lt infin
Lemma 5 Let 119906 isin 1198631015840(119872Λ119897) and 119889119906 isin 119871119901(119872Λ119897+1) Then119906 minus 119906119872 is in 119871119899119901(119899minus119901)(119872Λ119897) and(int119872
1003816100381610038161003816119906 minus 1199061198721003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 119862119901 (119899) (int119872|119889119906|119901 119889119909)1119901
(12)
where 119897 = 1 2 119899 1 lt 119901 lt 119899 and 119862119901(119899) is a constantindependent of 119906
The following lemma appears in [27]
Lemma 6 Let 120595 defined on [0 +infin) be a strictly increasingconvex function 120595(0) = 0 and119872 sub R119899 be a domain Assumethat 119906(119909) isin 1198631015840(119872Λ119897) satisfies 120595(119896(|119906| + |119906119872|)) isin 1198711(119872 120583)for any real number 119896 gt 0 and 120583(119909 isin 119872 |120583 minus 120583119872| gt 0) gt 0where 120583 is a Radon measure defined by 119889120583(119909) = 120596(119909)119889119909 witha weight 120596(119909) then for any 119886 gt 0 we obtain
int119872120595(12 1003816100381610038161003816119906 minus 1199061198721003816100381610038161003816) 119889120583 le 1198621 int119872120595 (119886 |119906|) 119889120583le 1198622 int
119872120595 (2119886 1003816100381610038161003816119906 minus 1199061198721003816100381610038161003816) 119889120583
(13)
where 1198621 gt 0 and 1198622 gt 0 are constantsChoose 120595(119905) = 119905119901 119901 gt 1 and119908(119909) = 1 and let119872 be a ball119861 in Lemma 6 we find that the norms 119906119901119861 and 119906 minus 119906119861119901119861
are comparable that is
1003817100381710038171003817119906 minus 1199061198611003817100381710038171003817119901119861 le 1198621 119906119901119861 le 1198622 1003817100381710038171003817119906 minus 1199061198611003817100381710038171003817119901119861 (14)
for any ball 119861 with |119909 isin 119861 |119906 minus 119906119861| gt 0| gt 0The covering lemma below belongs to [19]
Lemma 7 Each domain119872 has a modified Whitney cover ofcubesV = 119876119894 such that
cup119894119876119894 = 119872sum119876119894isinV
120594radic54119876119894 le 119873120594Ω (15)
and some 119873 gt 1 and if 119876119894 cap 119876119895 = 0 then there exists acube 119877 (this cube need not be a member ofV) in119876119894 cap 119876119895 suchthat 119876119894 cup 119876119895 sub 119873119877 Moreover if119872 is 120575-John then there is adistinguished cube 1198760 isinV which can be connected with everycube 119876 isin V by a chain of cubes 1198760 1198761 119876119896 = 119876 from V
and such that 119876 sub 120588119876119894 119894 = 0 1 2 119896 for some 120588 = 120588(119899 120575)
3 Higher Integrability
In this section we show the higher integrability of commu-tators of Calderon-Zygmund singular integral operators ondifferential forms We first concentrate on the local higherintegrability of the commutators We need the followinglemma appearing in [28]
Lemma 8 Let
119860 (119891) (119909) = lim120598997888rarr0int|119909minus119910|gt120598
119896 (119909 minus 119910) 119891 (119910) 119889119910 (16)
where 119909 119910 are points in n-dimensional Euclidean spaceR119899 and119896(119909) is a homogeneous function of degree minus119899 with mean valuezero on |119909| = 1 and let 119861(119891) = 119887(119909)119891(119909) If 119896 and 119887 aresufficiently smooth and 119887 is bounded then
1003817100381710038171003817100381710038171003817100381710038171003817120597120597119909119895 (119860119861 minus 119861119860) 119891
1003817100381710038171003817100381710038171003817100381710038171003817119901 le 11988810038171003817100381710038171198911003817100381710038171003817119901 (17)
for 1 lt 119901 lt infin where 119888 is a constant independent of 119891Lemma 9 Let 119906 isin 119871119901(R119899 Λ119897) 119897 = 1 2 119899 1 lt 119901 lt infin119879Ω be the Calderon-Zygmund singular integral operator ondifferential forms and 119887 be sufficiently smooth and boundedThen there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901R119899 le 119862 119906119901R119899 (18)
Proof For any differential 119897-form 119906(119909) by the definition ofcommutator of Calderon-Zygmund singular integral opera-tor on differential forms we have
[119887 119879Ω] (119906) (119909) = 119887 (119909) 119879Ω119906 (119909) minus 119879Ω (119887119906) (119909)= sum119868
(intR119899
Ω(119909 minus 119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899 (119887 (119909) minus 119887 (119910)) 119906119868 (119910) 119889119910)119889119909119868(19)
Let 119870(119909 minus 119910) = Ω(119909 minus 119910)|119909 minus 119910|119899 and then according to thedefinition of the exterior differential operator we obtain
119889 [119887 119879Ω] (119906) (119909)= sum119868
119899sum119896=1
120597 intR119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896 119889119909119896
and 119889119909119868(20)
Using the elementary inequality | sum119873119894=1 119905119894|119904 le 119873119904minus1sum119873119894=1 |119905119894|119904for constants 119873 119904 gt 0 we deduce1003817100381710038171003817119889 [119887 119879Ω] 119906 (119909)1003817100381710038171003817119901119901R119899le int
R119899(sum119868
119899sum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
10038161003816100381610038161003816100381610038161003816100381610038162)1199012
119889119909
4 Journal of Function Spaces
le 1198621 intR119899sum119868
119899sum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003816100381610038161003816100381610038161003816100381610038161003816119901 119889119909
= 1198621sum119868
119899sum119896=1
intR119899
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003816100381610038161003816100381610038161003816100381610038161003816119901 119889119909
= 1198621sum119868
119899sum119896=1
1003817100381710038171003817100381710038171003817100381710038171003817120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003817100381710038171003817100381710038171003817100381710038171003817119901
119901R119899
(21)
By Lemma 8 we obtain
1003817100381710038171003817100381710038171003817100381710038171003817120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003817100381710038171003817100381710038171003817100381710038171003817119901R119899le 1198622 1003817100381710038171003817119906119868 (119909)1003817100381710038171003817119901R119899
(22)
Substituting (22) into (21) and applying the fundamentalinequality 119886119904 + 119887119904 le (119886 + 119887)119904 119904 ge 1 yields that1003817100381710038171003817119889 [119887 119879Ω] 119906 (119909)1003817100381710038171003817119901119901R119899 le 1198623sum
119868
1003817100381710038171003817119906119868 (119909)1003817100381710038171003817119901119901R119899= 1198623sum
119868
intR119899
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816119901 119889119909= 1198623 int
R119899sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816119901 119889119909le 1198624 int
R119899(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816)119901 119889119909
(23)
Using the inequality | sum119873119894=1 119905119894|119904 le 119873119904minus1sum119873119894=1 |119905119894|119904 again weeasily have
(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816)2 le 1198625sum
119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162 (24)
that is
(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816) le (1198625sum119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162)12 (25)
Substituting (25) into (23) gives
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901119901R119899 le 1198626 intR119899(sum119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162)1199012 119889119909
= 1198626 intR119899|119906 (119909)|119901 119889119909 = 1198626 119906119901119901R119899
(26)
This completes the proof of Lemma 9
In Lemma 9 let 119906 = 119906 in bounded domain119872 and 119906 = 0inR119899 119872 we can easily obtain the following lemma
Lemma 10 Let 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt infin119879Ω be the Calderon-Zygmund singular integral operator on
differential forms and 119887 be sufficiently smooth and boundedThen there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901119872 le 119862 119906119901119872 (27)
where119872 sub R119899 is any bounded domain
Using Lemma 1 and the analogous method developed inLemma 9 we have the following estimate for [119887 119879Ω]Lemma 11 Let 119906 isin 119871119901(119872Λ119897) 119897 = 0 1 119899 1 lt 119901 lt infinand 119879Ω be a Calderon-Zygmund singular integral operator ondifferential forms Then given any function 119887 isin 119861119872119874(R119899)[119887 119879Ω] satisfies the strong (119901 119901) inequality
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901119872 le 119862 119887119861119872119874 119906119901119872 (28)
for any bounded domain 119872 sub R119899 where 119862 is a constantindependent of 119906
We now present the local higher integrability of commu-tators of Calderon-Zygmund singular integral operators ondifferential forms
Theorem 12 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt 119899then [119887 119879Ω]119906 isin 119871119904119897119900119888(119872Λ119897) for any 0 lt 119904 lt 119899119901(119899 minus 119901)Moreover there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 119862120583 (119861)1119904+1119899minus1119901 119906119901120590119861 (29)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof For any ball 119861 with 120590119861 sub 119872 for some 120590 gt 1 in thecase that the measure 120583119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0 = 0 We have obviously that [119887 119879Ω]119906 = ([119887 119879Ω]119906)119861 almosteverywhere in 119861 which implies [119887 119879]119906 is a closed form and asolution of the Dirac-harmonic equation (1) in119872 ApplyingLemmas 4 and 2 to [119887 119879Ω]119906 with any 119904 119901 gt 0 we have1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 = 1003817100381710038171003817([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861
le 1198621120583 (119861) 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861le 1198622120583 (119861)1+1119904minus1119901 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901120590119861
(30)
for all balls B with 120590119861 sub 119872 for some 120590 gt 1 We note that 119887 isbounded thus 119887 isin 119861119872119874(R119899) By Lemma 11 it follows that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 1198622120583 (119861)1+1119904minus1119901 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901120590119861le 1198623120583 (119861)1+1119904minus1119901 119887119861119872119874 119906119901120590119861le 1198624120583 (119861)1+1119904minus1119901 119906119901120590119861
(31)
Journal of Function Spaces 5
On the other hand if the measure 120583119909 isin 119861 |[119887 119879Ω]119906 minus([119887 119879Ω]119906)119861| gt 0 gt 0 Applying Lemma 5 to [119887 119879Ω]119906 we get(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198625 (119899 119901) (int119861
1003816100381610038161003816119889 [119887 119879Ω] 1199061003816100381610038161003816119901 119889119909)1119901 (32)
Then by Lemma 10 it follows that
(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198626 (119899 119901) (int119861|119906|119901 119889119909)1119901
(33)
Noticing that the measure |119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0| gt 0 we could use Lemma 6 Taking 120595(119905) = 119905119899119901(119899minus119901) inLemma 6 we have
(int119861|V|119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198627 (int119861
1003816100381610038161003816V minus V1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901(34)
for any differential form V and any ball B with |119909 isin 119861 |V minusV119861| gt 0| gt 0 Replacing V by [119887 119879Ω]119906 in (34) we obtain
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901 le 1198628 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
(35)
Combining (33) and (35) yields that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198629 (int119861|119906|119901 119889119909)1119901
(36)
In view of the monotonic property of the 119871119901-space with 0 lt119904 lt 119899119901(119899 minus 119901) we have( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)
1119904
le ( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
(37)
Substituting (36) and (37) gives
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904
le 11986210120583 (119861)1119904minus1119901+1119899 (int119861|119906|119901 119889119909)1119901
(38)
Therefore inequality (29) follows in both cases which indi-cates that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879]119906 isin 119871119904119897119900119888(119872Λ119897) Wehave completed the proof of Theorem 12
Moreover inequality (38) can be written as the followingintegral average inequality
( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904
le 11986211120583 (119861)1119899 ( 1120583 (119861) int120590119861 |119906|119901 119889119909)1119901
(39)
Clearly with 119901 close to 119899 the integral exponent 119904 on thelefthand side could bemuch larger than the integral exponent119901 on the right hand side since the condition 0 lt 119904 lt 119899119901(119899 minus119901) Hence the higher integrability of operator [119887 119879Ω] for thecase that 1 lt 119901 lt 119899 is obtained
Next we consider the higher integrability of [119887 119879Ω] forthe case 119901 ge 119899Theorem 13 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904119897119900119888(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 119862120583 (119861)1119904+1119899minus1119901 119906119901120590119861 (40)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof It is immediate by the same method developed in theproof of Theorem 12 that (40) holds for 120583119909 isin 119861 |[119887 119879Ω]119906 minus([119887 119879Ω]119906)119861| gt 0 = 0 Let us consider the remain case that120583119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt 0 gt 0 Select 120572 =max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Then it is easy to checkthat
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (41)
since 119899 minus 119901 le 0 Notice that 1 lt 119902 = 120572119899119901(119899 + 120572119901) lt 119899Using Lemma 5 to [119887 119879Ω]119906 and combining Lemma 10 and themonotonic property of the 119871119901-space we have(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198621 (int119861
1003816100381610038161003816119889 [119887 119879Ω] 1199061003816100381610038161003816119902 119889119909)1119902
le 1198622 (int119861|119906|119902 119889119909)1119902
le 1198623120583 (119861)1119902minus1119901 (int119861|119906|119901 119889119909)1119901
(42)
Notice that the measure |119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0| gt 0 Therefore we can select 120595(119905) = 119905119899119902(119899minus119902) in Lemma 6Then for any differential form 119908 we have
(int119861|119908|119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198624 (int119861
1003816100381610038161003816119908 minus 1199081198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 (43)
6 Journal of Function Spaces
Replacing 119908 by [119887 119879Ω]119906 in (43) we get
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 le 1198625 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
(44)
Taking into account the fact that 119899119902(119899 minus 119902) = 120572119901 gt 119904 we seefrom the monotonic property of the 119871119901-space (42) and (44)that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904 le 1198626120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902le 1198627120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198628120583 (119861)1119902minus1119901+1119904minus1120572119901 (int119861|119906|119901 119889119909)1119901
= 1198628120583 (119861)1119899+1119904minus1119901 (int119861|119906|119901 119889119909)1119901
(45)
which means that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879Ω]119906 isin119871119904119897119900119888(119872Λ119897)We have completed the proof ofTheorem 13
Now we are ready to assert the global higher integrabilityof the commutator of Calderon-Zygmund singular integraloperator on differential forms
Theorem 14 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 0 lt 119904 lt 119899119901(119899 minus 119901) Moreoverthere exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (46)
for any bounded domain119872 sub R119899Proof Notice that 1119904+1119899minus1119901 gt 0 since 0 lt 119904 lt 119899119901(119899minus119901)By Lemma 7 andTheorem 12 we have
[119887 119879] 119906119904119872 le sum119861isin][119887 119879] 119906119904119861
le sum119861isin](1198621120583 (119861)1119904+1119899minus1119901 119906119901120590119861)
le sum119861isin](1198621120583 (119872)1119904+1119899minus1119901 119906119901120590119861)
le 1198622120583 (119872)1119904+1119899minus1119901119873119906119901119872le 1198623 119906119901119872
(47)
which finishes the proof of Theorem 14
Using the similar method as we did in Theorem 14 andcombining with Theorem 13 we can deduce the followingglobal result for the case 119901 ge 119899Theorem 15 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant C independent of u such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (48)
for any bounded domain119872 sub R1198994 Higher Order Poincareacute-Type Inequalities
In this section we shall state the higher order Poincare-typeinequalities for commutator of Calderon-Zygmund singularintegral operator acting on the solutions of the Dirac-harmonic equations
Theorem 16 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (49)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof For any 119906 isin 119871119901119897119900119888(119872Λ119897) using the 119871119901-decompositionformula (9) to [119887 119879Ω]119906 we obtain
[119887 119879Ω] 119906 = 119889119879 ([119887 119879Ω] 119906) + 119879119889 ([119887 119879Ω] 119906) (50)
Noticing that 119889119879([119887119879Ω]119906) = ([119887 119879Ω]119906)119861 for any differentialform 119906 by (50) (10) and Lemma 10 we get1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901(119899minus119901)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198621 (119899 119901) 120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198622 (119899 119901) 120583 (119861)1+1119899 119906119899119901(119899minus119901)119861
(51)
Since 119906 is a solution of the Dirac-harmonic equation (1) byLemma 2 we have
119906119899119901(119899minus119901)119861 le 120583 (119861)minus1119899 119906119901120590119861 (52)
where120590 gt 1 is a constant Substituting (52) into (51) gives that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861le 1198623 (119899 119901) 120583 (119861) 119906119901120590119861 (53)
Journal of Function Spaces 7
Moreover for any 0 lt 119904 lt 119899119901(119899minus119901) applying the monotonicproperties of 119871119901-space it follows that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119901+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861 (54)
Combined with (53) the above inequality becomes
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198624 (119899 119901) 120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (55)
which finishes the proof of Theorem 16
Next we will prove the higher order Poincare-typeinequality still holds for the case that 119901 ge 119899Theorem 17 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of the Dirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then for any119904 gt 0 there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (56)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof Select 120572 = max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Thenwe easily obtain
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (57)
that is 119902 lt 119901 since 119899 minus 119901 le 0 We can also find 1 lt 119902 =120572119899119901(119899 + 120572119901) lt 119899 Using the same technique as in the proofof Theorem 16 we have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902(119899minus119902)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198621120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198622120583 (119861)1+1119899 119906119899119902(119899minus119902)119861le 1198623120583 (119861)1+1119899 120583 (119861)1119902minus1119899minus1119901 119906119901120590119861le 1198624120583 (119861)1+1119902minus1119901 119906119901120590119861
(58)
Noticing that 119899119902(119899 minus 119902) = 120572119901 gt 119904 and combining themonotonic property of the 119871119901-space we obtain1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119902+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902119899minus119902119861 (59)
Combining (58) and (59) we finally have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198625120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (60)
The proof of Theorem 17 is completed
Based onTheorems 16 and 17 we can obtain the followingglobal higher order Poincare-type inequalities for the com-mutator [119887 119879Ω] using the analogous method developed inTheorem 14
Theorem 18 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (61)
for any bounded domain119872 sub R119899Theorem 19 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then forany 119904 gt 0 there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (62)
for any bounded domain119872 sub R1198995 Applications
In this section we demonstrate some applications of ourmainresults established in the previous sections
Example 20 Let 119903 gt 0 and 119896 gt 0 be any constants and1198721 =(1199091 1199092 1199093) 11990921 + 11990922 + 11990923 le 1199032 sub R3 Consider the 1-form119906 (1199091 1199092 1199093) = 1199091119896 + 11990921 + 11990922 + 11990923 1198891199091
+ 1199092119896 + 11990921 + 11990922 + 11990923 1198891199092+ 1199093119896 + 11990921 + 11990922 + 11990923 1198891199093
(63)
defined in119872 such that1198721 sub 119872 It is easy to check that 119889119906 =0 Hence 119906 is a solution of the Dirac-harmonic equation (1)
8 Journal of Function Spaces
for any operators 119860 satisfying (2) Also it can be calculatedthat
|119906| = (( 1199091119896 + 11990921 + 11990922 + 11990923)2 + ( 1199092119896 + 11990921 + 11990922 + 11990923)
2
+ ( 1199093119896 + 11990921 + 11990922 + 11990923)2)12
= ( 11990921 + 11990922 + 11990923(119896 + 11990921 + 11990922 + 11990923)2)12 lt 1
(64)
for any 119909 = (1199091 1199092 1199093) isin 1198721 Thus
1199061199011198721 = (int1198721|119906|119901 119889119909)1119901
= (int1198721
100381610038161003816100381610038161003816100381610038161003816100381611990921 + 11990922 + 119909231003816100381610038161003816119896 + 11990921 + 11990922 + 1199092310038161003816100381610038162
10038161003816100381610038161003816100381610038161003816100381610038161199012 119889119909)
1119901
le (int1198721
1119889119909)1119901 = (41205873 1199033)1119901
(65)
Applying Theorem 14 for 1 lt 119901 lt 3 we have [119887 119879Ω]119906 isin119871119904(119872Λ119897) for any 0 lt 119904 lt 3119901(3 minus 119901) Analogously we have[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0when119901 ge 3 byTheorem 15In the meantime we have the higher estimate for [119887 119879Ω]119906that is
1003817100381710038171003817[119887 119879Ω] 11990610038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (66)
In addition we can evaluate the following integrals byTheorem 18 andTheorem 19 that
10038171003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721100381710038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (67)
We should notice that the above example can be extendedto the case of R119899 as follows
Example 21 We can check that the 1-form defined inR119899
119906 (1199091 119909119899) = 119899sum119894=1
119909119894119896 + 11990921 + sdot sdot sdot + 1199092119899 119889119909119894 119896 gt 0 (68)
is a solution of the Dirac-harmonic equation (1) for anyoperators 119860 satisfying (2) Hence Theorems 14 15 18 and 19are also applicable to 119906(1199091 119909119899)Remark It is worth pointing out that the global resultsobtained in this paper can be extended to larger classes ofdomains such as 119871119901-averaging domains and 119871120593(120583)-averagingdomains see [1 27] Also the techniques developed inthis paper provide an effective method to study the higherintegrability of bilinear commutators of singular integrals ondifferential forms which are defined in [25] We leave thestatements and proofs to the interested readers
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The work was supported by the National Natural ScienceFoundation of China (Grant no 11601105)
References
[1] R P Agarwal S Ding andCNolder Inequalities forDifferentialForms Springer New York NY USA 2009
[2] Y Xing and C Wu ldquoGlobal weighted inequalities for operatorsand harmonic forms on manifoldsrdquo Journal of MathematicalAnalysis and Applications vol 294 no 1 pp 294ndash309 2004
[3] J B Perot and C J Zusi ldquoDifferential forms for scientists andengineersrdquo Journal of Computational Physics vol 257 no partB pp 1373ndash1393 2014
[4] C Scott ldquoL119901-theory of differential forms onmanifolds theory ofdifferential forms on manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 6 pp 2075ndash2096 1995
[5] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004
[6] S Ding and B Liu ldquoA singular integral of the compositeoperatorrdquo Applied Mathematics Letters vol 22 no 8 pp 1271ndash1275 2009
[7] S Ding and B Liu ldquoGlobal estimates for singular integrals ofthe composite operatorrdquo Illinois Journal of Mathematics vol 53no 4 pp 1173ndash1185 2009
[8] S Ding and B Liu ldquoGlobal integrability of the Jacobian of acomposite mappingrdquo Journal of Inequalities and ApplicationsArt ID 89134 9 pages 2006
[9] H Bi and S Ding ldquoSome strong (pq)-type inequalities forthe homotopy operatorrdquo Computers and Mathematics withApplications An International Journal vol 62 no 4 pp 1780ndash1789 2011
[10] H Bi ldquoWeighted inequalities for potential operators on differ-ential formsrdquo Journal of Inequalities and Applications Art ID713625 13 pages 2010
[11] Y Ling and B Gejun ldquoSome local Poincare inequalities for thecomposition of the sharp maximal operator and the Greenrsquosoperatorrdquo Computers and Mathematics with Applications vol63 no 3 pp 720ndash727 2012
[12] S Ding ldquoIntegral estimates for the Laplace-Beltrami andGreenrsquos operators applied to differential forms on manifoldsrdquoJournal for Analysis and its Applications vol 22 no 4 pp 939ndash957 2003
[13] G Lu and R L Wheeden ldquoHigh order representation formulasand embedding theorems on stratified groups and generaliza-tionsrdquo Studia Mathematica vol 142 no 2 pp 101ndash133 2000
[14] G Lu ldquoPolynomials higher order Sobolev extension theo-rems and interpolation inequalities on weighted Folland-STEin
Journal of Function Spaces 9
spaces on stratified groupsrdquo Acta Mathematica Sinica vol 16no 3 pp 405ndash444 2000
[15] E W Stredulinsky ldquoHigher integrability from reverse Holderinequalitiesrdquo Indiana University Mathematics Journal vol 29no 3 pp 407ndash413 1980
[16] F W Gehring ldquoThe L119901-integrability of the partial derivativesof a quasiconformal mappingrdquo Acta Mathematica vol 130 pp265ndash277 1973
[17] W S Cohn G Lu and S Lu ldquoHigher order Poincare inequal-ities associated with linear operators on stratified groups andapplicationsrdquoMathematische Zeitschrift vol 244 no 2 pp 309ndash335 2003
[18] S Ding and B Liu ldquoDirac-harmonic equations for differentialformsrdquo Nonlinear Analysis Theory Methods and ApplicationsAn International Multidisciplinary Journal vol 122 pp 43ndash572015
[19] C A Nolder ldquoHardy-Littlewood theorems for 119860-harmonictensorsrdquo Illinois Journal of Mathematics vol 43 no 4 pp 613ndash632 1999
[20] A Carbonaro A McIntosh and A J Morris ldquoLocal Hardyspaces of differential forms on Riemannian manifoldsrdquo TheJournal of Geometric Analysis vol 23 no 1 pp 106ndash169 2013
[21] K Kodaira ldquoHarmonic fields in Riemannian manifolds (gener-alized potential theory)rdquo Annals of Mathematics Second Seriesvol 50 pp 587ndash665 1949
[22] R R Coifman R Rochberg and G Weiss ldquoFactorizationtheorems for Hardy spaces in several variablesrdquo Annals ofMathematics vol 103 no 3 pp 611ndash635 1976
[23] L Chaffee and D Cruz-Uribe ldquoNecessary conditions for theboundedness of linear and bilinear commutators on Banachfunction spacesrdquo Mathematical Inequalities and Applicationsvol 21 no 1 pp 1ndash16 2018
[24] C Segovia and J L Torrea ldquoWeighted inequalities for com-mutators of fractional and singular integralsrdquo PublicacionsMatematiques vol 35 no 1 pp 209ndash235 1991
[25] C Perez ldquoSharp estimates for commutators of singular integralsvia iterations of the Hardy-Littlewood maximal functionrdquoJournal of Fourier Analysis and Applications vol 3 no 6 pp743ndash756 1997
[26] T Iwaniec and A Lutoborski ldquoIntegral estimates for nullLagrangiansrdquo Archive for Rational Mechanics and Analysis vol125 no 1 pp 25ndash79 1993
[27] S Ding ldquoL120593(120583)-averaging domains and the quasi-hyperbolicmetricrdquo Computers and Mathematics with Applications AnInternational Journal vol 47 no 10-11 pp 1611ndash1618 2004
[28] A P Calderon ldquoCommutators of singular integral operatorsrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 53 pp 1092ndash1099 1965
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4 Journal of Function Spaces
le 1198621 intR119899sum119868
119899sum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003816100381610038161003816100381610038161003816100381610038161003816119901 119889119909
= 1198621sum119868
119899sum119896=1
intR119899
1003816100381610038161003816100381610038161003816100381610038161003816120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003816100381610038161003816100381610038161003816100381610038161003816119901 119889119909
= 1198621sum119868
119899sum119896=1
1003817100381710038171003817100381710038171003817100381710038171003817120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003817100381710038171003817100381710038171003817100381710038171003817119901
119901R119899
(21)
By Lemma 8 we obtain
1003817100381710038171003817100381710038171003817100381710038171003817120597 int
R119899(119887 (119909) minus 119887 (119910))119870 (119909 minus 119910) 119906119868 (119910) 119889119910120597119909119896
1003817100381710038171003817100381710038171003817100381710038171003817119901R119899le 1198622 1003817100381710038171003817119906119868 (119909)1003817100381710038171003817119901R119899
(22)
Substituting (22) into (21) and applying the fundamentalinequality 119886119904 + 119887119904 le (119886 + 119887)119904 119904 ge 1 yields that1003817100381710038171003817119889 [119887 119879Ω] 119906 (119909)1003817100381710038171003817119901119901R119899 le 1198623sum
119868
1003817100381710038171003817119906119868 (119909)1003817100381710038171003817119901119901R119899= 1198623sum
119868
intR119899
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816119901 119889119909= 1198623 int
R119899sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816119901 119889119909le 1198624 int
R119899(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816)119901 119889119909
(23)
Using the inequality | sum119873119894=1 119905119894|119904 le 119873119904minus1sum119873119894=1 |119905119894|119904 again weeasily have
(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816)2 le 1198625sum
119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162 (24)
that is
(sum119868
1003816100381610038161003816119906119868 (119909)1003816100381610038161003816) le (1198625sum119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162)12 (25)
Substituting (25) into (23) gives
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901119901R119899 le 1198626 intR119899(sum119868
1003816100381610038161003816119906119868 (119909)10038161003816100381610038162)1199012 119889119909
= 1198626 intR119899|119906 (119909)|119901 119889119909 = 1198626 119906119901119901R119899
(26)
This completes the proof of Lemma 9
In Lemma 9 let 119906 = 119906 in bounded domain119872 and 119906 = 0inR119899 119872 we can easily obtain the following lemma
Lemma 10 Let 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt infin119879Ω be the Calderon-Zygmund singular integral operator on
differential forms and 119887 be sufficiently smooth and boundedThen there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817119889 [119887 119879Ω] 1199061003817100381710038171003817119901119872 le 119862 119906119901119872 (27)
where119872 sub R119899 is any bounded domain
Using Lemma 1 and the analogous method developed inLemma 9 we have the following estimate for [119887 119879Ω]Lemma 11 Let 119906 isin 119871119901(119872Λ119897) 119897 = 0 1 119899 1 lt 119901 lt infinand 119879Ω be a Calderon-Zygmund singular integral operator ondifferential forms Then given any function 119887 isin 119861119872119874(R119899)[119887 119879Ω] satisfies the strong (119901 119901) inequality
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901119872 le 119862 119887119861119872119874 119906119901119872 (28)
for any bounded domain 119872 sub R119899 where 119862 is a constantindependent of 119906
We now present the local higher integrability of commu-tators of Calderon-Zygmund singular integral operators ondifferential forms
Theorem 12 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt 119899then [119887 119879Ω]119906 isin 119871119904119897119900119888(119872Λ119897) for any 0 lt 119904 lt 119899119901(119899 minus 119901)Moreover there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 119862120583 (119861)1119904+1119899minus1119901 119906119901120590119861 (29)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof For any ball 119861 with 120590119861 sub 119872 for some 120590 gt 1 in thecase that the measure 120583119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0 = 0 We have obviously that [119887 119879Ω]119906 = ([119887 119879Ω]119906)119861 almosteverywhere in 119861 which implies [119887 119879]119906 is a closed form and asolution of the Dirac-harmonic equation (1) in119872 ApplyingLemmas 4 and 2 to [119887 119879Ω]119906 with any 119904 119901 gt 0 we have1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 = 1003817100381710038171003817([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861
le 1198621120583 (119861) 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861le 1198622120583 (119861)1+1119904minus1119901 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901120590119861
(30)
for all balls B with 120590119861 sub 119872 for some 120590 gt 1 We note that 119887 isbounded thus 119887 isin 119861119872119874(R119899) By Lemma 11 it follows that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 1198622120583 (119861)1+1119904minus1119901 1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119901120590119861le 1198623120583 (119861)1+1119904minus1119901 119887119861119872119874 119906119901120590119861le 1198624120583 (119861)1+1119904minus1119901 119906119901120590119861
(31)
Journal of Function Spaces 5
On the other hand if the measure 120583119909 isin 119861 |[119887 119879Ω]119906 minus([119887 119879Ω]119906)119861| gt 0 gt 0 Applying Lemma 5 to [119887 119879Ω]119906 we get(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198625 (119899 119901) (int119861
1003816100381610038161003816119889 [119887 119879Ω] 1199061003816100381610038161003816119901 119889119909)1119901 (32)
Then by Lemma 10 it follows that
(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198626 (119899 119901) (int119861|119906|119901 119889119909)1119901
(33)
Noticing that the measure |119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0| gt 0 we could use Lemma 6 Taking 120595(119905) = 119905119899119901(119899minus119901) inLemma 6 we have
(int119861|V|119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198627 (int119861
1003816100381610038161003816V minus V1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901(34)
for any differential form V and any ball B with |119909 isin 119861 |V minusV119861| gt 0| gt 0 Replacing V by [119887 119879Ω]119906 in (34) we obtain
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901 le 1198628 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
(35)
Combining (33) and (35) yields that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198629 (int119861|119906|119901 119889119909)1119901
(36)
In view of the monotonic property of the 119871119901-space with 0 lt119904 lt 119899119901(119899 minus 119901) we have( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)
1119904
le ( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
(37)
Substituting (36) and (37) gives
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904
le 11986210120583 (119861)1119904minus1119901+1119899 (int119861|119906|119901 119889119909)1119901
(38)
Therefore inequality (29) follows in both cases which indi-cates that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879]119906 isin 119871119904119897119900119888(119872Λ119897) Wehave completed the proof of Theorem 12
Moreover inequality (38) can be written as the followingintegral average inequality
( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904
le 11986211120583 (119861)1119899 ( 1120583 (119861) int120590119861 |119906|119901 119889119909)1119901
(39)
Clearly with 119901 close to 119899 the integral exponent 119904 on thelefthand side could bemuch larger than the integral exponent119901 on the right hand side since the condition 0 lt 119904 lt 119899119901(119899 minus119901) Hence the higher integrability of operator [119887 119879Ω] for thecase that 1 lt 119901 lt 119899 is obtained
Next we consider the higher integrability of [119887 119879Ω] forthe case 119901 ge 119899Theorem 13 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904119897119900119888(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 119862120583 (119861)1119904+1119899minus1119901 119906119901120590119861 (40)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof It is immediate by the same method developed in theproof of Theorem 12 that (40) holds for 120583119909 isin 119861 |[119887 119879Ω]119906 minus([119887 119879Ω]119906)119861| gt 0 = 0 Let us consider the remain case that120583119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt 0 gt 0 Select 120572 =max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Then it is easy to checkthat
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (41)
since 119899 minus 119901 le 0 Notice that 1 lt 119902 = 120572119899119901(119899 + 120572119901) lt 119899Using Lemma 5 to [119887 119879Ω]119906 and combining Lemma 10 and themonotonic property of the 119871119901-space we have(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198621 (int119861
1003816100381610038161003816119889 [119887 119879Ω] 1199061003816100381610038161003816119902 119889119909)1119902
le 1198622 (int119861|119906|119902 119889119909)1119902
le 1198623120583 (119861)1119902minus1119901 (int119861|119906|119901 119889119909)1119901
(42)
Notice that the measure |119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0| gt 0 Therefore we can select 120595(119905) = 119905119899119902(119899minus119902) in Lemma 6Then for any differential form 119908 we have
(int119861|119908|119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198624 (int119861
1003816100381610038161003816119908 minus 1199081198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 (43)
6 Journal of Function Spaces
Replacing 119908 by [119887 119879Ω]119906 in (43) we get
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 le 1198625 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
(44)
Taking into account the fact that 119899119902(119899 minus 119902) = 120572119901 gt 119904 we seefrom the monotonic property of the 119871119901-space (42) and (44)that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904 le 1198626120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902le 1198627120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198628120583 (119861)1119902minus1119901+1119904minus1120572119901 (int119861|119906|119901 119889119909)1119901
= 1198628120583 (119861)1119899+1119904minus1119901 (int119861|119906|119901 119889119909)1119901
(45)
which means that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879Ω]119906 isin119871119904119897119900119888(119872Λ119897)We have completed the proof ofTheorem 13
Now we are ready to assert the global higher integrabilityof the commutator of Calderon-Zygmund singular integraloperator on differential forms
Theorem 14 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 0 lt 119904 lt 119899119901(119899 minus 119901) Moreoverthere exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (46)
for any bounded domain119872 sub R119899Proof Notice that 1119904+1119899minus1119901 gt 0 since 0 lt 119904 lt 119899119901(119899minus119901)By Lemma 7 andTheorem 12 we have
[119887 119879] 119906119904119872 le sum119861isin][119887 119879] 119906119904119861
le sum119861isin](1198621120583 (119861)1119904+1119899minus1119901 119906119901120590119861)
le sum119861isin](1198621120583 (119872)1119904+1119899minus1119901 119906119901120590119861)
le 1198622120583 (119872)1119904+1119899minus1119901119873119906119901119872le 1198623 119906119901119872
(47)
which finishes the proof of Theorem 14
Using the similar method as we did in Theorem 14 andcombining with Theorem 13 we can deduce the followingglobal result for the case 119901 ge 119899Theorem 15 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant C independent of u such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (48)
for any bounded domain119872 sub R1198994 Higher Order Poincareacute-Type Inequalities
In this section we shall state the higher order Poincare-typeinequalities for commutator of Calderon-Zygmund singularintegral operator acting on the solutions of the Dirac-harmonic equations
Theorem 16 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (49)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof For any 119906 isin 119871119901119897119900119888(119872Λ119897) using the 119871119901-decompositionformula (9) to [119887 119879Ω]119906 we obtain
[119887 119879Ω] 119906 = 119889119879 ([119887 119879Ω] 119906) + 119879119889 ([119887 119879Ω] 119906) (50)
Noticing that 119889119879([119887119879Ω]119906) = ([119887 119879Ω]119906)119861 for any differentialform 119906 by (50) (10) and Lemma 10 we get1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901(119899minus119901)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198621 (119899 119901) 120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198622 (119899 119901) 120583 (119861)1+1119899 119906119899119901(119899minus119901)119861
(51)
Since 119906 is a solution of the Dirac-harmonic equation (1) byLemma 2 we have
119906119899119901(119899minus119901)119861 le 120583 (119861)minus1119899 119906119901120590119861 (52)
where120590 gt 1 is a constant Substituting (52) into (51) gives that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861le 1198623 (119899 119901) 120583 (119861) 119906119901120590119861 (53)
Journal of Function Spaces 7
Moreover for any 0 lt 119904 lt 119899119901(119899minus119901) applying the monotonicproperties of 119871119901-space it follows that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119901+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861 (54)
Combined with (53) the above inequality becomes
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198624 (119899 119901) 120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (55)
which finishes the proof of Theorem 16
Next we will prove the higher order Poincare-typeinequality still holds for the case that 119901 ge 119899Theorem 17 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of the Dirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then for any119904 gt 0 there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (56)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof Select 120572 = max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Thenwe easily obtain
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (57)
that is 119902 lt 119901 since 119899 minus 119901 le 0 We can also find 1 lt 119902 =120572119899119901(119899 + 120572119901) lt 119899 Using the same technique as in the proofof Theorem 16 we have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902(119899minus119902)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198621120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198622120583 (119861)1+1119899 119906119899119902(119899minus119902)119861le 1198623120583 (119861)1+1119899 120583 (119861)1119902minus1119899minus1119901 119906119901120590119861le 1198624120583 (119861)1+1119902minus1119901 119906119901120590119861
(58)
Noticing that 119899119902(119899 minus 119902) = 120572119901 gt 119904 and combining themonotonic property of the 119871119901-space we obtain1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119902+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902119899minus119902119861 (59)
Combining (58) and (59) we finally have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198625120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (60)
The proof of Theorem 17 is completed
Based onTheorems 16 and 17 we can obtain the followingglobal higher order Poincare-type inequalities for the com-mutator [119887 119879Ω] using the analogous method developed inTheorem 14
Theorem 18 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (61)
for any bounded domain119872 sub R119899Theorem 19 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then forany 119904 gt 0 there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (62)
for any bounded domain119872 sub R1198995 Applications
In this section we demonstrate some applications of ourmainresults established in the previous sections
Example 20 Let 119903 gt 0 and 119896 gt 0 be any constants and1198721 =(1199091 1199092 1199093) 11990921 + 11990922 + 11990923 le 1199032 sub R3 Consider the 1-form119906 (1199091 1199092 1199093) = 1199091119896 + 11990921 + 11990922 + 11990923 1198891199091
+ 1199092119896 + 11990921 + 11990922 + 11990923 1198891199092+ 1199093119896 + 11990921 + 11990922 + 11990923 1198891199093
(63)
defined in119872 such that1198721 sub 119872 It is easy to check that 119889119906 =0 Hence 119906 is a solution of the Dirac-harmonic equation (1)
8 Journal of Function Spaces
for any operators 119860 satisfying (2) Also it can be calculatedthat
|119906| = (( 1199091119896 + 11990921 + 11990922 + 11990923)2 + ( 1199092119896 + 11990921 + 11990922 + 11990923)
2
+ ( 1199093119896 + 11990921 + 11990922 + 11990923)2)12
= ( 11990921 + 11990922 + 11990923(119896 + 11990921 + 11990922 + 11990923)2)12 lt 1
(64)
for any 119909 = (1199091 1199092 1199093) isin 1198721 Thus
1199061199011198721 = (int1198721|119906|119901 119889119909)1119901
= (int1198721
100381610038161003816100381610038161003816100381610038161003816100381611990921 + 11990922 + 119909231003816100381610038161003816119896 + 11990921 + 11990922 + 1199092310038161003816100381610038162
10038161003816100381610038161003816100381610038161003816100381610038161199012 119889119909)
1119901
le (int1198721
1119889119909)1119901 = (41205873 1199033)1119901
(65)
Applying Theorem 14 for 1 lt 119901 lt 3 we have [119887 119879Ω]119906 isin119871119904(119872Λ119897) for any 0 lt 119904 lt 3119901(3 minus 119901) Analogously we have[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0when119901 ge 3 byTheorem 15In the meantime we have the higher estimate for [119887 119879Ω]119906that is
1003817100381710038171003817[119887 119879Ω] 11990610038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (66)
In addition we can evaluate the following integrals byTheorem 18 andTheorem 19 that
10038171003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721100381710038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (67)
We should notice that the above example can be extendedto the case of R119899 as follows
Example 21 We can check that the 1-form defined inR119899
119906 (1199091 119909119899) = 119899sum119894=1
119909119894119896 + 11990921 + sdot sdot sdot + 1199092119899 119889119909119894 119896 gt 0 (68)
is a solution of the Dirac-harmonic equation (1) for anyoperators 119860 satisfying (2) Hence Theorems 14 15 18 and 19are also applicable to 119906(1199091 119909119899)Remark It is worth pointing out that the global resultsobtained in this paper can be extended to larger classes ofdomains such as 119871119901-averaging domains and 119871120593(120583)-averagingdomains see [1 27] Also the techniques developed inthis paper provide an effective method to study the higherintegrability of bilinear commutators of singular integrals ondifferential forms which are defined in [25] We leave thestatements and proofs to the interested readers
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The work was supported by the National Natural ScienceFoundation of China (Grant no 11601105)
References
[1] R P Agarwal S Ding andCNolder Inequalities forDifferentialForms Springer New York NY USA 2009
[2] Y Xing and C Wu ldquoGlobal weighted inequalities for operatorsand harmonic forms on manifoldsrdquo Journal of MathematicalAnalysis and Applications vol 294 no 1 pp 294ndash309 2004
[3] J B Perot and C J Zusi ldquoDifferential forms for scientists andengineersrdquo Journal of Computational Physics vol 257 no partB pp 1373ndash1393 2014
[4] C Scott ldquoL119901-theory of differential forms onmanifolds theory ofdifferential forms on manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 6 pp 2075ndash2096 1995
[5] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004
[6] S Ding and B Liu ldquoA singular integral of the compositeoperatorrdquo Applied Mathematics Letters vol 22 no 8 pp 1271ndash1275 2009
[7] S Ding and B Liu ldquoGlobal estimates for singular integrals ofthe composite operatorrdquo Illinois Journal of Mathematics vol 53no 4 pp 1173ndash1185 2009
[8] S Ding and B Liu ldquoGlobal integrability of the Jacobian of acomposite mappingrdquo Journal of Inequalities and ApplicationsArt ID 89134 9 pages 2006
[9] H Bi and S Ding ldquoSome strong (pq)-type inequalities forthe homotopy operatorrdquo Computers and Mathematics withApplications An International Journal vol 62 no 4 pp 1780ndash1789 2011
[10] H Bi ldquoWeighted inequalities for potential operators on differ-ential formsrdquo Journal of Inequalities and Applications Art ID713625 13 pages 2010
[11] Y Ling and B Gejun ldquoSome local Poincare inequalities for thecomposition of the sharp maximal operator and the Greenrsquosoperatorrdquo Computers and Mathematics with Applications vol63 no 3 pp 720ndash727 2012
[12] S Ding ldquoIntegral estimates for the Laplace-Beltrami andGreenrsquos operators applied to differential forms on manifoldsrdquoJournal for Analysis and its Applications vol 22 no 4 pp 939ndash957 2003
[13] G Lu and R L Wheeden ldquoHigh order representation formulasand embedding theorems on stratified groups and generaliza-tionsrdquo Studia Mathematica vol 142 no 2 pp 101ndash133 2000
[14] G Lu ldquoPolynomials higher order Sobolev extension theo-rems and interpolation inequalities on weighted Folland-STEin
Journal of Function Spaces 9
spaces on stratified groupsrdquo Acta Mathematica Sinica vol 16no 3 pp 405ndash444 2000
[15] E W Stredulinsky ldquoHigher integrability from reverse Holderinequalitiesrdquo Indiana University Mathematics Journal vol 29no 3 pp 407ndash413 1980
[16] F W Gehring ldquoThe L119901-integrability of the partial derivativesof a quasiconformal mappingrdquo Acta Mathematica vol 130 pp265ndash277 1973
[17] W S Cohn G Lu and S Lu ldquoHigher order Poincare inequal-ities associated with linear operators on stratified groups andapplicationsrdquoMathematische Zeitschrift vol 244 no 2 pp 309ndash335 2003
[18] S Ding and B Liu ldquoDirac-harmonic equations for differentialformsrdquo Nonlinear Analysis Theory Methods and ApplicationsAn International Multidisciplinary Journal vol 122 pp 43ndash572015
[19] C A Nolder ldquoHardy-Littlewood theorems for 119860-harmonictensorsrdquo Illinois Journal of Mathematics vol 43 no 4 pp 613ndash632 1999
[20] A Carbonaro A McIntosh and A J Morris ldquoLocal Hardyspaces of differential forms on Riemannian manifoldsrdquo TheJournal of Geometric Analysis vol 23 no 1 pp 106ndash169 2013
[21] K Kodaira ldquoHarmonic fields in Riemannian manifolds (gener-alized potential theory)rdquo Annals of Mathematics Second Seriesvol 50 pp 587ndash665 1949
[22] R R Coifman R Rochberg and G Weiss ldquoFactorizationtheorems for Hardy spaces in several variablesrdquo Annals ofMathematics vol 103 no 3 pp 611ndash635 1976
[23] L Chaffee and D Cruz-Uribe ldquoNecessary conditions for theboundedness of linear and bilinear commutators on Banachfunction spacesrdquo Mathematical Inequalities and Applicationsvol 21 no 1 pp 1ndash16 2018
[24] C Segovia and J L Torrea ldquoWeighted inequalities for com-mutators of fractional and singular integralsrdquo PublicacionsMatematiques vol 35 no 1 pp 209ndash235 1991
[25] C Perez ldquoSharp estimates for commutators of singular integralsvia iterations of the Hardy-Littlewood maximal functionrdquoJournal of Fourier Analysis and Applications vol 3 no 6 pp743ndash756 1997
[26] T Iwaniec and A Lutoborski ldquoIntegral estimates for nullLagrangiansrdquo Archive for Rational Mechanics and Analysis vol125 no 1 pp 25ndash79 1993
[27] S Ding ldquoL120593(120583)-averaging domains and the quasi-hyperbolicmetricrdquo Computers and Mathematics with Applications AnInternational Journal vol 47 no 10-11 pp 1611ndash1618 2004
[28] A P Calderon ldquoCommutators of singular integral operatorsrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 53 pp 1092ndash1099 1965
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Journal of Function Spaces 5
On the other hand if the measure 120583119909 isin 119861 |[119887 119879Ω]119906 minus([119887 119879Ω]119906)119861| gt 0 gt 0 Applying Lemma 5 to [119887 119879Ω]119906 we get(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198625 (119899 119901) (int119861
1003816100381610038161003816119889 [119887 119879Ω] 1199061003816100381610038161003816119901 119889119909)1119901 (32)
Then by Lemma 10 it follows that
(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198626 (119899 119901) (int119861|119906|119901 119889119909)1119901
(33)
Noticing that the measure |119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0| gt 0 we could use Lemma 6 Taking 120595(119905) = 119905119899119901(119899minus119901) inLemma 6 we have
(int119861|V|119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198627 (int119861
1003816100381610038161003816V minus V1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901(34)
for any differential form V and any ball B with |119909 isin 119861 |V minusV119861| gt 0| gt 0 Replacing V by [119887 119879Ω]119906 in (34) we obtain
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901 le 1198628 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
(35)
Combining (33) and (35) yields that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
le 1198629 (int119861|119906|119901 119889119909)1119901
(36)
In view of the monotonic property of the 119871119901-space with 0 lt119904 lt 119899119901(119899 minus 119901) we have( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)
1119904
le ( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119901(119899minus119901) 119889119909)(119899minus119901)119899119901
(37)
Substituting (36) and (37) gives
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904
le 11986210120583 (119861)1119904minus1119901+1119899 (int119861|119906|119901 119889119909)1119901
(38)
Therefore inequality (29) follows in both cases which indi-cates that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879]119906 isin 119871119904119897119900119888(119872Λ119897) Wehave completed the proof of Theorem 12
Moreover inequality (38) can be written as the followingintegral average inequality
( 1120583 (119861) int119861 1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904
le 11986211120583 (119861)1119899 ( 1120583 (119861) int120590119861 |119906|119901 119889119909)1119901
(39)
Clearly with 119901 close to 119899 the integral exponent 119904 on thelefthand side could bemuch larger than the integral exponent119901 on the right hand side since the condition 0 lt 119904 lt 119899119901(119899 minus119901) Hence the higher integrability of operator [119887 119879Ω] for thecase that 1 lt 119901 lt 119899 is obtained
Next we consider the higher integrability of [119887 119879Ω] forthe case 119901 ge 119899Theorem 13 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904119897119900119888(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119861 le 119862120583 (119861)1119904+1119899minus1119901 119906119901120590119861 (40)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof It is immediate by the same method developed in theproof of Theorem 12 that (40) holds for 120583119909 isin 119861 |[119887 119879Ω]119906 minus([119887 119879Ω]119906)119861| gt 0 = 0 Let us consider the remain case that120583119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt 0 gt 0 Select 120572 =max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Then it is easy to checkthat
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (41)
since 119899 minus 119901 le 0 Notice that 1 lt 119902 = 120572119899119901(119899 + 120572119901) lt 119899Using Lemma 5 to [119887 119879Ω]119906 and combining Lemma 10 and themonotonic property of the 119871119901-space we have(int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198621 (int119861
1003816100381610038161003816119889 [119887 119879Ω] 1199061003816100381610038161003816119902 119889119909)1119902
le 1198622 (int119861|119906|119902 119889119909)1119902
le 1198623120583 (119861)1119902minus1119901 (int119861|119906|119901 119889119909)1119901
(42)
Notice that the measure |119909 isin 119861 |[119887 119879Ω]119906 minus ([119887 119879Ω]119906)119861| gt0| gt 0 Therefore we can select 120595(119905) = 119905119899119902(119899minus119902) in Lemma 6Then for any differential form 119908 we have
(int119861|119908|119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198624 (int119861
1003816100381610038161003816119908 minus 1199081198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 (43)
6 Journal of Function Spaces
Replacing 119908 by [119887 119879Ω]119906 in (43) we get
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 le 1198625 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
(44)
Taking into account the fact that 119899119902(119899 minus 119902) = 120572119901 gt 119904 we seefrom the monotonic property of the 119871119901-space (42) and (44)that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904 le 1198626120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902le 1198627120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198628120583 (119861)1119902minus1119901+1119904minus1120572119901 (int119861|119906|119901 119889119909)1119901
= 1198628120583 (119861)1119899+1119904minus1119901 (int119861|119906|119901 119889119909)1119901
(45)
which means that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879Ω]119906 isin119871119904119897119900119888(119872Λ119897)We have completed the proof ofTheorem 13
Now we are ready to assert the global higher integrabilityof the commutator of Calderon-Zygmund singular integraloperator on differential forms
Theorem 14 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 0 lt 119904 lt 119899119901(119899 minus 119901) Moreoverthere exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (46)
for any bounded domain119872 sub R119899Proof Notice that 1119904+1119899minus1119901 gt 0 since 0 lt 119904 lt 119899119901(119899minus119901)By Lemma 7 andTheorem 12 we have
[119887 119879] 119906119904119872 le sum119861isin][119887 119879] 119906119904119861
le sum119861isin](1198621120583 (119861)1119904+1119899minus1119901 119906119901120590119861)
le sum119861isin](1198621120583 (119872)1119904+1119899minus1119901 119906119901120590119861)
le 1198622120583 (119872)1119904+1119899minus1119901119873119906119901119872le 1198623 119906119901119872
(47)
which finishes the proof of Theorem 14
Using the similar method as we did in Theorem 14 andcombining with Theorem 13 we can deduce the followingglobal result for the case 119901 ge 119899Theorem 15 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant C independent of u such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (48)
for any bounded domain119872 sub R1198994 Higher Order Poincareacute-Type Inequalities
In this section we shall state the higher order Poincare-typeinequalities for commutator of Calderon-Zygmund singularintegral operator acting on the solutions of the Dirac-harmonic equations
Theorem 16 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (49)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof For any 119906 isin 119871119901119897119900119888(119872Λ119897) using the 119871119901-decompositionformula (9) to [119887 119879Ω]119906 we obtain
[119887 119879Ω] 119906 = 119889119879 ([119887 119879Ω] 119906) + 119879119889 ([119887 119879Ω] 119906) (50)
Noticing that 119889119879([119887119879Ω]119906) = ([119887 119879Ω]119906)119861 for any differentialform 119906 by (50) (10) and Lemma 10 we get1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901(119899minus119901)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198621 (119899 119901) 120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198622 (119899 119901) 120583 (119861)1+1119899 119906119899119901(119899minus119901)119861
(51)
Since 119906 is a solution of the Dirac-harmonic equation (1) byLemma 2 we have
119906119899119901(119899minus119901)119861 le 120583 (119861)minus1119899 119906119901120590119861 (52)
where120590 gt 1 is a constant Substituting (52) into (51) gives that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861le 1198623 (119899 119901) 120583 (119861) 119906119901120590119861 (53)
Journal of Function Spaces 7
Moreover for any 0 lt 119904 lt 119899119901(119899minus119901) applying the monotonicproperties of 119871119901-space it follows that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119901+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861 (54)
Combined with (53) the above inequality becomes
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198624 (119899 119901) 120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (55)
which finishes the proof of Theorem 16
Next we will prove the higher order Poincare-typeinequality still holds for the case that 119901 ge 119899Theorem 17 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of the Dirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then for any119904 gt 0 there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (56)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof Select 120572 = max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Thenwe easily obtain
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (57)
that is 119902 lt 119901 since 119899 minus 119901 le 0 We can also find 1 lt 119902 =120572119899119901(119899 + 120572119901) lt 119899 Using the same technique as in the proofof Theorem 16 we have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902(119899minus119902)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198621120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198622120583 (119861)1+1119899 119906119899119902(119899minus119902)119861le 1198623120583 (119861)1+1119899 120583 (119861)1119902minus1119899minus1119901 119906119901120590119861le 1198624120583 (119861)1+1119902minus1119901 119906119901120590119861
(58)
Noticing that 119899119902(119899 minus 119902) = 120572119901 gt 119904 and combining themonotonic property of the 119871119901-space we obtain1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119902+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902119899minus119902119861 (59)
Combining (58) and (59) we finally have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198625120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (60)
The proof of Theorem 17 is completed
Based onTheorems 16 and 17 we can obtain the followingglobal higher order Poincare-type inequalities for the com-mutator [119887 119879Ω] using the analogous method developed inTheorem 14
Theorem 18 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (61)
for any bounded domain119872 sub R119899Theorem 19 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then forany 119904 gt 0 there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (62)
for any bounded domain119872 sub R1198995 Applications
In this section we demonstrate some applications of ourmainresults established in the previous sections
Example 20 Let 119903 gt 0 and 119896 gt 0 be any constants and1198721 =(1199091 1199092 1199093) 11990921 + 11990922 + 11990923 le 1199032 sub R3 Consider the 1-form119906 (1199091 1199092 1199093) = 1199091119896 + 11990921 + 11990922 + 11990923 1198891199091
+ 1199092119896 + 11990921 + 11990922 + 11990923 1198891199092+ 1199093119896 + 11990921 + 11990922 + 11990923 1198891199093
(63)
defined in119872 such that1198721 sub 119872 It is easy to check that 119889119906 =0 Hence 119906 is a solution of the Dirac-harmonic equation (1)
8 Journal of Function Spaces
for any operators 119860 satisfying (2) Also it can be calculatedthat
|119906| = (( 1199091119896 + 11990921 + 11990922 + 11990923)2 + ( 1199092119896 + 11990921 + 11990922 + 11990923)
2
+ ( 1199093119896 + 11990921 + 11990922 + 11990923)2)12
= ( 11990921 + 11990922 + 11990923(119896 + 11990921 + 11990922 + 11990923)2)12 lt 1
(64)
for any 119909 = (1199091 1199092 1199093) isin 1198721 Thus
1199061199011198721 = (int1198721|119906|119901 119889119909)1119901
= (int1198721
100381610038161003816100381610038161003816100381610038161003816100381611990921 + 11990922 + 119909231003816100381610038161003816119896 + 11990921 + 11990922 + 1199092310038161003816100381610038162
10038161003816100381610038161003816100381610038161003816100381610038161199012 119889119909)
1119901
le (int1198721
1119889119909)1119901 = (41205873 1199033)1119901
(65)
Applying Theorem 14 for 1 lt 119901 lt 3 we have [119887 119879Ω]119906 isin119871119904(119872Λ119897) for any 0 lt 119904 lt 3119901(3 minus 119901) Analogously we have[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0when119901 ge 3 byTheorem 15In the meantime we have the higher estimate for [119887 119879Ω]119906that is
1003817100381710038171003817[119887 119879Ω] 11990610038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (66)
In addition we can evaluate the following integrals byTheorem 18 andTheorem 19 that
10038171003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721100381710038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (67)
We should notice that the above example can be extendedto the case of R119899 as follows
Example 21 We can check that the 1-form defined inR119899
119906 (1199091 119909119899) = 119899sum119894=1
119909119894119896 + 11990921 + sdot sdot sdot + 1199092119899 119889119909119894 119896 gt 0 (68)
is a solution of the Dirac-harmonic equation (1) for anyoperators 119860 satisfying (2) Hence Theorems 14 15 18 and 19are also applicable to 119906(1199091 119909119899)Remark It is worth pointing out that the global resultsobtained in this paper can be extended to larger classes ofdomains such as 119871119901-averaging domains and 119871120593(120583)-averagingdomains see [1 27] Also the techniques developed inthis paper provide an effective method to study the higherintegrability of bilinear commutators of singular integrals ondifferential forms which are defined in [25] We leave thestatements and proofs to the interested readers
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The work was supported by the National Natural ScienceFoundation of China (Grant no 11601105)
References
[1] R P Agarwal S Ding andCNolder Inequalities forDifferentialForms Springer New York NY USA 2009
[2] Y Xing and C Wu ldquoGlobal weighted inequalities for operatorsand harmonic forms on manifoldsrdquo Journal of MathematicalAnalysis and Applications vol 294 no 1 pp 294ndash309 2004
[3] J B Perot and C J Zusi ldquoDifferential forms for scientists andengineersrdquo Journal of Computational Physics vol 257 no partB pp 1373ndash1393 2014
[4] C Scott ldquoL119901-theory of differential forms onmanifolds theory ofdifferential forms on manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 6 pp 2075ndash2096 1995
[5] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004
[6] S Ding and B Liu ldquoA singular integral of the compositeoperatorrdquo Applied Mathematics Letters vol 22 no 8 pp 1271ndash1275 2009
[7] S Ding and B Liu ldquoGlobal estimates for singular integrals ofthe composite operatorrdquo Illinois Journal of Mathematics vol 53no 4 pp 1173ndash1185 2009
[8] S Ding and B Liu ldquoGlobal integrability of the Jacobian of acomposite mappingrdquo Journal of Inequalities and ApplicationsArt ID 89134 9 pages 2006
[9] H Bi and S Ding ldquoSome strong (pq)-type inequalities forthe homotopy operatorrdquo Computers and Mathematics withApplications An International Journal vol 62 no 4 pp 1780ndash1789 2011
[10] H Bi ldquoWeighted inequalities for potential operators on differ-ential formsrdquo Journal of Inequalities and Applications Art ID713625 13 pages 2010
[11] Y Ling and B Gejun ldquoSome local Poincare inequalities for thecomposition of the sharp maximal operator and the Greenrsquosoperatorrdquo Computers and Mathematics with Applications vol63 no 3 pp 720ndash727 2012
[12] S Ding ldquoIntegral estimates for the Laplace-Beltrami andGreenrsquos operators applied to differential forms on manifoldsrdquoJournal for Analysis and its Applications vol 22 no 4 pp 939ndash957 2003
[13] G Lu and R L Wheeden ldquoHigh order representation formulasand embedding theorems on stratified groups and generaliza-tionsrdquo Studia Mathematica vol 142 no 2 pp 101ndash133 2000
[14] G Lu ldquoPolynomials higher order Sobolev extension theo-rems and interpolation inequalities on weighted Folland-STEin
Journal of Function Spaces 9
spaces on stratified groupsrdquo Acta Mathematica Sinica vol 16no 3 pp 405ndash444 2000
[15] E W Stredulinsky ldquoHigher integrability from reverse Holderinequalitiesrdquo Indiana University Mathematics Journal vol 29no 3 pp 407ndash413 1980
[16] F W Gehring ldquoThe L119901-integrability of the partial derivativesof a quasiconformal mappingrdquo Acta Mathematica vol 130 pp265ndash277 1973
[17] W S Cohn G Lu and S Lu ldquoHigher order Poincare inequal-ities associated with linear operators on stratified groups andapplicationsrdquoMathematische Zeitschrift vol 244 no 2 pp 309ndash335 2003
[18] S Ding and B Liu ldquoDirac-harmonic equations for differentialformsrdquo Nonlinear Analysis Theory Methods and ApplicationsAn International Multidisciplinary Journal vol 122 pp 43ndash572015
[19] C A Nolder ldquoHardy-Littlewood theorems for 119860-harmonictensorsrdquo Illinois Journal of Mathematics vol 43 no 4 pp 613ndash632 1999
[20] A Carbonaro A McIntosh and A J Morris ldquoLocal Hardyspaces of differential forms on Riemannian manifoldsrdquo TheJournal of Geometric Analysis vol 23 no 1 pp 106ndash169 2013
[21] K Kodaira ldquoHarmonic fields in Riemannian manifolds (gener-alized potential theory)rdquo Annals of Mathematics Second Seriesvol 50 pp 587ndash665 1949
[22] R R Coifman R Rochberg and G Weiss ldquoFactorizationtheorems for Hardy spaces in several variablesrdquo Annals ofMathematics vol 103 no 3 pp 611ndash635 1976
[23] L Chaffee and D Cruz-Uribe ldquoNecessary conditions for theboundedness of linear and bilinear commutators on Banachfunction spacesrdquo Mathematical Inequalities and Applicationsvol 21 no 1 pp 1ndash16 2018
[24] C Segovia and J L Torrea ldquoWeighted inequalities for com-mutators of fractional and singular integralsrdquo PublicacionsMatematiques vol 35 no 1 pp 209ndash235 1991
[25] C Perez ldquoSharp estimates for commutators of singular integralsvia iterations of the Hardy-Littlewood maximal functionrdquoJournal of Fourier Analysis and Applications vol 3 no 6 pp743ndash756 1997
[26] T Iwaniec and A Lutoborski ldquoIntegral estimates for nullLagrangiansrdquo Archive for Rational Mechanics and Analysis vol125 no 1 pp 25ndash79 1993
[27] S Ding ldquoL120593(120583)-averaging domains and the quasi-hyperbolicmetricrdquo Computers and Mathematics with Applications AnInternational Journal vol 47 no 10-11 pp 1611ndash1618 2004
[28] A P Calderon ldquoCommutators of singular integral operatorsrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 53 pp 1092ndash1099 1965
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6 Journal of Function Spaces
Replacing 119908 by [119887 119879Ω]119906 in (43) we get
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902 le 1198625 (int119861
1003816100381610038161003816[119887 119879Ω] 119906minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
(44)
Taking into account the fact that 119899119902(119899 minus 119902) = 120572119901 gt 119904 we seefrom the monotonic property of the 119871119901-space (42) and (44)that
(int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119904 119889119909)1119904 le 1198626120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 1199061003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902le 1198627120583 (119861)1119904minus1120572119901sdot (int119861
1003816100381610038161003816[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003816100381610038161003816119899119902(119899minus119902) 119889119909)(119899minus119902)119899119902
le 1198628120583 (119861)1119902minus1119901+1119904minus1120572119901 (int119861|119906|119901 119889119909)1119901
= 1198628120583 (119861)1119899+1119904minus1119901 (int119861|119906|119901 119889119909)1119901
(45)
which means that if 119906 isin 119871119901119897119900119888(119872Λ119897) then [119887 119879Ω]119906 isin119871119904119897119900119888(119872Λ119897)We have completed the proof ofTheorem 13
Now we are ready to assert the global higher integrabilityof the commutator of Calderon-Zygmund singular integraloperator on differential forms
Theorem 14 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 1 lt 119901 lt 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 0 lt 119904 lt 119899119901(119899 minus 119901) Moreoverthere exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (46)
for any bounded domain119872 sub R119899Proof Notice that 1119904+1119899minus1119901 gt 0 since 0 lt 119904 lt 119899119901(119899minus119901)By Lemma 7 andTheorem 12 we have
[119887 119879] 119906119904119872 le sum119861isin][119887 119879] 119906119904119861
le sum119861isin](1198621120583 (119861)1119904+1119899minus1119901 119906119901120590119861)
le sum119861isin](1198621120583 (119872)1119904+1119899minus1119901 119906119901120590119861)
le 1198622120583 (119872)1119904+1119899minus1119901119873119906119901119872le 1198623 119906119901119872
(47)
which finishes the proof of Theorem 14
Using the similar method as we did in Theorem 14 andcombining with Theorem 13 we can deduce the followingglobal result for the case 119901 ge 119899Theorem 15 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901(119872Λ119897) 119897 = 1 2 119899 119901 ge 119899 then[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0 Moreover there exists aconstant C independent of u such that
1003817100381710038171003817[119887 119879Ω] 1199061003817100381710038171003817119904119872 le 119862120583 (119872)1119904+1119899minus1119901 119906119901119872 (48)
for any bounded domain119872 sub R1198994 Higher Order Poincareacute-Type Inequalities
In this section we shall state the higher order Poincare-typeinequalities for commutator of Calderon-Zygmund singularintegral operator acting on the solutions of the Dirac-harmonic equations
Theorem 16 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (49)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof For any 119906 isin 119871119901119897119900119888(119872Λ119897) using the 119871119901-decompositionformula (9) to [119887 119879Ω]119906 we obtain
[119887 119879Ω] 119906 = 119889119879 ([119887 119879Ω] 119906) + 119879119889 ([119887 119879Ω] 119906) (50)
Noticing that 119889119879([119887119879Ω]119906) = ([119887 119879Ω]119906)119861 for any differentialform 119906 by (50) (10) and Lemma 10 we get1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901(119899minus119901)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198621 (119899 119901) 120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119901(119899minus119901)119861le 1198622 (119899 119901) 120583 (119861)1+1119899 119906119899119901(119899minus119901)119861
(51)
Since 119906 is a solution of the Dirac-harmonic equation (1) byLemma 2 we have
119906119899119901(119899minus119901)119861 le 120583 (119861)minus1119899 119906119901120590119861 (52)
where120590 gt 1 is a constant Substituting (52) into (51) gives that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861le 1198623 (119899 119901) 120583 (119861) 119906119901120590119861 (53)
Journal of Function Spaces 7
Moreover for any 0 lt 119904 lt 119899119901(119899minus119901) applying the monotonicproperties of 119871119901-space it follows that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119901+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861 (54)
Combined with (53) the above inequality becomes
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198624 (119899 119901) 120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (55)
which finishes the proof of Theorem 16
Next we will prove the higher order Poincare-typeinequality still holds for the case that 119901 ge 119899Theorem 17 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of the Dirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then for any119904 gt 0 there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (56)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof Select 120572 = max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Thenwe easily obtain
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (57)
that is 119902 lt 119901 since 119899 minus 119901 le 0 We can also find 1 lt 119902 =120572119899119901(119899 + 120572119901) lt 119899 Using the same technique as in the proofof Theorem 16 we have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902(119899minus119902)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198621120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198622120583 (119861)1+1119899 119906119899119902(119899minus119902)119861le 1198623120583 (119861)1+1119899 120583 (119861)1119902minus1119899minus1119901 119906119901120590119861le 1198624120583 (119861)1+1119902minus1119901 119906119901120590119861
(58)
Noticing that 119899119902(119899 minus 119902) = 120572119901 gt 119904 and combining themonotonic property of the 119871119901-space we obtain1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119902+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902119899minus119902119861 (59)
Combining (58) and (59) we finally have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198625120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (60)
The proof of Theorem 17 is completed
Based onTheorems 16 and 17 we can obtain the followingglobal higher order Poincare-type inequalities for the com-mutator [119887 119879Ω] using the analogous method developed inTheorem 14
Theorem 18 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (61)
for any bounded domain119872 sub R119899Theorem 19 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then forany 119904 gt 0 there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (62)
for any bounded domain119872 sub R1198995 Applications
In this section we demonstrate some applications of ourmainresults established in the previous sections
Example 20 Let 119903 gt 0 and 119896 gt 0 be any constants and1198721 =(1199091 1199092 1199093) 11990921 + 11990922 + 11990923 le 1199032 sub R3 Consider the 1-form119906 (1199091 1199092 1199093) = 1199091119896 + 11990921 + 11990922 + 11990923 1198891199091
+ 1199092119896 + 11990921 + 11990922 + 11990923 1198891199092+ 1199093119896 + 11990921 + 11990922 + 11990923 1198891199093
(63)
defined in119872 such that1198721 sub 119872 It is easy to check that 119889119906 =0 Hence 119906 is a solution of the Dirac-harmonic equation (1)
8 Journal of Function Spaces
for any operators 119860 satisfying (2) Also it can be calculatedthat
|119906| = (( 1199091119896 + 11990921 + 11990922 + 11990923)2 + ( 1199092119896 + 11990921 + 11990922 + 11990923)
2
+ ( 1199093119896 + 11990921 + 11990922 + 11990923)2)12
= ( 11990921 + 11990922 + 11990923(119896 + 11990921 + 11990922 + 11990923)2)12 lt 1
(64)
for any 119909 = (1199091 1199092 1199093) isin 1198721 Thus
1199061199011198721 = (int1198721|119906|119901 119889119909)1119901
= (int1198721
100381610038161003816100381610038161003816100381610038161003816100381611990921 + 11990922 + 119909231003816100381610038161003816119896 + 11990921 + 11990922 + 1199092310038161003816100381610038162
10038161003816100381610038161003816100381610038161003816100381610038161199012 119889119909)
1119901
le (int1198721
1119889119909)1119901 = (41205873 1199033)1119901
(65)
Applying Theorem 14 for 1 lt 119901 lt 3 we have [119887 119879Ω]119906 isin119871119904(119872Λ119897) for any 0 lt 119904 lt 3119901(3 minus 119901) Analogously we have[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0when119901 ge 3 byTheorem 15In the meantime we have the higher estimate for [119887 119879Ω]119906that is
1003817100381710038171003817[119887 119879Ω] 11990610038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (66)
In addition we can evaluate the following integrals byTheorem 18 andTheorem 19 that
10038171003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721100381710038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (67)
We should notice that the above example can be extendedto the case of R119899 as follows
Example 21 We can check that the 1-form defined inR119899
119906 (1199091 119909119899) = 119899sum119894=1
119909119894119896 + 11990921 + sdot sdot sdot + 1199092119899 119889119909119894 119896 gt 0 (68)
is a solution of the Dirac-harmonic equation (1) for anyoperators 119860 satisfying (2) Hence Theorems 14 15 18 and 19are also applicable to 119906(1199091 119909119899)Remark It is worth pointing out that the global resultsobtained in this paper can be extended to larger classes ofdomains such as 119871119901-averaging domains and 119871120593(120583)-averagingdomains see [1 27] Also the techniques developed inthis paper provide an effective method to study the higherintegrability of bilinear commutators of singular integrals ondifferential forms which are defined in [25] We leave thestatements and proofs to the interested readers
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The work was supported by the National Natural ScienceFoundation of China (Grant no 11601105)
References
[1] R P Agarwal S Ding andCNolder Inequalities forDifferentialForms Springer New York NY USA 2009
[2] Y Xing and C Wu ldquoGlobal weighted inequalities for operatorsand harmonic forms on manifoldsrdquo Journal of MathematicalAnalysis and Applications vol 294 no 1 pp 294ndash309 2004
[3] J B Perot and C J Zusi ldquoDifferential forms for scientists andengineersrdquo Journal of Computational Physics vol 257 no partB pp 1373ndash1393 2014
[4] C Scott ldquoL119901-theory of differential forms onmanifolds theory ofdifferential forms on manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 6 pp 2075ndash2096 1995
[5] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004
[6] S Ding and B Liu ldquoA singular integral of the compositeoperatorrdquo Applied Mathematics Letters vol 22 no 8 pp 1271ndash1275 2009
[7] S Ding and B Liu ldquoGlobal estimates for singular integrals ofthe composite operatorrdquo Illinois Journal of Mathematics vol 53no 4 pp 1173ndash1185 2009
[8] S Ding and B Liu ldquoGlobal integrability of the Jacobian of acomposite mappingrdquo Journal of Inequalities and ApplicationsArt ID 89134 9 pages 2006
[9] H Bi and S Ding ldquoSome strong (pq)-type inequalities forthe homotopy operatorrdquo Computers and Mathematics withApplications An International Journal vol 62 no 4 pp 1780ndash1789 2011
[10] H Bi ldquoWeighted inequalities for potential operators on differ-ential formsrdquo Journal of Inequalities and Applications Art ID713625 13 pages 2010
[11] Y Ling and B Gejun ldquoSome local Poincare inequalities for thecomposition of the sharp maximal operator and the Greenrsquosoperatorrdquo Computers and Mathematics with Applications vol63 no 3 pp 720ndash727 2012
[12] S Ding ldquoIntegral estimates for the Laplace-Beltrami andGreenrsquos operators applied to differential forms on manifoldsrdquoJournal for Analysis and its Applications vol 22 no 4 pp 939ndash957 2003
[13] G Lu and R L Wheeden ldquoHigh order representation formulasand embedding theorems on stratified groups and generaliza-tionsrdquo Studia Mathematica vol 142 no 2 pp 101ndash133 2000
[14] G Lu ldquoPolynomials higher order Sobolev extension theo-rems and interpolation inequalities on weighted Folland-STEin
Journal of Function Spaces 9
spaces on stratified groupsrdquo Acta Mathematica Sinica vol 16no 3 pp 405ndash444 2000
[15] E W Stredulinsky ldquoHigher integrability from reverse Holderinequalitiesrdquo Indiana University Mathematics Journal vol 29no 3 pp 407ndash413 1980
[16] F W Gehring ldquoThe L119901-integrability of the partial derivativesof a quasiconformal mappingrdquo Acta Mathematica vol 130 pp265ndash277 1973
[17] W S Cohn G Lu and S Lu ldquoHigher order Poincare inequal-ities associated with linear operators on stratified groups andapplicationsrdquoMathematische Zeitschrift vol 244 no 2 pp 309ndash335 2003
[18] S Ding and B Liu ldquoDirac-harmonic equations for differentialformsrdquo Nonlinear Analysis Theory Methods and ApplicationsAn International Multidisciplinary Journal vol 122 pp 43ndash572015
[19] C A Nolder ldquoHardy-Littlewood theorems for 119860-harmonictensorsrdquo Illinois Journal of Mathematics vol 43 no 4 pp 613ndash632 1999
[20] A Carbonaro A McIntosh and A J Morris ldquoLocal Hardyspaces of differential forms on Riemannian manifoldsrdquo TheJournal of Geometric Analysis vol 23 no 1 pp 106ndash169 2013
[21] K Kodaira ldquoHarmonic fields in Riemannian manifolds (gener-alized potential theory)rdquo Annals of Mathematics Second Seriesvol 50 pp 587ndash665 1949
[22] R R Coifman R Rochberg and G Weiss ldquoFactorizationtheorems for Hardy spaces in several variablesrdquo Annals ofMathematics vol 103 no 3 pp 611ndash635 1976
[23] L Chaffee and D Cruz-Uribe ldquoNecessary conditions for theboundedness of linear and bilinear commutators on Banachfunction spacesrdquo Mathematical Inequalities and Applicationsvol 21 no 1 pp 1ndash16 2018
[24] C Segovia and J L Torrea ldquoWeighted inequalities for com-mutators of fractional and singular integralsrdquo PublicacionsMatematiques vol 35 no 1 pp 209ndash235 1991
[25] C Perez ldquoSharp estimates for commutators of singular integralsvia iterations of the Hardy-Littlewood maximal functionrdquoJournal of Fourier Analysis and Applications vol 3 no 6 pp743ndash756 1997
[26] T Iwaniec and A Lutoborski ldquoIntegral estimates for nullLagrangiansrdquo Archive for Rational Mechanics and Analysis vol125 no 1 pp 25ndash79 1993
[27] S Ding ldquoL120593(120583)-averaging domains and the quasi-hyperbolicmetricrdquo Computers and Mathematics with Applications AnInternational Journal vol 47 no 10-11 pp 1611ndash1618 2004
[28] A P Calderon ldquoCommutators of singular integral operatorsrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 53 pp 1092ndash1099 1965
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Journal of Function Spaces 7
Moreover for any 0 lt 119904 lt 119899119901(119899minus119901) applying the monotonicproperties of 119871119901-space it follows that1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119901+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119901119899minus119901119861 (54)
Combined with (53) the above inequality becomes
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198624 (119899 119901) 120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (55)
which finishes the proof of Theorem 16
Next we will prove the higher order Poincare-typeinequality still holds for the case that 119901 ge 119899Theorem 17 Let 119879Ω be the Calderon-Zygmund singular inte-gral operator on differential forms and 119887 be sufficiently smoothand bounded If 119906 isin 119871119901119897119900119888(119872Λ119897) is a solution of the Dirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then for any119904 gt 0 there exists a constant 119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 119862120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (56)
for all balls 119861 with 120590119861 sub 119872 for some 120590 gt 1Proof Select 120572 = max1 119904119901 and 119902 = 120572119899119901(119899 + 120572119901) Thenwe easily obtain
119902 minus 119901 = 119901 (120572 (119899 minus 119901)) minus 119899119899 + 120572119901 lt 0 (57)
that is 119902 lt 119901 since 119899 minus 119901 le 0 We can also find 1 lt 119902 =120572119899119901(119899 + 120572119901) lt 119899 Using the same technique as in the proofof Theorem 16 we have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902(119899minus119902)119861= 1003817100381710038171003817119879119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198621120583 (119861) 119889119894119886119898 (119861) 1003817100381710038171003817119889 ([119887 119879Ω] 119906)1003817100381710038171003817119899119902(119899minus119902)119861le 1198622120583 (119861)1+1119899 119906119899119902(119899minus119902)119861le 1198623120583 (119861)1+1119899 120583 (119861)1119902minus1119899minus1119901 119906119901120590119861le 1198624120583 (119861)1+1119902minus1119901 119906119901120590119861
(58)
Noticing that 119899119902(119899 minus 119902) = 120572119901 gt 119904 and combining themonotonic property of the 119871119901-space we obtain1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 120583 (119861)1119904minus1119902+1119899 1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119899119902119899minus119902119861 (59)
Combining (58) and (59) we finally have
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198611003817100381710038171003817119904119861le 1198625120583 (119861)1+1119904minus1119901+1119899 119906119901120590119861 (60)
The proof of Theorem 17 is completed
Based onTheorems 16 and 17 we can obtain the followingglobal higher order Poincare-type inequalities for the com-mutator [119887 119879Ω] using the analogous method developed inTheorem 14
Theorem 18 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 1 lt 119901 lt 119899 Thenfor any 0 lt 119904 lt 119899119901(119899 minus 119901) there exists a constant 119862 gt 0independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (61)
for any bounded domain119872 sub R119899Theorem 19 Let 119879Ω be the Calderon-Zygmund singularintegral operator on differential forms and 119887 be sufficientlysmooth and bounded If 119906 isin 119871119901(119872Λ119897) is a solution of theDirac-harmonic equation (1) 119897 = 1 2 119899 119901 ge 119899 Then forany 119904 gt 0 there exists a constant119862 independent of 119906 such that
1003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721003817100381710038171003817119904119872le 119862120583 (119872)1119899+1119904minus1119901 119906119901119872 (62)
for any bounded domain119872 sub R1198995 Applications
In this section we demonstrate some applications of ourmainresults established in the previous sections
Example 20 Let 119903 gt 0 and 119896 gt 0 be any constants and1198721 =(1199091 1199092 1199093) 11990921 + 11990922 + 11990923 le 1199032 sub R3 Consider the 1-form119906 (1199091 1199092 1199093) = 1199091119896 + 11990921 + 11990922 + 11990923 1198891199091
+ 1199092119896 + 11990921 + 11990922 + 11990923 1198891199092+ 1199093119896 + 11990921 + 11990922 + 11990923 1198891199093
(63)
defined in119872 such that1198721 sub 119872 It is easy to check that 119889119906 =0 Hence 119906 is a solution of the Dirac-harmonic equation (1)
8 Journal of Function Spaces
for any operators 119860 satisfying (2) Also it can be calculatedthat
|119906| = (( 1199091119896 + 11990921 + 11990922 + 11990923)2 + ( 1199092119896 + 11990921 + 11990922 + 11990923)
2
+ ( 1199093119896 + 11990921 + 11990922 + 11990923)2)12
= ( 11990921 + 11990922 + 11990923(119896 + 11990921 + 11990922 + 11990923)2)12 lt 1
(64)
for any 119909 = (1199091 1199092 1199093) isin 1198721 Thus
1199061199011198721 = (int1198721|119906|119901 119889119909)1119901
= (int1198721
100381610038161003816100381610038161003816100381610038161003816100381611990921 + 11990922 + 119909231003816100381610038161003816119896 + 11990921 + 11990922 + 1199092310038161003816100381610038162
10038161003816100381610038161003816100381610038161003816100381610038161199012 119889119909)
1119901
le (int1198721
1119889119909)1119901 = (41205873 1199033)1119901
(65)
Applying Theorem 14 for 1 lt 119901 lt 3 we have [119887 119879Ω]119906 isin119871119904(119872Λ119897) for any 0 lt 119904 lt 3119901(3 minus 119901) Analogously we have[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0when119901 ge 3 byTheorem 15In the meantime we have the higher estimate for [119887 119879Ω]119906that is
1003817100381710038171003817[119887 119879Ω] 11990610038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (66)
In addition we can evaluate the following integrals byTheorem 18 andTheorem 19 that
10038171003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721100381710038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (67)
We should notice that the above example can be extendedto the case of R119899 as follows
Example 21 We can check that the 1-form defined inR119899
119906 (1199091 119909119899) = 119899sum119894=1
119909119894119896 + 11990921 + sdot sdot sdot + 1199092119899 119889119909119894 119896 gt 0 (68)
is a solution of the Dirac-harmonic equation (1) for anyoperators 119860 satisfying (2) Hence Theorems 14 15 18 and 19are also applicable to 119906(1199091 119909119899)Remark It is worth pointing out that the global resultsobtained in this paper can be extended to larger classes ofdomains such as 119871119901-averaging domains and 119871120593(120583)-averagingdomains see [1 27] Also the techniques developed inthis paper provide an effective method to study the higherintegrability of bilinear commutators of singular integrals ondifferential forms which are defined in [25] We leave thestatements and proofs to the interested readers
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The work was supported by the National Natural ScienceFoundation of China (Grant no 11601105)
References
[1] R P Agarwal S Ding andCNolder Inequalities forDifferentialForms Springer New York NY USA 2009
[2] Y Xing and C Wu ldquoGlobal weighted inequalities for operatorsand harmonic forms on manifoldsrdquo Journal of MathematicalAnalysis and Applications vol 294 no 1 pp 294ndash309 2004
[3] J B Perot and C J Zusi ldquoDifferential forms for scientists andengineersrdquo Journal of Computational Physics vol 257 no partB pp 1373ndash1393 2014
[4] C Scott ldquoL119901-theory of differential forms onmanifolds theory ofdifferential forms on manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 6 pp 2075ndash2096 1995
[5] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004
[6] S Ding and B Liu ldquoA singular integral of the compositeoperatorrdquo Applied Mathematics Letters vol 22 no 8 pp 1271ndash1275 2009
[7] S Ding and B Liu ldquoGlobal estimates for singular integrals ofthe composite operatorrdquo Illinois Journal of Mathematics vol 53no 4 pp 1173ndash1185 2009
[8] S Ding and B Liu ldquoGlobal integrability of the Jacobian of acomposite mappingrdquo Journal of Inequalities and ApplicationsArt ID 89134 9 pages 2006
[9] H Bi and S Ding ldquoSome strong (pq)-type inequalities forthe homotopy operatorrdquo Computers and Mathematics withApplications An International Journal vol 62 no 4 pp 1780ndash1789 2011
[10] H Bi ldquoWeighted inequalities for potential operators on differ-ential formsrdquo Journal of Inequalities and Applications Art ID713625 13 pages 2010
[11] Y Ling and B Gejun ldquoSome local Poincare inequalities for thecomposition of the sharp maximal operator and the Greenrsquosoperatorrdquo Computers and Mathematics with Applications vol63 no 3 pp 720ndash727 2012
[12] S Ding ldquoIntegral estimates for the Laplace-Beltrami andGreenrsquos operators applied to differential forms on manifoldsrdquoJournal for Analysis and its Applications vol 22 no 4 pp 939ndash957 2003
[13] G Lu and R L Wheeden ldquoHigh order representation formulasand embedding theorems on stratified groups and generaliza-tionsrdquo Studia Mathematica vol 142 no 2 pp 101ndash133 2000
[14] G Lu ldquoPolynomials higher order Sobolev extension theo-rems and interpolation inequalities on weighted Folland-STEin
Journal of Function Spaces 9
spaces on stratified groupsrdquo Acta Mathematica Sinica vol 16no 3 pp 405ndash444 2000
[15] E W Stredulinsky ldquoHigher integrability from reverse Holderinequalitiesrdquo Indiana University Mathematics Journal vol 29no 3 pp 407ndash413 1980
[16] F W Gehring ldquoThe L119901-integrability of the partial derivativesof a quasiconformal mappingrdquo Acta Mathematica vol 130 pp265ndash277 1973
[17] W S Cohn G Lu and S Lu ldquoHigher order Poincare inequal-ities associated with linear operators on stratified groups andapplicationsrdquoMathematische Zeitschrift vol 244 no 2 pp 309ndash335 2003
[18] S Ding and B Liu ldquoDirac-harmonic equations for differentialformsrdquo Nonlinear Analysis Theory Methods and ApplicationsAn International Multidisciplinary Journal vol 122 pp 43ndash572015
[19] C A Nolder ldquoHardy-Littlewood theorems for 119860-harmonictensorsrdquo Illinois Journal of Mathematics vol 43 no 4 pp 613ndash632 1999
[20] A Carbonaro A McIntosh and A J Morris ldquoLocal Hardyspaces of differential forms on Riemannian manifoldsrdquo TheJournal of Geometric Analysis vol 23 no 1 pp 106ndash169 2013
[21] K Kodaira ldquoHarmonic fields in Riemannian manifolds (gener-alized potential theory)rdquo Annals of Mathematics Second Seriesvol 50 pp 587ndash665 1949
[22] R R Coifman R Rochberg and G Weiss ldquoFactorizationtheorems for Hardy spaces in several variablesrdquo Annals ofMathematics vol 103 no 3 pp 611ndash635 1976
[23] L Chaffee and D Cruz-Uribe ldquoNecessary conditions for theboundedness of linear and bilinear commutators on Banachfunction spacesrdquo Mathematical Inequalities and Applicationsvol 21 no 1 pp 1ndash16 2018
[24] C Segovia and J L Torrea ldquoWeighted inequalities for com-mutators of fractional and singular integralsrdquo PublicacionsMatematiques vol 35 no 1 pp 209ndash235 1991
[25] C Perez ldquoSharp estimates for commutators of singular integralsvia iterations of the Hardy-Littlewood maximal functionrdquoJournal of Fourier Analysis and Applications vol 3 no 6 pp743ndash756 1997
[26] T Iwaniec and A Lutoborski ldquoIntegral estimates for nullLagrangiansrdquo Archive for Rational Mechanics and Analysis vol125 no 1 pp 25ndash79 1993
[27] S Ding ldquoL120593(120583)-averaging domains and the quasi-hyperbolicmetricrdquo Computers and Mathematics with Applications AnInternational Journal vol 47 no 10-11 pp 1611ndash1618 2004
[28] A P Calderon ldquoCommutators of singular integral operatorsrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 53 pp 1092ndash1099 1965
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Journal of Function Spaces
for any operators 119860 satisfying (2) Also it can be calculatedthat
|119906| = (( 1199091119896 + 11990921 + 11990922 + 11990923)2 + ( 1199092119896 + 11990921 + 11990922 + 11990923)
2
+ ( 1199093119896 + 11990921 + 11990922 + 11990923)2)12
= ( 11990921 + 11990922 + 11990923(119896 + 11990921 + 11990922 + 11990923)2)12 lt 1
(64)
for any 119909 = (1199091 1199092 1199093) isin 1198721 Thus
1199061199011198721 = (int1198721|119906|119901 119889119909)1119901
= (int1198721
100381610038161003816100381610038161003816100381610038161003816100381611990921 + 11990922 + 119909231003816100381610038161003816119896 + 11990921 + 11990922 + 1199092310038161003816100381610038162
10038161003816100381610038161003816100381610038161003816100381610038161199012 119889119909)
1119901
le (int1198721
1119889119909)1119901 = (41205873 1199033)1119901
(65)
Applying Theorem 14 for 1 lt 119901 lt 3 we have [119887 119879Ω]119906 isin119871119904(119872Λ119897) for any 0 lt 119904 lt 3119901(3 minus 119901) Analogously we have[119887 119879Ω]119906 isin 119871119904(119872Λ119897) for any 119904 gt 0when119901 ge 3 byTheorem 15In the meantime we have the higher estimate for [119887 119879Ω]119906that is
1003817100381710038171003817[119887 119879Ω] 11990610038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (66)
In addition we can evaluate the following integrals byTheorem 18 andTheorem 19 that
10038171003817100381710038171003817[119887 119879Ω] 119906 minus ([119887 119879Ω] 119906)1198721100381710038171003817100381710038171199041198721 le 119862(41205873 1199033)1119901 (67)
We should notice that the above example can be extendedto the case of R119899 as follows
Example 21 We can check that the 1-form defined inR119899
119906 (1199091 119909119899) = 119899sum119894=1
119909119894119896 + 11990921 + sdot sdot sdot + 1199092119899 119889119909119894 119896 gt 0 (68)
is a solution of the Dirac-harmonic equation (1) for anyoperators 119860 satisfying (2) Hence Theorems 14 15 18 and 19are also applicable to 119906(1199091 119909119899)Remark It is worth pointing out that the global resultsobtained in this paper can be extended to larger classes ofdomains such as 119871119901-averaging domains and 119871120593(120583)-averagingdomains see [1 27] Also the techniques developed inthis paper provide an effective method to study the higherintegrability of bilinear commutators of singular integrals ondifferential forms which are defined in [25] We leave thestatements and proofs to the interested readers
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The work was supported by the National Natural ScienceFoundation of China (Grant no 11601105)
References
[1] R P Agarwal S Ding andCNolder Inequalities forDifferentialForms Springer New York NY USA 2009
[2] Y Xing and C Wu ldquoGlobal weighted inequalities for operatorsand harmonic forms on manifoldsrdquo Journal of MathematicalAnalysis and Applications vol 294 no 1 pp 294ndash309 2004
[3] J B Perot and C J Zusi ldquoDifferential forms for scientists andengineersrdquo Journal of Computational Physics vol 257 no partB pp 1373ndash1393 2014
[4] C Scott ldquoL119901-theory of differential forms onmanifolds theory ofdifferential forms on manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 6 pp 2075ndash2096 1995
[5] S Ding ldquoTwo-weight Caccioppoli inequalities for solutionsof nonhomogeneous 119860-harmonic equations on Riemannianmanifoldsrdquo Proceedings of the American Mathematical Societyvol 132 no 8 pp 2367ndash2375 2004
[6] S Ding and B Liu ldquoA singular integral of the compositeoperatorrdquo Applied Mathematics Letters vol 22 no 8 pp 1271ndash1275 2009
[7] S Ding and B Liu ldquoGlobal estimates for singular integrals ofthe composite operatorrdquo Illinois Journal of Mathematics vol 53no 4 pp 1173ndash1185 2009
[8] S Ding and B Liu ldquoGlobal integrability of the Jacobian of acomposite mappingrdquo Journal of Inequalities and ApplicationsArt ID 89134 9 pages 2006
[9] H Bi and S Ding ldquoSome strong (pq)-type inequalities forthe homotopy operatorrdquo Computers and Mathematics withApplications An International Journal vol 62 no 4 pp 1780ndash1789 2011
[10] H Bi ldquoWeighted inequalities for potential operators on differ-ential formsrdquo Journal of Inequalities and Applications Art ID713625 13 pages 2010
[11] Y Ling and B Gejun ldquoSome local Poincare inequalities for thecomposition of the sharp maximal operator and the Greenrsquosoperatorrdquo Computers and Mathematics with Applications vol63 no 3 pp 720ndash727 2012
[12] S Ding ldquoIntegral estimates for the Laplace-Beltrami andGreenrsquos operators applied to differential forms on manifoldsrdquoJournal for Analysis and its Applications vol 22 no 4 pp 939ndash957 2003
[13] G Lu and R L Wheeden ldquoHigh order representation formulasand embedding theorems on stratified groups and generaliza-tionsrdquo Studia Mathematica vol 142 no 2 pp 101ndash133 2000
[14] G Lu ldquoPolynomials higher order Sobolev extension theo-rems and interpolation inequalities on weighted Folland-STEin
Journal of Function Spaces 9
spaces on stratified groupsrdquo Acta Mathematica Sinica vol 16no 3 pp 405ndash444 2000
[15] E W Stredulinsky ldquoHigher integrability from reverse Holderinequalitiesrdquo Indiana University Mathematics Journal vol 29no 3 pp 407ndash413 1980
[16] F W Gehring ldquoThe L119901-integrability of the partial derivativesof a quasiconformal mappingrdquo Acta Mathematica vol 130 pp265ndash277 1973
[17] W S Cohn G Lu and S Lu ldquoHigher order Poincare inequal-ities associated with linear operators on stratified groups andapplicationsrdquoMathematische Zeitschrift vol 244 no 2 pp 309ndash335 2003
[18] S Ding and B Liu ldquoDirac-harmonic equations for differentialformsrdquo Nonlinear Analysis Theory Methods and ApplicationsAn International Multidisciplinary Journal vol 122 pp 43ndash572015
[19] C A Nolder ldquoHardy-Littlewood theorems for 119860-harmonictensorsrdquo Illinois Journal of Mathematics vol 43 no 4 pp 613ndash632 1999
[20] A Carbonaro A McIntosh and A J Morris ldquoLocal Hardyspaces of differential forms on Riemannian manifoldsrdquo TheJournal of Geometric Analysis vol 23 no 1 pp 106ndash169 2013
[21] K Kodaira ldquoHarmonic fields in Riemannian manifolds (gener-alized potential theory)rdquo Annals of Mathematics Second Seriesvol 50 pp 587ndash665 1949
[22] R R Coifman R Rochberg and G Weiss ldquoFactorizationtheorems for Hardy spaces in several variablesrdquo Annals ofMathematics vol 103 no 3 pp 611ndash635 1976
[23] L Chaffee and D Cruz-Uribe ldquoNecessary conditions for theboundedness of linear and bilinear commutators on Banachfunction spacesrdquo Mathematical Inequalities and Applicationsvol 21 no 1 pp 1ndash16 2018
[24] C Segovia and J L Torrea ldquoWeighted inequalities for com-mutators of fractional and singular integralsrdquo PublicacionsMatematiques vol 35 no 1 pp 209ndash235 1991
[25] C Perez ldquoSharp estimates for commutators of singular integralsvia iterations of the Hardy-Littlewood maximal functionrdquoJournal of Fourier Analysis and Applications vol 3 no 6 pp743ndash756 1997
[26] T Iwaniec and A Lutoborski ldquoIntegral estimates for nullLagrangiansrdquo Archive for Rational Mechanics and Analysis vol125 no 1 pp 25ndash79 1993
[27] S Ding ldquoL120593(120583)-averaging domains and the quasi-hyperbolicmetricrdquo Computers and Mathematics with Applications AnInternational Journal vol 47 no 10-11 pp 1611ndash1618 2004
[28] A P Calderon ldquoCommutators of singular integral operatorsrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 53 pp 1092ndash1099 1965
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Journal of Function Spaces 9
spaces on stratified groupsrdquo Acta Mathematica Sinica vol 16no 3 pp 405ndash444 2000
[15] E W Stredulinsky ldquoHigher integrability from reverse Holderinequalitiesrdquo Indiana University Mathematics Journal vol 29no 3 pp 407ndash413 1980
[16] F W Gehring ldquoThe L119901-integrability of the partial derivativesof a quasiconformal mappingrdquo Acta Mathematica vol 130 pp265ndash277 1973
[17] W S Cohn G Lu and S Lu ldquoHigher order Poincare inequal-ities associated with linear operators on stratified groups andapplicationsrdquoMathematische Zeitschrift vol 244 no 2 pp 309ndash335 2003
[18] S Ding and B Liu ldquoDirac-harmonic equations for differentialformsrdquo Nonlinear Analysis Theory Methods and ApplicationsAn International Multidisciplinary Journal vol 122 pp 43ndash572015
[19] C A Nolder ldquoHardy-Littlewood theorems for 119860-harmonictensorsrdquo Illinois Journal of Mathematics vol 43 no 4 pp 613ndash632 1999
[20] A Carbonaro A McIntosh and A J Morris ldquoLocal Hardyspaces of differential forms on Riemannian manifoldsrdquo TheJournal of Geometric Analysis vol 23 no 1 pp 106ndash169 2013
[21] K Kodaira ldquoHarmonic fields in Riemannian manifolds (gener-alized potential theory)rdquo Annals of Mathematics Second Seriesvol 50 pp 587ndash665 1949
[22] R R Coifman R Rochberg and G Weiss ldquoFactorizationtheorems for Hardy spaces in several variablesrdquo Annals ofMathematics vol 103 no 3 pp 611ndash635 1976
[23] L Chaffee and D Cruz-Uribe ldquoNecessary conditions for theboundedness of linear and bilinear commutators on Banachfunction spacesrdquo Mathematical Inequalities and Applicationsvol 21 no 1 pp 1ndash16 2018
[24] C Segovia and J L Torrea ldquoWeighted inequalities for com-mutators of fractional and singular integralsrdquo PublicacionsMatematiques vol 35 no 1 pp 209ndash235 1991
[25] C Perez ldquoSharp estimates for commutators of singular integralsvia iterations of the Hardy-Littlewood maximal functionrdquoJournal of Fourier Analysis and Applications vol 3 no 6 pp743ndash756 1997
[26] T Iwaniec and A Lutoborski ldquoIntegral estimates for nullLagrangiansrdquo Archive for Rational Mechanics and Analysis vol125 no 1 pp 25ndash79 1993
[27] S Ding ldquoL120593(120583)-averaging domains and the quasi-hyperbolicmetricrdquo Computers and Mathematics with Applications AnInternational Journal vol 47 no 10-11 pp 1611ndash1618 2004
[28] A P Calderon ldquoCommutators of singular integral operatorsrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 53 pp 1092ndash1099 1965
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
