# Commutators of Pseudodifferential Operators on Weighted Hardy Spaces

Authors: Yu-long Deng
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we establish an endpoint estimate for the commutator,b,T, of a class of pseudodifferential operators T with symbols in Hörmander class Sρ,δmRn. In particular, there exists a nontrivial subspace of BMORn such that, when b belongs to this subspace, the commutators b,T is bounded from Hω1Rn into Lω1Rn, which we extend the well-known result of Calderón-Zygmund operators.


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## Body

## 1. Introduction

The purpose of this paper is to find out a proper subspace ofBMORn such that, the commutators of pseudodifferential operators T is bounded on weighted Hardy space Hω1Rn, where the operators T associated with the symbols in the Hölmander class Sρ,δmRn and ω∈ApRn. As in [1], we firstly recall some notations and lemmas. For m∈R and ρ,δ∈0,1, a symbol ax,ξ∈Sρ,δmRn is a smooth function defined on Rn×Rn such that(1)∂xα∂ξβax,ξ≤Cα,β1+ξm−ρβ+δα,holds for all multi-indices α,β∈Nn, where Cα,β is independent of x and ξ (see, e.g., [2]).Given an infinitely differentiable functionf∈Rn with compact supports and symbol ax,ξ∈Sρ,δmRn, the pseudodifferential operator T is defined by(2)Tfx=∫Rnax,ξe2πix⋅ξf^ξdξ,where f^ is the Fourier transform of f and we write T∈Lρ,δm . Moreover, the operator T can be expressed by a distribution kernel Kx,y as (see, e.g., [3])(3)Tfx=∫Kx,yfydy.Letb∈BMORn and T be a Calderón-Zygmund operator. A classical result in [4] stated that the commutator operators b,T, defined by(4)b,Tfx=bxTfx−Tbfx,is bounded on LpRn for p∈1,∞. However, it fails to be of weak type 1,1 and of type H1Rn,L1Rn when b∈BMORn (see, [5, 6]). Instead, some endpoint theories are provided.Remark that if the symbolax,ξ satisfies some particular assumptions, pseudodifferential operator T in Lρ,δm is a Calderón-Zygmund operator (see, [7]). Correspondingly, when b∈BMORn and T∈Lρ,δm, the boundness of b,T on Lebesgue space LpRn for p∈1,∞ was considered (see, e.g., [8–10]).It is widely known thatH1Rn is an advantageous substitute for L1Rn. The behavior of commutator b,T on H1Rn has also attracted a lot of interest. For example, when b∈LMO∞Rn (see, [11]), Yang et al. [12] obtained that b,T is bounded from H1Rn into L1Rn, where T∈L1,δ0 with δ∈0,1; Hung and Ky [13] established an estimate for b,T on local Hardy space h1Rn. Very recently, Deng and Long [14] got an estimate for b,T from H1Rn into weak L1Rn, where b∈BMORn.Forω∈ApRn, there are numerous papers dealing with the weighted Lp boundedness of the commutators b,T for p∈1,∞ and we refer to [15–20] for more details, where T∈Lρ,δm and b∈BMORn. A nature question is that can one establish an estimate for b,T on weighted Hardy spaces Hω1Rn?In general, the commutatorsb,T is not bounded from the weighted Hardy spaces Hω1Rn into the weighted Lebesgue spaces Lω1Rn if b∈BMORn is not a constant function, even T is a Calderón-Zygmund operator. It is worthy to pointing out that in [21], Liang et al. found a proper subspaces of BMORn, such that, the commutators of Calderón-Zygmund operator is bounded on weighted Hardy spaces. Motivated by this result, we wonder whether there exists a nontrivial subspace of BMORn such that when b belongs to this subspace, the commutators b,T of pseudodifferential operator is bounded on Hω1Rn.The main concern of this paper is to give an answer to the above question. For this purpose, we recall the definition of the Muckenhoupt weightsApRn. A nonnegative measurable function ω is said to be in the Muckenhoupt class ApRn for p∈1,∞, if(5)ωApRn=supB⊂Rn1B∫Bωxdx1B∫Bωx−1/p−1dxp−1<∞,and for p=1, if(6)ωA1Rn=supB⊂Rn1B∫Bωxdxesssupx∈Bωx−1<∞,where the supremum is taken over all balls B⊂Rn and A∞Rn=⋃p≥1ApRn.As known, ifω∈Ap, then ω∈Aq for some q∈1,p. We thus write qω=infp≥1:ω∈Ap to denote the critical index of ω. For a measurable set E, we denote ωE=∫Eωxdx. The following lemma provides a way to compare E and ωE of a set E(see [22]).Lemma 1.
Letω∈Ap and p≥1. Then, there exists a constant C>0 such that(7)CEBp≤ωEωB,for all balls B and measurable subsets E⊂B.Definition 1.
Letω∈A∞Rn and ∫Rnωx/1+xndx<∞. A locally integrable function b is said to belong to BMOωRn if(8)bℬℳOωℝn=supB⊂ℝn1ωB∫ℝn∖Bωx|x−x0|ndx∫Bby−bBdy<∞.
Here,bB=1/B∫Bbxdx and the supremum is taken over all balls B=Bx0,r⊂Rn with center x0 and radius r.
We point out that the spaceBMOωRn has been studied in [21, 23, 24]. A locally integrable function b is said to be in BMORn if(9)bBMORn=supB⊂ℝn1B∫Bbx−bB|dx<∞,where the supremum is taken over all balls B⊂Rn.
In [21], the space BMOωRn is proved to be a subspace of BMORn, and not be a trivial space since it contains the Lipschitz function with compact support and also that,Lemma 2.
Letω∈A∞Rn and p∈1,∞. Then, there exists a constant C>0 such that, for any f∈BMORn and any ball B⊂Rn, (10)1ωB∫Bfx−fBpωxdx1/p≲fBMOℝn.
The first main result is stated as follow.Theorem 1.
Letϵ=min1,1+m+n/ρ, ω∈A1+ϵ/n satisfies ∫Rnωx/1+xndx<∞ and b∈BMOωRn. Assume that the pseudodifferential operator T∈Lρ,δm with ρ∈0,1, δ∈0,1 and(11)m∈−n+1,−n+11−ρ.
Then, the commutatorb,T is bounded from Hω1Rn into Lω1Rn; i.e., there exists a constant C>0 such that, for all f∈Hω1Rn,(12)b,TfLω1ℝn≤CbℬℳOωℝn.
Finally, we make some conventions on notations.C denotes a positive constant may change from line to line and we write a≲b as shorthand for a≤Cb. If a≲b and b≲a, we mean a∼b. For a measurable set A, A denotes the Lebesgue measure of A. B will always denote a ball and tBt>0 denotes the ball B dilated by t.


## 2. Notations and Technical Lemmas

In this section, we begin our story by presenting an estimate about the pseudodifferential operatorT associated with the kernel Kx,y. Let SRn be the class of Schwartz functions and S′Rn be its dual space. The space of C∞-function with compact support is denoted by C0∞Rn. Pseudodifferential operators are bounded from SRn to SRn and so possess distribution kernels Kx,y∈S′Rn×Rn. Then, the following formula for the kernel is useful (cf. Proposition 1 in [25], see also [26]).Lemma 3.
Letax,ξ∈Sρ,δmRn with 0<ρ≤1,0≤δ<1 and associate with the pseudodifferential operator T∈Lρ,δm. Then, the distribution kernel Kx,y of T is smooth away from the diagonal x,x:x∈Rn and is given by(13)Kx,y=limε⟶0∫Rne2πix−y⋅ξax,ξψεξdξ,where ψ∈C0∞Rn satisfies ψξ=1 for ξ≤1 and the limit is taken in S′Rn and independent of the choice of ψ. If M∈N and M+m+n>0, Kx,y satisfies the estimates(14)supα+βDxαDyβKx,y|≤CM1x−yM+m+n/ρ,x≠y.
Moreover, for any multi-indexα,β∈Nn and N∈N,(15)supx−y≥1/2x−yNDxαDyβKx,y≤Cα,β,N.
In [20], the following is derived from Lemma 3.Lemma 4.
Letρ∈0,1,δ∈0,1 and m∈−n+1,−n+11−ρ and the pseudodifferential operator T∈Lρ,δm associated with the distribution kernel Kx,y. Then, for any y∈B=Bx0,r and every x∈2j+1B∖2jB, we have(16)Kx,y−Kx,x0≲2−jn+εr−jn,where ϵ=min1,1+n+m/ρ.
Letω∈A∞Rn and p∈0,∞. We denote by LωpRn the weighted Lebesgue space of all measurable functions f satisfying(17)fLωpRn=∫Rnfx|pωxdx1/p<∞.
Whenp=∞, Lω∞Rn is defined to be the same as L∞Rn, the following useful Lωp bounds for pseudodifferential operator T∈Lρ,δm are due to Michalowski et al. [15].Lemma 5.
LetT∈Lρ,δm with ρ∈0,1,δ∈0,1 and m∈−∞,−n1−ρ. Then, for each p∈1,∞ and ω∈Ap, there exists a constant C>0 such that(18)b−bBTaLω1ℝn≤CbBMOℝn,
Letϕ∈SRn with ∫ϕdx≠0. Then, for any x∈Rn, the maximal function of a distribution f∈S′Rn is defined by(19)ϕ∗fx=supt>0ϕt∗fx,where ϕty=1/tnϕy/t for any t>0. Let p∈1,∞. Then, the maximal function is bounded on LωpRn if and only if ω∈Ap. Analogous to the classical Hardy space, the weighted Hardy space Hω1Rn can be defined in terms of maximal functions.Definition 2.
Letω∈A∞. The weighted Hardy space Hω1Rn is defined by(20)Hω1Rn=f∈S′Rn:φ∗fx=supt>0φt∗fx∈Lω1Rn,which is independent of the choice of φ∈SRn. Moreover, we define f‖Hω1Rn=‖φ∗f‖Lω1Rn.Definition 3.
Letω be a weight with critical index qω. Anω,1,∞-atom is a function a satisfying(21)suppa⊂B,aL∞Rn≤ωB−1,and ∫Rnaxxαdx=0 for every multi-index α with α≤nqω−1. Conventionally, T∗1=0 means ∫RnTaxdx=0 for all ω,1,∞-atoms a.
The Hardy spaceHω1Rn is spanned by all of ω,1,∞-atoms (see [22]). Namely,(22)f=∑jλjaj,in the sense of S′, where each aj is an ω,1,∞-atom and λj satisfies(23)∑jλj<∞.
Moreover,f‖Hω1Rn=inf∑j=1∞λj:f=∑j=1∞λjaj.
Deng et al. [1] got some sufficient conditions for the boundedness of pseudodifferential operators T∈Lρ,δm on weighted Hardy space Hω1Rn.Lemma 6.
Letϵ=min1,1+m+n/ρ, p∈1,1+ϵ/n, ω∈Ap, and T∈Lρ,δm with ρ∈0,1, δ∈0,1. If m∈−n+1,−n+11−ρ, then T is bounded from Hω1Rn into Lω1Rn, i.e., there exists a constant C>0 such that(24)TfLω1ℝn≤CfHω1ℝn.


## 3. The Proof of Theorem 1

In this section, we establish the sufficient condition for the boundedness ofb,T from Hω1Rn into Lω1Rn. As in [21], we need the following proposition.Proposition 1.
Writeϵ=min1,1+m+n/ρ. Let p∈1,1+ϵ/n and ω∈ApRn. Assume that the pseudodifferential operator T∈Lρ,δm with ρ∈0,1, δ∈0,1 and m∈−n+1,−n+11−ρ. Then, there exists a constant C>0 such that, for any b∈BMORn and ω,1,∞-atom a,(25)b−bBTaLω1ℝn≤CbBMOℝn,where suppa⊂B=Bx0,r.Proof.
It suffices to show that(26)I1=∫2Bbx−bBTaxωxdx≤CbBMOℝn,and(27)I2=∫2B∁bx−bBTax|ωxdx≤CbBMORn.
ForI1, it is easy to see that(28)I1≤∫2Bb2B−bBTaxωxdx+∫2B|bx−b2B‖Tax|ωxdx≤|b2B−bB|‖Ta‖Lω1ℝn+∫2B|bx−b2B‖Tax|ωxdx=I11+I12.
Then, by Lemma6, the boundedness of the operator T from Hω1Rn to Lω1Rn, we conclude that(29)I11≤CbBMOℝn.
Also, by Hölder inequality, Lemma2 and the boundedness of T on LωpRn (Lemma 5), we have(30)I12≤∫2Bbx−b2Bq′ωxdx1/q′∫2BTaxqωxdx1/q≤Cω2B1−1/qbBMOℝnTaLωqℝn≤Cω2B1−1/qbBMOℝnaLωqℝn≤CbBMOℝn.
Thus, (26) holds.
ForI2, by the moment condition of ω,1,∞-atoms a, we have(31)I2=∫2B∁bx−bB|∫BayKx,y−Kx,x0dyωxdx≤∫Bay∫2B∁bx−bBKx,y−Kx,x0ωxdxdy=∫Bay∑k=1∞∫2k+1B∖2kBbx−bBKx,y−Kx,x0ωxdxdy.
Then, we apply Lemma4 to get(32)I2≤C∫Bay|∑k=1∞∫2k+1B∖2kB2−kn+ϵr−knbx−bBωxdxdy≲BωB∑k=1∞2−kϵ2k+1B∫2k+1Bbx−bB|ωxdx.
Finally, by using Lemma2 again and combing the inequality(33)b2k+1B−bB≤∑j=0kb2j+1B−b2jB≲k+1bBMOℝn,with Lemma 1 and the condition p∈1,1+ϵ/n, we deduce(34)I2≤CBωB∑k=1∞2−kεkω2k+1B2k+1BbBMOℝn≤CbBMOℝn∑k=1∞k2−kε+n−pn≤CbBMOℝn,and which suggests that (27) holds. Thus, we finish the proof of Proposition 1.
Now, we are ready to give the proofs of Theorem1.Proof.
Writeϵ=min1,1+m+n/ρ. Let ω∈A1+ϵ/n and f∈Hω1Rn. According to the atomic characterization of f and Proposition 1, it is reduced to showing that(35)Tb−bBaLω1ℝn≤CbℬℳOωℝn,holds for each ω,1,∞-atom a related to some ball B=Bx0,r.
Then, by the boundedness ofT from Hω1Rn to Lω1Rn as in Lemma 6, we just need to prove(36)b−bBaHω1ℝn≲bℬℳOωℝn.
Finally, (36) is equivalent to establishing(37)ϕ∗b−bBaLω1ℝn≲bℬℳOωℝn,for φ∈SRn with ∫φdx≠0, since b−bBa‖Hω1Rn=‖φ∗b−bBa‖Lω1Rn as in Definition 2.
In order to get (37), we consider(38)I3=∫2Bϕ∗b−bBaxωxdx,I4=∫2B∁ϕ∗b−bBaxωxdx.
ForI3, combining Hölder’s inequality with the weighted Lp boundedness of the maximal function and Lemma 2, we have(39)I3≤ω2B1/q′ϕ∗b−bBaLωqRn≤ω2B1/q′b−bBaLωqRn≤C1ωB∫Bbx−bB|qωxdx1/q≤CbBMORn≲bBMOωRn.
ForI4, noting that x−y∼x−x0 for every x∈2B∁ and any y∈B, we get(40)ϕ∗b−bBa=supt>01tn∫Bby−bBay|ϕx−ytdy≲1x−x0n∫Bby−bBay|dy.
Hence,I4≲b‖BMOωRn and it completes the proof of Theorem 1.



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