Nonlinear and multiplicative inequalities in Sobolev spaces associated with Lie algebras

Nonlinear Analysis 70 (2009) 1574–1609www.elsevier.com/locate/na

Nonlinear and multiplicative inequalities in Sobolev spacesassociated with Lie algebras

Vladimir Georgieva, Sandra Lucenteb,∗

a Dipartimento di Matematica, Università degli Studi di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa, Italyb Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona 4, I-70125 Bari, Italy

Received 20 July 2007; accepted 14 February 2008

Abstract

This paper is devoted to multiplicative inequalities in some generalized Sobolev spaces associated with Lie algebras. TheseLie algebras are generated by the differential operator of variable coefficients or by pseudo-differential operators having non-regular symbols. Under geometrical assumptions we show that the norms of two suitable classes of generalized Sobolev spaces areequivalent. This leads to the proof that the composition operator u → |u|p acts on such spaces.c© 2008 Elsevier Ltd. All rights reserved.

MSC: 47H30; 35S05

Keywords: Nonlinear inequalities; Generalized Sobolev norms

1. Introduction

1.1. Motivation

Let A1, . . . , Am be Banach spaces of real functions on a given set Ω . Fixing G : Rm→ R, one can associate an

operator TG on A1 × · · · × Am such that

TG( f1, . . . , fm)(x) = G( f1(x), . . . , fm(x)), x ∈ Ω .

This very general and standard construction plays a crucial role in nonlinear analysis. In particular it is necessary toestablish the mapping properties of TG dependent on the properties of G. It is important to find in which functionspace A one has

TG(A1 × · · · × Am) → A.

Hence one investigates other properties of TG such as the boundedness, the continuity, the weak-continuity, thecompactness, the differentiability, the analyticity and so on. Such kinds of results are scattered in a lot of papers

∗ Corresponding author. Tel.: +39 0805442275; fax: +39 0805443610.E-mail addresses: [email protected] (V. Georgiev), [email protected] (S. Lucente).

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V. Georgiev, S. Lucente / Nonlinear Analysis 70 (2009) 1574–1609 1575

and monographs in which this operator is called the composition operator or superposition operator or substitutionoperator or Nemitskij operator.

The interested reader may consult the survey [1] and its very detailed bibliography which cover the period1918–1988. In this survey and in [2], J. Appell considered the case where A j are Lebesgue, Orlicz, Hölder or Sobolevspaces. The corresponding results in Besov–Triebel–Lizorkin-type spaces can be found in [12].

Obviously, in the past years, other results on the analytical and topological properties of TG have been establishedaccording to the involved map G and the employed spaces in the study of nonlinear partial differential equations.

The final aim of this paper is to establish the continuity and the local boundedness of the superposition operatorrelated to the function

G : x 7→ |x |p, p > 1

in a particular class of generalized Sobolev spaces which arises when studying nonlinear hyperbolic first ordersystems.

Before stating this result, let us briefly describe in which context these spaces occur.A great deal of work has been devoted to study nonlinear hyperbolic Cauchy problems in the form

Pku = f (u, u(1), . . . , u(k)), u : Rt × Rn→ Rm

∂(h)t u(0, ·) = uh .

Here Pk is a hyperbolic operator of order k ∈ N∗ and f is a nonlinear function of the solution u and of all its derivativesup to the order k, for h = 0, . . . , k − 1 the functions uh : Rn

→ Rm are the given data.The main ingredient for the existence theory of such problems are the energy estimates and the dispersive

properties.Roughly speaking, the energy estimates can be written as

‖u(t, ·)‖H ≤ C‖data‖H

with C > 0 and H an appropriate Hilbert space, for example H = L2(Rn) or H = W N ,2(RN ). Such an estimate canbe easily found for a large class of operators Pk .

On the contrary, the dispersive properties has been established for a smaller class of operators and taking f = 0,these inequalities have the general form

|u(t, x)| ≤ Ct−α‖data‖B

where C > 0, α ≥ 0 and B is a Banach space, for example, B = L1(Rn) or B = W N ,1(RN ).Unfortunately, the L1 norm is unusable in the applications. In fact trying to combine the energy estimate (in

H = L2) with the dispersive inequality (in B = L1), one could lose the information on the decay rate.S. Klainerman in [10] considered the wave operator P2 = ∂t t − ∆x and he succeeded in obtaining a dispersive

estimate in an appropriate Hilbert space H . This modification allowed to improve previous results on the small datasolutions. Such important L∞ − H estimates depend on the invariance properties of the operator under investigation.For example the wave operator in Rn is invariant under the action of the Poincaré group

P = ∂t , ∂x j , x j∂xk − xk∂x j , t∂x j + x j∂t j,k=1,...,n .

On the other hand, this family of first order operators generates a Lie algebra; hence one can associate a Hilbert spaceH N (Γ ) obtained as the closure of C∞0 (R

n) with respect to the norm

‖u‖H N (Γ ) =∑|α|≤N

‖Γ αu‖L2(Rn)

with Γ ∈ P. This is one of the spaces in which Klainerman settled the decay estimates. In order to use such estimatesfor proving the existence of small amplitude solutions of nonlinear wave equations, the contraction mapping principlecomes into play. Hence, one needs to manipulate the nonlinear term in H N (Γ ), by studying the boundedness of theNemitskij operator in these generalized Sobolev spaces. Since P contains differential operators, the mapping propertyof the superposition operators in such spaces can be deduced from the corresponding one in the usual Sobolev spaces.

1576 V. Georgiev, S. Lucente / Nonlinear Analysis 70 (2009) 1574–1609

In a more general setting, these ideas can be found in a contemporary paper [9], where L. Hörmander consideredother generalized Sobolev spaces defined by differential operators of kind ∂ j A(x)∂xk − ∂k A(x)∂x j where A is a non-degenerate quadratic form in Rn . Another example is given in [7], where the present authors obtain some nonlinearinequalities in weighted H s,δ spaces associated with a Lie algebra of differential operators. This was a crucial step forproving a decay result for the nonlinear Klein Gordon equation.

We recall that Klainerman’s method (known as the commuting vector fields method) has been applied toother equations which present invariance properties, for example Dirac equations, Maxwell systems, Wave maps,Maxwell–Klein–Gordon system, Yang–Mills equations and so on. In [8], the authors extended Klainerman’s approachto a class of symmetric first order systems with constant coefficients. In this way one gains a dispersive property for alarger class of operators. Unfortunately, it is very difficult to apply this decay result in the nonlinear context. In fact inthese L∞ −H N (Γ ) estimates, the involved Γ vector fields are pseudo-differential operators.

The group of symmetry of a pseudo-differential operator has been firstly introduced in [4]; in that work the authorsobtain a Sobolev embedding theorem in such spaces.

According to our knowledge, no result has been established for the continuity of Nemitskij operators in generalizedSobolev spaces defined by means of pseudo-differential operators. This paper is a first attempt in this direction. Inparticular we deal with non-regular symbols which appears for example in crystal optics problems.

1.2. Statement of the main results

Let µ(x) ∈ C∞(Rn\ 0) be a homogeneous function of degree 1, such that the level set

Σµ := x ∈ Rn| µ(x) = 1

is diffeomorphic to the unit sphere. By using the embedding Σµ ⊂ Rn , one can locally extend any vector field Ztangent to Σµ to a vector field Z∗ in Rn . Locally Z∗ can be represented as a differential operator

Z∗(x, Dx ) =∑

j=1,...,n

a j (x)Dx j . (1.1)

By using the notation D = −i∂ , the differential operator Z∗ has symbol

Z∗(x, ξ) =∑

j=1,...,n

a j (x)ξ j ,

where a j (x) are homogeneous functions of degree 1 in x . Furthermore, one can construct a Lie algebra A(µ) generatedby vector fields having form (1.1) and satisfying

Z∗(x, Dx )µ(x) = 0, x 6= 0.

In the simplest case, µ(x) = |x |, one has A(µ) = so(n) with generators of type (1.1) and coefficients a j (x)polynomials of degree 1 in x .

We notice that for any two generators of A(µ), the commutator [Z∗j , Z∗k ] is a linear combination of the generatorsof the same algebra with homogeneous coefficients of degree zero in x . The corresponding Sobolev spaces can bedefined by means of the norm

‖ f ‖H N (A(µ)) =

N∑k=0

‖Z∗1 Z∗2 · · · Z∗

k f ‖L2(Rn),

where Z∗1 , Z∗2 , . . . , Z∗N are chosen among the generators of A(µ), eventually they can coincide. For more details seeDefinitions 1.2 and 1.3 and Section 4.

While studying the nonlinear time dependent hyperbolic equations, one needs the continuity of the compositionoperator u → |u|p on H N ( A(µ)). Here and the sequel, the Lie algebra A(µ) is generated by Z ,∇. In this case, forany given t > 0, the sets x ∈ Rn

| t = µ(x) describe the characteristic surfaces and the Z fields give informationon the invariance properties of the equation.

On the other hand, it is possible to reduce a symmetric first order system to pseudo-differential equations (see [8]).The characteristic surfaces of these equations are given in dual coordinates: for any given t > 0 one considersξ ∈ Rn

| t = λ(ξ). For this reason, we look for continuity of the composition operator u → |u|p in the dual setting.


More precisely, assuming that µ(x) is a norm on Rn , its dual norm is given by

λ(ξ) = supω∈Σµ

|ω · ξ |.

Proceeding as before, we can introduce symbols of type

Y ∗(x, ξ) =n∑

j=1

b j (ξ)x j ,

where b j (ξ) are homogeneous functions of degree 1 in ξ, and Y ∗(ξ, Dξ )λ(ξ) = 0 for all ξ 6= 0. Then the operatorshaving form Y ∗(x, Dx ) generate a Lie algebra A(λ). We also introduce the corresponding Sobolev spaces defined bymeans of

‖ f ‖H N (A(λ)) =

N∑k=0

‖Y ∗1 Y ∗2 · · · Y∗

k f ‖L2(Rn)

where Y ∗1 , Y ∗2 , . . . , Y ∗N are chosen among the generators of A(λ) (see Definitions 1.2 and 1.3 and Section 4). We canalso consider the Lie algebra A(λ) generated by Y,∇ and we ask if the composition operator u → |u|p acts onH N (A(λ)).

It may be surprising that this result will require a geometric assumption on λ.

Definition 1.1. Let Σλ := ξ ∈ Rn| λ(ξ) = 1. We say that Σλ is strictly convex if ξ ∈ Rn

| λ(ξ) ≤ 1 is a convexset and TξΣλ ∩ Σλ = ξ for all ξ ∈ Σλ. Here TξΣλ is the tangent plane to Σλ at ξ .

The final aim of this paper is to gain the following results.

Theorem 1.1. Let N ∈ N such that N ≥ 2[(n − 1)/2] + 1. Take p ∈ N∗ or p ∈ R, with p ≥ N. Assume that Σµ isstrictly convex. For any given f ∈ H N (A(µ)) one has

‖| f |p‖H N (A(µ))≤ C‖ f ‖p

H N (A(µ)).

Theorem 1.2. Let N ∈ N such that N ≥ 2[(n − 1)/2] + 1. Take p ∈ N∗ or p ∈ R, with p ≥ N. Assume that Σλ isstrictly convex. For any given f ∈ H N (A(λ)), one has

‖| f |p‖H N (A(λ))≤ C‖ f ‖p

H N (A(λ)).

Note that for the particular case λ(ξ) = |ξ |, µ(x) = |x | we have A(λ) = A(µ). In general A(λ) 6= A(µ). Thereader can find an analysis of H N (A(µ)) in [9] and the pseudo-differential counterpart H N (A(λ)) in [4]. Here weprove that under the strictly convexity assumption for Σλ, these spaces coincide in the sense of the equivalence of thecorresponding norms.

Theorem 1.3. If the hypersurface Σλ is strictly convex, then for any given N ∈ N, one has

H N (A(λ)) = H N (A(µ)).

By using this equivalence, we can carry out the proof of Theorem 1.2 from Theorem 1.1.The proof of Theorem 1.3 is the core of this paper and it will follow from a more general result. In order to state

this result, we shall introduce suitable classes of pseudo-differential operators whose symbols are represented by finitelinear combinations of homogeneous functions p(x)q(ξ). In particular, these locally bounded symbols can lose thesmoothness either in x = 0 or in ξ = 0.

Definition 1.2. We say that y(x, ξ) ∈ C∞ (Rn× (Rn

\ 0)) belongs to the symbol class S1,1P H if y(x, ξ) is given by

y(x, ξ) = q0(ξ) + y1(x, ξ) where q0(ξ) ∈ C∞(Rn\ 0) is a homogeneous function of degree zero and y1(x, ξ)

is a finite linear combination of functions p(x)q(ξ), where p(x) is a homogeneous polynomial of degree 1, whileq(ξ) ∈ C∞(Rn

\ 0) is a homogeneous function of degree 1.


In a similar way, we introduce the symbol class S1,1H P of the functions z(x, ξ) ∈ C∞ ((Rn

\ 0)× Rn) such thatz(x, ξ) = p0(x)+ z1(x, ξ) with p0(x) ∈ C∞(Rn

\ 0) a homogeneous function of degree zero and z1(x, ξ) a finitelinear combination of functions p(x)q(ξ), where p(x) ∈ C∞(Rn

\ 0) is a homogeneous function of degree 1, whileq(ξ) is a homogeneous polynomial of degree 1.

Passing to the corresponding operators, we have that OPS1,1P H contains pseudo-differential operators with non-

regular symbols in the ξ -variable, while OPS1,1H P contains differential operators with non-regular coefficients. For

example,

A(λ) ⊂ OPS1,1P H and A(µ) ⊂ OPS1,1

H P .

Usually, smoothness is needed for having a good composition rule among the operators. In Lemma 4.1, we shallestablish that the composition of the above defined operators works as expected. In particular, we will observe thatOPS1,1

H P and OPS1,1P H are endowed with a Lie algebra structure with the standard bracket

[A, B] = AB − B A.

Moreover we shall consider suitable Lie subalgebras of OPS1,1P H , (respectively of OPS1,1

P H ); this means we deal with

subspaces A ⊆ OPS1,1P H such that if a1, a2 ∈ A then [a1, a2] ∈ A.

Definition 1.3. Take a finite set of elements Y j j=1,...,N ⊂ OPS1,1P H , such that

[Y j , Yl ] =∑

c jlp(D)Yp, (1.2)

where c jlp(ξ) are smooth functions in Rn\ 0, homogeneous of degree zero.

Let A(Y1, . . . , YN ) be the set of all linear combinations c1(D)Y1+· · ·+ cN (D)YN , where c j (ξ) are homogeneousfunctions of degree zero. Then A(Y1, . . . , YN ) is a subalgebra in OPS1,1

P H called the subalgebra generated byY1, . . . , YN .

Similarly, take a finite set of elements Z j j=1,...,N ⊂ OPS1,1H P , such that

[Z j , Zl ] =∑

d jlp(x)Z p, (1.3)

where d jlp(x) are smooth functions in Rn\ 0, homogeneous of degree zero. Let A(Z1, . . . , Z N ) be the set of all

linear combinations d1(x)Z1 + · · · + dN (x)Z N , where d j (x) are homogeneous of degree zero. Then A(Z1, . . . , Z N )

is a subalgebra of OPS1,1H P called the subalgebra generated by Z1, . . . , Z N .

We are ready to introduce Sobolev spaces associated with the above subalgebras.

Definition 1.4. Let A(Y1, . . . , YN ) be the subalgebra in OPS1,1P H generated by Y1, . . . , YN . Given any integer m ≥ 0

and any smooth compactly supported function f , we can define the norm

‖ f ‖Hm (A(Y )) =∑|α|≤m

‖Y α f ‖2,

here Y α = Y α11 · · · Y

αNN with α = (α1, . . . , αN ). Taking the closure (with respect to this norm) of all smooth

compactly supported functions, we obtain the generalized Sobolev space Hm(A(Y1, . . . , YN )). By using (1.3), asimilar construction can be done by using OPS1,1

H P operators. Given A(Z1, . . . , Z N ), we can associate the spaceHm(A(Z1, . . . , Z N )) endowed with the norm

‖ f ‖Hm (A(Z)) =∑|α|≤m

‖Zα f ‖2.

Due to (1.2), respectively (1.3), we see that different orders of the above generators give equivalent norms inHm(Y1, . . . , YN ), respectively in Hm(Z1, . . . , Z N ).

While considering the algebras A(Y1, . . . , YN ) and A(Z1, . . . , Z N ) we shall make the following crucialassumption.


Assumption 1.1. The generators Y j j=1,...,N ⊂ OPS1,1P H , which satisfy the condition (1.2), can be represented as∑N

k=1 ck(x, ξ)Zk(x, ξ), where the functions ck(x, ξ) ∈ C∞ ((Rn\ 0)× (Rn

\ 0)) are homogeneous of degreezero in x and ξ.

Similarly, the generators Z j j=1,...,N ⊂ OPS1,1H P , which satisfy the condition (1.3), can be represented as∑N

k=1 dk(x, ξ)Yk(x, ξ), where dk(x, ξ) ∈ C∞ ((Rn\ 0)× (Rn

\ 0)) are homogeneous of degree zero in x and ξ.

The key point in the proof of Theorem 1.3 is the following more general result.

Theorem 1.4. Let A(Y1, . . . , YN ) be the subalgebra of OPS1,1P H having generators Y1, . . . , YN which satisfy (1.2).

Let A(Z1, . . . , Z N ) be the subalgebra of OPS1,1H P generated by Z1, . . . , Z N which satisfy (1.3). If Assumption 1.1

holds, then, for all integer m ≥ 1, we have

Hm(A(Y )) = Hm(A(Z)).

1.3. Plan of the work

In Section 2 we prove that if Σλ is strictly convex, then the generators of A(λ) and A(µ) satisfy Assumption 1.1.We present some examples of these “equivalent symbols” which appear in the hyperbolic PDE theory. In Section 3we recall some definitions and properties of Paley–Littlewood partitions of unity and we collect some basic facts ofpseudo-differential operators with smooth symbols. In Section 4 we generalize Definition 1.2 introducing the OPSH Pand OPSP H classes of pseudo-differential operators with non-regular symbols which form Lie algebras of operators.Section 5 is devoted to commutator estimates used in Section 6 to prove that “equivalent symbols” lead to “equivalentnorms”, that is Theorem 1.4. As a consequence, Theorem 1.3 follows. In Section 7 we establish some nonlinearestimates, in particular we prove Theorems 1.1 and 1.2. In the Appendix, for completeness, we collect some technicalresults on the Lie bracket.

1.4. Notation

– We assume that n ≥ 3. The inner product of x, y ∈ Rn will be denoted by x · y.– Let α = (α1, α2, . . . , αn) ∈ Nn be a multi-index of length |α| = α1 + α2 + αn . We put ∂α = ∂α1

x1 ∂α2x2 . . . ∂

αnxn and

Dα= i−|α|∂α . Sometimes we denote ∂α = ∇α .

– Our choice for the coefficients of Fourier transform is the following:

f (ξ) = (2π)−n/2∫Rn

e−ix ·ξ f (x)dx =:∫Rn

e−ix ·ξ f (x)dx .

– With C k0(R

n) we denote the space of functions belonging to C k(Rn) with compact support. Let s ∈ R. The Sobolevspaces H s(Rn) is the closure of C∞0 (R

n) with respect to the norm ‖ f ‖H s (Rn) := ‖(1 + |ξ |2)s/2 f ‖L2(Rn). FinallyLi p(Rn) stands for the space of Lipschitzian functions with finite seminorm:

‖ f ‖Li p := supx 6=y

| f (x)− f (y)|

|x − y|.

We omit to write Rn when considering spaces of functions defined on Rn . Moreover, we put ‖ · ‖p := ‖ · ‖L p(Rn).– With L(H s, H r ) we denote all the linear bounded operators from H s in H r .– Let A(x), B(x) be two positive functions on suitable domains. We write A ' B if there exist C1,C2 > 0 such that

C1 A(x) ≤ B(x) ≤ C2 A(x) for all x in the intersection of the domains of A, B. Similarly A(x) . B(x) means thatthere exists C > 0, independent of x , such that A(x) ≤ C B(x).

– We shall often use cut-off functions, in particular we denote with ϕ0 a smooth compactly supported function suchthat ϕ0(x) = 1 close to zero and ϕ0(x) = 0 for large x . Moreover we put ϕ∞(x) = (1− ϕ0(x)).

– The scaling operator will be denoted by Sa , so that Sa f (x) = f (ax). If there is no danger of confusion, for brevity,we shall put S j ( f ) := S2− j ( f ).

2. Equivalent symbols related to hyperbolic first order systems

In this section we exhibit a class of operators to which the result of this work applies. In particular, this exampleappears while studying hyperbolic systems.


Let us assume that λ(ξ) : Rn→ R is a norm with corresponding unit sphere

Σλ := ξ ∈ Rn| λ(ξ) = 1.

Let µ be its dual norm

µ(x) = supω∈Σλ

|ω · x | = supω∈Σλ

ω · x

with unit sphere Σµ := x ∈ Rn| µ(x) = 1. For simplicity, we take λ,µ ∈ C∞(Rn

\ 0).The following operators are naturally related to Σλ, Σµ:

Ω jk(λ)(x, D) = x j∂kλ2(D)− xk∂ jλ

2(D);

Ω jk(µ)(x, D) = ∂ jµ2(x)∂k − ∂kµ

2(x)∂ j ,

with symbols

Ω jk(λ)(x, ξ) = x j∂kλ2(ξ)− xk∂ jλ

2(ξ);

Ω jk(µ)(x, ξ) = ∂ jµ2(x)ξk − ∂kµ

2(x)ξ j .

In particular we can choose n − 1 fields amongΩ j,k(µ)(x, D)

j<k which span the tangent space to Σµ in a fixed

x ∈ Σp.

Lemma 2.1. There exist some symbols arsjklm(ξ) homogeneous of degree zero functions, smooth on Rn

\ 0, such that

[Ω jk(λ)(x, D),Ωlm(λ)(x, D)] =∑r,s

arsjklm(D)Ωrs(λ)(x, D). (2.1)

There exist some brsjklm(x) ∈ C∞(Rn

\ 0) homogeneous of degree zero functions, such that

[Ω jk(µ)(x, D),Ωlm(µ)(x, D)] =∑r,s

brsjklm(x)Ωrs(µ)(x, D). (2.2)

Proof. Applying the relation [a(D), x j ] = (D j a)(D), we get

[Ω j,k(λ),Ωl,m(λ)] = ∂2k,lλ

2(D)Ω j,m(λ)− ∂2m,kλ

2(D)Ω j,l(λ)− ∂2j,lλ

2(D)Ωk,m(λ)+ ∂2j,mλ

2(D)Ωk,l(λ).

This gives (2.1). The proof of (2.2) is similar.

Our aim is to find suitable conditions on λ,µ such that

Ω jk(µ)(x, ξ) =∑r,s

arsjk(x, ξ)Ωrs(λ)(x, ξ), (2.3)

Ω jk(λ)(x, ξ) =∑r,s

brsjk(x, ξ)Ωrs(µ)(x, ξ), (2.4)

with arsjk, brs

jk homogeneous of degree zero in both variables. Our goal is to show that Lemma 2.1 and the previousequivalence imply that∑

|α|≤N

‖Ω(λ)α f ‖2 '∑|α|≤N

‖Ω(µ)α f ‖2,

for all N ∈ N. This means that Theorem 1.3 holds.A typical example arises in geometrical optics (see [8]). In that case

λ(ξ) = (a1ξ21 + a2ξ

22 + a3ξ

23 )

1/2

with a j > 0, j = 1, 2, 3. It is clear that µ(x) = (a−11 ξ2

1 + a−12 ξ2

2 + a−13 ξ2

3 )1/2. The relations (2.3) and (2.4) hold, in

fact

Ω j,k(λ)(x, ξ) = a j akΩ j,k(µ)(x, ξ).


The situation is not so evident for other norms on Rn , for example for

λ(ξ) = |ξ |p = (|ξ1|p+ · · · + |ξn|

p)1/p, µ(x) = |x |q = (|x1|q+ · · · + |xn|

q)1/q (2.5)

with p−1+ q−1

= 1 and p 6= 2. An abstract argument can help us to determine a class of norms such that (2.3) and(2.4) hold.

Definition 2.1. Let λ ∈ C(Rn) be a positive function such that λ ∈ C∞(Rn\ 0), λ(0) = 0 and ∇λ(ξ) 6= 0 for all

ξ 6= 0. We say that Σλ is strictly convex if λ(ξ) ≤ 1 is a convex set and for any given ξ ∈ Σλ one has

TξΣλ ∩ Σλ = ξ.

Here TξΣλ means the affine tangent space to Σλ at ξ , that is the hyperplane orthogonal to ∇λ(ξ) passing through ξ .

Example 2.1. Let λ(ξ) = |ξ1|p+ · · · + |ξn|

p with 1 < p < +∞. The hypersurface Σp := ξ ∈ Rn| λ(ξ) = 1 is

strictly convex if 1 < p < +∞.In fact the set Dp := ξ ∈ Rn

| |ξ1|p+ · · · + |ξn|

p≤ 1 is convex and the outward normal N (ξ) :=

(|ξ1|p−2ξ1, . . . , |ξn|

p−2ξn) is well defined. It remains to show that for any ξ ∈ Σp, we have TξΣp ∩ Σp = ξ.

If this does not hold, then there exist ξ , ¯ξ ∈ TξΣp ∩ Σp with ξ 6= ¯ξ . Due to the strict convexity of the function

r ∈ R+ → r p∈ R, one sees that λ((ξ + ¯ξ)/2) < (λ(ξ ) + λ( ¯ξ))/2 = 1 that is (ξ + ¯ξ)/2 ∈ Dp \ Σp. On the other

hand ¯ξ ∈ TξΣp, hence we have (ξ + ¯ξ)/2 ∈ TξΣp. Since Dp is convex, this is absurd since an internal point of aconvex set cannot belong to a tangent plane of a boundary point.

A sufficient condition for the strict convexity is given by the strict positivity of the Gaussian curvature. We noticethat for p > 2, the hypersurface Σp is strictly convex even though in some points the Gaussian curvature vanishes.However, the strict convexity assumption for Σλ, Σµ will be enough for proving the equivalence of symbols Ω(λ),Ω(µ).

Lemma 2.2. Given a pair of dual norms (λ, µ), for all x, ξ ∈ Rn\ 0, we have

µ(∇λ(ξ)) = 1, λ(∇µ(x)) = 1. (2.6)

Moreover, if Σλ is strictly convex, then Σµ has the same property and

∇λ(∇µ(x)) =x

µ(x); ∇µ(∇λ(ξ)) =

ξ

λ(ξ). (2.7)

Proof. Sinceµ(x) = supω∈Σλω·x , in order to proveµ(∇ξλ) = 1, we look for the stationary point of L : Rn

×R→ Rdefined by

L(ω, α) = ω · ∇λ(ξ)+ α(λ(ω)− 1).

We observe that the couples (ωM (ξ), αM (ξ)) = (ξ/λ(ξ),−1) and (ωm(ξ), αm(ξ)) = (−ξ/λ(ξ), 1) solve the system∇w,αL = 0 and give respectively a maximum and a minimum of (ω · ∇λ(ξ)) on Σλ. In particular µ(∇λ(ξ)) =ωM (ξ) · ∇λ(ξ) = 1. In a similar way, one sees that λ(∇µ(x)) = 1.

Put Φx (ω) = ω · x . Since Σλ strictly convex, the function Φx (ω) has only two critical points on Σλ. Let us callthem ω+(x), ω−(x). Since µ(x) = supω∈Σλ

ω · x , one has Φx (ω+(x)) = µ(x). On the other hand, by the Eulerformula one has Φx (∇µ(x)) = µ(x), it follows that ω+(x) = ∇µ(x). Due to the Lagrange multiplier theorem, thereexists α+(x) ∈ R such that x = −α+(x)∇λ(∇µ(x)). Applying µ to both sides of this relation, due to (2.6), we obtainµ(x) = |α+(x)|. This implies that ∇λ(∇µ(x)) = x/µ(x) or ∇λ(∇µ(x)) = −x/µ(x). From (2.6) we deduce that∇µ(x) ·∇λ(∇µ(x)) = 1, hence we can exclude the second possibility and gain the first identity of (2.7) with ω = ω+.In order to consider ω = ω−, we recall that

Φx (ω−) = infω ∈ Σλ | ω · x = − supω ∈ Σλ | ω · x = −µ(x)

and we proceed as before.


Finally we prove by absurd that also Σµ is strictly convex and hence the second identity of (2.7) holds. Assumingthat Σµ is a not strictly convex set, then there exist two different elements x, ¯x ∈ Σµ, such that ¯x ∈ TxΣµ. Let us put

ξ = ∇µ(x) and ¯ξ = ∇µ( ¯x). By using the first relation of (2.7) we have ξ 6= ¯ξ . Since ∇µ is homogeneous of degreezero, ¯x ∈ TxΣµ means that 0 = ( ¯x− x) ·∇µ(x) = ¯x ·∇µ(x). In turn, this implies that ∇λ( ¯ξ) · ξ = 0, that is ¯ξ ∈ TξΣλagainst the strict convexity assumption for Σλ. This concludes the proof.

Theorem 2.3. Suppose λ is a norm such that Σλ is strictly convex. Consider µ its dual norm. Then (2.3) and (2.4)hold. Moreover the homogeneous functions ars

jk(x, ξ), brsjk(x, ξ) are smooth on the manifold Σµ × Σλ.

Proof. Since Ω jk(λ)(x, ξ) and Ω jk(µ)(x, ξ) are homogeneous of degree one with respect to x and ξ , it is sufficientto establish our result for x ∈ Σµ and ξ ∈ Σλ. Consider the function

ξ → F(ξ) = (∇λ2(ξ))/2.

This is a well-defined and smooth function on the hypersurface Σλ. Since Σλ and Σµ are strictly convex, the relations(2.7), imply

∇µ(F(ξ)) = ξ, µ(F(ξ)) = 1

for any ξ ∈ Σλ. Hence, F maps Σλ into Σµ. The function F is invertible and its inverse is

x → G(x) = (∇µ2(x))/2.

This yields that F is a diffeomorphism of Σλ onto Σµ with inverse diffeomorphism G.We shall prove only (2.3), since the argument for (2.4) is similar. Making the change of variables y = G(x) in

(2.3), we see that

Ω jk(µ)(F(y), ξ) = 2(y jξk − ykξ j ).

So fixing j, k, we reduce to establish the following identity

∂ jλ(y)∂kλ(ξ)− ∂kλ(y)∂ jλ(ξ) =∑r<s

brsjk(y, ξ)(yrξs − ysξr ), y, ξ ∈ Σλ (2.8)

with brsjk(y, ξ) being smooth functions on Σλ × Σλ. Fixing y ∈ Σλ, we prove (2.8) for any given ξ ∈ Σλ. We

distinguish two cases, according to the size of

‖y ∧ ξ‖2 :=∑j<k

(y jξk − ykξ j )2.

When ‖y ∧ ξ‖2 ≥ C0 > 0, we have (2.8) with

brsjk(ξ, y) =

(∂ jλ(y)∂kλ(ξ)− ∂kλ(ξ)∂ jλ(y))(y ∧ ξ)rs

‖y ∧ ξ‖2.

For any C0 > 0, such coefficients are smooth on Σλ in both variables and homogeneous of degree zero. The constantC0 > 0 will be chosen during the proof.

Since ξ, y ∈ Σλ, the relation ‖y ∧ ξ‖2 ≤ C0 means ξ is close to y or ξ is close to −y. We shall consider only thefirst case, since the other is similar. We construct a smooth orthonormal basis

e1(y) =y

‖y‖, e2(y), . . . , en(y).

In this local coordinate system, a given point z ∈ Rn will be written as

z := (z1, z′), z′ ∈ Rn−1.

In particular y = (‖y‖, 0Rn−1). We represent ξ ∈ Σλ in the form

ξ = y + z.

The condition ξ ∈ Σλ gives λ(y + z) = 1.


We introduce the function Fy(z1, z′) := λ(y + z) − 1, we have Fy(0, 0) = 0 and ∂z1 Fy(0, 0) = (∂z1λ)(y) 6= 0.We can apply the implicit function theorem that yields W ⊂ Rn−1 a sufficiently small neighborhood of the origin ofRn−1 and a smooth function W 3 z′ 7→ z1(y, z′) such that λ(y + (z1(y, z), z′)) = 1. By using the identity

z(y, z′) = z1(y, z′)e1(y)+ z2e2(y)+ · · · + znen(y),

we can take C0 > 0 such that (ξ − y)′ ∈ W . Hence we define the parametrization

ξ = y + z(y, z′).

Moreover ζ : z′ ∈ W → ξ(y, z′) gives a diffeomorphism of W onto ζ(W ) ∩Σλ. For any given f : ζ(W ) ∩Σλ→ R,we can associate f : W → R such that f (ξ) = f (y + z(y, z′)) =: f (z′). By using the Taylor expansion of f at thepoint z′ = 0, we have

f (ξ) = f (y + z(y, 0))+n∑

s=2

asj (y, z′)zs

where as(y, z′) are smooth functions in y and z′. Note also that z(y, 0) = 0, hence taking f = ∂ jλ, we deduce

∂ jλ(ξ) = ∂ jλ(y + z(y, z′)) = ∂ jλ(y)+n∑

s=2

asj (y, z′)zs .

This relation implies

∂ jλ(y)∂kλ(ξ)− ∂kλ(y)∂ jλ(ξ) =

n∑s=2

(∂ jλ(y)ask(y, z′)− ∂kλ(y)a

sj (y, z′))zs . (2.9)

It is clear that the coefficients are smooth functions in (y, z′) ∈ Σλ ×W . Taking advantage of the relations

(y ∧ ξ(y, z′))1s = (y ∧ z(y, z′))1s = ‖y‖zs, s = 2, . . . n,(y ∧ ξ(y, z′))rs = 0, r < s, r 6= 1,

we see that (2.9) corresponds to (2.8) in the new coordinates (y, z′).In order to prove the smoothness of the coefficients in the intersection of ζ(W ) with the set ‖ξ ∧ y‖ ≥ C0, we take

C0 > diamζ(W ). Hence it suffices to rewrite (2.8) in z′ coordinates and to use the uniqueness of the decompositionin this basis.

Remark 2.1. The relation (2.8) can be easily proved for the norms (2.5) when p ∈ N, p ≥ 2. In this case we have

brsjk = (yrξs)

p−2+ (yrξs)

p−3(ysξr )+ · · · + (ysξr )p−3(yrξs)+ (ysξr )

p−2, r = j, s = kbrs

jk = 0 otherwise.

For the case p 6∈ N, the abstract proof of the previous theorem gives the equivalence without determining explicitlythe coefficients brs

jk .

3. The basic tools

The notation a(x, D) means a pseudo-differential operator with symbol a(x, ξ):

a(x, D) f (x) =∫Rn

eix ·ξa(x, ξ) f (ξ)dξ.

3.1. Convolution-type operators

A convolution operator has symbol a(x, ξ) = a(ξ). We say that a(D) is a convolution operator since a(D) f (x) =(2π)−n/2(a∨ ∗ f )(x). We will often use convolution operators whose symbols have compact support in a ring thatdoes not contain zero.


Definition 3.1. Let Φ : Rn→ R be a smooth function such that Φ(ξ) ≥ 0 and

supp Φ ⊂ 0 < B < |ξ | < C.

We shall denote by C the class of such symbols with corresponding operators Φ(D) ∈ OPC.

By means of C symbols we can construct a Paley–Littlewood partition of unity.

Definition 3.2. A Paley–Littlewood partition of unity is a sequence of smooth functions Φ j j∈Z which satisfy0 ≤ Φ j ≤ 1, moreover

∑j∈Z Φ j (ξ) = 1 for any ξ ∈ Rn

\ 0 and there exist two positive constants 0 < B < C suchthat supp Φ j ⊂ 2 j B ≤ |ξ | ≤ 2 j C for any j ∈ Z.

Lemma 3.1. Given B,C > 0, we consider Φ(ξ) ∈ C with supp Φ ⊂ B < |ξ | < C. Then

(i) there exists Φ j j∈Z a Paley–Littlewood partition of unity such that for all j ∈ Z, one has supp Φ j ⊂ 2 j B ≤|ξ | ≤ 2 j C and supp Φ0 = supp Φ;

(ii) there exists Ψ j j∈Z a Paley–Littlewood partition of unity such that Ψ0(ξ) ≡ 1 on the support of Φ(ξ).

Proof. For any given ξ 6= 0, we put

F(ξ) =∑

j∈Z Φ(2− jξ).

We observe that there exists N ∈ N, independent of ξ 6= 0, such that this sum contains only N non-vanishing terms.Moreover, for all k ∈ Z one has F(2−kξ) = F(ξ). In order to obtain Φ j j∈Z, we put Φ0(ξ) = Φ(ξ)/F(ξ) andΦ j (ξ) = Φ0(2− jξ). This gives (i). In the same manner, fixing another f ∈ C0(Rn) with supp f ⊂ B < |ξ | < C,we associate F =

∑j∈Z f (2− jξ) and the partition of unity Ψ j such that

Ψ0(ξ) = f (ξ)/F(ξ); Ψ j (ξ) = Ψ0(2− jξ).

It remains to impose some conditions on B,C, B, C so that Ψ0(ξ) ≡ 1 if B < |ξ | < C . This means f (ξ) = F(ξ)for B < |ξ | < C . On the other hand, this is equivalent to obtaining f (2− jξ) = 0 for all j ∈ Z, j 6= 0 and for allB < |ξ | < C . This holds true whenever |ξ | ≥ C/2 and |ξ | ≤ 2B. Hence, we require

maxB, C/2 ≤ B < C ≤ min2B, C.

These conditions are fulfilled for example by taking B = 1 + 2ε, B = 1 + ε, C = 1 + δ, C = 1 + 2δ with0 < 2ε < δ < 1/2+ 2ε. This concludes the proof.

We point out the following property of the Paley–Littlewood partitions of unity:

‖ f ‖22 '∑j∈Z‖Φ j f ‖22. (3.1)

We can rewrite the previous lemma and the previous equivalence in terms of operators.

Lemma 3.2. Let Φ ∈ OPC. There exists a sequence of operators Φ j j∈Z ⊂ OPC such that supp Φ0(ξ) =

supp Φ(ξ), Φ j (D) = S2 j Φ0(D)S2− j and for any given f ∈ L2(Rn) one has

‖ f ‖22 '∑j∈Z‖Φ j (D) f ‖22 '

∑j∈Z

2− jn‖Φ0(D)S2− j f ‖22. (3.2)

Moreover, one has∑j∈Z

2− jn‖Φ(D)S2− j f ‖22 . ‖ f ‖22.

Proof. With the same notation of the proof of Lemma 3.1, given Φ, we associate a Paley–Littlewood partition ofunity such that Φ0(ξ) = Φ(ξ)/F(ξ) and Φ j (ξ) = Φ0(2− jξ). Applying the formula f (ax) = a−n(Sa−1 f )∧(x) and


the Plancherel identity to (3.1) we obtain (3.2). Finally, since F(ξ) ≤ N supξ |Φ(ξ)|, we have∑j∈Z

2− jn‖Φ(D)S2− j f ‖22 .

∑j∈Z

2− jn‖Φ0(D)S2− j f ‖22

and we get the required inequality.

By abuse of notation, we say that Φ j (D) j∈Z is a Paley–Littlewood partition of unity.

3.2. Pseudo-differential operators with smooth symbols

Definition 3.3. Let k ∈ R. The Hörmander’s class OPSk consists of pseudo-differential operators having symbolsa(x, ξ) ∈ C∞(Rn

× Rn) in the Sk symbol class:

|∂αξ ∂βx a(x, ξ)| ≤ Cα,β(1+ |ξ |)k−|α| ∀x ∈ Rn, ∀ξ ∈ Rn,∀α, β ∈ Nn .

The set ∩k Sk will be denoted by S−∞.

In the previous definition an uniform estimate in x-variable is required. In order to cover the case of symbols withsome growth in the x-variable, it is convenient to deal with a larger class of symbols introduced by Parenti in [11].The corresponding operators are often called SG-operators. A complete study of this subject can be found in [6].

Definition 3.4. Let m, k ∈ R. The functions a ∈ C∞(Rn×Rn) belong to the symbol class Sm,k , if for all multi-indexes

α, β ∈ Nn , one has

|∂αξ ∂βx a(x, ξ)| ≤ C(1+ |x |)m−|β|(1+ |ξ |)k−|α| ∀x ∈ Rn, ∀ξ ∈ Rn .

In this case, we write a(x, D) ∈ OPSm,k . Further, we set

Sm,−∞= ∩k Sm,k, S−∞,k = ∩m Sm,k, S−∞,−∞ = ∩m,k Sm,k .

We note that for m ≤ 0, we have Sm,k⊂ Sk .

Now we recall some properties for the calculus of pseudo-differential operators. For the proof of the followingresults we refer to [6,13,14].

Lemma 3.3 ([14] Proposition 0.3.C). Let k1, k2 ∈ R. Let p(x, D) ∈ OPSk1 and q(x, D) ∈ OPSk2 . Thenp(x, D)q(x, D) ∈ OPSk1+k2 and p(x, D)q(x, D) has symbol p(x, ξ)q(x, ξ)+r(x, ξ) where the operator r(x, D) ∈OPSk1+k2−1.

Lemma 3.4 ([6] Chapter I, Sections 5, 6 and 7). Let m1,m2, k1, k2 ∈ R. Let p(x, D) ∈ OPSm1,k1 and q(x, D) ∈OPSm2,k2 . Then p(x, D)q(x, D) ∈ OPSm1+m2,k1+k2 and p(x, D)q(x, D) has symbol

p(x, ξ)q(x, ξ)+ r(x, ξ)

where r(x, D) ∈ OPSm1+m2−1,k1+k2−1.

Remark 3.1. In some cases the previous results can be simplified. If q(x, D) = q(D) is a convolution-type operatorthen p(x, D)q(D) has symbol p(x, ξ)q(ξ). Similarly if p(x, D) = p(x) is a multiplicative operator then p(x)q(x, D)has symbol p(x)q(x, ξ).

Lemmas 3.3 and 3.4 imply the following.

Lemma 3.5. Let p(x, D) ∈ OPSk1 , q(x, D) ∈ OPSk2 . One has [P, Q] ∈ OPSk1+k2−1.Let p(x, D) ∈ OPSm1,k1 , q(x, D) ∈ OPSm2,k2 . One has [P, Q] ∈ OPSm1+m2−1,k1+k2−1.

We quote the following classical result on the L2 boundedness of these operators.

Lemma 3.6 ([14] Proposition 0.5.E). If P ∈ OPS0, then ‖P(x, D) f ‖2 . ‖ f ‖2 for any given f ∈ L2.


4. Lie algebras of pseudo-differential operators with non-regular symbols

Here we consider symbols represented as a linear combination of homogeneous functions p(x)q(ξ). In particular,these symbols are locally bounded but they can lose smoothness either in x = 0 or in ξ = 0.

Definition 4.1. Let m, k ∈ N. We define the symbol class

(P H)m,k

that consists of all functions a(x, ξ) ∈ C∞ (Rn× (Rn

\ 0)) such that a(x, ξ) is a finite linear combinationof functions p(x)q(ξ), where p(x) is a homogeneous polynomial of degree m, while q(ξ) ∈ C∞(Rn

\ 0) ishomogeneous function of degree k.

In a similar way, we introduce the space

(H P)m,k

of all functions a(x, ξ) ∈ C∞ ((Rn\ 0)× Rn) which are finite linear combination of functions p(x)q(ξ) with

p(x) ∈ C∞(Rn\ 0) a homogeneous function of degree m and q(ξ) a homogeneous polynomial of degree k.

Definition 4.2. The symbol a(x, ξ) belongs to Sm,kP H if and only if a = am + am−1 + · · · + a0 where am− j ∈

(P H)m− j,k− j , for all j = 0, . . . ,m. In other words

Sm,kP H =

m⊕j=0

(P H)m− j,k− j .

In a similar way, we set

Sm,kH P =

k⊕j=0

(H P)m− j,k− j .

Finally, we shall denote by OPSm,kP H (respectively OPSm,k

H P ) the class of operators associated with symbols in Sm,kP H

(respectively in Sm,kH P ).

Example 4.1. With the same notation of Section 2, we have Ω jk(λ)(x, D) ∈ OPS1,1P H . On the contrary,

Ω jk(µ)(x, D) ∈ OPS1,1H P . More precisely Ω jk(λ)(x, ξ) ∈ (P H)1,1 and Ω jk(µ)(x, ξ) ∈ (H P)1,1.

Remark 4.1. We notice that OPSm,kP H contains pseudo-differential operators with non-regular symbol in the ξ -variable,

while OPSm,kH P contains differential operators with non-regular coefficients. It is evident that OPS0,0

P H ∈ L(L2, L2) and

OPS0,0H P ∈ L(L2, L2).

Lemma 4.1. Let m,m1,m2, k, k1, k2 ∈ N and α, β ∈ Nn .

(i) If a(x, ξ) ∈ Sm,kP H , then Dα

x Dβξ a(x, ξ) ∈ Sm−|α|,k−|β|

P H .

Likewise, if a(x, ξ) ∈ Sm,kH P then Dα

x Dβξ a(x, ξ) ∈ Sm−|α|,k−|β|

H P .

(ii) Let a(x, D) ∈ OPSm,kP H , a(x, ξ) =

∑mj=0 p j (x)q j (ξ). For any given f ∈ C∞0 we have

(a(x, D) f )∧ (ξ) =m∑

j=0

(−1)m− j p j (Dξ )(q j (ξ) f (ξ)).

(iii) If a1 ∈ OPSm1,k1P H and a2 ∈ OPSm2,k2

P H , then a1a2 ∈ OPSm1+m2,k1+k2P H .

(iv) If a1 ∈ OPSm1,k1H P and a2 ∈ OPSm2,k2

H P then a1a2 ∈ OPSm1+m2,k1+k2H P .

Proof. (i) This is obvious.(ii) It suffices to take a(x, ξ) = p(x)q(ξ). Since p(x) is a homogeneous polynomial of degree m, we have

eix ·ξ p(x) = p(Dξ )(eix ·ξ ).


Integration by parts implies

a(x, D) f (x) = (−1)m∫Rn

eix ·ξ p(Dξ )(q(ξ) f (ξ))dξ. (4.1)

(iii) We need only to check the assertion for a j (x, ξ) = p j (x)q j (ξ), j = 1, 2 with p j (x) being a homogeneouspolynomial of degree m j and q j (ξ) homogeneous of degree k j . From (4.1) we get

a1a2( f )(x) =∫Rn

eix ·ξa1(x, ξ) (a2(x, D) f )∧ (ξ)dξ = (−1)m2

∫Rn

eix ·ξa1(x, ξ)p2(Dξ )(

q2(ξ) f (ξ))

dξ.

Let p t2(Dξ ) be the transpose of p2(Dξ ), we have p t

2(Dξ ) = (−1)m2 p2(Dξ ), hence

a1a2( f )(x) =∫Rn

eix ·ξ p(x, ξ) f (ξ)dξ, p(x, ξ) = e−ix ·ξq2(ξ)p2(Dξ )(

eix ·ξa1(x, ξ)).

Moreover, one has

e−ix ·ξ p2(Dξ )(

eix ·ξa1(x, ξ))=

∑|α|+|β|=m2

Cα,βxαDβξ a1(x, ξ). (4.2)

It is clear that q2(ξ)xαDβξ a1(x, ξ) ∈ (P H)m1+|α|,k1+k2−|β|; since m2 = |α| + |β|, we obtain

p(x, ξ) ∈m2⊕j=0

(P H)m1+m2− j,k1+k2− j⊂ Sm1+m2,k1+k2

P H .

(iv) Let x 6= 0. For a composition rule in OPSm,kH P , we directly get

a1(x, Dx )a2(x, Dx ) f (x) = p1(x)∑

|α|+|β|=k1

Cα,βDαx p2(x)D

βx q2(Dx ) f (x).

Since p1(x)Dαx p2(x) is homogeneous of degree m1 + m2 − |α| and Dβ

x q2(Dx ) is a differential operator of orderk2 + |β| = k1 + k2 − |α|, we can conclude

a1a2 ∈

k1⊕j=0

OPSm1+m2− j,k1+k2− jH P ⊂ OPSm1+m2,k1+k2

H P .

This completes the proof.

A trivial but useful result concerns the composition of OPSP H operators with OPC ones.

Lemma 4.2. Let Y (x, D) ∈ OPSm,kP H and Ψ(D) ∈ OPC. Then Y (x, D)Ψ(D) ∈ OPSm,−∞.

Now we extend Lemma 3.5 to OPSH P and OPSP H operators.

Lemma 4.3. If a1 ∈ OPSm1,k1P H and a2 ∈ OPSm2,k2

P H then [a1, a2] ∈ OPSm1+m2−1,k1+k2−1P H .

If a1 ∈ OPSm1,k1H P and a2 ∈ OPSm2,k2

H P then [a1, a2] ∈ OPSm1+m2−1,k1+k2−1H P .

Proof. In order to treat the P H case, we follow the same line of the proof of Lemma 4.1(iii). Let a j (x, ξ) =p j (x)q j (ξ), for j = 1, 2; we can express

[a1, a2] f (x) =∫Rn

eix ·ξ p1,2(x, ξ) f (ξ)dξ

with

p1,2(x, ξ) = e−ix ·ξq2(ξ)p2(Dξ )(

eix ·ξa1(x, ξ))− e−ix ·ξq1(ξ)p1(Dξ )

(eix ·ξa2(x, ξ)

).


From (4.2) we obtain

p1,2(x, ξ) = q2(ξ)p2(x)p1(x)q1(ξ)+ q2(ξ)∑

|α|+|β|=m2|β|6=0

Cα,βxαDβξ a1(x, ξ)

− q1(ξ)p1(x)p2(x)q2(ξ)+ q1(ξ)∑

|γ |+|δ|=m1|δ|6=0

Cγ,δxγ Dδ

ξa2(x, ξ).

The first and third terms are opposite, so that

p1,2(x, ξ) ∈m2−1⊕

j=0

(P H)m1+m2−1− j,k1+k2−1− j⊕

m1−1⊕j=0

(P H)m1+m2−1− j,k1+k2−1− j;

it follows p1,2 ∈ OPSm1+m2−1,k1+k2−1P H .

In the H P case we can proceed as follows:

[p1(x)q1(D), p2(x)q2(D)] = p1(x)[q1(D), p2(x)]q2(D)+ p2(x)[p1(x), q2(D)]q1(D).

By using the Leibnitz rule we see that [ph(x), q j (D)] ∈ OPSmh−1,k j−1H P . Recalling Lemma 4.1(iv), we conclude.

We can deduce that OPS1,1P H and OPS1,1

H P have Lie algebra structure on R. Now we construct suitable Lie

subalgebras of OPS1,1P H , (respectively of OPS1,1

P H ); this means we deal with subspaces A ⊆ OPS1,1P H such that if

a1, a2 ∈ A then [a1, a2] ∈ A.

Definition 4.3. Take a finite set of elements Y j j=1,...,N ⊂ OPS1,1P H , such that

[Y j , Yl ] =∑

p=1,...,N

c jlp(D)Yp, (4.3)

where c jlp(ξ) are smooth functions in Rn\0 homogeneous of degree zero. Let A(Y1, . . . , YN ) be the set of all linear

combinations

c1(D)Y1 + · · · + cN (D)YN ,

where c j (ξ) are smooth functions in Rn\ 0 homogeneous of degree zero. Then A(Y1, . . . , YN ) is a subalgebra in

OPS1,1P H called subalgebra generated by Y1, . . . , YN .

Definition 4.4. Take a finite set of elements Z j j=1,...,N ⊂ OPS1,1H P , such that

[Z j , Zl ] =∑

p=1,...,N

d jlp(x)Z p, (4.4)

with d jlp(x) being smooth functions in Rn\ 0 homogeneous of degree zero. Let A(Z1, . . . , Z N ) be the set of all

linear combinations

d1(x)Z1 + · · · + dN (x)Z N ,

where d j (x) are homogeneous of degree zero and smooth in Rn\0. Then A(Z1, . . . , Z N ) is a subalgebra of OPS1,1

H Pcalled subalgebra generated by Z1, . . . , Z N .

Example 4.2. Let n ≥ 3, j, k = 1, . . . , n with j < k. Recalling Lemma 2.1, we deduce that the (n − 1)(n − 2)/2operators Ω jk(λ)(x, D) ∈ OPS1,1

H P generate a subalgebra of OPS1,1P H .

In the same way, by using (2.2), we see that the (n− 1)(n− 2)/2 operators Ω jk(µ)(x, D) generate a subalgebra ofOPS1,1

H P .


Let A(Y ) = A(Y1, . . . , YN ) be the subalgebra in OPS1,1P H generated by Y1, . . . , YN . Given any integer m ≥ 0 and

any smooth compactly supported function f (x), we define the norm

‖ f ‖Hm (A(Y )) =∑|α|≤m

‖Y α f ‖2.

Here Y α = Y α11 · · · Y

αNN , with α = (α1, . . . , αN ) ∈ NN . The property (4.3) shows that a different order of the

generators Y j gives equivalent norms.Taking the closure (with respect to this norm) of all smooth compactly supported functions, we obtain the Hilbert

space Hm(A(Y )).A similar construction can be done in OPS1,1

H P . Namely, to the subalgebra A(Z) = A(Z1, . . . , Z N ) we canassociate the space Hm(A(Z)) endowed with the norm

‖ f ‖Hm (A(Z)) =∑|α|≤m

‖Zα f ‖2

where Zα = Zα11 · · · Z

αNN with α = (α1, . . . , αN ).

5. Estimates for the commutators

The L2 boundedness of [P, Q] with P ∈ OPSm and Q ∈ OPS1−m is a consequence of Lemmas 3.6 and 3.5.This result is useless for non-regular symbols. If a lack of smoothness involves an x-point, then from Proposition 7,Chapter IV in [5], one can deduce the following.

Lemma 5.1. ( [14] Proposition 3.6.B) Let 0 ≤ σ ≤ 1 and P ∈ OPS0. Then

‖[P, g] f ‖Hσ . ‖g‖Li p‖ f ‖Hσ−1

for all f, g : Rn→ R such that the right-hand side is finite.

A simple consequence of this lemma is given by

‖[P, g]∂k f ‖2 . ‖g‖Li p‖ f ‖2, k = 1, . . . , n. (5.1)

We shall also need a variant of (5.1) where ∂k is replaced by Q(D) and P ∈ OPS0 is replaced by P(D). We shallrequire that P(ξ) and Q(ξ) have disjoint compact supports. Before stating this variant, we recall the next result.

Lemma 5.2. Let n ≥ 1 and N > 0. Let gN , ϕ : Rn→ R be such that ϕ ∈ C∞(Rn) and gN ∈ C∞(Rn

\ 0) is ahomogeneous function of degree N. Let γ ∈ N and |y| 6= 0. There exists C(ϕ, γ ) > 0, such that the estimate∣∣∣∣∫ eix ·yϕ(x)gN (x)dx

∣∣∣∣ . C(ϕ, γ )|y|−γ (5.2)

holds under one of the following assumptions

(i) ϕ ∈ C∞0 (Rn) and 0 6∈ suppϕ;

(ii) ϕ ∈ C∞(Rn), ∇ϕ ∈ C∞0 (Rn), 0 6∈ suppϕ and γ > n + N;

(iii) ϕ ∈ C∞0 (Rn), 0 ∈ suppϕ and γ ≤ n + N.

Proof. We reduce our problem to showing∣∣∣∣yγk ∫ eix ·y gN (x)ϕ(x)dx

∣∣∣∣ . C(γ, ϕ), k = 1, . . . , n.

A simple integration by parts gives∣∣∣∣yγk ∫ eix ·y gN (x)ϕ(x)dx

∣∣∣∣ . ∑|α|+|β|≤γ

∫suppϕ

|DβgN (x)||Dαϕ(x)|dx . (5.3)


Since |x ||β|−N|DβgN (x)| ∈ L∞, we get the case (i) with

C(ϕ, γ ) = C(γ, n)∑

|α|+|β|=γ

‖Dαϕ‖∞

∫suppϕ

|x |−|β|+N dx (5.4)

for suitable C(γ, n) > 0. The same computation gives the case (ii). The restriction γ > n+N assures the convergenceof the last integral for any |β| = γ when |x | → +∞. If |β| 6= γ , for the convergence of the corresponding integralthe condition ∇ϕ ∈ C∞0 (R

n) comes into play.Finally, we discuss (iii). For γ ≤ n + N − 1 we use again (5.3) and we obtain (5.2) with C(γ, ϕ) > 0 given by

(5.4). It remains to discuss the case γ = n + N . To this aim we take χ0 ∈ C∞([0,+∞)) with suppχ0 ⊂ [0, 2] andχ0(r) = 1 on [0, 1]. As usual, we put χ∞ = 1− χ0, hence∫

eix ·y gN (x)ϕ(x)dx =∫

eix ·y gN (x)ϕ(x)χ0(|x‖y|)dx +∫

eix ·y gN (x)ϕ(x)χ∞(|x‖y|)dx = I1 + I2.

Let R = diam suppϕ. In order to estimate I1, we use the fact that |x | ≤ 2/|y| in suppχ0, hence

|I1| . |y|−n−N+1

∫|x |≤minR,2|y|−1

|x |N |y|n+N−1dx . |y|−n−N+1∫|x |≤2|y|−1

|x |−n+1dx . |y|−n−N .

For I2 we integrate by parts n + N + 1 times. We put α = (α1, α2, α3); we get

|I2| .∑

|α|=n+N+1

|y|−(n+N+1)‖Dα2ϕDα3χ∞‖∞

∫|y|−1<|x |<maxR,2|y|−1

|x |−|α1|+N|y||α3|dx

. |y|−(n+N+1)∫|y|−1<|x |<maxR,2|y|−1

|x |−(n+N+1)+1dx . |y|−(n+N ).

This concludes our proof.

Lemma 5.3. Let P(D), Q(D) ∈ OPC with symbols P(ξ), Q(ξ), which satisfy

supp P ⊆ 2 j−1≤ |ξ | ≤ 2 j+1

, supp Q ⊆ 2k−1≤ |ξ | ≤ 2k+1

, (5.5)

where j, k are integers with | j − k| ≥ 3. Assume that gN (x) ∈ C∞(Rn\ 0) is homogeneous of degree N and

ϕ ∈ C∞(Rn). There exists C(ϕ, r) > 0 such that for any f ∈ S(Rn), one has

‖[P(D), gNϕ]Q(D) f ‖2 ≤ C(ϕ, r)2−|k− j |n/2−r max j,k‖P(ξ)‖L∞ξ

‖Q(D) f ‖2, (5.6)

where

(i) r ∈ N if ϕ ∈ C∞0 and 0 6∈ suppϕ;(ii) r ∈ N, r ≥ N + 1 if ϕ ∈ C∞, ∇ϕ ∈ C∞0 and 0 6∈ suppϕ;

(iii) r = 0, . . . , N if ϕ ∈ C∞0 , 0 ∈ suppϕ.

Assume that (5.5) holds for j = 0 and |k| ≥ 3. Under the previous hypotheses on r, N , ϕ, one has

‖[P(D), gNϕ]∇αQ(D) f ‖2 ≤ C(ϕ, r)2−|k|n/2+(|α|−r)k

‖P(ξ)‖L∞ξ‖Q(D) f ‖2, α ∈ Nn . (5.7)

Proof. We prove (5.6) since the proof of (5.7) is similar. Set

d = dist(supp P, supp Q) = 2min j,k(2|k− j |−1− 2).

Let g = gNϕ; notice that [P(D), g]Q(D) f = P(D)gQ(D) f . From this, we have

(P(D)gQ(D) f )∧(ξ) = P(ξ)∫Rn

∫Rn

eix(η−ξ)g(x)Q(η) f (η)dηdx . (5.8)

Under our assumption on r ∈ N, the previous lemma gives∣∣∣∣∫Rn

eix(η−ξ)g(x)dx

∣∣∣∣ ≤ C(ϕ, n + r)|ξ − η|−n−r .


Combining this estimate, the relation (5.8) and the Young inequality, we get

‖P(D)gQ(D) f ‖22 . ‖P‖2L2(|ξ |∼2 j )

‖Q f ‖2L2(|η|∼2k )sup|ξ |∼2 j

∫|η|∼2k

|ξ − η|−2(n+r)dη

. 2( j+k)nd−2n−2r‖P‖2∞‖Q(D) f ‖22.

Since | j − k| ≥ 3, we have d ≥ 2max j,k−2 and

2( j+k)nd−2n−2r . 2( j+k)n2−2 max j,k(n+r)= 2−| j−k|n−2r max j,k.

This completes the proof.

Another consequence of (5.1) is that [Ψ(D), g∇] ∈ L(L2, L2), whenever g ∈ Li p and Ψ(D) ∈ OPS0 is aconvolution operator. We need a recursive version of this relation. This will be proved by using cut-off functions andthe trivial relation

f ∈ L∞, P ∈ L(L2, L2)⇒ [ f, P] ∈ L(L2, L2). (5.9)

Lemma 5.4. Fix k ∈ N. Let p11, p2

1, . . . pk1 be homogeneous functions of degree one and Ψ(D) ∈ OPS0,−∞. Then

[p11∇, [p

21∇, [. . . , [p

k1∇,Ψ(D)] . . .] ∈ L(L2, L2).

Proof. Let j ≤ k. We decompose each p j1∇ as a sum of two operators:

p j1∇ = ϕ0 p j

1∇ + ϕ∞ p j1∇.

Here ϕ0 p j1∇ is localized close to the singularity and ϕ∞ p j

1∇ is zero near the singularity.

We observe that ϕ0 p j1 ∈ C(Rn)∩C 1(Rn

\ 0) has compact support and ∇(ϕ0 p j1) ∈ L∞(Rn). We neglect the index

j , denoting by f0 any function of this kind. With f∞, we denote smooth functions such that 0 6∈ supp f∞, | f∞| . |x |with bounded derivatives. In particular, ϕ∞ p1 is of f∞ type. Finally we set f0,∞ for a smooth compactly supportedfunction such that 0 6∈ supp f0,∞. We have

[ f0∇, f∞∇] = f0,∞∇. (5.10)

After the decomposition p j1∇ = f0∇ + f∞∇, the commutator [p1

1∇, [p21∇, [. . . , [p

k1∇,Ψ(D)] . . .] gives a sum

of terms with f0∇ or f∞∇. Since f∞∇ ∈ OPS1,1 and Ψ(D) ∈ OPS0,−∞, from Lemma 3.5 it follows that[ f∞∇,Ψ(D)] ∈ OPS0,−∞. By iteration, we have

[ f∞∇, [ f∞∇, [. . . , [ f∞∇,Ψ(D)] . . .] ∈ L(L2, L2).

If f0∇ appears as the first term, recalling that [AC, B] = A[C, B] + [A, B]C , we have

[ f0∇,OPS0,−∞] = f0OPS0,−∞

+ [ f0,OPS0,−∞]∇. (5.11)

Since f0 ∈ L∞ multiplies an OPS0,−∞ operator, the first term is L2 bounded. By using (5.1) it results that the secondterm is also L2 bounded.

Assume that f0∇ appears after some f∞∇ terms: we are dealing with

[ f∞∇, [. . . , [ f0∇, [ f∞∇, [. . . , [ f∞,Ψ(D)] . . .].

We can reduce to the previous case by the aid of Lemma A.2. Here we take A = f0∇, C =

[ f∞∇, [. . . , [ f∞,Ψ(D)] . . .] and B j = f∞∇. We only require that [A, B] = f∞∇ as given by (5.10). In conclusion,we only need to prove

[ f0∇, [. . . , [ f0∇,OPS0,−∞] . . .] ∈ L(L2, L2). (5.12)

Here OPS0,−∞ stands for [ f∞∇, [. . . , [ f∞,Ψ(D)] . . .] or Ψ(D). In order to gain this boundedness, we apply aninductive argument on K , the number of elements of the form f0∇. In (5.11), we have already discussed the case


K = 1. Let (5.12) be true when f0∇ appears K − 1 times, we prove the same for K times. Let g a smooth functionand P a certain operator, we observe that

[g∇, P] = g[∇, P] + [g, P]∇ = g[∇, P] + [g, P∇] − P(∇g) = g[∇, P] + [g, P∇] + [(∇g), P] − (∇g)P.

In our case we have [ f0∇, [. . . , [ f0∇,OPS0,−∞] . . .] = T1 + T2 + T3 + T4 where

T1 = [ f0∇, [. . . , [ f0∇, f0[∇,OPS0,−∞] . . .]

T2 = [ f0∇, [. . . , [ f0,OPS0,−∞∇] . . .]

T3 = [ f0∇, [. . . , [(∇ f0),OPS0,−∞] . . .]

T4 = [ f0∇, [. . . , [ f0∇, (∇ f0)OPS0,−∞] . . .].

In all these terms f0∇ appears K − 1 times. In order to get T1 ∈ L(L2, L2), we use Lemma A.3 withB = f0, C = [∇,OPS0,−∞

] = OPS0,−∞, A j = f0∇. By inductive assumption we can control terms like[ f0∇, [. . . [ f0∇,OPS0,−∞

] . . .]. It remains to check that

f0∇( f0∇ . . . ( f0(∇ f0) . . .) = [ f0∇, [. . . [ f0∇, f0] . . .] ∈ L(L2, L2).

Since f0 behaves like |x | close to x = 0, we can conclude that f0∇( f0∇ . . . ( f0(∇ f0) . . .) ∈ L∞. Hence the estimatefor T1 follows.

A similar argument works for T4 since f0∇( f0∇ . . . ( f0(∇2 f0) . . .) ∈ L∞.

In order to estimate T2, we use Lemma A.4 with A j = f0∇, B = f0 functions and C = OPS0,−∞∇ = OPS0,−∞.

First we notice that [ f0∇, f0] = f0, hence such a lemma is available. Then we have terms like [ f0,OPS0,−∞] and

[ f0, [ f0∇, [. . . , [ f0∇,OPS0,−∞] . . .] that can be estimated by means of inductive assumptions and (5.9). Similarly,

taking B = ∇ f0 functions, we have T3 ∈ L(L2, L2).

The next result shows that any OPS1,1H P and OPS1,1

P H operators commute with dilations.

Lemma 5.5. Let p, q be homogeneous of degree k and a ∈ R. Then [p(x)q(D), Sa] = 0.

Proof. Since q(D) is a convolution operator, with symbol q(ξ), the operator q(D/a), with symbol q(ξ/a), satisfiesthe relation q(D/a) = Saq(D)Sa−1 for all a ∈ R \ 0. Moreover q(ξ) is homogeneous of degree k, thenSaq(D) = a−kq(D)Sa . Finally, we observe that [p, Sa] f (x) = (1− ak)p(x)(Sa f )(x). Our thesis follows.

In order to deal with OPS1,1P H operators, the following simple commutator estimate is very useful:

[a(D), x j ] = (D j a)(D). (5.13)

From now on, we shall use the notation S j ( f ) := S2− j ( f ). In particular, Lemma 5.5 implies that

S j (Z f ) = Z S j ( f ) and S j (Y f ) = Y S j ( f ).

Lemma 5.6. Let Y (x, D) ∈ OPS1,1P H , and Φ(D) ∈ OPC. For any f ∈ S(Rn), one has∑

j∈Z2− jn‖[Φ(D), Y (x, D)]S j ( f )‖22 . ‖ f ‖22. (5.14)

Proof. It suffices to use the expression Y (x, D) = q0(D)+∑n

k=1 ck xk Qk(D). Hence

[Φ(D), Y (x, D)] =n∑

k=1

ck[Φ(D), xk]Qk = −in∑

k=1

ck(∂kΦ)(D)Qk(D).

In particular, we have

[Φ(D), Y (x, D)] ∈ OPC. (5.15)

Lemma 3.2 gives the thesis.


Let Φ(D) ∈ OPC be such that Φ(2− j D) j∈Z is a Paley–Littlewood partition of unity. Combining Lemma 5.5with (3.2) and (5.14), we see that

‖Y f ‖22 + ‖ f ‖22 ' ‖ f ‖22 +∑j∈Z

2− jn‖Φ(D)Y (x, D)S j ( f )‖22

' ‖ f ‖22 +∑j∈Z

2− jn‖Y (x, D)Φ(D)S j ( f )‖22. (5.16)

Our next aim is to prove the OPS1,1H P version of this equivalence:

‖Z f ‖22 + ‖ f ‖22 ' ‖ f ‖22 +∑j∈Z

2− jn‖Φ(D)Z(x, D)S j ( f )‖22

' ‖ f ‖22 +∑j∈Z

2− jn‖Z(x, D)Φ(D)S j ( f )‖22. (5.17)

In order to gain the last equivalence, we shall employ Lemmas 5.1 and 5.3 and the complex interpolation method.

Lemma 5.7. Let Ai , Bi , be Banach spaces, i = 0, 1. Let T be a linear operator from D(T ) into A0 ∩ A1 and P aninvertible operator from D(P) into B0 ∩ B1. Let D(P) ⊂ D(T ). If

‖T f ‖A0 . ‖P f ‖B0 , and ‖T f ‖A1 . ‖P f ‖B1 ,

then, for any given 0 < θ < 1, one has

‖T f ‖(A0,A1)θ . ‖P f ‖(B0,B1)θ .

Proof. It suffices to consider f = P−1g and use the standard interpolation result.

Lemma 5.8. Let σ1, σ2 ∈ R, K , N ∈ N. Let T : L2(Rn)→ L2(Rn) be a linear operator. Let Y j j=1,...,K ⊂ OPS1,1P H ,

be such that (4.3) is satisfied. Let Φ(D) be a convolution-type operator with Φ ≥ 0. Let α = (α1, . . . , αK ) andY α := Y α1 . . . Y αK . If the relations

supj∈Z

2− jσ1‖T f j‖2 . supm∈Z

2−mσ2∑|α|≤N

‖Φ(D)Y α fm‖2, (5.18)

∑j∈Z

2− jσ1‖T f j‖2 .∑m∈Z

2−mσ2∑|α|≤N

‖Φ(D)Y α fm‖2, (5.19)

hold, then∑j∈Z

2−2 jσ1‖T f j‖22 .

∑m∈Z

2−2mσ2∑|α|≤N

‖Φ(D)Y α fm‖22. (5.20)

Here fmm∈Z is a sequence of functions such that the right-hand sides of (5.18)–(5.20) are finite.

Proof. Following [3, Section 5.6], we denote by lσq (A), 1 ≤ q < +∞, σ ∈ R, the space of all the sequence fmm∈Zin a Banach space A such that

‖f‖qlσq (A)=

∑m∈Z

2qmσ‖ fm‖

qA, f := fmm∈Z

is finite. Moreover the norm of lσ∞(A) is given by

‖f‖qlσ∞(A)

= supm∈Z

2mσ‖ fm‖A.

Theorem 5.6.3 in [3] assures that (lσ∞(A), lσ1 (A))[1/2] = lσ2 (A). Given T a linear operator on A, we put Tf :=

T fmm∈Z; hence we obtain a linear operator on the sequences of A.


A direct consequence of Lemma 4.1(ii) is that if Y ∈ OPS1,1P H then there exists Z ∈ OPS1,1

H P such that (Y f )∧ = Z f .

A recursive argument assures that for any given Y j j=1,...,K ⊂ OPS1,1P H , there exists Z j j=1,...,K ⊂ OPS1,1

H P suchthat

(Y α f )∧ = (Y α11 . . . Y αK

K f )∧ = Zα11 . . . ZαK

K f = Zα f .

With this notation we have∑|α|≤N

‖Φ(D)Y αg‖L2(Rnx )=

∑|α|≤N

(∫Φ2(ξ)|Zα g|2dξ

)1/2

. (5.21)

Finally we introduce the Hilbert space H N (Z ,Φ) as the closure of C∞0 with respect to the previous norm. We canrewrite (5.18) and (5.19) as

‖Tf‖lσ1∞ (L2)

. ‖F f‖lσ2∞ (H N (Z ,Φ)) and ‖Tf‖lσ1

1 (L2). ‖F f‖lσ2

1 (H N (Z ,Φ)).

Applying the previous lemma, we get ‖Tf‖lσ12 (L2)

. ‖F f‖lσ22 (H N (Z ,Φ)). Having in mind (5.21), we see that the last

inequality coincides with our thesis (5.20).

We are ready to prove (5.17). By virtue of (3.2), it suffices to establish the next result.

Lemma 5.9. Let Z(x, D) ∈ OPS1,1H P and Ψ(D) ∈ OPC. For any f ∈ S(Rn), one has∑

j∈Z

2− jn‖[Ψ(D), Z(x, D)]S j f ‖22 . ‖ f ‖22;

Proof. Let Z(x, D) = p0(x)+∑n

k=1 pk(x)∂k . Due to Lemma 3.2, we can directly estimate∑j∈Z

2− jn‖[Ψ(D), p0]S j ( f )‖22 .

∑j∈Z

2− jn(‖Ψ(D)S j (p0 f )‖22 + ‖p0Ψ(D)S j ( f )‖22

). ‖p0 f ‖22 + ‖ f ‖22 . ‖ f ‖22 .

It remains to gain∑j∈Z

2− jn‖A(x, D)S j ( f )‖22 .

∑m∈Z‖Φm(D) f ‖22 ,

where

A(x, D) =n∑

k=1

[Ψ(D), pk(x)∂k] =

n∑k=1

[Ψ(D), pk(x)]∂k

and Φmm∈Z = Φ(2−m D)m∈Z is a Paley–Littlewood partition of unity. Due to the interpolation result ofLemma 5.8, it is sufficient to show that

supj∈Z

2− jn/2‖A(x, D)S j ( f )‖2 . sup

m∈Z‖Φm(D) f ‖2; (5.22)∑

j∈Z2− jn/2

‖A(x, D)S j ( f )‖2 .∑m∈Z‖Φm(D) f ‖2. (5.23)

We split ‖A(x, D)S j ( f )‖2 as

‖A(x, D)S j ( f )‖2 .m= j+2∑m= j−2

‖A(x, D)S j (Φm(D) f )‖2 +∑|m− j |≥3

‖A(x, D)S j (Φm(D) f )‖2.

By using (5.1), we get

‖A(x, D)S j (Φm(D) f )‖2 . maxk=1,...,n

‖pk‖Li p‖S j (Φm(D) f )‖2 . 2 jn/2‖Φm(D) f ‖2.


Applying this estimate, we have

supj∈Z

2− jn/2m= j+2∑m= j−2

‖A(x, D)S j (Φm(D) f )‖2 . supm∈Z‖Φm(D) f ‖2;

∑j∈Z

2− jn/2m= j+2∑m= j−2

‖A(x, D)S j (Φm(D) f )‖2 .∑m∈Z‖Φm(D) f ‖2.

It remains to consider the case | j − m| ≥ 3. From

S2− j Φm(D) = S2− j+m Φ(D)S2−m = Φm− j (D)S2− j , (5.24)

it follows that∑|m− j |≥3

‖A(x, D)S j (Φm(D) f )‖2 .∑|k|≥3

‖A(x, D)Φk(D)S j ( f )‖2.

Let ϕ0(x) ∈ C∞0 , ϕ0(x) = 1 if |x | ≤ 1 and suppϕ0 ⊂ |x | ≤ 2. Hence ϕ∞ = 1 − ϕ0 satisfies suppϕ∞ ⊂ |x | ≥ 1and supp (∇ϕ∞) ⊂ 1 ≤ |x | ≤ 2. We set

A(x, D) =n∑

h=1

[Φ(D), ϕ0(x)ph(x)]∂h +

n∑h=1

[Φ(D), ϕ∞(x)ph(x)]∂h =: A0(x, D)+ A∞(x, D).

Since ϕ0 satisfies the assumption (iii) of Lemma 5.3, from (5.7) we get

‖A0(x, D)Φk(D)S j ( f )‖2 . 2−|k|n/2‖Φk(D)S j ( f )‖2.

Moreover ϕ∞ satisfies the assumption (ii) of Lemma 5.3, this yields

‖A∞(x, D)Φk(D)S j ( f )‖2 . 2−|k|n/2−k‖Φk(D)S j ( f )‖2.

We can conclude that

supj∈Z

2− jn/2∑|k|≥3

‖A(x, D)Φk(D)S j ( f )‖2 . supj∈Z

2− jn/2∑|k|≥3

2−|k|n/2(1+ 2−k)‖Φk(D)S j ( f )‖2

. supj∈Z

∑|k|≥3

2−|k|n/2(1+ 2−k)‖Φk+ j (D) f ‖2 . supm∈Z‖Φm(D) f ‖2

∑|k|≥3

2−|k|n/2(1+ 2−k).

Since n ≥ 3, for negative values of k the previous series converges. This gives (5.22). In a similar way, by means ofelementary properties of the numerical series, we obtain (5.23):∑

j∈Z2− jn/2

∑|m− j |≥3

‖A(x, D)S j (Φm(D) f )‖2 .∑j∈Z

∑|k|≥3

2−|k|n/2(1+ 2−k)‖Φk+ j (D) f ‖2 .∑m∈Z‖Φm(D) f ‖2.

This concludes our proof.

Our next step is to generalize Lemmas 5.6 and 5.9 for the higher orders. Again the OPS1,1P H case is simpler than the

OPS1,1H P case.

Lemma 5.10. Let α ∈ Nn . Let Y1(x, D), . . . , YK (x, D) ∈ OPS1,1P H and Φ(D) ∈ OPC. For any f ∈ S(Rn), one has∑

j∈Z2− jn‖[Y α,Φ(D)]S j ( f )‖22 .

∑|β|≤|α|−1

‖Y β f ‖22. (5.25)

Proof. Recalling Lemma A.1, we can write

[Y α,Φ(D)] = [Y1, [Y2, [. . . , [YK ,Φ(D)] . . .] + · · · +K−1∑j=2

[Yi ,Φ(D)]Y1 . . . Yi−1Yi+1 . . . YK .


Due to Lemma 3.2, it suffices to show that [Y1, [. . . , [Y j ,Φ(D)] . . .] ∈ OPC for any j = 1, . . . , K . The relation(5.13) assures [Y,Φ(D)] ∈ OPC for all Y ∈ OPS1,1

H P . After j step we see that the same properties hold for[Y1, [. . . , [Y j ,Φ(D)] . . .]. This concludes the proof.

Remark 5.1. Let Φ(D) ∈ OPC. The previous lemma and Lemma 3.2 imply that∑|β|≤|α|

∑j∈Z

2− jn‖Y β(x, D)Φ(D)S j ( f )‖22 .

∑|β|≤|α|

‖Y β f ‖22. (5.26)

The converse inequality is true whenever Φ(2− j D) j∈Z is a Paley–Littlewood partition of unity. Combining (3.2)with (5.25), we see that∑

|β|≤|α|

∥∥Y β f∥∥2

2 .∑|β|≤|α|

∑j∈Z

2− jn‖Y β(x, D)Φ(D)S j ( f )‖22 +

∑|β|≤|α|−1

‖Y β f ‖22. (5.27)

Lemma 5.11. Let α ∈ Nn . Let Z(x, D) ∈ OPS1,1H P and Y (x, D) ∈ OPS1,1

P H which satisfies∑|β|≤|α|−1

‖Zβ f ‖2 .∑

|β|≤|α|−1

‖Y β f ‖2. (5.28)

Let Ψ(D) ∈ OPC. For any f ∈ S(Rn), one has∑j∈Z

2− jn‖[Zα(x, D),Ψ(D)]S j ( f )‖22 .

∑|β|≤|α|−1

‖Y β f ‖22.

Proof. We proceed as in the proof of Lemma 5.9. Due to (5.24), we can write

‖[Zα,Ψ(D)]S j ( f )‖2 .m= j+2∑m= j−2

‖[Zα,Ψ(D)]S j (Φm(D) f )‖2 +∑|m− j |≥3

‖[Zα,Ψ(D)]S j (Φm(D) f )‖2

.2∑

k=−2

‖[Zα,Ψ(D)]Φk(D)S j ( f )‖2 +∑|k|≥3

‖[Zα,Ψ(D)]Φk(D)S j ( f )‖2 =: A j + B j .

From Lemma 4.1(iv), we deduce that Zα ∈ OPS|α|,|α|H P . This means that there exists pβ(x) a homogeneous function ofdegree |β| ≤ |α|, such that Zα =

∑|β|≤|α| pβ(x)∇β . In particular, we have

B j .∑|β|≤|α|

∑|k|≥3

‖[pβ(x),Ψ(D)]∇βΦk(D)S j ( f )‖2

.∑|β|≤|α|

∑|k|≥3

‖[ϕ0(x)pβ(x),Ψ(D)]∇βΦk(D)S j ( f )‖2

+

∑|β|≤|α|

∑|k|≥3

‖[ϕ∞(x)pβ(x),Ψ(D)]∇βΦk(D)S j ( f )‖2.

We are in a position to apply Lemma 5.3. More precisely, we apply (iii) with r = |β| to the first term and (ii) withr = |β| + 1 to the second one. Employing (5.7), we arrive at

B j .∑

|β|≤|α|−1

∑|k|≥3

2−|k|n/2(1+ 2−k)‖Φk(D)S j ( f )‖2.

As in Lemma 5.9, this gives

supj∈Z

2− jn/2∑|k|≥3

‖[Zα,Ψ(D)]Φk(D)S j ( f )‖2 . supm∈Z‖Φm(D) f ‖2;∑

j∈Z2− jn/2

∑|k|≥3

‖[Zα,Ψ(D)]Φk(D)S j ( f )‖2 .∑m∈Z‖Φm(D) f ‖2.


In order to conclude the proof by means of interpolation, we need to check

supj∈Z

2− jn/2∑|k|≤2

‖[Zα,Ψ(D)]Φk(D)S j ( f )‖2 . supm∈Z

2−mn/2∑

|β|≤|α|−1

‖Φ(D)Y β Sm( f )‖2;∑j∈Z

2− jn/2∑|k|≤2

‖[Zα,Ψ(D)]Φk(D)S j ( f )‖2 .∑m∈Z

2−mn/2∑

|β|≤|α|−1

‖Φ(D)Y β Sm( f )‖2.

If these hold, then Lemma 5.8 yields∑j∈Z

2− jn‖[Zα,Ψ(D)]S j ( f )‖22 .

∑m∈Z

2−mn∑

|β|≤|α|−1

‖Φ(D)Y β Sm( f )‖22. (5.29)

This corresponds to our thesis. Since |k| ≤ 2, it suffices to show

‖[Zα,Ψ(D)]Φk(D)S j ( f )‖2 . 2−kn/2∑

|β|≤|α|−1

‖Φ(D)Y β Sk+ j ( f )‖2.

By using the same argument of the proof of Lemma 5.10, we reduce to showing

‖[Zα,Ψ(D)]g‖2 .∑

|β|≤|α|−1

‖Y βg‖2. (5.30)

Let us see this reduction in detail. If (5.30) holds, then

‖[Zα,Ψ(D)]Φk(D)S j ( f )‖2 .∑

|β|≤|α|−1

‖Y βΦk(D)S j ( f )‖2 = 2−kn/2∑

|β|≤|α|−1

‖Y βΦ(D)S j+k f ‖2.

It remains to show

‖[Y β ,Φ(D)]S j+k( f )‖2 .∑|γ |≤|β|

‖Φ(D)Y γ S j+k( f )‖2. (5.31)

In the proof of Lemma 5.10, we established

[Y β ,Φ(D)] = Ξ0(D)+ Ξ1Y + · · · + Ξ (D)Y β−1Y j+1 · · · Y|β| (5.32)

with Ξi ∈ OPC and β − 1 any multi-index of length |β| − 1. This gives (5.31).Now, we show that (5.30) holds. By using Lemma A.1, we have

‖[Zα,Ψ(D)]g‖2 . ‖[Z1, [. . . , [Z|α|,Ψ(D)] . . .]g‖2 + · · ·

+ ‖[Z1,Ψ(D)]Z2 · · · Z|α|g‖2 +n−1∑j=2

‖[Z j ,Ψ(D)]Z1 · · · Z j−1 Z j+1 · · · Z|α|g‖2.

Suppose, we know

[Z1, [. . . , [Z N ,Ψ(D)] . . .] ∈ L(L2, L2), (5.33)

the assumption (5.28) will imply that (5.30) holds. The proof of (5.33) requires long calculations. In order to simplifythe notation, we denote with p0 any homogeneous function of degree zero and with p1∇ any (H P)1,1 symbols. Forexample we put

Z = p0 +∑

jp j∂ j =: p0 + p1∇, and [Z , p0] =

∑j

p j (∂ j p0) =: p0,

similarly [Z , p1∂] = p0+ p1∇, even though we are considering different p(x) functions. We underline that, with thisnotation, the commutators like [p1∇, p1∇] are not zero.

We prove (5.33) by induction. For N = 1 we used (5.1) and the property (5.9) obtaining

‖[Z ,Ψ(D)]g‖2 . ‖[p0,Ψ(D)]g‖2 + ‖[p1,Ψ(D)]∇g‖2 . ‖g‖2.

Let N ≥ 2. We can write

[Z1, [Z2, [. . . , [Z N ,Ψ(D)] . . .] = [p0, [Z2, [. . . , [Z N ,Ψ(D)] . . .] + [p1∇, [Z2, [. . . , [Z N ,Ψ(D)] . . .].


The first term can be estimated by means of (5.9) and of the inductive assumption. In order to deal with the secondone, we decompose Z2, obtaining

[p1∇, [Z2, [. . . , [Z N ,Ψ(D)] . . .] = [p1∇, [p0, [. . . , [Z N ,Ψ(D)] . . .]+ [p1∇, [p1∇, [. . . , [Z N ,Ψ(D)] . . .]. (5.34)

To the first term, we apply the Jacobi identity and the relation [p0, p1∇] = p0; we have

[p1∇, [p0, [. . . , [Z N ,Ψ(D)] . . .] = [p0, [Z3, [. . . , [Z N ,Ψ(D)] . . .] + [p0, [p1∇, [Z3, [. . . , [Z N ,Ψ(D)] . . .]= [p0, [Z3, [. . . , [Z N ,Ψ(D)] . . .] − [p0, [p0, [Z3, . . . , [Z N ,Ψ(D)] . . .]+ [p0, [Z1, [Z3, [. . . , [Z N ,Ψ(D)] . . .].

By means of inductive arguments we can control all these terms. It remains to estimate the second term of (5.34).Applying N − 2 times the previous arguments, we reduce to showing

[p1∇, [p1∇, [. . . , [p1∇,Ψ(D)] . . .] ∈ L(L2, L2).

This has been proved in Lemma 5.4.

Finally we analyze the commutators between an OPS1,1P H operator and an OPS0,0 operator. In order to do this, first

we state the following result.

Lemma 5.12. Let Y1, . . . , YK ∈ OPS1,1P H and Ψ(D) ∈ OPC. There exist K (K − 1)/2 convolution operators

Ψ j,k(D) ∈ OPC, with j, k = 1, . . . , K , j ≤ k, such that

Y1 · · · YkΨ(D) =K∑

j=1

Y1Ψ1, j (D) · · · Y jΨ j, j (D).

Proof. Let Φ ∈ OPC such that Φ(ξ) ≡ 1 on supp Ψ . Then

Y1Y2Ψ(D) = Y1Y2Φ(D)Ψ(D) = Y1Φ(D)Y2Ψ(D)+ Y1[Y2,Φ(D)]Ψ(D).

Putting Ψ1,2 = Φ(D), Ψ1,2 = Ψ(D) and Ψ1,1(D) = [Y2,Φ(D)]Ψ(D) we get the case K = 2. In particular, we used(5.15). For the higher order case, we have

Y1 · · · YK Ψ(D) = Y1 · · · YK−2(YK−1Φ(D)YK Ψ(D)+ YK−1ΨK−1(D)).

In order to conclude, it suffices to apply inductive arguments to the terms Y1 · · · YK−2YK−1ΨK−1 andY1 · · · YK−2YK−1Φ(D).

Lemma 5.13. Fix K ∈ N. Consider Yk ∈ OPS1,1P H and Υk ∈ OPC, with k = 1, . . . , K . Let α = (α1, . . . , αK ) ∈ NK .

Put (YΥ)α = (Y1Υ1)α1 · · · (YK ΥK )

αK . There exists a function Ξ ∈ OPC such that for any g ∈ S(Rn) one has

‖(YΥ(D))αg‖2 .∑|β|≤|α|

‖Y βΞ (D)g‖2.

Proof. As usual an inductive argument gives the thesis. The case |α| = 1 is trivial. By means of inductive assumption,we have

‖(YΥ(D))αg‖2 = ‖(YΥ(D))α−1YK ΥK (D)g‖2 .∑

|β|≤|α|−1

‖Y βΞ (D)YK ΥK (D)g‖2.

Here α − 1 denotes a multi-index of length |α| − 1 and K a suitable element of 1, . . . , K . We write

Ξ (D)YK ΥK (D) = [Ξ (D), YK ]ΥK (D)+ YK Ξ (D)ΥK (D).

On the other hand, (5.15) gives Ξ (D)ΥK (D) ∈ OPC and [Ξ (D), YK ]ΥK (D) ∈ OPC. Hence we have theconclusion.


Lemma 5.14. Let α ∈ Nn . Let Y1(x, D), . . . , YK (x, D) ∈ OPS1,1P H . Fix Φ(D) ∈ OPC and r(x, D) ∈ OPS0,0. There

exists a function Ξ ∈ OPC such that for any g ∈ S(Rn), one has

‖Y αΦ(D)r(x, D)g‖2 .∑|β|≤|α|

‖Y βΞ (D)g‖2. (5.35)

Proof. Lemma 5.12 guarantees that we can write

‖Y αΦ(D)r(x, D)g‖2 .∑|β|≤|α|

‖(YΥ(D))βr(x, D)g‖2.

Commuting and using Lemma 3.6, we have

‖(YΥ(D))βr(x, D)g‖2 . ‖(YΥ(D))βg‖2 + ‖[(YΥ(D))β , r(x, D)]g‖2. (5.36)

For the second term we apply Lemma A.1 obtaining

[(YΥ(D))β , r(x, D)] = [Y1Υ1(D), [. . . , [Y|β|Υ|β|(D), r(x, D)] . . .] + · · ·

+

|β|−1∑j=2

[Y jΥ j (D), r(x, D)]Y1Υ1(D) . . . Y j−1Υ j−1(D)Y j+1Υ j+1(D) . . . Y|β|Υβ(D).

Since YΥ(D) ∈ OPS1,1 and r(x, D) ∈ OPS0,0, all the previous commutators belong to L(L2, L2). In particular, thesecond term of (5.36) can be absorbed in the first one. The conclusion follows by means of Lemma 5.13.

6. Equivalent norms in generalized Sobolev spaces

In this section, we shall take Y1, . . . , YL ∈ OPS1,1P H and Z1, . . . , Z M ∈ OPS1,1

H P which satisfy (4.3) and (4.4) andAssumption 1.1. This means that given L ,M ∈ N, for all l = 1, . . . , L , m = 1, . . . ,M , one has

Yl(x, ξ) =∑

j=1,...,M

c j,l(x, ξ)Z j (x, ξ), (6.1)

Zm(x, ξ) =∑

j=1,...,L

d j,m(x, ξ)Y j (x, ξ), (6.2)

with suitable functions c j,l(x, ξ), d j,m(x, ξ) homogeneous of degree one in both variables. Our next aim is to provethe equivalence of the corresponding norms∑

|α|+|β|≤k

‖Y α f ‖2 '∑

|α|+|β|≤k

‖Zα f ‖2, k ∈ N. (6.3)

6.1. First order equivalence

Theorem 6.1. Let Y (x, D) ∈ OPS1,1P H and Z(x, D) ∈ OPS1,1

H P . Let c(x, ξ) be a homogeneous function of degree zeroin x, ξ such that Z(x, ξ) = c(x, ξ)Y (x, ξ). For any f ∈ S(Rn), one has

‖Z(x, D) f ‖2 . ‖Y (x, D) f ‖2 + ‖ f ‖2.

Proof. Let us consider Φ(D) ∈ OPC such that Φ(2− j D) j∈Z is a Paley–Littlewood partition of unity. From (5.17)we have

‖Z(x, D) f ‖22 . ‖ f ‖22 +∑j∈Z

2− jn‖Z(x, D)Φ(D)S j ( f )‖22.

Due to the lack of smoothness of Z(x, ξ) in x = 0, we take a cut-off function ϕ0 : Rn→ R supported near the origin

and write

Z(x, D)Φ(D)g(x) = ϕ0(x)Z(x, D)Φ(D)g(x)+ ϕ∞(x)Z(x, D)Φ(D)g(x).


By using the explicit expression of Z , and Lemma 3.2, we get∑j∈Z

2− jn‖ϕ0(x)Z(x, D)Φ(D)S j ( f )‖22 .

∑j∈Z

2− jn‖Φ(D)S j ( f )‖22 +

∑j∈Z

2− jn‖Φ(D)∇S j ( f )‖22 . ‖ f ‖22.

The operator ϕ∞(x)Z(x, D) is a differential operator with unbounded coefficients. We write:

ϕ∞(x)Z(x, D)Φ(D) f (x) =∫

eix ·ξϕ∞(x)c(x, ξ)Y (x, ξ)Φ(ξ) f (ξ)dξ

=

∫eix ·ξϕ∞(x)c(x, ξ)Φ(ξ)Y (x, ξ)Ψ2(ξ) f (ξ)dξ

where Ψ(D) ∈ OPC with Ψ(ξ) ≡ 1 on supp Φ(ξ). Hence Y (x, D)Ψ(D) ∈ OPS1,1 and a(x, D) = ϕ∞(x)c(x, D)Φ(D) ∈ OPS0,0. There exists r(x, D) ∈ OPS0,0 such that

ϕ∞(x)Z(x, D)Φ(D) f (x) = a(x, D)Y (x, D)Ψ2(D) f (x)+ r(x, D)Ψ(D) f (x). (6.4)

Combining Lemmas 3.6 and 3.2, we have∑j∈Z

2− jn‖r(x, D)Ψ(D)S j ( f )‖22 . ‖ f ‖22.

Similarly, by using (5.14), we get∑j∈Z

2− jn‖a(x, D)Y (x, D)Ψ2(D)S j ( f )‖22 .

∑j∈Z

2− jn‖Y (x, D)Ψ2(D)S j ( f )‖22 . ‖Y (x, D) f ‖22 + ‖ f ‖22.

In the last line we used again Lemma 3.2. From this, we get our thesis.

The converse of the last theorem is the following.

Theorem 6.2. Let Y (x, D) ∈ OPS1,1P H and Z(x, D) ∈ OPS1,1

H P . Let Y (x, ξ) = c(x, ξ)Z(x, ξ) with c(x, ξ) ∈C∞ ((Rn

\ 0)× (Rn\ 0)) homogeneous of degree zero in x, ξ . For any f ∈ S(Rn), one has

‖Y (x, D) f ‖2 . ‖Z(x, D) f ‖2 + ‖ f ‖2.

Proof. We take Φ(D) ∈ OPC such that Φ(2− j D) j∈Z is a Paley–Littlewood partition of unity. From (5.16) wehave

‖Y (x, D) f ‖22 . ‖ f ‖22 +∑j∈Z

2− jn‖Y (x, D)Φ(D)S j ( f )‖22.

Let us consider a cut-off function ϕ0(x) = 1 close to x = 0. Then

Y (x, D)Φ(D)g(x) = ϕ0(x)Y (x, D)Φ(D)g(x)+ ϕ∞(x)Y (x, D)Φ(D)g(x). (6.5)

We write Y (x, D) = Q0(D)+∑

k ck xk Qk(D) =: q0(D)+ x Q1(D), Lemma 3.2 leads to∑j∈Z

2− jn‖ϕ0(x)YΦ(D)S j ( f )‖22 .

∑j∈Z

2− jn(‖Q0(D)Φ(D)S j ( f )‖22 + ‖Q1(D)Φ(D)S j ( f )‖22

). ‖ f ‖22.

In order to carry out the estimate of ϕ∞(x)Y (x, D)Φ(D)g(x), we choose another cut-off function ϕ∞(x) = 0 nearzero and ϕ∞ = 1 on the support of ϕ∞. In particular

ϕ∞(x)Y (x, D)Φ(D)g(x) =∫

eix ·ξϕ∞(x)ϕ∞(x)c(x, ξ)Z(x, ξ)Φ(ξ)Ψ(ξ)g(ξ)dξ,

where Ψ(D) ∈ OPC is chosen such that Ψ(2− j D) j∈Z is a Paley–Littlewood partition of unity and Ψ(ξ) ≡ 1on the support of Φ. Thanks to Lemma 3.1, such a function exists. We are in a position to apply Lemma 3.4 to theoperators a(x, D) having symbol

a(x, ξ) = ϕ∞(x)c(x, ξ)Φ(ξ) ∈ S0,0


and ϕ∞(x)Z(x, D) ∈ OPS1,1. It follows that there exists r(x, D) ∈ OPS0,0 such that

ϕ∞(x)Y (x, D)Φ(D) = a(x, D)ϕ∞(x)Z(x, D)Ψ(D)+ r(x, D)Ψ(D). (6.6)

By using Lemma 3.6, it follows that∑j∈Z

2− jn‖ϕ∞(x)Y (x, D)Φ(D)S j ( f )‖22

.∑j∈Z

2− jn‖Z(x, D)Ψ(D)S j ( f )‖22 +

∑j∈Z

2− jn‖Ψ(D)S j ( f )‖22 . ‖Z(x, D) f ‖22 + ‖ f ‖22.

In the last line we apply (5.17). The conclusion follows.

Remark 6.1. Previous theorems still hold under the more general assumption that (6.1) and (6.2) hold. This meansthat (6.3) holds when k = 1.

6.2. Higher order equivalence

Theorem 6.3. Let α ∈ Nn . Let Y1, . . . , YL ∈ OPS1,1P H , Z1, . . . , Z M ∈ OPS1,1

H P , such that (6.2) holds. For anyf ∈ S(Rn), one has∑

|β|≤|α|

‖Zβ(x, D) f ‖2 .∑|β|≤|α|

‖Y β(x, D) f ‖2.

Proof. We apply an inductive argument on the length of α. Theorem 6.1 and Remark 6.1 give the claim when |α| = 1.Let Φ(D) ∈ OPC be such that Φ(2− j D) j∈Z is a Paley–Littlewood partition of unity. Let Ξ (D) ∈ OPC be suchthat Ξ (ξ) = 1 on supp Φ(·), then∥∥Zα f

∥∥22 '

∑j∈Z

2− jn∥∥Φ(D)Zα f

∥∥22 '

∑j∈Z

2− jn∥∥Φ(D)Ξ (D)ZαS j ( f )

∥∥22

.∑j∈Z

2− jn∥∥Φ(D)ZαΞ (D)S j ( f )

∥∥22 +

∑j∈Z

2− jn∥∥Φ(D)[Ξ (D), Zα]S j ( f )

∥∥22 := I1 + I2.

By using the inductive assumption and Lemma 5.11, we can control I2. It remains to estimate I1. We split the sum as

I1 .∑j∈Z

2− jn(∥∥∥Φ(D)Zα−1ϕ0(x)ZΞ (D)S j ( f )

∥∥∥2

2+

∥∥∥Φ(D)Zα−1ϕ∞(x)ZΞ (D)S j ( f )∥∥∥2

2

):= I1,1 + I1,2.

Here α − 1 denotes any multi-index of length |α| − 1. For I1,1 we commute again obtaining∥∥∥Φ(D)Zα−1ϕ0 ZΞ (D)S j ( f )∥∥∥2

2.∥∥∥Φ(D)[Zα−1, ϕ0(x)]ZΞ (D)S j ( f )

∥∥∥2

2+

∥∥∥Φ(D)ϕ0 Zα−1 ZΞ (D)S j ( f )∥∥∥2

2.

Applying Lemma A.1, we get

[Zα−1, ϕ0(x)] = [Z , [. . . , [Z , ϕ0(x)] . . .] + · · · +∑

j=1,...,|α|−1

[Zi , ϕ0(x)]Z1 . . . Zi−1 Zi+1 . . . Z|α|−1.

A recursive argument gives

[Z , [. . . , [Z , ϕ0(x)] . . .] ∈ L(L2, L2).

In fact [Z , ϕ0] = p1(∇ϕ0) ∈ C∞0 so that for [Z , [Z , ϕ0]] we can apply the same argument and proceed in a similarway. It follows that∑

j∈Z2− jn

∥∥∥Φ(D)[Zα−1, ϕ0(x)]ZΞ (D)S j ( f )∥∥∥2

2.

∑|β|≤|α|−2

∑j∈Z

2− jn‖Zβ ZΞ (D)S j ( f )‖22.

Again we can control the last term by means of Lemmas 5.11 and 3.2.


In order to conclude the estimate for I1,1, we deal with∑j∈Z

2− jn∥∥∥Φ(D)ϕ0 Zα−1 ZΞ (D)S j ( f )

∥∥∥2

2

.∑j∈Z

2− jn(∥∥∥∇Zα−2 ZΞ (D)S j ( f )

∥∥∥2

2+

∥∥∥Zα−2 ZΞ (D)S j ( f )∥∥∥2

2

)

.∑j∈Z

2− jn(∥∥∥[Zα−1,∇]Ξ (D)S j ( f )

∥∥∥2

2+

∥∥∥Zα−1∇Ξ (D)S j ( f )

∥∥∥2

2+

∥∥∥Zα−1Ξ (D)S j ( f )∥∥∥2

2

).

Lemma 5.11 can be used to control the last two terms. For the first one, Lemma A.1 gives

[Zα−1,∇] = [Z , [. . . , [Z ,∇] . . .] + · · · +∑

j=1...|α|−1

[Zi ,∇]Z1 . . . Zi−1 Zi+1 . . . Z|α|−1.

By virtue of Lemma 4.3, all previous commutators are OPS0,1H P operators, in particular

‖[Z , [. . . , [Z ,∇] . . .]ZβΞ (D)S j ( f )‖2 . ‖∇ZβΞ (D)S j ( f )‖2.

We can repeat this argument and after a finite number of steps we can absorb this term in∑j∈Z

2− jn∥∥Zβ∇Ξ (D)S j ( f )

∥∥22 with |β| ≤ |α| − 1.

We can finally estimate the leading term I1,2. As costumary, we use a commutator argument. In this way we can applyinductive assumption and (5.29) obtaining

I1,2 .∑j∈Z

2− jn(∥∥∥[Φ(D), Zα−1

]ϕ∞ZΞ (D)S j ( f )∥∥∥2

2+

∥∥∥Zα−1Φ(D)ϕ∞ZΞ (D)S j ( f )∥∥∥2

2

)

.∑j∈Z

2− jn(∥∥∥[Φ(D), Zα−1

]S j (ϕ∞(2 j·)ZΞ (2 j D) f )

∥∥∥2

2+

∥∥∥Zα−1Φ(D)ϕ∞ZΞ (D)S j ( f )∥∥∥2

2

)

.∑

|β|≤|α|−1

∑j∈Z

2− jn(∥∥∥Φ(D)Y β S j (ϕ∞(2 j

·)ZΞ (2 j D) f )∥∥∥2

2+∥∥Y βΦ(D)ϕ∞ZΞ (D)S j ( f )

∥∥22

):= I + II.

We see that the first term can be absorbed in the second one. To this aim it suffices to get

[Φ, Y β ] =∑

|γ |≤|β|−1

Y γΞγ (D)

with suitable Ξγ (D) ∈ OPC. Due to (5.13) this holds true when |β| = 1. Thanks to (5.32) an inductive argumentapplies.

In order to estimate II, we use the relation (6.4):∑j∈Z

2− jn∥∥Y βΦ(D)ϕ∞(x)ZΞ (D)S j ( f )

∥∥22

.∑j∈Z

2− jn∥∥∥Y βΦ(D)a(x, D)YΨ2(D)S j ( f )

∥∥∥2

2+

∑j∈Z

2− jn∥∥Y βΦ(D)r(x, D)Ψ(D)S j ( f )

∥∥22 .

Since a(x, D), r(x, D) ∈ OPS0,0, we can use (5.35) and obtain∑j∈Z

2− jn∥∥Y βΦ(D)ϕ∞(x)ZΞ (D)S j ( f )

∥∥22

.∑|γ |≤|β|

∑j∈Z

2− jn∥∥∥Y γΞ (D)YΨ2(D)S j ( f )

∥∥∥2

2+

∑|γ |≤|β|

∑j∈Z

2− jn∥∥Y γΞ (D)S j ( f )

∥∥22


for suitable Ξ (D) ∈ OPC. Since Ξ (D)[Y,Ψ2(D)] ∈ OPC, we can absorb the second term in the first one. Finally,for the last term we use (5.26) and get the conclusion.

Theorem 6.4. Let α ∈ Nn . Let Y1, . . . , YK ∈ OPS1,1P H , Z1, . . . , Z H ∈ OPS1,1

H P . Assume that (6.1) and (6.2) hold. Forany f ∈ S(Rn), one has∑

|β|≤|α|

‖Y β(x, D) f ‖2 .∑|β|≤|α|

‖Zβ(x, D) f ‖2.

Proof. We apply an inductive argument on the length of α. Theorem 6.2 and Remark 6.1 give the claim when |α| = 1.Combining inductive assumption with (5.27), we reduce our aim to estimate∑

|β|≤|α|

∑j∈Z

2− jn‖Y β(x, D)Φ(D)S j ( f )‖22

with Φ(D) ∈ OPC such that Φ(2− j D) j∈Z is a Paley–Littlewood partition of unity. Due to (5.26), the inductiveassumption implies that∑

|β|≤|α|−1

∑j∈Z

2− jn‖Y β(x, D)Ξ (D)S j ( f )‖22 .

∑|β|≤|α|−1

‖Zβ(x, D) f ‖2 (6.7)

for any given Ξ (D) ∈ OPC. Hence, it suffices to consider only the case |β| = |α|. Let Ξ (D) ∈ OPC be such thatΞ (ξ) = 1 on supp Φ, then∑

j∈Z2− jn‖Y βΦ(D)S j ( f )‖22 .

∑j∈Z

2− jn(∥∥∥Y α−1Ξ (D)YΦ(D)S j ( f )

∥∥∥2

2+

∥∥∥Y α−1Ξ (D)[Φ(D), Y ]S j ( f )∥∥∥2

2

).

As usual α− 1 denotes any multi-index of length |α| − 1. Since (5.15) gives Ξ (D)[Φ(D), Y ] ∈ OPC, we can apply(6.7) to the second term. By means of (6.5) and (6.6), we decompose the first term as∑

j∈Z2− jn

∥∥∥Y α−1Ξ (D)YΦ(D)S j ( f )∥∥∥2

2.∑j∈Z

2− jn∥∥∥Y α−1Ξ (D)ϕ0(x)YΦ(D)Ξ (D)S j ( f )

∥∥∥2

2

+

∑j∈Z

2− jn∥∥∥Y α−1Ξ (D)r(x, D)Φ(D)S j ( f )

∥∥∥2

2

+

∑j∈Z

2− jn∥∥∥Y α−1Ξ (D)a(x, D)ϕ∞(x)ZΦ(D)S j ( f )

∥∥∥2

2=: I1 + I2 + I3

with a(x, D), r(x, D) ∈ OPS0,0.

Since Yα−1∈ OPS|α|−1,|α|−1

P H , Lemma 4.2 implies that Yα−1

Ξ (D) ∈ OPS|α|−1,−∞. On the other handϕ0(x)YΦ(D) ∈ OPS−∞,−∞ so that the commutator of these operators is L2 bounded. We get

I1 . ‖ f ‖22 +∑j∈Z

2− jn∥∥∥ϕ0(x)YΦ(D)Y α−1Ξ 2(D)S j ( f )

∥∥∥2

2

+

∑j∈Z

2− jn∥∥∥[ϕ0(x)YΦ(D), Y α−1Ξ (D)]Ξ (D)S j ( f )

∥∥∥2

2.∑j∈Z

2− jn∥∥∥Y α−1Ξ 2(D)S j ( f )

∥∥∥2

2.

We are in a position to apply (6.7) and conclude the estimate for I1. In order to estimate I2, it suffices to combine (6.7)with Lemma 5.14.

It remains to estimate I3. As for I2, we commute Y α−1 and Ξ (D)a(x, D)ϕ∞ ∈ OPS0,0, obtaining

I3 .∑

|β|≤|α|−1

∑j∈Z

2− jn∥∥Y β ZΦ(D)S j ( f )

∥∥22 .

∑|β|≤|α|−1

∑j∈Z

2− jn∥∥Zβ ZΦ(D)S j ( f )

∥∥22 .


In the last line we use the inductive assumption. By virtue of Theorem 6.3, we can apply Lemma 5.11 and we obtain∑|β|≤|α|−1

∑j∈Z

2− jn‖Zβ+1Φ(D)S j ( f )‖22 .

∑|β|≤|α|−1

∑j∈Z

2− jn‖Φ(D)Zβ+1S j ( f )‖22 +

∑|β|≤|α|−1

‖Y β f ‖22.

SinceΦ(2− j D) j∈Z is a Paley–Littlewood partition of unity, we conclude our proof.

The last two theorems prove (6.3) that corresponds to Theorem 1.4. We note that in Theorem 6.4 we require moreassumptions (both (6.1) and (6.2)) than in Theorem 6.3. This is connected with the difficulty of treating pseudo-differential operators with non-regular symbols in the ξ -variable.

Combining Theorem 1.4 with Theorem 2.3, we have Theorem 1.3.

7. Nonlinear multiplicative inequalities

In this section we shall prove Theorems 1.1 and 1.2. This means we establish the continuity of Nemitskij operatorassociated with the function G : x 7→ |x |p in the generalized Sobolev spaces defined in the previous sections. Thiscan be done, once we know some multiplicative estimates in suitable Sobolev norms. Such kind of estimates are oftencalled Moser-type inequalities.

First, we recall the classical results (see [12] 4.6.4, 5.3.2, 5.2.5, 5.4.3).

Theorem 7.1. Let s ≥ 0. For all f, g ∈ H s∩ L∞, we have

‖ f g‖H s . (‖ f ‖H s‖g‖L∞ + ‖ f ‖L∞‖g‖H s ).

Theorem 7.2. Let s ≥ 0. Assume that one of the following conditions holds true:

(i) p ∈ N∗;(ii) s ∈ N and p > 1 such that s < p + 1/2;

(iii) p > 1 such that s < p.

Then, for any given f ∈ H s∩ L∞, one has ‖| f |p‖H s . ‖ f ‖p−1

L∞ ‖ f ‖H s .

Combining the previous result with the Sobolev embedding theorem, we establish the continuity of the Nemitskijoperator in H s(Rn).

Corollary 7.3. Let s ≥ n/2. Assume that one of the conditions (i)–(iii) of Theorem 7.2 is satisfied. Then, for anygiven f ∈ H s , one has ‖| f |p‖H s . ‖ f ‖p

H s .

7.1. First order estimates

One can easily find multiplicative and nonlinear estimates in generalized Sobolev spaces of order one.

Theorem 7.4. Let Z1, . . . , Z H ∈ OPS1,1H P be such that (4.4) holds. Let H1(A(Z)) be the Sobolev space with norm

‖ f ‖H1(A(Z)) '∑|α|≤1

‖Zα f ‖2.

Let p ≥ 1. Given f, g ∈ H1(A(Z)) ∩ L∞(Rn), we have

‖ f g‖H1(A(Z)) . (‖ f ‖H1(A(Z))‖g‖L∞ + ‖ f ‖L∞‖g‖H1(A(Z))); (7.1)

‖| f |p‖H1(A(Z)) . ‖ f ‖p−1L∞ ‖ f ‖H1(A(Z)). (7.2)

Proof. In order to gain (7.1), we observe that

Z( f g) = (Z f )g + f (Zg)− p0 f g (7.3)

where Z = p0 + p1∇ ∈ OPS1,1H P . Hence, we derive

‖ f g‖H1(A(Z)) . ‖ f ‖∞‖g‖2 + ‖Z f ‖2‖g‖∞ + ‖ f ‖∞‖Zg‖2 + ‖p0‖∞‖ f ‖2‖g‖∞.

Similarly, since Z(| f |p) = p| f |p−2 f (Z f )+ (1− p)p0| f |p, we obtain (7.2).


Combining this theorem with (6.3), we have the following nonlinear pseudo-differential inequalities.

Theorem 7.5. Let Y1, . . . , YK ∈ OPS1,1P H be such that (4.3) holds. Let H1(A(Y )) be the Sobolev space with norm

‖ f ‖H1(A(Y )) '∑|α|≤1

‖Y α f ‖2.

Assume that there exist Z1, . . . , Z N ∈ OPS1,1H P such that (4.4), (6.1) and (6.2) hold.

Let p ≥ 1. Given f, g ∈ H1(A(Y )) ∩ L∞(Rn), we have

‖ f g‖H1(A(Y )) . (‖ f ‖H1(A(Y ))‖g‖L∞ + ‖ f ‖L∞‖g‖H1(A(Y ))); (7.4)

‖| f |p‖H1(A(Y )) . ‖ f ‖p−1L∞ ‖ f ‖H1(A(Y )). (7.5)

7.2. Higher order estimates

While proving Theorem 7.1 for s ∈ N, s ≥ 2, Gagliardo–Nirenberg inequalities come into play. This is the essentialdifficulty for proving our estimates in the higher order case. Due to this lack of sharp interpolation inequalities for ourgeneralized Sobolev norms, we shall obtain another kind of estimate.

Theorem 7.6. Let Z1, . . . , Z H ∈ OPS1,1H P be such that (4.4) is true. Given N ∈ N, one has

‖ f g‖H N (A(Z)) . ‖g‖H N (A(Z)) sup|α|≤[N/2]

‖Zα f ‖∞ + ‖ f ‖H N (A(Z)) sup|α|≤[N/2]

‖Zαg‖∞.

for any f, g such that the right-hand side of the previous inequality is finite.

Proof. Iterating (7.3), we get

Zα( f g) =∑

|β|+|γ |≤|α|

dβ,γ (x)Zβ f Zγ g,

with dβ,γ (x) being homogeneous functions of degree zero. Passing to the norm, we get

‖Zα( f g)‖2 .∑

|β|+|γ |≤|α|

|β|≤[|α|/2]

‖dβ,γ ‖∞‖Zβ f ‖∞‖Z

γ g‖2 +∑

|β|+|γ |≤|α|

|γ |≤[|α|/2]

‖dβ,γ ‖∞‖Zβ f ‖2‖Z

γ g‖∞.

This gives the claim.

In terms of Y operators, we can use (6.3) for the L2 norm, but we have no information on L∞ norm, so we canonly state the following.

Theorem 7.7. Let Y1, . . . , YK ∈ OPS1,1P H be such that (4.3) holds. Let H N (A(Y )) be the corresponding Sobolev

space for a fixed order N ∈ N. Assume that there exist Z1, . . . , Z H ∈ OPS1,1H P such that (4.4), (6.1) and (6.2) hold. It

results

‖ f g‖H N (A(Y )) . ‖g‖H N (A(Y )) sup|α|≤[N/2]

‖Zα f ‖∞ + ‖ f ‖H N (A(Y )) sup|α|≤[N/2]

‖Zαg‖∞.

for any f, g such that the right-hand side of the previous inequality is finite.

The following theorems are the nonlinear counterpart of the previous ones.

Theorem 7.8. Let Z1, . . . , Z H ∈ OPS1,1H P be such that (4.4) is true. Let N ∈ N. Consider p ∈ N∗ or p ∈ R with

p ≥ N. One has

‖| f |p‖H N (A(Z)) . ‖ f ‖H N (A(Z)) sup|α|≤[N/2]

‖Zα f ‖p−1∞

for any given f such that the right-hand side of the previous inequality is finite.


Proof. First one derives a composition rule for Z fields applied to G( f ) = | f |p. The classical case is known as Faadi Bruno’s formula (see for example [12] 5.2.1.). Having in mind that f G(`+1)( f ) = C`G(`)( f ) with C` ∈ R, byinductive arguments, one has

Zα(G( f )) = d0G( f )+|α|∑`=1

G(`)( f )∑

`≤|α1|+···+|α`|≤|α|

dα1,...,α` Zα1 f . . . Zα` f

with d0, dα1,...,α` being homogeneous functions of degree zero, the multi-indexes α1, . . . , α` ∈ NH and for any given

αi , we put Zαi = Zα1

i1 · · · Z

αHi

H . In the previous expression only one length |αi | can be larger than [|α|/2]. Moreoverone has: |G(`)( f )| . | f |p−`. The desired inequality follows.

Theorem 7.9. Let Y1, . . . , YK ∈ OPS1,1P H be such that (4.3) holds. Let H N (A(Y )) be the corresponding Sobolev space

for a fixed N ∈ N. Assume that there exist Z1, . . . , Z H ∈ OPS1,1H P such that (4.4), (6.1) and (6.2) hold. Consider p ∈ N

or p ∈ R with p ≥ N. One has

‖| f |p‖H N (A(Y )) . ‖ f ‖H N (A(Y )) sup|α|≤[N/2]

‖Zα f ‖p−1∞

for any given f such that right-hand side is finite.

With the same technique it is possible to generalize previous results in the following manner.

Theorem 7.10. Let Y1, . . . , YK ∈ OPS1,1P H and Z1, . . . , Z H ∈ OPS1,1

H P such that (4.3), (4.4), (6.1) and (6.2) hold. LetF ∈ C N (Rn) such that F(0) = 0, then

‖F( f )‖H N (A(Z)) . C(

N , n, ‖F‖C N (B1(0))

)‖ f ‖H N (A(Z)) if f ∈ H N (A(Z));

‖F( f )‖H N (A(Y )) . C(

N , n, ‖F‖C N (B1(0))

)‖ f ‖H N (A(Y )) if f ∈ H N (A(Y ));

whenever sup|α|≤[N/2] ‖Zα f ‖∞ ≤ 1.

7.3. Sobolev embedding theorems

According to Section 2, we can apply Theorems 7.6 and 7.8 for the Lie algebra generated by

Ω j,k(µ) = ∂ jµ2(x)∂k − ∂kµ

2(x)∂ j , 1 ≤ j < k ≤ n,

where µ is a positively homogeneous function of degree one.For µ(x) = |x | Theorems 7.6 and 7.10 have been stated by Klainerman in [10]. Assuming that Σλ is strictly

convex, we have an extension of such results not only to other norms µ(x), but also to pseudo-differential operatorsof the kind Ω jk(λ) = x j∂kλ

2(D)− xk∂ jλ2(D).

In order to conclude the proof of Theorem 1.1, it suffices to combine Theorem 7.8 with a Sobolev embeddingtheorem, which will assure that

‖g‖∞ . ‖g‖H K (A(µ)). (7.6)

We will derive this differential embedding from its pseudo-differential counterpart.

Lemma 7.11. Let A(λ) be the Lie algebra generated by Ω j,k(λ), ∂ j j,k=1,...,n . Let K ∈ N, K ≥ [(n − 1)/2] + 1.Assume that Σλ is a strictly convex hypersurface. For any given g ∈ H K (A(λ)), one has ‖g‖∞ . ‖g‖H K (A(λ))

.

This result has been proved in [4] by means of stationary phase methods. It can be also deduced by Theorem 2.3,Lemma 2.4 and Corollary 2.5 in [8].

On the other hand, Theorem 1.3 gives

H N (A(µ)) = H N (A(λ)) (7.7)

in the sense of the equivalence of the norms. Hence we can deduce a Sobolev embedding for H K (A(µ)).


Lemma 7.12. Let A(µ) be the Lie algebra generated by Ω j,k(µ), ∂ j j,k=1,...,n . Let K ∈ N, K ≥ [(n − 1)/2] + 1.Assume that Σµ is a strictly convex hypersurface. For any given g ∈ H K (A(µ)), the inequality (7.6) holds.

Proof of Theorem 1.1. Due to Theorem 7.8, if p ∈ N∗ or p ∈ R with p ≥ N , one has

‖| f |p‖H N (A(µ)) . ‖ f ‖H N (A(µ)) sup|α|≤[N/2]

‖Ωα(µ) f ‖p−1∞ .

By using (7.6), we arrive at

‖| f |p‖H N (A(µ)) . ‖ f ‖H N (A(µ)) sup|α|≤[N/2]

‖Ωα(µ) f ‖p−1H K (A(µ))

with K ≥ [(n − 1)/2] + 1. Combining this with the embedding H N (Rn) → L∞(Rn), we have

‖| f |p‖H N (A(µ)). ‖ f ‖H N (A(µ))

‖ f ‖p−1H K+[N/2](A(µ))

provided N > n/2. It remains to choose N such that [N/2] + K ≤ N . Since K ≥ [(n − 1)/2] + 1, we take N ≥ nfor even n and N ≥ n − 1 in the odd case like in the hypotheses of Theorem 1.1.

The proof of Theorem 1.2 is a direct consequence of Theorem 1.1 and (7.7).

Acknowledgments

This paper started as part of a joint project with Dott. Guido Ziliotti and we benefited from various discussionswith him especially on Section 2. The authors also wish to thank the anonymous referee for many useful corrections.

This work was partially supported by the PRIN programme “Buona positira e stime di decadimento per equazionidispersive e sistemi iperbolici”.

Appendix. Linear algebra lemmas

In this appendix X is an associative algebra on R, endowed with the natural Lie product

[A, B] = AB − B A, A, B ∈ X.

Lemma A.1. Let n ≥ 2 and A, B1, B2, . . . , Bn ∈ X. Then

[B1 B2 . . . Bn, A] = [B1, [. . . [Bn−1[Bn, A]] . . .]] +n∑

j=1

[B1, [. . . [Bn−1, [Bn, A]] . . .]]︸︷︷︸n−1 terms except B j

B j

+

n∑k=1

∑j<k

[B1, . . . [Bn−1, [Bn, A]] . . .]]︸︷︷︸n−2 terms except B j and Bk

B j Bk

+ · · · + [B1, A]B2 . . . Bn +

n−1∑j=2

[B j , A]B1 B2 . . . B j−1 B j+1 . . . Bn . (A.1)

Proof. Let n = 2. Since [B1 B2, A] = B1[B2, A] + [B1, A]B2 , we have

[B1 B2, A] = [B1, [B2, A]] + [B2, A]B1 + [B1, A]B2, (A.2)

according to the required formula. Assuming that (A.1) holds with n = N , we inductively prove that it holds forn = N + 1. By using (A.2), we get

[B1 . . . BN BN+1, A] = [B1 . . . BN , [BN+1, A]] + [B1 . . . BN , A]BN+1 + [BN+1, A]B1 . . . BN .

Applying the inductive assumption on the second term, we recognize all the addends with BN+1 out from the bracket.For the first term, we apply the inductive argument with A = [BN+1, A].


Lemma A.2. Let A,C ∈ X. Let B be a linear subspace of X. If

[A, B] ∈ B ∀B ∈ B,

then for all B1, B2, . . . , Bn,∈ B there exist K ∈ N and B j j≤K ⊂ B such that

[A, [B1, [. . . , [Bn,C] . . .] =∑

j1,... jn≤K

[B j1 , [. . . , [B jn ,C] . . .] + [B1, [. . . , [Bn, [A,C] . . .].

Proof. We proceed inductively. Let n = 1. From the Jacobi identity, it follows that

[A, [B1,C]] = −[C, [A, B1]] − [B1, [C, A]] = [B1,C] + [B1, [A,C]] (A.3)

where B1 = [A, B1]. Applying the inductive assumptions, we have

[A, [B1, [. . . , [Bn,C] . . .] =n−1∑j=1

[B1, [. . . , [B j , [Bn,C] . . .] + [B1, [. . . , [Bn−1, [A, [Bn,C] . . .].

In order to conclude, it suffices to apply (A.3) to the last term.

Lemma A.3. Let n ≥ 2 and A1, . . . , An, B,C ∈ X. Then

[A1, [. . . , [An, BC] . . .] = B[A1, [. . . , [An,C] . . .] + [A1, [. . . , [An, B] . . .]C +

+

∑h≤n

[A j1 , [. . . , [A jh , B] . . .][A jh+1 , [. . . , [A jn ,C] . . .].

Proof. We use an inductive argument. Let n = 2, we have

[A1, [A2, BC]] = [A1, B[A2,C]] + [A1, [A2, B]C]

= B[A1, [A2,C]] + [A1, B][A2,C] + [A2, B][A1,C] + [A1, [A2, B]]C.

Similarly

[A1, [. . . , [An, BC] . . .] = [A1, [. . . , B[An,C] . . .] + [A1, [. . . , [An, B]C] . . .].

The inductive assumption applies to the right-hand side. This concludes our proof.

Lemma A.4. Let A1, . . . , An,C ∈ X. Let B be a linear subspace of X and B ∈ B. If

[A j , B] ∈ B ∀ j = 1, . . . , n,

then there exists Bhh=1,...,n+1 ⊂ B such that

[A1, [. . . , [An, [B,C] . . .] =∑h≤n

[Bh, [A j1 . . . , [A jh ,C] . . .] + [Bn+1,C].

Proof. We use an inductive argument. If n = 1, the relation (A.3) directly gives our thesis. Assume that the requiredidentity holds fixed n − 1 elements chosen in A j j=1...n , then

[A1, [. . . , [An, [B,C] . . .]

=

∑h=1,...,n−1

[A1, [Bh, [A j1 . . . , [A jh ,C] . . .] + [A1, [Bn,C]] =∑

h=1,...,n−1

[[A1, Bh], [A j1 . . . , [A jh ,C] . . .]

+

∑h=1,...,n−1

[Bh, [A1, [A j1 . . . , [A jh ,C] . . .] + [[A1, Bn],C] + [Bn, [A1,C]].

Putting [A1, Bh] = Bh+1, after renaming the indexes we obtain our thesis.


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Nonlinear and multiplicative inequalities in Sobolev spaces associated with Lie algebras

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