Introductiongrupo.us.es/anaresba/trabajos/carlosperez/CRM-Proceedings-Versio… · , the maximal...

to appear in Birkauser “Advanced courses in Mathematics C.R.M. Barcelona”

A COURSE ON SINGULAR INTEGRALS AND WEIGHTS

CARLOS PEREZ

Abstract. This article is an expanded version of the material covered in a mini-course given at the Centre de Recerca Matematica in Barcelona during the weekMay 4-8, 2009. We provide details and different proofs of known results as well asnew ones. We also survey on several recent results related to the core of this course,namely weighted optimal bounds for Calderon-Zygmund operators with weights. Thebasic topics covered by these lecture revolved around the Rubio de Francia iterationalgorithm, the extrapolation theorem with optimal bounds, the Coifman-Feffermanestimate, the Besicovith covering lemma and the rearrangements of functions. Thepaper can be seen as a modern introduction to the Ap theory of weights.

1. Introduction

The Hardy-Littlewood maximal function is the operator defined by

Mf(x) = supx∈Q

1

|Q|

∫Q

|f(y)| dy

where the supremum is taken over all the cubes containing x, and where f is anylocally integrable function. It is clear that M is not a linear operator but it is a sortof self-dual operator since the following inequality holds

(1.1) supλ>0

λwx ∈ Rn : Mf(x) > λ ≤ c

∫Rn|f |Mwdx,

for any nonnegative functions f and w. It is crucial here that the constant c is indepen-dent of both functions f and w. Here we use the standard notation w(E) =

∫Ew(x)dx

where E is any measurable set. The inequality (1.1) is interesting on its own because

1991 Mathematics Subject Classification. Primary 42B20, 42B25. Secondary 46B70, 47B38.Key words and phrases. Maximal operators, weighted norm inequalities, multilinear singular inte-

grals, Calderon-Zygmund theory, commutators.The author would like to thank Professors Joan Mateu and Joan Orobitg for coordinating two

special minicourses at the Centre de Recerca Mathematica on Multilinear Harmonic Analysis andWeights presented by Professor Loukas Grafakos and by the author during the period May 4-8, 2009.These mincourses were part of a special research program for the academic year 2008-2009 entitled“Harmonic Analysis, Geometric Measure Theory, and Quasiconformal Mappings” coordinated byProfessors Xavier Tolsa and Joan Verdera. The author would also like to thank the Centre de RecercaMatematica for the invitation to spend the semester there and to give this course.

The author would like to acknowledge the support of the grant Spanish Ministry of Science and In-novation grant MTM2009-08934 and the grant from the Junta de Andalucıa, “proyecto de excelencia”FQM-4745 .

1

2 CARLOS PEREZ

it is an improvement of the classical weak-type (1, 1) property of the Hardy-Littlewoodmaximal operator M . However, the crucial new point of view is that it can be seen asa sort of duality for M since the following Lp inequality holds

(1.2)

∫Rn

(Mf)pw dx ≤ cp

∫Rn|f |pMwdx f,w ≥ 0.

This estimate follows from the classical interpolation theorem of Marcinkiewicz. Bothresults (1.1) and (1.2) were proved by C. Fefferman and E.M. Stein in [29] to derive thefollowing vector-valued extension of the classical Hardy-Littlewood maximal theorem:for every 1 < p, q <∞, there is a finite constant c = cp,q such that

(1.3)

∥∥∥∥(∑j

(Mfj)q) 1q

∥∥∥∥Lp(Rn)

≤ c

∥∥∥∥(∑j

|fj|q) 1q

∥∥∥∥Lp(Rn)

.

This is a very deep theorem and has been used a lot in modern harmonic analysisexplaining the central role of inequality (1.1). Nevertheless, the proof of (1.1) does notfollow from the classical maximal theorem (corresponding to the case w ≡ 1) but theproof is nearly identical and is based on a covering lemma of Vitali type as can be seenfor instance in [31] or [27]. We show in Section 2 a simpler and direct proof based onthe classical Besicovith covering lemma.

• The Muckenhoupt-Wheeden conjecture

In these lectures we are mainly interested in corresponding estimates for Calderon-Zygmund operators T instead of M . Here we use the standard concept of Calderon-Zygmund operator as can be found in many places as for instance in [33].

Conjecture 1.1 (The Muckenhoupt-Wheeden conjecture). There exists a constant csuch that for any function f and any weight w

(1.4) supλ>0

λwx ∈ Rn : |Tf(x)| > λ ≤ c

∫Rn|f |Mwdx.

We remark that the author was studying this problem during the 90’s, only muchlater found out that was studied by B. Muckenhoupt and R. Wheeden during the 70’s.In particular, it seems that these authors conjectured that (1.4) should hold for T = H,namely the simplest singular integral operator, the Hilbert transform:

Hf(x) = pv

∫R

f(y)

x− ydy,

We could think that to obtain a vector-valued extension of the classical Calderon-Zygmund tgeorem for singular integral operators similar to (1.3), namely∥∥∥∥(∑

j

|Tfj|q) 1q

∥∥∥∥Lp(Rn)

≤ c

∥∥∥∥(∑j

|fj|q) 1q

∥∥∥∥Lp(Rn)

.

A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 3

we would need to prove (1.4). However, less refined estimates such as

supλ>0

λwx ∈ Rn : |Tf(x)| > λ ≤ cr

∫Rn|f |Mrwdx

would do the job where Mrw = (M(wr))1/r, r > 1. This was shown in [16] by A.Cordoba and C. Fefferman being the key fact that Mrw ∈ A1 if r > 1 (Coifman-Rochberg estimate (5.1)). This result shows that inequalities with weights havingsome additional properties yield interesting results as well.

It seems that the best result obtained is due to the author and can be found in [62]where M is replaced by ML(logL)ε is a “ε-logarithmically” bigger maximal type operatorthan M . If ε = 0 we recover M but the constant cε blows up as ε → 0. The preciseresult is the following.

Theorem 1.2 (The L(logL)ε theorem). There exists a constant c depending on T suchthat for any ε > 0, any function f and any weight w

(1.5) supλ>0

λwx ∈ Rn : |Tf(x)| > λ ≤ c

ε

∫Rn|f |ML(logL)ε(w)dx w ≥ 0.

We remark that the operator ML(logL)ε is pointwise smaller than Mr, r > 1.

Remark 1.3. We remark that very recently and in a joint work with T. Hytonen wehave improved this theorem replacing T by T∗, the maximal singular integral operator.The main difficulty is that T∗ is not linear and the approach used in these notes cannotbe applied. See [39].

The author conjectured in [62] that inequality (1.4) would be false. This was con-firmed in [9] in the case of fractional integrals Iα which are positive operators. How-ever, there were some evidences suggesting that the conjecture could be true. Indeed,the conjecture was confirmed by Chanillo and Wheeden in [11] for certain continuosLttlewood-Paley square function S (see (1.23)) was considered instead of H, namely

(1.6) supt>0

t wx ∈ Rn : Sf(x) > t ≤ c

∫Rn|f |Mwdx w ≥ 0.

Also in [65] a similar result was obtained for the vector-valued maximal function (see(1.3)):

(1.7) supt>0

t w(x ∈ Rn :

(∞∑i=1

(Mfi(x))q

)1/q

> t) ≤ cq

∫Rn|f(x)|qMw(x)dx,

where w ≥ 0, 1 < q <∞. Since many evidences show that the vector-valued maximal

function (∑∞

i=1(Mfi(x))q)1/q

behaves somehow like a singular integral, both inequalities(1.7) and (1.6) suggested that it would be the same for the case of singular integrals.

Muckenhoupt-Wheeden conjecture has been opened even for the Hilbert transformuntil the end of 2010 when was disproved for the Hilbert transform by M. C. Regueraand C. Thiele [74]. Previously, Reguera had shown that (1.4) is false for dyadic type

4 CARLOS PEREZ

singular operators T , more precisely for a special Haar multiplier operator. This wasthe main result of Reguera’s PhD’s thesis [71]. Haar multipliers can be seen as dyadicversions of singular integrals and are used as models to understand them. It is remark-able that Reguera disproved an stronger weighted L2 result of special type, namely ofthe form

(1.8)

∫R|Tf |2w dx ≤ c

∫R|f |2 (

Mw

w)2w dx w ≥ 0.

Indeed, it was shown in [22] that if (1.4) holds for an arbitrary operator T then thisweighted L2 estimate holds for T . In other words, Reguera disproved an strongerinequality than the original one of Muckenhoupt-Wheeden at least for these specialdyadic singular integrals. This result gave strong evidence for a negative answer of theMuckenhoupt-Wheeden conjecture.

This scheme was used in the subsequent paper [74] by Reguera and Thiele wherethey gave a simplified construction of the weight given in [72] and finally showed that(1.4) result is really false for the Hilbert transform H by showing again that (1.8) isfalse for H. This result is really interesting and is related to what is called the A1 con-jecture that we will discuss below and which are really the main motivation of theselecture notes.

• The A1 conjectureThis is a variant of Conjecture 1.1 and the idea is to assume an a priori condition

on the weight w. This condition can be read directly from Fefferman-Stein’s inequality(1.1) and in fact was already introduced by these authors in that paper: the weight wis an A1 weight or satisfies the A1 condition if there is a finite constant c such that

(1.9) Mw ≤ cw a.e.

It is standard to denote by [w]A1 the smallest of these constants c. Then if w ∈ A1

(1.10) supt>0

t wx ∈ Rn : Mf(x) > t ≤ cn [w]A1

∫Rn|f |wdx

and it is natural to ask wether the corresponding inequality holds for singular integrals(say for the Hilbert transform):

Conjecture 1.4 (The A1 conjecture). Let w ∈ A1, then

(1.11) supλ>0

λwx ∈ Rn : |Tf(x)| > λ ≤ c [w]A1

∫Rn|f |wdx.

However, this inequality seems to be false too (see [57]) for T = H, the Hilberttransform.

In this paper we will survey on some recent progress in connection with this conjec-ture exhibiting an extra logarithmic growth in (1.11) which in view of [57] could bethe best possible result. To prove this logarithmic growth result we have to study firstthe corresponding weighted Lp(w) estimates with 1 < p < ∞ and w ∈ A1 being theresult this time fully sharp. The final part of the proofs of both theorem can be found

in Sections 6 and 7 and are essentially taken from [52] and [50].

• Strong type estimatesTo study inequality (1.11), it is natural to ask first the dependence of ‖T‖Lp(w),

p > 1, in terms of [w]A1

. We discuss briefly some results before the papers [50] and

[52].Denote by δ the best possible exponent in the inequality

(1.12) ‖T‖Lp(w)

≤ cn,p[w]δA1,

in the case when p = 2 and T = H is the Hilbert transform, R. Fefferman and J. Pipher[28] established that δ = 1. The proof is based on a sharp A1 bounds for appropriatesquare functions on L2(w) from the works [10, 11], in particular, the following celebratedinequality of Chang-Wilson-Wolff was used:∫

Rn(Sf)2w dx ≤ C

∫Rn|f |2M(w) dx

where S is any of the classical Littlewood-Paley square function as for instance (1.23)(compare with inequality (1.6)). One can show that this approach yields δ = 1 also forp > 2. However, for 1 < p < 2 the same approach gives the estimate δ ≤ 1/2 + 1/p.Also, that approach works only for smooth singular integrals of convolution type andrecall that Calderon-Zygmund operators are non-convolution operators with a veryminimal regularity condition.

In [50] and [52] a different approach was used to show that for any Calderon-Zygmundoperator, the sharp exponent in (1.12) is δ = 1 for all 1 < p < ∞. The method inthis paper has its root in the classical Calderon-Zygmund theory but with several extrarefinements. We believe that the circle of ideas in these papers may lead to anotherproof of the A2 conjecture 1.11 below.

We state now our main theorems. From now on T will always denote any Calderon-Zygmund operator and we assume that the reader is familiar with the classical un-weighted theory.

Theorem 1.5. [The linear growth theorem] Let T be a Calderon-Zygmund operatorand let 1 < p <∞. Then

(1.13) ‖T‖Lp(w) ≤ c pp′ [w]A1

where c = c(n, T ).

As an application of this result we obtain the following endpoint estimate.

Theorem 1.6. [The logarithmic growth theorem] Let T be a Calderon-Zygmund oper-ator. Then

(1.14) ‖T‖L1,∞(w) ≤ c[w]A1(1 + log[w]A1),

where c = c(n, T ).

6 CARLOS PEREZ

Remark 1.7. The result in (1.13) is best possible and, as already mentioned, [57]strongly suggests that (1.14) could also be the best possible result. On the other handand very recently, a new improvement of these two theorems have been found by theauthor and T. Hytonen [38] in terms of mixed A1 − A∞ constants. See Section 9 forsome details, in particular Theorem 9.3 and also Theorem 1.13.

Remark 1.8. As in remark (1.3), these two theorems can be further improved byreplacing T by T∗ the maximal singular integral operator. Again, the method presentedin these notes cannot be applied because is based on the fact that T is linear while T∗is not. See [39].

• The weak (p, p) conjecture and the Rubio de Francia’s algorithm

If we could improve (1.14) by removing the log term, namely if the A1 conjecturewere to hold then we had the following result.

Conjecture 1.9. [The weak (p, p) conjecture] Let 1 < p <∞ and let T be a Calderon-Zygmund singular integral operator. There is a constant c = c(n, T ) such that for anyAp weight w,

(1.15) ‖T‖Lp,∞(w)

≤ c p [w]Ap.

We recall that a weight w satisfies the Muckenhoupt Ap condition if

[w]Ap≡ sup

Q

(1

|Q|

∫Q

w(x)dx

)(1

|Q|

∫Q

w(x)−1/(p−1)dx

)p−1

<∞.

[w]Ap

is usually called the Ap constant (or often called characteristic or norm) of the

weight. The case p = 1 is understood by replacing the right hand side by (infQw)−1

which is equivalent to the definition given above, (1.9).Observe the duality relationship:

[w]1p

Ap= [σ]

1p′

Ap′

where we use the standard notation σ = w−1p−1 = w1−p′ . Also observe that [w]

Ap≥ 1.

In section 4.1 we will prove this conjecture assuming that the A1 conjecture weretrue. The argument will be based on an application of the Rubio de Francia’s algorithmor scheme. The same argument applied to inequality (1.14) yields the following result.

Corollary 1.1. Let 1 < p < ∞ and let T be a Calderon-Zygmund operator. Also letw ∈ Ap, then

(1.16) ‖Tf‖Lp,∞(w) ≤ c p [w]Ap(1 + log[w]Ap)‖f‖Lp(w),

where c = c(n, T ).

Observe that for p close to one, the behavior of the constant is much better thanin (1.21). The advantage here is that this method works for any Calderon-Zygmundoperator.


Rubio de Francia’s algorithm is a technique which has become ubiquitous in themodern theory of weights and beyond. It is also very flexible as will be shown in Sec-tions 3 and 4 as it will be applied to five different scenarios of interest for these lecturenotes. We refer the reader to the monograph [19] for a full account of the technique.

• The A2 conjecture

In [54] B. Muckenhoupt proved the fundamental result characterizing all the weightsfor which the Hardy–Littlewood maximal operator is bounded on Lp(w); the surpris-ingly simple necessary and sufficient condition is the celebrated Ap condition of Muck-enhoupt. Of course the operator norm ‖M‖

Lp(w)will depend on the Ap condition of

w but it seems that the first precise result was proved by S. Buckley [8] as part of hisPh.D. thesis.

Theorem 1.10. Let w ∈ Ap, then the Hardy-Littlewwod maximal function satisfies thefollowing operator estimate:

‖M‖Lp(w)

≤ cn p′ [w]

1p−1

Ap

namely,

(1.17) supw∈Ap

1

[w]1p−1

Ap

‖M‖Lp(w)

≤ cn p′.

Furthermore the result is sharp in the sense that: for any θ > 0

(1.18) supw∈Ap

1

[w]1p−1−θ

Ap

‖M‖Lp(w)

=∞

In fact, we cannot replace the function ψ(t) = t1p−1 by a “smaller” function ψ :

[1,∞)→ (0,∞) in the sense that

inft>1

ψ(t)

t1p−1

= 0

(or limt→∞ψ(t)tβ

= 0, or supt>1tβ

ψ(t)=∞ or limt→∞

tβ

ψ(t)=∞) since then

(1.19) supw∈Ap

1

ψ([w]Ap)‖M‖

Lp(w)=∞

The original proof of Buckley is delicate because is based on a sharp version ofthe so called Reverse Holder Inequality for Ap weights. However, very recently, A.Lerner [46] has found a very nice and simple proof of this result that we will given insection 2. It is based on the Besicovich lemma which can be avoided when M is dyadicmaximal function. To see the power of this new proof we remark that the constant cpin Buckley’s proof cannot be that precise. In fact in these lecture notes we avoid theuse of The Besicovich by considering first the dyadic case and finally “shifting” thedyadic cubes.

8 CARLOS PEREZ

(1.17) should be compared with the weak-type bound

(1.20) ‖M‖Lp(w)→Lp,∞(w) ≤ cn [w]1/pAp

whose proof is much simpler and will be shown in Lemma 2.3.Buckley’s theorem attracted renewed attention after the work of Astala, Iwaniec and

Saksman [3] on the theory of quasirregular mapppings. They proved sharp regularityresults for solutions to the Beltrami equation, assuming that the operator norm of theBeurling-Ahlfors transform grows linearly in terms of the Ap constant for p ≥ 2. Thislinear growth was proved by S. Petermichl and A. Volberg in [70]. This result openedup the possibility of considering some other operators such as the classical HilbertTransform. Finally S. Petermichl [68, 69] has proved the corresponding results for theHilbert transform and the Riesz Transforms. To more precise, in [70] [68, 69] it hasbeen shown that if T is either the Ahlfors-Beurling, Hilbert or Riesz Transforms and1 < p <∞, then

(1.21) ‖T‖Lp(w)

≤ cp,n[w]max1, 1

p−1

Ap.

Furthermore the exponent max1, 1p−1 is best possible by examples similar to the one

related to Theorem 1.10.It should be compared this result with the linear growth theorem 1.5. Indeed, recall

that A1 ⊂ Ap, and that

[w]Ap≤ [w]

A1.

Therefore, (1.21) clearly gives that δ = 1 in (1.12) when p ≥ 2. However, (1.21) cannotbe used in Theorem 1.5 to get the sharp exponent δ in the range 1 < p < 2, becomingthe exponent worst when p gets close to 1.

In view of these results and others (for instance in the case of paraproducts [?] dueto O. Beznosova) It was then believed that the conjecture that should be true is thefollowing.

Conjecture 1.11 (the A2 conjecture). Let 1 < p < ∞ and let T be a Calderon-Zygmund singular integral operator. Then, there is a constant c = c(n, T ) such thatfor any Ap weight w,

(1.22) ‖T‖Lp(w)

≤ c p p′ [w]max

1, 1p−1

Ap

.

This conjecture has been proved by Tuomas Hytonen in [36]. We will briefly mentionin next paragraphs some of the previous steps done toward this conjecture althoughthe main topic of these lectures is more related to the A1 conjecture already mentioned.

The maximum in the exponent reflects the duality of T , namely that T ∗ is alsoa Calderon-Zygmund operator. In fact it can be shown that if T is selfadjoint (oressentially like Calderon-Zygmund operators) and if (1.22) is proved for p > 2 thenthe case 1 < p < 2 follows by duality (see Corollary 3.1). What it is more interestingis that by the sharp Rubio de Francia extrapolation theorem obtained in [26] (or see

Corollary 3.1 again) it is enough to prove (1.22) only for p = 2. This is the reason whythe Ap result has been called the A2 conjecture. Observe that in this case the growthof the constant is simply linear. In these lecture notes we will prove an special case ofthe extrapolation theorem which is enough for our purposes, namely Theorem 3.3 andCorollary 3.1.

In each of the previously known cases, the proof of (1.22) (again, just in the case p = 2by the sharp Rubio de Francia extrapolation theorem) is based on a technique developedby S. Petermichl [68] reducing the problem to proving the analogous inequality for acorresponding Haar shift operator. The norm inequalities for these dyadic operatorswere then proved using Bellman function techniques. Much more recently, Lacey,Petermichl and Reguera-Rodriguez [42] gave a proof of the A2 conjecture for a largefamily of Haar shift operators that includes all the dyadic operators needed for theabove results. Their proof avoids the use of Bellman functions, and instead uses a deep,two-weight testing type “Tb theorem” for Haar shift operators due to Nazarov, Treil andVolberg [56]. A bit later and motivated by [42], a completely different proof was foundby the author together with D. Cruz-Uribe and J.M. Martell in [21] which avoids bothBellman functions and two-weight norm inequalities such as the Tb theorem. Instead,it is used a very interesting and flexible decomposition formula for general functions fdue to Andrei Lerner [48] whose main ideas go back to the work of Fuji [30]. The mainnew ingredient is the use of the local mean oscillation instead of the usual oscillationwhich was the one considered by Fuji. We remit the reader to [21] for details of how toapply Lerner’s formula to generalized Haar shift operators. An important advantageof this approach (again by means of Lerner’s formula) is that it also yields the optimalsharp one weight norm inequalities for other operators such as dyadic square functionsand paraproducts for the vector-valued maximal function of C. Fefferman-Stein as wellas some very sharp two weight “Ap bump” type conditions. All these results can befound in [20].

As a sample we mention the following result for dyadic square functions. Let ∆

denote the collection of dyadic cubes in Rn. Given Q ∈ ∆, let Q be its dyadic parent:the unique dyadic cube containing Q whose side-length is twice that of Q. The dyadicsquare function is the operator

(1.23) Sdf(x) =

(∑Q∈∆

(fQ − fQ)2χQ(x)

)1/2

,

where fQ = 1|Q|

∫Qf. For the properties of the dyadic square function we refer the reader

to Wilson [78].

Theorem 1.12. Given p, 1 < p <∞, then for any w ∈ Ap,

‖Sdf‖Lp(w) ≤ Cn,p[w]max 1

2, 1p−1

Ap‖f‖Lp(w).

Further, the exponent max

12, 1p−1

is the best possible.

10 CARLOS PEREZ

The exponent in Theorem 1.12 was first conjectured by Lerner [45] for the continuoussquare function; he also showed it was the best possible. An interesting fact concerningthe proof of this theorem is that its proof is based again on the sharp Rubio de Franciaextrapolation theorem but the novelty is that the extrapolation hypothesis is for thecase p = 3, instead of p = 2 as in the case of singular integrals. Again, details can befound in [20].

The results mentioned above for generalized Haar shifts could be used to prove the A2

conjecture when the kernel of the Calderon-Zygmund operator was sufficiently smooth(for instance C2 would be enough by applying for instance the approximating result byA. Vagharshakyan [75]). However, this was not enough to prove the full A2 conjecturesince it is assumed that the kernel satisfies merely a Holder-Lipschitz condition. Finallythe conjecture was proved, as already mentioned, by T. Hytonen in August 2010 [36].The proof was based on an important reduction obtained by the author with S. Treiland A. Volberg in [67]. Very roughy this reduction says that a weighted L2 weak typeestimate is essentially equivalent to prove the corresponding strong type. A bit latera direct proof, avoiding this reduction, was found in [40]. One of the key points is touse a probabilistic representation formula due to Hytonen. Then the generalized shiftoperators act as building blocks of this representation. Therefore, an important andhard part of the proof of the A2 conjecture was to obtain bounds for appropriate Haarshifts operators with “complexity (m,n)” that depend at most polynomially on thecomplexity (the problem in [20] is that the method is very flexible but the complexity’sdependence is of exponential type). The estimate obtained in [40] is of polynomialdegree k = 3 that was further improved to linear by [73]. Last result was based on theBellman method using some ideas from another argument given in [58] with a slightworst estimate but with the advantage that can be transferred to the context of dou-bling metric spaces.

• Improving the A2 conjecture, now theorem

On the other hand, and in a direction that we think is more interesting, the A2

conjecture, which is now a theorem, has been improved by the author and T. Hytonen[38] in terms of mixed A2−A∞ constants (see remark 1.7) and in [37] in the general p.We state this new result whose proof is based on a new sharp reverse Holder propertyfor A∞ weights (Theorem 9.1) that can also be found in [38]. The new idea is to deriveresults fractioning the A2 constant in two fractions, one involves the A2 constant assuch and the other involves the A∞.

Theorem 1.13. Let T be a Calderon-Zygmund operator. Then there is a constantdepending on T such that

‖T‖L2(w) ≤ c [w]1/2A2

max[w]A∞ , [w−1]A∞1/2.

See Section 9 for some details of these mixed Ap −A∞ constants mainly in the casep = 1.

• The “fractional”A2 conjecture

We finish this introductory section by mentioning briefly some results about frac-tional integrals that confirm both Conjectures 1.9 and 1.11. Of course these operatorsare different from singular integrals but still is a good indication.

Recall that for 0 < α < n, the fractional integral operator or Riesz potential Iα isdefined, except perhaps for a constant, by

Iαf(x) =

∫Rn

f(y)

|x− y|n−αdy.

In [55], B. Muckenhoupt and R. Wheeden characterized the weighted strong-type in-equality for fractional operators in terms of the so-called Ap,q condition. For 1 < p < n

α

and q defined by 1q

= 1p− α

n, they showed that,

(1.24)

(∫Rn

(wIαf)q dx

)1/q

≤ c

(∫Rn

(wf)p dx

)1/p

f ≥ 0

if and only if w ∈ Ap,q:

[w]Ap,q ≡ supQ

(1

|Q|

∫Q

wq dx

)(1

|Q|

∫Q

w−p′dx

)q/p′<∞.

In [43] it has been shown the following estimates:

Suppose that p, q, α are as above. Then• (The weak estimate)

‖Iαf‖Lq,∞(wq) ≤ c [w]1−α

nAp,q‖w f‖Lp(Rn) f ≥ 0.

• (The strong estimate)

‖wIαf‖Lq,∞(Rn) ≤ c [w](1−α

n) max1, p

′q

Ap,q‖w f‖Lp(Rn) f ≥ 0.

Furthermore both exponents are sharp.

Note that if we formally put α = 0 in these results we may think that the fractionalintegral becomes a singular integral operator and we recover the two conjectures al-ready mentioned.

2. Three applications of the Besicovitch covering lemma to the maximalfunction

In this section we take the maximal function as a model example to understand moredifficult operators. Before considering the strong case, Theorem 1.10, we will show firsta corresponding result for the weak type case. To prove both theorems we will use theclassical Besicovitch covering lemma.

12 CARLOS PEREZ

Lemma 2.1 (The Besicovitch covering lemma). Let K de a bounded set in Rn andsuppose that for every x ∈ K there is an (open) cube Q(x) with center at x. Then wecan find a sequence (possible finite) of points xj in K such that

K ⊂⋃j

Q(xj)

and ∑j

χQ(xj)

≤ cn

where cn is a finite dimensional constant.

The proof of this result can be found in several places such as the classical lecturenotes by M. De Guzman [25], also in [33].

We distinguish two cases, p = 1, and 1 0

λw(x ∈ Rn : Mf(x) > λ) ≤ c

∫Rn|f(x)|Mw(x)dx

and hence

‖M‖L1,∞(w)

≤ cn [w]A1.

Proof. The proof is just an application of the Besicovitch covering lemma. Indeed,assuming that w is bounded as we may, the first observation is that (2.1) is equivalentto

supλ>0

λw(x ∈ Rn : M(fw

Mw)(x) > λ) ≤ C

∫Rn|f(x)|w(x)dx λ > 0.

The second is that we trivially have the pointwise inequality

M(fw

Mw)(x) ≤ cnM

cwf(x),

where M cw is the weighted centered maximal function

(2.2) M cwf(x) = sup

r>0

1

w(Qr(x))

∫Qr(x)

|f(y)|w(y)dy.

Therefore (2.1) follows from

supλ>0

λw(x ∈ Rn : M cwf(x) > λ) ≤ C

∫Rn|f(x)|w(x)dx

which is a consequence of the Besicovitch covering lemma where C si a dimensionalconstant.

Lemma 2.3. Let w ∈ Ap, 1 < p <∞.There is a constant c = cn such that for any Apweight w,

(2.3) ‖M‖Lp,∞(w)

≤ cn [w]1p

Ap,

This result is known but the point we want to make is to compare the exponentof the Ap constant with the other exponents appearing in these lectures notes. Forinstance, compare it with both exponents in Conjecture 1.9 and Theorem 1.5.

Another interesting observation here is that if we consider the dual estimate, theconstant is essentially the same, namely:

(2.4) ‖M‖Lp′,∞(σ)

≤ cn [σ]1p′

Ap′= cn [w]

1p

Ap.

where recall σ = w1−p′ .

Proof of the Lemma. Since w ∈ Ap for each cube Q and nonnegative function f(1

|Q|

∫Q

f(y) dy

)pw(Q) ≤ [w]

Ap

∫Q

f(y)pw(y)dy

and hence

Mf(x) ≈M cf(x) ≤ [w]1p

ApM c

w(fp)(x)1p .

We finish by applying again the Besicovtich covering lemma:

‖Mf‖Lp,∞(w)

≤ cn[w]1p

Ap‖M c

w(fp)1p‖

Lp,∞(w)≤ cn[w]

1p

Ap‖M c

w(fp)‖1p

L1,∞(w)

≤ cn[w]1p

Ap

(∫Rnfpw dx

)1/p

We remark that there are other proofs of these lemmas without appealing the Besi-covitch lemma, just by a Vitali type covery lemma. We leave the proof to the interestedreader.

We now prove Buckley’s Theorem 1.10 based on Lerner’s proof [46]. This proof hasas a bonus an improvement of the original constant cn,p.

Proof of Theorem 1.10. To prove (1.17) we set for any cube Q

Ap(Q) =w(Q)

|Q|

(σ(Q)

|Q|

)p−1

14 CARLOS PEREZ

where as usual σ = w1−p′ . Now, we will consider first case of the dyadic maximalfunction.

1

|Q|

∫Q

|f |=Ap(Q)1p−1

|Q|w(Q)

( 1

σ(Q)

∫Q

|f |)p−1

1p−1

≤ [w]1p−1

Ap

1

w(Q)

∫Q

Mdσ(fσ−1)p−1dx

1p−1

.

where Mdσ is the weighted dyadic maximal function. Hence

Mdf(x) ≤ [w]1p−1

Ap

Md

w

(Md

σ(fσ−1)p−1w−1)(x) 1p−1

We conclude by using that

‖Mdµ‖Lp(µ) ≤ p′

with bounds independent of µ which follows from the improved version of the Marcinkewiczinterpolation theorem in the following form:

‖T‖Lp(µ) ≤ p′ ‖T‖1/p

L1,∞(µ) ‖T‖1/p′

L∞(µ) 1 < p <∞

where T is any sublinear operator bounded on L∞(µ) and of weak type (1, 1) withnorms ‖T‖L∞(µ) and ‖T‖L1,∞(µ) respectively. (see for instance [32] p. 42 exercise 1.3.3).This gives the estimate

‖Md‖Lp(w) ≤ [w]1p−1

Appp′/p p′ ≤ [w]

1p−1

Ape p′

finishing the proof in the case of the dyadic maximal function. Observe that theAp constant here is the dyadic Ap constant. The general situation follows easily by“shifting” the dyadic network applying Minkowski’s inequality to the following well-known Fefferman-Stein shifting lemma that can be found (cf. [31] p. 431):

For each integer k

M2kf(x) ≤ 23n+1

|Q2k+2(0)|

∫Q

2k+2 (0)

(τ−t Md τt

)f(x) dt x ∈ Rn

where τtg(x) = g(x − t), Qr(0) is the cube centered at the origin withside length r, and M δ, δ > 0, is the operator defined as M but withcubes having side length smaller than δ.

For the sharpness we consider n = 1 and 0 < ε < 1. Let

w(x) = |x|(1−ε)(p−1).

It is easy to check that

[w]1p−1

Ap≈ 1

ε.

Let also

f(y) = y−1+ε(p−1) χ(0,1)(y)

and observe that:

‖f‖pLp(w) ≈1

ε

To estimate now ‖Mf‖Lp(w) we pick 0 < x < 1, hence

Mf(x) ≥ 1

x

∫ x

0

f(y) dy = cp1

εf(x)

and hence

‖Mf‖Lp(w) ≥ cp1

ε‖f‖Lp(w)

From which the sharpness (1.18) follows easily.

3. Two applications of the Rubio de Francia algorithm: Optimalfactorization and extrapolation.

3.1. The sharp factorization theorem.Muckenhoupt already observed in [54] that it follows from the definition of the A1

class of weights that if w1, w2 ∈ A1, then the weight

w = w1w1−p2

is an Ap weight. Furthermore we have

(3.1) [w]Ap ≤ [w1]A1 [w2]p−1A1

He conjectured that any Ap weight can be written in this way. This conjecture wasproved by P. Jones’s, namely if w ∈ Ap then there are A1 weights w1, w2 such that

w = w1w1−p2 .

It is also well known that the modern approach to this question uses completelydifferent path and it is due to J. L. Rubio de Francia as can be found in [33] wherewe remit the reader for more information about the Ap theory of weights. Here wepresent a variation which appears in [36]. Here we give a proof of this result usingsharp constants. To be more precise we have the following result.

Lemma 3.1. Let 1 < p <∞ and let w ∈ Ap, then there are A1 weights u, v ∈ A1,such that

w = u v1−p

in such a way that

(3.2) [u]A1 ≤ cn[w]Ap & [v]A1 ≤ cn[w]1p−1

Ap

and hence [w]Ap ≤ [u]A1 [v]p−1A1≤ cn [w]2Ap

16 CARLOS PEREZ

Proof. We use Rubio de Francia’s iteration scheme or algorithm to our situation. Define

S1(f)p′ ≡ w1/pM(

|f |p′

w1/p),

and

S2(f)p ≡ 1

w1/pM(|f |pw1/p),

Observe that Si : Lpp′(Rn)→ Lpp

′(Rn) with constant

‖Si‖Lpp′ (Rn) ≤ cn[w]1/pAp

i = 1, 2

by Buckley’s theorem.Now, the operator S = S1 + S2 is bounded on Lpp

′(Rn) with

‖S‖Lpp′ (Rn) ≤ cn[w]1/pAp.

Define the Rubio de Francia algorithm R by

R(h) ≡∞∑k=0

1

2kSk(h)

(‖S‖Lpp′ (Rn))k.

Observe that R is also bounded on Lpp′(Rn) . Now, if h ∈ Lpp′(Rn) is fixed, R(h) ∈

A1(S). More precisely

S(R(h)) ≤ 2 ‖S‖Lpp′ (Rn) ≤ cn[w]1/pAp.

In particular R(h) ∈ A1(Si) i = 1, 2, with

Si(R(h)) ≤ cn[w]1/pApR(h) i = 1, 2

Hence

M(R(h)p′w−1/p) ≤ cn[w]

p′/pAp

R(h)p′w−1/p

and

M(R(h)pw1/p) ≤ cn[w]Ap R(h)pw1/p.

Finally, letting

u ≡ R(h)pw1/p & v ≡ R(h)p′w−1/p

we have u, v ∈ A1 w = uv1−p with

[u]A1 ≤ cn[w]Ap & [v]A1 ≤ cn[w]1p−1

Ap

3.2. The sharp extrapolation theorem.One of the main results in modern Harmonic Analysis is the extrapolation theorem

of J.L. Rubio de Francia for Ap weights. This result is very useful because reducesmatters to study one special exponent typically p = 2. We refer the reader to themonograph [19] for a new proof and for a full account of the theory. On the otherhand, in [26] it has been shown a version of the extrapolation theorem with sharpconstants which turns out to be very useful. The proof follows the classical method asexposed in [31] and it is based on the Rubio de Francia’s algorithm. Here we will givethis proof.

Theorem 3.2. Let T be any operator such that for some exponent α > 0

(3.3) ‖T‖L2(w) ≤ c [w]αA2

w ∈ A2

then

(3.4) ‖T‖Lp(w) ≤ cp [w]αAp

p > 2, w ∈ Ap

In [26] a corresponding result for 1 < p < 2 can be found, but since all the applica-tions we have in these lecture notes deal with linear operators whose adjoints behavelike the operator itself we have the following corollary.

Corollary 3.1. Let T be a linear operator satisfying (3.3). Suppose also that theadjoint operator T ∗ (with respect to the Lebesgue measure) also satisfies (3.3). Then

‖T‖Lp(w) ≤ cp [w]αmax1, 1

p−1

App > 1, w ∈ Ap

The proof is simply a duality argument. All we have to do is to check the case1 2 and we can apply the theoremto T ∗ because it verifies (3.3) obtaining

‖T ∗‖Lp′ (σ) ≤ c [σ]αAp′ = c [w]αp−1

Ap.

Proof. The proof that follows is taken from Garcia-Cuerva and Rubio de Francia’s book[31]. We only need to be absolutely precise with the exponents. Let

t =p− 2

p− 1

so that

(3.5) ‖T (f)‖2

Lp(w)= sup

h

∫Rn|T (f)(x)|2 h(x)w(x) dx,

where the supremum runs over all 0 ≤ h ∈ Lp′/t(w) with ‖h‖Lp′/t(w) = 1.

18 CARLOS PEREZ

We run the Rubio de Francia algorithm now as follows. Define the operator

Sw(h) =

(M(h1/tw)

w

)th ≥ 0.

It is easy to see that by Muckenhoupt’s theorem Sw is bounded on Lp′/t(w) if w ∈ Ap.

Furthermore if we use Buckley’s theorem 1.10 we have

‖Sw‖Lp′/t(w)≤ cp [w]tAp

Define now

D(h) =∞∑k=0

1

2kSkw(h)

‖Sw‖kLp′/t(w)

Then we have(A) h ≤ D(h)(B) ‖D(h)‖Lp′/t(w) ≤ 2 ‖h‖Lp′/t(w)

(C) Sw(D(h)) ≤ 2 ‖Sw‖Lp′/t(w) D(h) and hence:

(3.6) [D(h).w]A2 ≤ c [w]Ap .

(A) and (B) and the first part of (C) are immediate. It only remains to prove (3.6).First we claim: for any h ≥ 0

[hw, Sw(h)w]A2 = supQ

(1

|Q|

∫Q

hw(x)dx

)(1

|Q|

∫Q

(Sw(h)w)−1dx

)≤ 2 [w]1−tAp

.

Indeed, since for x ∈ Q it verifies that

M(h1/tw)(x) ≥ 1

|Q|

∫Q

h1/tw(x)dx,

we have that

I =

(1

|Q|

∫Q

hw(x)dx

)(1

|Q|

∫Q

(Sw(h)w)−1dx

)=

(1

|Q|

∫Q

hw(x)dx

)(1

|Q|

∫Q

M(h1/tw)−tw−1/(p−1)

)≤(

1

|Q|

∫Q

h1/tw(x)dx

)t(1

|Q|

∫Q

w(x)dx

)1−t

×(

1

|Q|

∫Q

h1/tw(x)dx

)−t(1

|Q|

∫Q

w(x)−1/(p−1)dx

)≤ [w]1−tAp

,

where we used Holder’s inequality and then that (1− t)(1− p) = 1. Hence

[D(h)w]A2 = supQ

(1

|Q|

∫Q

D(h)w(x)dx

)(1

|Q|

∫Q

(D(h)w)−1dx

)≤ 2 ‖Sw‖Lp′/t(w) sup

Q

(1

|Q|

∫Q

D(h)w(x)dx

)(1

|Q|

∫Q

(Sw(D(h))w)−1dx

)= 2 ‖Sw‖Lp′/t(w) [D(h)w, Sw(D(h))w]A2

≤ 2 ‖Sw‖Lp′/t(w) [w]1−tAp

≤ c [w]Ap .

We are ready now to conclude the proof of the theorem using finally the extrapolationhypothesis (3.3). Indeed, fixing one of the h’s from (3.5) we continue with∫

Rn|T (f)(x)|2 h(x)w(x) dx≤

∫Rn|T (f)(x)|2D(h)(x)w(x) dx

≤ [D(h)w]2αA2

∫Rn|f(x)|2D(h)(x)w(x) dx

≤ c [w]2αAp

∫Rn|f(x)|2D(h)(x)w(x) dx

≤ c [w]2αAp ‖f‖2

Lp(w)‖D(h)‖L(p/2)′ (w)

= c [w]2αAp ‖f‖2

Lp(w)‖D(h)‖Lp′/t(w)

≤ c [w]2αAp ‖f‖2

Lp(w)‖h‖Lp′/t(w)

= c [w]2αAp ‖f‖2

Lp(w).

This proves (3.4).

4. Three more applications of the Rubio de Francia algorithm.

We have seen in the previous section two applications of the the Rubio de Franciaalgorithm. These application were already known except for the use of the sharpexponents. In this section we show some other applications, perhaps more technical,but they play a crucial role in the works [50] and [52]. It should be mentioned that otherapplications can be found in [17], [18], [24] and [53]. Also we remit to the monograph[19] for more information.

4.1. Building A1 weights from duality.The following lemma, a variation of the Rubio de Francia iteration scheme, is the

key link for proving Conjecture 1.9 assuming that the A1 conjecture holds.

Lemma 4.1. Let 1 < p <∞ and let w ∈ Ap. Then there exists a nonnegative sublinearoperator D bounded on Lp

′(w) such that for any nonnegative h ∈ Lp′(w):

20 CARLOS PEREZ

(a) h ≤ D(h)(b) ‖D(h)‖Lp′ (w) ≤ 2 ‖h‖Lp′ (w)

(c) D(h) · w ∈ A1 with[D(h) · w]

A1≤ c p [w]

Ap

where the constant c is a dimensional constant.

Proof. To define the algorithm we consider the operator

Sw(f) =M(fw)

wand observe that for any 1 < p <∞, by Muckenhoupt’s theorem

Sw : Lp′(w)→ Lp

′(w) w ∈ Ap

However we need the sharp version in both the constant and the Ap constant (1.17):

‖Sw‖Lp′ (w)≤ cp [w1−p′ ]p−1

Ap′= cp [w]

Ap

Define now for any nonnegative h ∈ Lp′(w)

D(h) =∞∑k=0

1

2kSkw(h)

‖Sw‖kLp′ (w)

Hence properties (a) and (b) are immediate and for (c) simply observe that

Sw(D(h)) ≤ 2 ‖Sw‖Lp′ (w) D(h) ≤ 2c p [w]ApD(h)

or what is the same D(h) · w ∈ A1 with

[D(h) · w]A1≤ 2 c p [w]

Ap

As an application of this lemma we prove Conjecture 1.9 assuming that the A1

conjecture 1.4 holds.

Proof of Conjecture 1.9. Let w ∈ Ap and let f ∈ C∞(Rn) with compact support. Foreach t > 0, let

Ωt = x ∈ Rn : |Tf(x)| > t.This set is bounded, so w(Ωt) < ∞. By duality, there exists a non-negative functionh ∈ Lp′(w) such that ‖h‖Lp′ (w) = 1 and

w(Ωt)1/p = ‖χΩt‖Lp(w) =

∫Ωt

hwdx.

We consider now the operator D associated to this weight from Lemma 4.1. Hencethe operator D satisfies

(a) h ≤ D(h)(b) ‖Dh‖Lp′ (w) ≤ 2 ‖h‖Lp′ (w) = 2

(c) [D(h)w]A1 ≤ c p [w]Ap

hence assuming that the weak Muckenhoupt conjecture holds, then

w(Ωt)1/p ≤

∫Ωt

D(h)w dx = (D(h)w)(Ωt)

≤ c [D(h)w]A1

∫Rn

|f |tD(h)w dx

≤ c

tp[w]Ap

(∫Rn|f |pw dx

)1/p(∫RnD(h)p

′w dx

) 1p′

≤ cp

t[w]Ap

(∫Rn|f |pw dx

)1/p

.

This completes the proof.

4.2. Improving inequalities with A∞ weights.In harmonic analysis, there are a number of important inequalities of the form

(4.1)

∫Rn|Tf(x)|pw(x) dx ≤ C

∫Rn|Sf(x)|pw(x) dx,

where T and S are operators. Typically, T is an operator with some degree of singu-larity (e.g., a singular integral operator), S is an operator which is, in principle, easierto handle (e.g., a maximal operator), and w is in some class of weights.

The standard technique for proving such results is the so-called good-λ inequality ofBurkholder and Gundy. These inequalities compare the relative measure of the levelsets of S and T : for every λ > 0 and ε > 0 small,

(4.2) w(y ∈ Rn : |Tf(y)| > 2λ, |Sf(y)| ≤ λε) ≤ C εw(y ∈ Rn : |Sf(y)| > λ).Here, the weight w is usually assumed to be in the Muckenhoupt class A∞ = ∪p>1Ap.Given inequality (4.2), it is easy to prove the strong-type inequality (4.1) for any p,0 < p <∞, as well as the corresponding weak-type inequality

(4.3) ‖Tf‖Lp,∞(w) ≤ C ‖Sf‖Lp,∞(w).

In these notes the special case of

(4.4) ‖Tf‖Lp(w)

≤ c ‖Mf‖Lp(w)

where T is a Calderon-Zygmund operator and M is the maximal function, will playa central role. This theorem was proved by Coifman-Fefferman in the celebratedpaper[14]. Here, the weight w also satisfies the A∞ condition but the problem isthat the behavior of the constant is too rough. We need a more precise result for veryspecific weights.

Lemma 4.2 (the tricky lemma). Let w be any weight and let 1 ≤ p, r < ∞.Then, there is a constant c = c(n, T ) such that:

‖Tf‖Lp(Mrw)1−p)

≤ cp ‖Mf‖Lp(Mrw)1−p)

22 CARLOS PEREZ

This is the main improvement in [52] of [50] where we had obtained logarithmicgrowth on p. It is an important step towards the proof of the the linear growthTheorem 1.5.

The above mentioned good λ of Coifman-Fefferman is not sharp because instead ofc p gives C(p) ≈ 2p because

[(Mrw)1−p)]Ap≈ (r′)p−1

There is another proof by R. Bagby and D. Kurtz using rearrangements given in themiddle of the 80’s which gives better estimates on p but not in terms of the weightconstant.

The proof of this lemma is tricky and it combines another variation the of Rubio deFrancia algorithm together with a sharp L1 version of (4.4):

(4.5) ‖Tf‖L1(w) ≤ c[w]Aq‖Mf‖L1(w) w ∈ Aq, 1 ≤ q <∞

The original proof given in [52] of this estimate was based on an idea of R. Fefferman-Pipher from [28] which combines a sharp version of the good-λ inequality of S. Buckleytogether with a sharp reverse Holder property of the weights (Lemma 8.1). The result ofBuckley establishes a very interesting exponential improvement of the good-λ estimateof above mentioned Coifman-Fefferman estimate as can be found in [8]:

(4.6) |x ∈ Rn : T ∗(f) > 2λ,Mf < γλ| ≤ c1e−c2/γ|T ∗(f) > λ| λ, γ > 0

where T ∗ is the maximal singular integral operator. This approach is interesting on itsown but we will present in these lecture notes a ‘more efficient approach based on thefollowing estimate: Let 0 t| <∞. Here,

M#δ f(x) = M#(|f |δ)(x)1/δ

and M# is the usual sharp maximal function of Fefferman-Stein:

M#(f)(x) = supQ3x

1

|Q|

∫Q

|f(y)− fQ| dy,

fQ = 1|Q|

∫Qf(y) dy. We present this theory in section 11.

To prove (4.5) we combine (4.7) with the following pointwise estimate [2]:

Lemma 4.3. Let T be any Calderon-Zygmund singular integral operator and let 0 <δ < 1 then there is a constant c such that

M#δ (T (f ))(x) ≤ cMf(x)

and in fact we have.

Corollary 4.1. Let 0 t| <∞.

Hence the heart of the matter is estimate (4.7). It will proved in Section 11 (seeCorollary 11.1) by a completely different path using instead properties of rearrangementof functions and corresponding local sharp maximal operator.

We now finish this section by proving the “tricky” Lemma 4.2. The proof is basedon the following lemma which is another variation of the Rubio de Francia algorithm.

Lemma 4.4. Let 1 < s <∞ and let w be a weight. Then there exists a nonnegativesublinear operator R satisfying the following properties:(a) h ≤ R(h)(b) ‖R(h)‖Ls(w) ≤ 2‖h‖Ls(w)

(c) R(h)w1/s ∈ A1 with

[R(h)w1/s]A1 ≤ cs′

Proof. We consider the operator

S(f) =M(f w1/s)

w1/s

Since ‖M‖Ls ∼ s′, we have

‖S(f)‖Ls(w) ≤ cs′‖f‖Ls(w).

Now, define the Rubio de Francia operator R by

R(h) =∞∑k=0

1

2kSk(h)

(‖S‖Ls(w))k.

It is very simple to check that R satisfies the required properties.

Proof of Lemma 4.2. We are now ready to give the proof of the “tricky” Lemma,namely to prove ∥∥∥∥ Tf

Mrw

∥∥∥∥Lp(Mrw)

≤ cp

∥∥∥∥ Mf

Mrw

∥∥∥∥Lp(Mrw)

By duality we have,∥∥∥∥ Tf

Mrw

∥∥∥∥Lp(Mrw)

= |∫RnTf h dx| ≤

∫Rn|Tf |h dx

for some ‖h‖Lp′ (Mrw) = 1. By Lemma 4.4 with s = p′ and v = Mrw there exists anoperator R such that(A) h ≤ R(h)(B) ‖R(h)‖Lp′ (Mrw) ≤ 2‖h‖Lp′ (Mrw)

(C) [R(h)(Mrw)1/p′ ]A1 ≤ cp.

24 CARLOS PEREZ

We want to make use of property (C) combined with the following two facts: First,if w1, w2 ∈ A1, and w = w1w

1−p2 ∈ Ap, then by (3.1)

[w]Ap ≤ [w1]A1 [w2]p−1A1

Second, if r > 1 then (Mf)1r ∈ A1 by Coifman-Rochberg theorem, furthermore we

need to be more precise (5.1)

[(Mf)1r ]A1≤ cn r

′.

Hence combining we obtain

[R(h)]A3

= [R(h)(Mrw)1/p′((Mrw)1/2p′

)−2]A3

≤ [R(h)(Mrw)1/p′ ]A1 [(Mrw)1/2p′ ]2A1

≤ cp.

Therefore, by Corollary 4.1 and by properties (A) and (B),∫Rn|Tf |h dx≤

∫Rn|Tf |R(h) dx

≤ c[R(h)]A3

∫RnM(f)R(h) dx

≤ cp∥∥∥∥ Mf

Mrw

∥∥∥∥Lp(Mrw)

‖h‖Lp′ (Mrw).

5. The sharp reverse Holder property of the A1 weights

We already encountered with weights of the form Mrw, 1 < r <∞. As we are goingto see they play an important role in the theory. Of course, it is well known that theseweights satisfy the A1 condition by the theorem of Coifman-Rochberg [15]. We will beusing the following quantitative version of it:

Let µ be a positive Borel and let 1 < r <∞(Mµ)

1r ∈ A1

and furthermore

(5.1) [(Mµ)1r ]A1≤ cn r

′.

In fact they proved that any A1 weight can be essentially written in this way

Recall that these weights satisfy an special important property, namely that if w ∈A1, then there is a constnat r > 1 such that(

1

|Q|

∫Q

wr)1/r

≤ c

|Q|

∫Q

w


However there is a bad dependence on the constant c = c(r, [w]A1). To prove our resultswe need a more precise estimate.

Lemma 5.1. Let w ∈ A1, and let rw = 1 + 12n+1[w]A1

. Then for each Q(1

|Q|

∫Q

wrw)1/rw

≤ 2

|Q|

∫Q

w

i.e. Mrww(x) ≤ 2 [w]A1 w(x).

Recall that Mrw = M(wr)1/r

Proof. We start with the “layer cake formula”∫X

ϕ(f) dν =

∫ ∞0

ϕ′(t) ν(x ∈ X : f(x) > t) dt

We fix a cube Q and denote by MdQ to the dyadic maximal operator restricted to the

cube Q. Also we denote wQ = 1|Q|

∫Qw. Hence,

1

|Q|

∫Q

w1+δ dx≤ 1

|Q|

∫Q

MdQ(w)(x)δ wdx

=δ

|Q|

∫ ∞0

tδ−1w(x ∈ Q : MdQ(w)(x) > t) dt

≤ δ

|Q|

∫ wQ

0


+δ

|Q|

∫ ∞wQ


≤ (wQ)δ+1 +δ

|Q|

∫ ∞wQ


We know use a sort of the “reverse” weak type (1, 1) estimate for the maximal function:if t > wQ

wx ∈ Q : Md

Qw(x) > t≤ 2nt |x ∈ Q : Md

Qw(x) > t|that can be found, for instance, in [32]. Hence,

1

|Q|

∫Q

(MdQw)δwdx≤ (wQ)δ+1 +

2nδ

δ + 1

1

|Q|

∫Q

(MdQw)δ+1dx

≤ (wQ)δ+1 +2nδ[w]A1

δ + 1

1

|Q|

∫Q

(MdQw)δ wdx

Setting here δ = 12n+1[w]A1

, we obtain

1

|Q|

∫Q

(MdQw)δwdx ≤ 2(wQ)δ+1

26 CARLOS PEREZ

6. The main lemma and the proof of the linear growth theorem

In this section we combine all the previous information to finish the proof of thelinear growth Theorem 1.5. We need a a lemma which immediately gives the proof.

Lemma 6.1. Let T be any Calderon-Zygmund singular integral operator and let w beany weight. Also let 1 < p <∞ and 1 < r < 2.

Then, there is a c = cn such that:

‖Tf‖Lp(w)

≤ cp′( 1

r − 1

)1−1/pr

‖f‖Lp(Mrw)

In applications we will often use the following consequence

‖Tf‖Lp(w)

≤ cp′ (r′)1/p′‖f‖Lp(Mrw)

since t1/t ≤ 2, t ≥ 1.We are now ready to finish the proof of the linear growth theorem 1.5.

Proof of Theorem 1.5. Indeed, apply the lemma to w ∈ A1 with sharp Reverse Holder’sexponent r = rw = 1 + 1

2n+1[w]A1obtaining

‖T‖Lp(w)

≤ c p′ [w]A1

Proof of the lemma. We consider to the equivalent dual estimate:

‖T ∗f‖Lp′ (Mrw)1−p′ )

≤ cp′( 1

r − 1

)1−1/pr

‖f‖Lp′ (w1−p′ )

Then use the “tricky” Lemma 4.2 since T ∗ is also a Calderon-Zygmund operator

‖ T∗f

Mrw‖Lp′ (Mrw))

≤ p′ c ‖ Mf

Mrw‖Lp′ (Mrw))

Next we note that by Holder’s inequality with exponent pr,

1

|Q|

∫Q

fw−1/pw1/p ≤(

1

|Q|

∫Q

wr)1/pr (

1

|Q|

∫Q

(fw−1/p)(pr)′)1/(pr)′

and hence,

(Mf)p′ ≤ (Mrw)p

′−1M(

(fw−1/p)(pr)′)p′/(pr)′

From this, and by the classical unweighted maximal theorem with the sharp constant,∥∥∥∥ Mf

Mrw

∥∥∥∥Lp′ (Mrw)

≤ c( p′

p′ − (pr)′

)1/(pr)′∥∥∥∥ fw∥∥∥∥Lp′ (w)

= c(rp− 1

r − 1

)1−1/pr∥∥∥∥ fw∥∥∥∥Lp′ (w)

≤ cp( 1

r − 1

)1−1/pr∥∥∥∥ fw∥∥∥∥Lp′ (w)

.

7. Proof of the logarithmic growth theorem.

Proof of Theorem 1.6. The proof is based on ideas from [62]. Applying the Calderon-Zygmund decomposition to f at level λ, we get a family of pairwise disjoint cubesQj such that

λ <1

|Qj|

∫Qj

|f | ≤ 2nλ

Let Ω = ∪jQj and Ω = ∪j2Qj . The “good part” is defined by

g =∑j

fQjχQj(x) + f(x)χΩc(x)

and the “bad part” b as

b =∑j

bj

where

bj(x) = (f(x)− fQj)χQj(x)

Then, f = g + b.However, it turns out that b is “excellent” and g is really “ugly”. It is so good the b

part that we have the full Muckenhoupt-Wheeden conjecture:

wx ∈ (Ω)c : |Tb(x)| > λ ≤ c

λ

∫Rn|f |Mwdx

by a well known argument using the cancellation of the bj and that we omit. Also

the term w(Ω) is the level set of the maximal function and the Fefferman-Stein applies(again we have the full Muckenhoupt conjecture).

Combining we have

wx ∈ Rn : |Tf(x)| > λ≤w(Ω) + wx ∈ (Ω)c : |Tb(x)| > λ/2+wx ∈ (Ω)c : |Tg(x)| > λ/2.

28 CARLOS PEREZ

and the first two terms are already controlled:

w(Ω) + wx ∈ (Ω)c : |Tb(x)| > λ/2 ≤ c

λ

∫Rn|f |Mwdx ≤ c[w]A1

λ

∫Rn|f |wdx

Now, by Chebyschev and the Lemma, for any p > 1 we have

wx ∈ (Ω)c : |Tg(x)| > λ/2

≤ c(p′)p( 1

r − 1

)p− 1r 1

λp

∫Rn|g|pMr(wχ(Ω)c)dx

≤ c(p′)p( 1

r − 1

)p− 1r 1

λ

∫Rn|g|Mr(wχ(Ω)c)dx.

By more or less standard arguments we have∫Rn|g|Mr(wχ(Ω)c)dx ≤ c

∫Rn|f |Mrwdx.

Combining this estimate with the previous one, and then taking the sharp reverseHolder’s exponent r = 1 + 1

2n+1[w]A1, by the reverse Holder’s inequality lemma we get

wx ∈ (Ω)c : |Tg(x)| > λ/2 ≤ c(p′[w]A1)p

λ

∫Rn|f |wdx.

Setting here

p = 1 +1

log(1 + [w]A1)

gives

wx ∈ (Ω)c : |Tg(x)| > λ/2 ≤ c[w]A1(1 + log[w]A1)

λ

∫Rn|f |wdx.

This estimate combined with the previous one completes the proof.

8. Properties of the Ap weights

We have seen how important is the sharp reverse Holder exponent for A1 weights,Lemma 5.1, for the proof of Theorem 1.5 and hence for that of Theorem 1.6. A naturalquestion is then to find a similar result for the Ap class of weights. The question isinteresting on its own but it turns out that it is very useful as well as can be seen inthe proof of the quadratic estimates for commutators given in Section 10.

Recall that if w ∈ Ap there are constants r > 1 and c ≥ 1 such that for any cube Q

(8.1)( 1

|Q|

∫Q

wrdx) 1r ≤ c

|Q|

∫Q

w

In the standard proofs both constants c, r depend upon the Ap constant of the weight.We prove here a more precise version of (8.1).

Lemma 8.1. Let w ∈ Ap, 1 < p <∞ and let rw = 1 + 122p+n+1[w]Ap

. Then for any Q

(8.2)( 1

|Q|

∫Q

wrwdx) 1rw ≤ 2

|Q|

∫Q

w

Remark 8.2. We remark that these result has been considerably improved in [38], itis stated below in Theorem 9.1 but we skip the proof which is different from the one wepresent here after next corollary.

The classical proofs of this property for any Ap weights produce non linear growthconstants.

We also remark that this result was stated and used in [8] with no proof. The authormentioned instead the work by Coifman-Fefferman [14] where no explicit statementcan be found. Since this lemma plays an important role (specially the case p = 2) wesupply below a proof.

As a corollary we deduce the following useful result.

Corollary 8.1. Let 1 < p <∞ and let w ∈ Ap. Denote

pw =p

1 + p−1

1+ 1

22p′+n+1[w]

p′−1Ap

Then w ∈ Ap/pw and furthermore

[w]Ap/pw ≤ 2p−1[w]Ap

or what is equivalent w ∈ Ap−ε where ε = p−1

1+22p′+n+1[w]p′−1Ap

≈ p−1

[w]p′−1Ap

The proof of the corollary is as follows, since w ∈ Ap, σ ∈ Ap′ and hence by thelemma

rσ = 1 +1

22p′+n+1[σ]Ap′= 1 +

1

22p′+n+1[w]p′−1Ap

with ( 1

|Q|

∫Q

σrσdx) 1rσ ≤ 2

|Q|

∫Q

σ

and hence(1

|Q|

∫Q

w(x)dx

)(1

|Q|

∫Q

σ(x)rσdx

) p−1rσ

≤ 2p−1

(1

|Q|

∫Q

w(x)dx

)(2

|Q|

∫Q

w1−p′ dx

)p−1

namely

[w]Ap/pw ≤ 2p−1[w]Apsince

p− 1

rσ=

p

pw− 1

30 CARLOS PEREZ

This theorem plays an important role in deriving a sharp version of the Coifman-Fefferman estimate as stated in Corollary 11.2 which plays a central role in (1.5).

Proof. Let wQ = 1|Q|

∫Qw.

1

|Q|

∫Q

w(x)δ w(x)dx=δ

|Q|

∫ ∞0

tδw(x ∈ Q : w(x) > t) dtt

=δ

|Q|

∫ wQ

0


+δ

|Q|

∫ ∞wQ


= I + II.

Observe that I ≤ (wQ)δ+1.For II we make first the following observation: for any Q we let

EQ = x ∈ Q : w(x) ≤ 1

2p−1[w]ApwQ

Then we claim

(8.3) |EQ| ≤1

2|Q|.

Indeed, by Holder’s inequality we have for any f ≥ 0(1

|Q|

∫Q

f(y) dy

)pw(Q) ≤ [w]Ap

∫Q

f(y)pw(y)dy

and hence if E ⊂ Q, (|E||Q|

)p≤ [w]Ap

w(E)

w(Q)

and in particular,(|EQ||Q|

)p≤ [w]Ap

w(EQ)

w(Q)≤ [w]Ap

wQw(Q)

|EQ|1

2p−1[w]Ap=

1

2p−1

|EQ||Q|

,

from which the claim follows.The second claim is the followingfor every λ > wQ

(8.4) w(x ∈ Q : w(x) > λ) ≤ 2n+1λ |x ∈ Q : w(x) >λ

2p−1[w]ApwQ|.

Consider the CZ decomposition of w at level λ, we find a family of disjoint cubes Qicontained in Q satisfying

λ < wQi ≤ 2n λ

for each i. Indeed, since except for a null set we have

x ∈ Q : w(x) > λ ⊂ x ∈ Q : MdQw(x) > λ = ∪iQi,

32 CARLOS PEREZ

This constant was introduced by Hruscev [35] (see also [31]) and has been the stan-dard A∞ constant until very recently since a “new” A∞ constant has found to be bettersuited. This new constant is defined as

(9.1) [w]A∞ := supQ

1

w(Q)

∫Q

M(wχQ).

However, this constant was introduced by M. Wilson long time ago (see [76, 77, 78])with a different notation. This constant is more relevant since there are examples ofweights w ∈ A∞ so that [w]A∞ is much smaller than ‖w‖A∞ . Indeed, first it can beshown that

(9.2) cn [w]A∞ ≤ ‖w‖A∞ ≤ [w]Ap , 1 < p <∞,where cn is a constant depending only on the dimension. The first inequality is theonly nontrivial part and can be found in [38] where it is also shown a more interestingfact, namely that this inequality can be strict. More precisely, the authors exhibit afamily of weights wt such that [wt]A∞ ≤ 4 log(t) and ‖wt‖A∞ ∼ t/ log(t) for t 1.

It should be mentioned that this constant has also been used by Lerner [49, Sec-tion 5.5] where it was coined for first time the term “A∞ constant”.

In this section, we state a new result obtained by the author with T. Hytonen (see[38]). This sharper version of the RHI plays a central role in the proofs of all thesenew results.

Theorem 9.1 (A new sharp reverse Holder inequality). Define rw := 1 +1

τn [w]A∞,

where τn is a dimensional constant that we may take to be τn = 211+n. Note thatr′w ≈ [w]A∞.

a) If w ∈ A∞, then (1

|Q|

∫Q

wrw)1/rw

≤ 2

|Q|

∫Q

w.

b) Furthermore, the result is optimal up to a dimensional factor: If a weight wsatisfies the RHI, i.e., there exists a constant K such that(

1

|Q|

∫Q

wr)1/r

≤ K

|Q|

∫Q

w,

then there exists a dimensional constant c = cn, such that [w]A∞ ≤ cnK r′.

Remark 9.2. Results analogous to the left hand side of inequality (9.2) and Theo-rem 9.1 have been independently obtained by O. Beznosova and A. Reznikov [7]. Theirformulation is slightly different, and involves yet another weight constant closely relatedto [w]A∞.

We finish this section by stating an improvement of Theorem 1.5 obtained in [38]using this new sharp reverse Holder property. The new idea is to derive results mixingboth constants A1 and A∞ in the final result.

Theorem 9.3. Let T be a Calderon-Zygmund operator and let 1 < p <∞. Then

‖T‖Lp(w) ≤ c pp′ [w]1/pA1

[w]1/p′

A∞, w ∈ A1, 1 < p <∞,

and

‖T‖L1,∞(w) ≤ c[w]A1 log(e+ [w]A∞)‖f‖L1(w)

In view of [57] seems to be the best possible result.

10. Quadratic estimates for commutators

In this section we consider commutators of singular integrals with BMO functions.When T is a singular integral operator, these operators were considered by Coifman,Rochberg and Weiss in [23]. Formally these operators are defined by

[b, T ]f(x) = b(x)T (f)(x)− T (b f)(x) =

∫Rn

(b(x)− b(y))K(x, y)f(y) dy,

where K is a kernel satisfying the standard Calderon-Zygmund estimates. Althoughthe original interest in the study of such operators was related to generalizations ofthe classical factorization theorem for Hardy spaces many other applications have beenfound.

The main result from [23] states that [b, T ] is a bounded operator on Lp(Rn),1 < p < ∞, when b is a BMO function and T is a singular integral operator. Infact, the BMO condition of b is also a necessary condition for the Lp-boundedness ofthe commutator when T is the Hilbert transform. We may think that these operatorsbehave as Calderon-Zygmund operators, however there are some differences. For in-stance, simple examples show that in general [b, T ] fails to be of weak type (1, 1) whenb ∈ BMO. This was observed by the the author in [63] where it is also shown thatthere is an appropriate weak-L(logL) type estimate replacement. To stress this pointof view it is also shown by the author [64] that the right operator controlling [b, T ] isM2 = M M , instead of the Hardy-Littlewood maximal function M . We pursue inthis way by showing that commutators have an extra “bad” behavior from the pointof view of Ap weights when trying to derive theorems such as Theorems 1.5 or 1.6.

10.1. A preliminary result: a sharp connection between the John-Nirenbergtheorem and the A2 class. In this section we outline the main results from [13].We want to stress how important are the reverse Holder property of the A2 weightsin conjunction with the following sharp version of the classical and well known John-Nirenberg theorem.

For a locally integrable b : Rn → R we define

‖b‖BMO = supQ

1

|Q|

∫Q

|b(y)− bQ| dy <∞

34 CARLOS PEREZ

where the supremum is taken over all cubes Q ∈ Rn with sides parallel to the axes,and

bQ =1

|Q|

∫Q

b(y) dy

The main relevance of BMO is due to the fact that has an exponential self-improvingproperty, namely the celebrated John-Nirenberg’s Theorem. We need a very preciseversion of it, as follows:

Theorem 10.1. [Sharp John-Nirenberg inequality] There are dimensional constants0 ≤ αn < 1 < βn such that

(10.1) supQ

1

|Q|

∫Q

exp

(αn

‖b‖BMO

|b(y)− bQ|)dy ≤ βn

In fact we can take αn = 12n+2 .

For the proof of this we remit to the Lecture Notes by J.L. Journe [41] p. 31-32.The result there is not so explicit but it follows from the proof which is very interestingand different from the usual ones that can be found in many references.

We derive from Theorem 10.1 the following relationship between BMO and the A2

class of weights Lemma 10.2 that will be used in the proof of the main theorem of thissection. Indeed, it is well known that if w ∈ A2 then b = logw ∈ BMO. A partialconverse also holds, if b ∈ BMO there is an s0 > 0 such that w = esb ∈ Ap, |s| ≤ s0.We have now a more precise version of this converse.

Lemma 10.2. Let b ∈ BMO and let αn < 1 < βn be the dimensional constantsfrom (10.1). Then

s ∈ R, |s| ≤ αn‖b‖BMO

=⇒ es b ∈ A2 and [es b]A2 ≤ β2n

Proof. By Theorem 10.1, if |s| ≤ αn‖b‖BMO

and if Q is fixed

1

|Q|

∫Q

exp(|s||b(y)− bQ|) dy ≤1

|Q|

∫Q

exp(αn

‖b‖BMO

|b(y)− bQ|) dy ≤ βn

and then1

|Q|

∫Q

exp(s(b(y)− bQ)) dy ≤ βn

and1

|Q|

∫Q

exp(−s(b(y)− bQ)) dy ≤ βn.

If we multiply the inequalities, the bQ parts cancel out:(1

|Q|

∫Q

exp(s(b(y)− bQ)) dy

) (1

|Q|

∫Q

exp(s(bQ − b(y))) dy

)=

(1

|Q|

∫Q

exp(sb(y)) dy

) (1

|Q|

∫Q

exp(−sb(y)) dy

)≤ β2

n

namely es b ∈ A2 with

[es b]A2 ≤ β2n

10.2. Results in the Ap context.In this section we use the extrapolation theorem, Corollary 3.1 to get the optimal

estimates for commutators. We remark that next results are stated for very generaloperators. These results can be found in [13] as well some generalizations.

Theorem 10.3. Let T be a linear operator such that

(10.2) ‖T‖L2(w) ≤ c [w]A2 w ∈ A2.

Then there is a constant c independent of w and b such that

(10.3) ‖[b, T ]‖L2(w) ≤ c [w]2A2‖b‖BMO.

Observe the quadratic exponent which makes it different from the non commutatorcase. As an easy consequence of the extrapolation theorem mentioned above we havethe following.

Corollary 10.1. Let T be a linear operator satisfying (10.2). Let 1 < p < ∞, thethere is a constant cn,p such that

(10.4) ‖[b, T ]‖Lp(w)

≤ cn,p [w]max1, 1

p−1

Ap‖b‖BMO.

In [12], the special cases of the Hilbert, Beurling and Riesz transforms, were obtained.

Sketch of the proof of Theorem 10.3. We “conjugate” the operator as follows: if z isany complex number we define

Tz(f) = ezbT (e−zbf).

Then, a computation gives (for instance for ”nice” functions),

[b, T ](f) =d

dzTz(f)|z=0 =

1

2πi

∫|z|=ε

Tz(f)

z2dz , ε > 0

by the Cauchy integral theorem, see [23], [1].Now, by Minkowski’s inequality

(10.5) ‖[b, T ](f)‖L2(w) ≤1

2π ε2

∫|z|=ε‖Tz(f)‖L2(w)|dz| ε > 0.

The key point is to find the appropriate radius ε. To do this we look at the innernorm ‖Tz(f)‖L2(w)

‖Tz(f)‖L2(w) = ‖T (e−zbf)‖L2(we2Rez b) ,

36 CARLOS PEREZ

and try to find appropriate bounds on z. To do this we use the main hypothesis,namely that T is bounded on L2(w) if w ∈ A2 with

‖T‖L2(w) ≤ c [w]A2 .

Hence we should estimate [we2Rez b]A2 . But since w ∈ A2 we can use Lemma 8.1 withp = 2 to obtain:

[we2Rez b]A2 ≤ 4 [w]A2 [e2Rez r′ b]1r′A2

where r = rw = 1 + 12n+5[w]A2

< 2. Now, since b ∈ BMO we are in a position to apply

Lemma 10.2,

if |2Rez r′| ≤ αn‖b‖BMO

then [e2Rez r′ b]A2 ≤ β2n.

Hence for these z,

[we2Rez b]A2 ≤ 4 [w]A2 β2r′n ≤ 4 [w]A2 βn,

since 1 < r < 2.Using this estimate and for these z we have

‖Tz(f)‖L2(w) ≤ 4βn [w]A2 ‖f‖L2(w).

Finally, choosing the radius

ε =αn

2r′‖b‖BMO

,

finish the proof of the theorem.All the details can be found in [13].

10.3. Examples.Below there is an example that the quadratic estimate is sharp for p = 2 in dimension

one.Consider the Hilbert transform

Hf(x) = pv

∫R

f(y)

x− ydy,

and consider the BMO function b(x) = log |x| and

[b,H]f(x) = b(x)H(f)(x)−H(bf)(x).

Consider the BMO function b(x) = log |x|. We know that

(10.6) ‖[b,H]‖L2(w) ≤ c [w]2A2

and we show that the result is sharp:

(10.7) supw∈A2

1

[w]2−θ2

‖[b, T ]‖L2(w) =∞ θ > 0

For 0 < δ < 1 we letw(x) = |x|1−δ

and it is easy to see that

[w]A2 ≈1

δ

We now consider the function

f(x) = x−1+δ χ(0,1)(x)

and observe that f is in L2(w).To estimate ‖[b,H]f‖L2(w) we claim

|[b,H]f(x)| ≥ 1

δ2f(x)

and hence

‖[b,H]f‖L2(w) ≥1

δ2‖f‖L2(w)

from which the sharpness (10.7) will follow.We now prove the claim: if 0 < x < 1,

[b,H]f(x) =

∫ 1

0

log(x)− log(y)

x− yy−1+δ dy =

∫ 1

0

log(xy)

x− yy−1+δ dy

= x−1+δ

∫ 1/x

0

log(1t)

1− tt−1+δ dt

Now, ∫ 1/x

0

log(1t)

1− tt−1+δ dt =

∫ 1

0

log(1t)

1− tt−1+δ dt+

∫ 1/x

1

log(1t)

1− tt−1+δ dt

and sincelog( 1

t)

1−t is positive for (0, 1) ∪ (1,∞) we have for 0 < x < 1

|[b,H]f(x)| > x−1+δ

∫ 1

0

log(1t)

1− tt−1+δ dt.

But since ∫ 1

0

log(1t)

1− tt−1+δ dt >

∫ 1

0

log(1

t) t−1+δ dt =

∫ ∞0

s e−sδ ds =1

δ2

and the claim

|[b,H]f(x)| ≥ 1

δ2f(x)

follows.

38 CARLOS PEREZ

10.4. The A1 case.In this section we will describe some recent work by C. Ortiz-Caraballo [60]. The

first result is a version of the linear growth Theorem 1.5 for commutators.

Theorem 10.4. Let T be a Calderon–Zygmund operator and let b be in BMO. Alsolet 1 < p, r < ∞. Then there exists a constant cn such that for any weight We claimthat the following inequality holds for any 1 < p, r <∞

(10.8) ‖b, T ]f‖Lp(w) ≤ cp(p′)2 ‖b‖BMO (r′)

1+ 1p′ ‖f‖Lp(Mrw).

In particular if w ∈ A1, we have

(10.9) ‖[b, T ]‖Lp(w) ≤ cn ‖b‖BMO p(p′)2[w]2A1

.

Furthermore this result is sharp in terms of the [w]A1 constant and in terms of p.

Observe again the quadratic exponent. We remit to [60] for the proof of the theorem.The second main result of [59] is the following endpoint version of Theorem 10.4

similar in spirit as in the logarithmic growth Theorem 1.6.

Theorem 10.5. Let T and b as above. Then there exists a constant c = c(n, ‖b‖BMO)such that for any weight w ∈ A1 and f ∈ L∞c (Rn)

(10.10) w(x ∈ Rn : |[b, T ]f(x)| > λ ≤ CΦ([w]A1)2

∫Rn

Φ

(|f(x)|λ

)w(x) dx

where Φ(t) = t(1 + log+ t).

More recently, C. Ortiz-Caraballo has obtained in her PhD dissertation [59] an im-provement of these theorems in terms of mixed A1–A∞ norms for the commutator andfor its iterations in the spirit of Theorems 9.3 and 1.13 above derived in [38]. For anyk ∈ N, the k-th iterated commutator T kb of a BMO function b and a C-Z operator T isdefined by

T kb := [b, T k−1b ].

Theorem 10.6. Let T and b as above and let 1 < p, r <∞. Consider the higher ordercommutators T kb , k = 1, 2, · · · ,. Then there exists a constant c = cn,T such that forany weight w the following inequality holds

(10.11) ‖T kb f‖Lp(w) ≤ c ‖b‖kBMO (pp′)k+1

(r′)k+1/p′ ‖f‖Lp(Mrw).

In particular if w ∈ A1, we have that

‖T kb ‖Lp(w) ≤ c ‖b‖kBMO(pp′)k+1[w]1/pA1

[w]k+1/p′

A∞.

Theorem 10.7. Let T and b as above, and let 1 < p, r < ∞. Then there exists aconstant c = cn,T such that for any w(10.12)

w(x ∈ Rn : |T kb f | > λ) ≤ c (pp′)(k+1)p(r′)(k+1)p−1

∫Rn

Φ

(‖b‖BMO

|f |λ

)Mrw dx,

where Φ(t) = t(1 + log+ t)k.

If w ∈ A1 we obtain that

w(x ∈ Rn : |T kb f(x)| > λ) ≤ cn β

∫Rn

Φ

(‖b‖BMO

|f(x)|λ

)w(x)dx,

where β = [w]A1 [w]kA∞(1 + log+[w]A∞)k+1 and Φ(t) = t(1 + log+ t)k.

The very first results of these type can be found in [63] (see also [66]).These results reflect the idea that as higher is the commutator the more singular it

is. Again as above, a crucial role is played by the sharp reverse Holder’s property forA∞ weights as stated in Theorem 9.1 See [59] for details.

11. Rearrangements type estimates

The main purpose of this section is to give a new proof of (4.5) avoiding any sort ofgood-λ type arguments like 4.6 used in [50] and [52]. Indeed, as already explained insection 4.2, the heart of the matter of the proof of Theorem 1.5 and hence of Theorem1.6 is Lemma 4.2. We plan to give a proof based on the idea of rearrangements.

Recall that the non-increasing rearrangement f ∗ of a measurable function f is definedby

f ∗(t) = infλ > 0 : µf (λ) < t

, t > 0,

where

µf (λ) = |x ∈ Rn : |f(x)| > λ|, λ > 0,

is the distribution function of f . More generally, for a weight w (or measure) we define

f ∗w(t) = infλ > 0 : wf (λ) < t

, t > 0,

where

wf (λ) = wx ∈ Rn : |f(x)| > λ|, λ > 0,

An important fact is that ∫Rn|f |pwdx =

∫ ∞0

f ∗w(t)p dt,

or more generally if E is any measurable function:∫E

|f |pwdx =

∫ w(E)

0

f ∗(t)p dt.

Similarly

‖f‖Lp,∞(w)

= supt>0

t wx ∈ Rn : |f(x)| > t1/p = supt>0

t1/p f ∗w(t)

If f is only a measurable function and if Q is a cube then we define the followingquantity:

(fχQ

)∗(λ|Q|) 0 < λ ≤ 1.

40 CARLOS PEREZ

We can think of this expression as another way of averaging the function. Indeed forany δ > 0, and 0 < λ ≤ 1

(fχQ

)∗(λ|Q|) ≤(

1

λ|Q|

∫Q

|f |δ dx)1/δ

.

The (dyadic) local maximal operator mλf defined for any measurable function f by

mλf(x) = supx∈Q∈D

(fχQ)∗(λ|Q|) (0 < λ < 1),

where f ∗ denotes the non-increasing rearrangement of f .We may use the following property

(11.1) |f(x)| ≤ mλf(x) a.e.x

(see [44, Lemma 6]).Given a cube Q, define the median value mf (Q) of f overQ as a, possibly non-unique,

number such that|x ∈ Q : f(x) > mf (Q)| ≤ |Q|/2

and|x ∈ Q : f(x) < mf (Q)| ≤ |Q|/2.

It follows easily from the definition that

|mf (Q)| ≤ (fχQ)∗(|Q|/2

)Also, it is easy to see that if f is non-negative

mf (Q) = (fχQ)∗(|Q|/2)

and for any constant c

(11.2) mf (Q)− c = mf−c(Q)

which in turns gives f f is positive and δ > 0.We will use

(11.3)

|mf (P )−mf (Q)| = |mf−mf (Q)(P )| ≤((f−mf (Q))χP

)∗(|P |/2) ≤ ( 2

|P |

∫P

|f −mf (Q)|δ dx)1/δ

.

Theorem 11.1. Let and w ∈ Aq and let 0 < δ < 1 and 0 < γ < 1. There is a constantc = cn,q,γ,δ, such that for any measurable function:

(11.4) f ∗w(t) ≤ c[w]Aq(M#

δ f)∗w

(γ t) + f ∗w(2t) t > 0

The term f ∗w(2t) is a sort of “error” term.These sort of estimates goes back to the work of R. Bagby and D. Kurtz in the

middle of the 80’s [4] and [5].We recall that the sharp maximal operator: if δ > 0

M#δ f(x) = M#(|f |δ)(x)1/δ

where M# is the usual sharp maximal function of Fefferman-Stein:

M#(f)(x) = supQ3x

1

|Q|

∫Q

|f(y)− fQ| dy,

and where fQ = 1|Q|

∫Qf(y) dy

An interesting observation is that the function can be non locally integrable in theinteresting case, 0 < δ < 1.

If we iterate (11.4) we have:

f ∗w(t) ≤ c [w]Aq

∞∑k=0

(M#δ f)∗w(2kγt) + f ∗w(+∞) ≤

[w]Aqlog 2

∫ ∞tγ/2

(M#δ f)∗w(s)

ds

s+ f ∗w(+∞).

Hence if we assume that

f ∗w(+∞) = 0

the inequality we obtain is

(11.5) f ∗w(t) ≤ c [w]Aq

∫ ∞tγ/2

(M#δ f)∗w(s)

ds

s.

We continue using the Hardy operator. Recall that if f : (0,∞)→ [0,∞)

Af(x) =1

x

∫ x

0

f(t)dt x > 0

is called the Hardy operator. The dual operator is given by

Sf(x) =

∫ ∞x

f(s)ds

s.

Hence above estimate can be expressed as:

f ∗w(t) ≤ c [w]Aq S((M#δ f)∗w)(tγ/2)

Finally since it is well known that these operators are bounded on Lp(0,∞) and fur-thermore, it is known that

‖S‖Lp(0,∞) = p p ≥ 1.

we have

‖f‖Lp(w)

= ‖f ∗w‖Lp(0,∞)≤ c [w]Aq ‖S((M#

δ f)∗w)‖Lp(0,∞)

≤ c p [w]Aq ‖(M#δ f)∗w‖Lp(0,∞)

= c p [w]Aq ‖M#δ f‖Lp(w)

This concludes the strong estimate in the case p ≤ 1. The triangle inequality can beused when 0 < p < 1 and the weak estimate is easier. This concludes the proof of thefollowing corollary except for the proof of Theorem 11.1.

42 CARLOS PEREZ

Corollary 11.1. Let 0 t| <∞, t > 0.

Then

(11.7) ‖f‖Lp(w)

≤ c p [w]Aq‖M#δ f‖Lp(w)

and

(11.8) ‖f‖Lp,∞(w)

≤ c p [w]Aq ‖M#δ f‖Lp,∞(w)

.

These estimate have been improved in [61] where [w]Aq was replaced by [w]A∞ . Asbefore this is Wilson’s constant (9.1).

To understand the reason of assuming (11.6) we observe that f ∗w(+∞) = 0 is equiv-alent to saying that for each t > 0

w(x : |f(x)| > t) <∞.This condition was already used in [5] (see also [47]). Now, since w ∈ Aq and sinceminw,N ∈ Aq, N = 1, 2, . . . , with Aq constant independent of N we may that w isbounded. This explains why it is enough to consider functions satisfying (11.6).

Finally, combining this corollary with Lemma 4.3 we derive the sharp Coifman-Fefferman inequality in both p and the Aq constant which was needed for the proof ofthe main results in these lecture notes.

Corollary 11.2. Let 0 0, |x : |Tf(x)| >t| <∞. Then

(11.9) ‖Tf‖Lp(w)

≤ c p [w]Aq‖Mf‖Lp(w)

and

(11.10) ‖Tf‖Lp,∞(w)

≤ c p [w]Aq ‖Mf‖Lp,∞(w)

.

(11.9) was already pointed out in (4.1).

12. Proof of the Theorem 11.1

Proof. As we already explained in the previous section we may assume that the weightis bounded.

We may assume that f is nonnegative Fix t > 0 and let A = [w]Aq . By definition ofrearrangement, (11.4) will follow if we prove that for each t > 0,

(12.1) wx ∈ Rn : f(x) > cA

(M#

δ f)∗w

(γ t) + f ∗w(2t)≤ t

We split the left hand side L as follows:

L ≤ wx ∈ Rn : cAM#δ f(x) > cA

(M#

δ f)∗w

(γ t)

+wx ∈ Rn : f(x) > cAM#δ f(x) + f ∗w(2t) ≡ I + II.

Observe that

I = wx ∈ Rn : M#

δ f(x) >(M#

δ f)∗w

(γ t)≤ γ t

by definition of rearrangement and hence the heart of the matter is to prove that

II = wx ∈ Rn : f(x) > cAM#δ f(x) + f ∗w(2t) ≤ (1− γ) t

The set x ∈ Rn : f(x) > cAM#δ f(x) + f ∗w(2t) is contained in the set E = x ∈

Rn : f(x) > f ∗w(2t) which has w-measure at most 2t by definition of rearrangement.Now, by the regularity of the measure we can find an open set Ω containing E suchthat w(Ω) < 3t. We claim that for large c dimensional constant c

II = wx ∈ Ω : f(x) > cAM#δ f(x) + f ∗w(2t) ≤ (1− γ) t.

Since we may assume that |Ω| is also finite we can consider the Calderon-Zygmundcubes of the function χ

Ωat level 0 < α < 1. Hence, there are dyadic cubes Qj,

maximal with respect to inclusion, satisfying

Ω ⊂ ∪jQj

and

(12.2) α <|Ω ∩Qj||Qj|

≤ 2nα

for each j. Observe that if we further choose α such that 2nα < 12

we have the followingimportant property:

(12.3)|Ωc ∩Qj||Qj|

≥ 1

2

Now, we continue estimating II

II ≤∑j

wx ∈ Qj : f(x) > cAM#

δ f(x) + f ∗w(2t)

Fix one of these j. If |mf (Qj)| = mf (Qj) ≤ f ∗w(2t) we have (recall that mf (Q) is themedian value of f) that

x ∈ Qj : f(x) > cAM#δ f(x) + f ∗w(2t)

⊂ x ∈ Qj : |f(x)−mf (Qj)| > cAM#

δ f(x).

Hence, assuming mf (Qj) > f ∗w(2t) and since f(x) ≤ f ∗w(2t) if x ∈ Ωc we have

1

|Qj|

∫Qj

|f(y)−mf (Qj)|δ dy ≥1

|Qj|

∫Ωc∩Qj

∣∣∣|mf (Qj)| − |f(y)|∣∣∣δ dy

≥ |Ωc ∩Qj||Qj|

(|mf (Qj)| − f ∗w(2t)

)δ≥ 1

2

(|mf (Qj)| − f ∗w(2t)

)δHence,

|mf (Qj)| ≤ |mf (Qj)− f ∗w(2t)|+ f ∗w(2t) ≤( 2

|Qj|

∫Qj

|f(y)−mf (Qj)|δ dy)1/δ

+ f ∗w(2t)

44 CARLOS PEREZ

≤ 21/δM#δ f(x) + f ∗w(2t)

if x ∈ Qj. Combining estimates we have that if c > 1 + 21/δ

x ∈ Qj : f(x) > cAM#δ f(x) + f ∗w(2t)

⊂ x ∈ Qj : |f(x)−mf (Qj)| > cAM#

δ f(x),

for any j. Now, if we denote

(12.4) Ej = x ∈ Qj : |f(x)−mf (Qj)| > cAM#δ f(x)

and observe that by Lemma 13.1 below we have

|Ej||Qj|

≤ c1e−c2A

since recall that A = [w]Aq . Then, to conclude if r = rw is the sharp reverse exponentfor w: rw = 1 + 1

22q+n+1A(see Lemma 8.1),( 1

|Q|

∫Q

wrwdx) 1rw ≤ 2

|Q|

∫Q

w

we have by Holder’s inequality and Lemma 13.1

w(Ej)

w(Qj)≤ 2 (

|Ej||Qj|

)1/r′ ≤ 2c1

ec c3

since r′ ≈ [w]Aq = A where c3 depends on the dimension and q. If we finally we choose

c such that 2c1ec3 c

< 1−γ3

we have

II ≤∑j

w(Ej) ≤1− γ

3

∑j

w(Qj) =1− γ

3w(Ω) < (1− γ) t,

since w(Ω) < 3t.

13. The exponential decay lemma

Let f ∈ Lδloc measurable. For t > 0 we define

ϕ(t) = supQ∈D

1

|Q|

∣∣∣x ∈ Q : |f(x)−mf (Q)| > tM#δ f(x)

∣∣∣which is finite since is bounded by 1 and by 1/tδ by Chebyshev. We prove that thereis an exponential decay on t.

Lemma 13.1. There is a dimensional constants c1, c2 such that

(13.1) ϕ(t) ≤ c1

ec2t.

Proof. Observe that we may assume that f is positive by the lattice property.We denote

oscδ(Q, f) =( 1

|Q|

∫Q

|f(y)−mf (Q)|δ dy)1/δ

It is enough to find t0 > 0 such that for t > t0, ϕ(t) ≤ cect

. Fix one cube Q andconsider the ratio

(13.2)1

|Q|

∣∣∣x ∈ Q : |f(x)−mf (Q)| > tM#δ f(x)

∣∣∣.Consider the maximal type operator related to Q

NQf(x) = supx∈P∈D(Q)

∣∣∣mf (P )−mf (Q)∣∣∣

and consider the subset of Q

Γ =x ∈ Q : NQf(x) > σ oδ(Q, f)

,

and where the parameter σ > 1 is going to be chosen. Observe that if Γ is empty thenusing that mf (P ) = (fχP )∗

(|P |/2) when f is positive, by the Lebesgue differentiation

theorem, shrinking P to x we have

|f(x)−mf (Q)| ≤ σ oscδ(Q, f) ≤ σM#δ f(x), x ∈ Q

so that if t > σ then the ratio (13.2) is zero. So we will assume that t > σ and thatΓ is not empty. Hence, by adapting the usual CZ covering lemma there is a collectionQi of dyadic disjoint subcubes of Q, with the usual properties,

Γ = ∪iQi

and each is maximal with respect to

(13.3) σ oscδ(Q, f) <∣∣∣mf (Qi)−mf (Q)

∣∣∣and hence

(13.4)∣∣∣mf (Q

′i)−mf (Q)

∣∣∣ ≤ σ oscδ(Q, f).

Observe that as before

(13.5) |f(x)−mf (Q)| ≤ σ oδ(Q, f) x ∈ Q \ Ω.

Also observe that by (11.3) and (13.3)

(13.6) |Γ| ≤ 2

σδ

Combining and if we asume that t > σ∣∣∣x ∈ Q : |f(x)−mf (Q)| > tM#δ f(x)

∣∣∣ =∣∣∣x ∈ Ω : |f(x)−mf (Q)| > tM#

δ f(x)∣∣∣

=∑i

∣∣∣x ∈ Qi : |f(x)−mf (Q)| > tM#δ f(x)

∣∣∣

46 CARLOS PEREZ

By (13.4)

|mf (Q′i)−mf (Q)| ≤ σ oscδ(Q, f) ≤ σM#

δ f(x)

we can continue with

≤∑i

∣∣∣x ∈ Qi : |f(x)−mf (Q′i)| > (t− σ)M#

δ f(x)∣∣∣.

Now, recalling that by definition if t > 0

ϕ(t) = supQ

1

|Q|

∣∣∣x ∈ Q : |f(x)−mf (Q)| > tM#f(x)∣∣∣

and hence if t > σ

1

|Q|

∣∣∣x ∈ Q : |f(x)−mf (Q)| > tM#δ f(x)

∣∣∣ ≤ 1

|Q|∑i

|Q′i|ϕ(t− σ) ≤ 2n+1

σδϕ(t− σ),

by (13.6). Hence, if we choose σ such that σδ = 2n+1e, for any t > σ, we have

ϕ(t) ≤ 1

eϕ(t− σ).

Reiterating, we get for k = 1, 2, · · · ,

ϕ(t) ≤ 1

ekϕ(t− σ k),

until k gets exhausted, namely k ≈ tσ

we have

ϕ(t) ≤ c1

ec2t.

(the case t ≤ σ is trivial).

References

[1] J. Alvarez, R. Bagby, D. Kurtz, C. Perez, Weighted estimates for commutators of linear operators,Studia Mat. 104 (2) (1994) 195-209.

[2] J. Alvarez and C. Perez, Estimates with A∞ weights for various singular integral operators,Bollettino U.M.I. (7) 8-A (1994), 123-133.

[3] K. Astala, T. Iwaniec, E. Saksman, Beltrami operators in the plane, Duke Mathematical Journal.107, no. 1, (2001), 27-56.

[4] R. J. Bagby and D. S. Kurtz, Covering lemmas and the sharp function, Proc. Amer. Math. Soc.93 (1985), 291-296.

[5] R. J. Bagby and D. S. Kurtz, A rearranged good-λ inequality, Trans. Amer. Math. Soc. 293(1986), 71-81.

[6] O. Beznosova, Linear bound for dyadic paraproduct on weighted Lebesgue space L2(w). J. Func.Anal. 255 4 (2008) 994-1007

[7] O. Beznosova and A. Reznikov, Sharp estimates involving A∞ and L logL constants, and theirapplications to PDE, Preprint (2011) available at Preprint, arXiv:1107.1885.

[8] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities,Trans. Amer. Math. Soc., 340 (1993), no. 1, 253–272.


[9] M. J. Carro, C. Perez, F. Soria y J. Soria, Examples and counterexamples for fractional operators,Indiana Univ. Math. J., 54 (2005), 627–644.

[10] S.-Y.A. Chang, J.M. Wilson and T. Wolff, Some weighted norm inequalities concerning theSchrodinger operator, Comm. Math. Helv., 60 (1985), 217–246.

[11] S. Chanillo and R.L. Wheeden, Some weighted norm inequalities for the area integral, IndianaUniv. Math. J., 36 (1987), 277–294.

[12] D. Chung, Sharp estimates for the commutators of the Hilbert, Riesz and Beurling-Ahlfors trans-forms on weighted Lebesgue spaces, Preprint (2010) available at http://arxiv.org/abs/1001.0755Indiana University Mathematics Journal (to appear).

[13] D. Chung, M.C. Pereyra and C. Perez Sharp bounds for general commutators on weighted Lebesguespaces, Trans. Amer. Math. Soc. 364 (2012), 1163-1177.

[14] R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singularintegrals, Studia Math. 51 (1974), 241–250.

[15] R.R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc., 79(1980), 249–254.

[16] A. Cordoba and C. Fefferman, A weighted norm inequality for singular integrals , Studia Math.57 (1976), 97–101.

[17] D. Cruz-Uribe, J.M. Martell y C. Perez, Extrapolation results for A∞ weights and applications,J. of Funct. Anal., 2, 213 (2004) 412–439.

[18] D. Cruz-Uribe, SFO, J.M. Martell and C. Perez, Weighted weak-type inequalities and a conjectureof Sawyer, Int. Math. Res. Not. 30 2005, 1849-1871.

[19] D. Cruz-Uribe, SFO, J.M. Martell and C. Perez, Weights, Extrapolation and the Theory of Ru-bio de Francia, monograph, Series: Operator Theory: Advances and Applications, Vol. 215,Birkauser, Basel, (http://www.springer.com/mathematics/analysis/book/978-3-0348-0071-6).

[20] D. Cruz-Uribe,SFO, J.M. Martell, C. Perez, Sharp weighted estimates for classical operators,Advances in Mathematics, 229, (2012), 408-441.

[21] D. Cruz-Uribe,SFO, J.M. Martell, C. Perez, Sharp weighted estimates for approximating dyadicoperators, Electronic Research Announcements in the Mathematical Sciences, 17 (2010),12-19.

[22] D. Cruz-Uribe y C.Perez, Two weight extrapolation via the maximal operator, J. Journal ofFunctional Analysis (1) 174 (2000), 1–17.

[23] R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables,Ann. of Math. 103 (1976), 611-635.

[24] G.P. Curbera, J. Garcıa-Cuerva, J.M. Martell and C. Perez, Extrapolation with weights, Re-arrangement Invariant Function Spaces, modular inequalities and applications to Singular Inte-grals, Advances in Mathematics, 203 (2006) 256-318.

[25] M. de Guzman, Differentiation of Integrals in Rn, Lect. Notes Math. 481, Springer Verlag, (1975).[26] O. Dragicevic, L. Grafakos, M.C. Pereyra and S. Petermichl, Extrapolation and sharp norm

estimates for classical operators on weighted Lebesgue spaces, Publ. Math., 49 (2005), no. 1,73–91.

[27] J. Duoandikoetxea, Fourier Analysis, American Math. Soc., Grad. Stud. Math. 29, Providence,RI, 2000.

[28] R. Fefferman and J. Pipher, Multiparameter operators and sharp weighted inequalities, Amer. J.Math. 119 (1997), no. 2, 337–369.

[29] C. Fefferman and E.M. Stein, Some maximal inequalities, Amer. J. Math., 93 (1971), 107–115.[30] N. Fujii, A proof of the Fefferman-Stein-Stromberg inequality for the sharp maximal functions,

Proc. Amer. Math. Soc., 106(2):371–377, 1989.[31] J. Garcıa-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics,

North Holland Math. Studies 116, North Holland, Amsterdam, 1985.

48 CARLOS PEREZ

[32] L. Grafakos, Classical Fourier Analysis, Springer-Verlag, Graduate Texts in Mathematics 249,Second Edition, (2008).

[33] L. Grafakos, Modern Fourier Analysis, Springer-Verlag, Graduate Texts in Mathematics 250,Second Edition, (2008).

[34] E. Hernandez, Fatorization and extrapolization of pairs of weights, Studia Math, 95 (1989), 179-193.

[35] Sergei Hruscev, A description of weights satisfying the A∞ condition of Muckenhoupt. Proc.Amer. Math. Soc., 90(2), 253–257, 1984.

[36] T. Hytonen, The sharp weighted bound for general Calderon-Zygmund operators, Annals of Math-ematics (to appear).

[37] T. Hytonen and M. T. Lacey, The Ap−A∞ inequality for general Calderon–Zygmund operators,Indiana Math J. (to appear).

[38] T. Hytonen and C. Perez, Sharp weighted bounds involving A∞, Journal of Analysis and PartialDifferential Equations, (2011), (to appear).

[39] T. Hytonen and C. Perez, Work in progress, 2012.[40] Tuomas Hytonen, Carlos Perez, Sergei Treil, and Alexander Volberg. Sharp weighted estimates

for dyadic shifts and the A2 conjecture, Preprint, arXiv:1010.0755 (2010).[41] J. L. Journe, Calderon–Zygmund operators, pseudo–differential operators and the Cauchy integral

of Calderon, Lect. Notes Math. 994, Springer Verlag, (1983).[42] M. Lacey, S. Petermichl, M. C. Reguera, Sharp A2 inequality for Haar shift operators, Math.

Ann., 348(1):127–141, 2010.[43] M. Lacey, K. Moen, C. Perez and R. Torres, The sharp bound the fractional operators on weighted

Lp spaces and related Sobolev inequalities, Journal of Functional Analysis, 259, (2010), 1073-1097.[44] A.K. Lerner, On some pointwise inequalities, J. Math. Anal. Appl. 289 (2004), no. 1, 248–259.[45] A.K. Lerner, On some sharp weighted norm inequalities, J. Funct. Anal., 232, 477–494, (2006).[46] A.K. Lerner An elementary approach to several results on the Hardy-Littlewood maximal operator,

Proc. Amer. Math. Soc. 136 (2008) no 8, 2829-2833.[47] A. K. Lerner, Weighted rearrangement inequalities for local sharp maximal functions, Trans.

Amer. Math. Soc. 357 (2004), 2445-2465.[48] A. K. Lerner, A pointwise estimate for local sharp maximal function with applications to singular

integrals, Bull. Lond. Math. Soc. 42 (2010), no.5, 843–856.[49] A.K. Lerner, Sharp weighted norm inequlities for Littlewood-Paley operators and singular inte-

grals, Advances in Mathematics (2011), 226, (5), 3912 – 3926.[50] A. K. Lerner, S. Ombrosi and C. Perez, Sharp A1 bounds for Calderon-Zygmund operators and the

relationship with a problem of Muckenhoupt and Wheeden, International Mathematics ResearchNotices, 2008, no. 6, Art. ID rnm161, 11 pp. 42B20.

[51] A. Lerner, S. Ombrosi and C. Perez, Weak type estimates for Singular Integrals related to a dualproblem of Muckenhoupt-Wheeden, J Fourier Anal Appl (on line May 2008) DOI 10.1007/s00041-008-9032-2.

[52] A. Lerner, S. Ombrosi and C. Perez, A1 bounds for Calderon-Zygmund operators related to aproblem of Muckenhoupt and Wheeden, Mathematical Research Letters (2009), 16, 149–156.

[53] J.M. Martell, C. Perez y R. Trujillo-Gonzalez, Lack of natural weighted estimates for some sin-gular integral operators, Trans. of the A.M.S., 357 (2005), 385-396.

[54] B. Muckenhoupt, Weighted norm inequalities for the Hardy–Littlewood maximal function, Trans.Amer. Math. Soc. 165 (1972), 207–226.

[55] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans.Amer. Math. Soc., 192 (1974), 261-274.

[56] F. Nazarov, S. Treil, and A. Volberg. Two weight inequalities for individual Haar multipliers andother well localized operators, Math. Res. Lett., 15 #3, 583–597, 2008.


[57] Nazarov, F., Reznikov, A. Vasuynin V. and Volberg, A., A1 conjecture: weak norm estimates ofweighted singular operators and Bellman functions, htpp://sashavolberg.wordpress.com. (2010)

[58] F. Nazarov, A. Volberg, Bellman function, polynomial estimates of weighted dyadic shifts, andA2 conjecture. Preprint (2011).

[59] C. Ortiz-Caraballo, PhD thesis, Universidad de Sevilla, October, 2011.[60] C. Ortiz-Caraballo. Quadratic A1 bounds for commutators of singular integrals with BMO func-

tions, Indiana Univ. Math. J. (to appear) (2011).[61] C. Ortiz-Caraballo, Perez, E. Rela, Improving bounds for singular operators via Sharp Reverse

Holder Inequality for A∞, preprint.[62] C. Perez, Weighted norm inequalities for singular integral operators, J. London Math. Soc., 49

(1994), 296–308.[63] C. Perez, Endpoint Estimates for Commutators of Singular Integral Operators, Journal of Func-

tional Analysis, (1) 128 (1995), 163–185.[64] C. Perez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-

Littlewood maximal function, J. Fourier Anal. Appl. 3 (1997), 743-756.[65] C. Perez Sharp Weighted Inequalities For The Vector-Valued Maximal Function, Trans. Amer.

Math. Soc., 352 (2000), 3265–3288.[66] C. Perez and G. Pradolini, Sharp weighted endpoint estimates for commutators of singular inte-

grals, Michigan Mathematical Journal 49 (2001).[67] C. Perez, S. Treil, and A. Volberg. On A2 conjecture and corona decomposition of weights,

Preprint, arXiv:1006.2630 (2010).[68] S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms

of the classical Ap- characteristic, Amer. J. Math., Amer. J. Math. 129 (2007), no. 5, 1355–1375.[69] S. Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc., 136

(2008), no. 4, 1237–1249.[70] S. Petermichl and A. Volberg, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps

on the plane are quasiregular, Duke Math. J. 112 (2002), no. 2, 281–305.[71] M.C. Reguera, PhD thesis, Georgia Institute of Technology, Spring 2011.[72] M.C. Reguera, On Muckenhoupt-Wheeden Conjecture, Advances in Math. 227 (2011), 1436–1450.[73] S. Treil, Sharp A2 estimates of Haar shifts via Bellman function. Preprint (2011) available at

arXiv:1105.2252.[74] M.C. Reguera and C. Thiele, The Hilbert transform does not map L1(Mw) to L1,∞(w), Math.

Res. Letters (to appear).[75] A. Vagharshakyan, Recovering singular integrals from Haar shifts, Proc. Amer. Math. Soc.,

138(12):4303–4309, 2010.[76] Wilson, J. Michael, Weighted inequalities for the dyadic square function without dyadic A∞, Duke

Math. J., 55(1), 19–50, 1987.[77] Wilson, J. Michael, Weighted inequalities for the continuous square function, Trans. Amer. Math.

Soc., 314(2), 661–692, 1989.[78] Wilson, J. Michael, Weighted Littlewood-Paley theory and exponential-square integrability, volume

1924 of Lecture Notes in Mathematics. Springer, Berlin, 2008.

Carlos Perez, Departamento De Analisis Matematico, Facultad de Matematicas,Universidad De Sevilla, 41080 Sevilla, Spain.

E-mail address: [email protected]

Introductiongrupo.us.es/anaresba/trabajos/carlosperez/CRM-Proceedings-Versio… · , the maximal...

Documents

Transcript of Introductiongrupo.us.es/anaresba/trabajos/carlosperez/CRM-Proceedings-Versio… · , the maximal...