# Pseudodifferential Operators on Weighted Hardy Spaces

Authors: Yu-long Deng; Shun-chao Long
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We study two sufficient conditions for the boundedness of a class of pseudodifferential operatorsT with symbols in the Hölmander class Sρ,δmℝn on weighted Hardy spaces Hω1ℝn, where ω belongs to Muckenhoupt class Ap. The first one is an estimate from Hω1ℝn into Lω1ℝn. We get a better range of admissible p and m. The second one is a weighted version bounded for the operators T on Hω1ℝn, and it is an addition to the literature.


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## Body

## 1. Introduction

The purpose of this paper is to study some sufficient conditions for the boundedness of pseudodifferential operatorsT on weighted Hardy space Hω1ℝn, where the operators T have symbols in the Hölmander class Sρ,δmℝn. As in [1], 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+δαholds for all multi-indices α,β∈ℕn, where Cα,β is independent of x and ξ. We now assume that the symbol ax,ξ is smooth in both the spatial variable x and the frequency variable ξ.Givenf∈C0∞ℝn, the pseudodifferential operator T∈ℒρ,δm associated with the symbol ax,ξ∈Sρ,δmℝn is given by(2)Tfx=∫ℝnax,ξe2πix⋅ξf^ξdξ,where f^ denotes the Fourier transform of f. Moreover, we can express T by a kernel Kx,y as (see, e.g., [2])(3)Tfx=∫Kx,yfydy.Pseudodifferential operators play an important role in the theory of partial differential equations. It is well known that the Hardy spacesHpℝn coincide with the Lebesgue spaces Lpℝn when p>1. The Lp and weighted Lp boundedness of the operator T∈ℒρ,δm have been extensively studied. We refer to [1, 2, 3, 4] for the Lp bounds and [5, 6, 7, 8] for the weighted Lp bounds.Forp∈0,1, there is an estimate from L1ℝn into weak L1ℝn for the pseudodifferential operator T∈ℒρ,δm (cf. [5]). As known, the Hardy space Hpℝn is an advantageous substitute for Lpℝn. The behavior of the pseudodifferential operator T on Hpℝn has attracted a lot of interest. For example, Alvarez and Hounie [5] have found that the pseudodifferential operator T with symbol in Sρ,δmℝn is bounded from H1ℝn into L1ℝn, where 0<ρ≤1, 0≤δ<1 and m≤−n/21−ρ+min0,nρ−δ/2. Hounie and Kapp [9] have shown that the operator T with 0≤δ≤ρ<1 and m=−n1−ρ/2 is bounded from the local Hardy space h1ℝn into L1ℝn. Yabuta [10] has proved the operator T involving a modulus of continuity wt is bounded from H1ℝn into L1ℝn.The bounds of the pseudodifferential operatorT from the weighted Hardy space Hω1ℝn into the weighted Lebesgue space Lω1ℝn have also been studied. Yabuta [11] has found that the operator T is bounded from Hω1ℝn into Lω1ℝn, where T∈ℒ1,δ0 and ω∈A1. In view of this, it is natural to look for a wide range of operator T in ℒρ,δm to study the bounds on the weighted Hardy space Hω1ℝn.In this paper, we establish two estimates for the pseudodifferential operatorT with symbols in Sρ,δmℝn. The first one is an estimate from Hω1ℝn into Lω1ℝn. We extend the result in Yabuta [11] to ω∈Ap with 1≤p<1+ε/n and the operator T with 0<ρ≤1, 0≤δ<1, and −n+1<m≤−n+11−ρ. Our first main result is stated as follows.Theorem 1.
Letε=min1,1+m+n/ρ, p∈1,1+ε/n, ω∈Ap and T∈ℒρ,δm with 0<ρ≤1, 0≤δ<1. If −n+1<m≤−n+11−ρ, then T is bounded from Hω1ℝn into Lω1ℝn, i.e., there exists a constant C>0 such that(4)TfLω1ℝn≤CfHω1ℝn.The second one is an estimate on weighted Hardy spacesHω1ℝn for the pseudodifferential operator T. It is well known that under certain conditions of m,ρ,δ, the operator T is bounded on h1ℝn (cf. [9, 12]). Alvarez and Hounie [5] have found that the pseudodifferential operator T is bounded on H1ℝn, where T∈ℒρ,δm with 0<ρ≤1, 0≤δ<1, and m≤−λ+min0,nρ−δ/2 for some n1−ρ/2≤λ<n/2. It is natural to look for a weighted version estimate on Hω1ℝn. We now state our second main result.Theorem 2.
Letμ=1+n+m/ρ−n, p∈1,1+μ/n, ω∈Ap, and T∈ℒρ,δm with 0<ρ≤1, 0≤δ<1. Assume ρn−n+1<m≤−n+11−ρ and T∗1=0. Then, T is bounded on Hω1ℝn; i.e., there exists a constant C>0 such that(5)TfHω1ℝn≤CfHω1ℝn.The remainder of this paper is organized as follows. In Section2, we present some definitions and well-known results we use later. The aim of Section 3 is to set up the estimate from Hω1ℝn into Lω1ℝn for pseudodifferential operators T in ℒρ,δm. We develop a method to handle TaLω12k+1B/2kB (see Proposition 1). The aim of Section 4 is to establish the estimate on weighted Hardy spaces Hω1ℝn for pseudodifferential operators T in ℒρ,δm.Most of the notations we use are standard.C denotes a constant that 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 and χA the characteristic function. B will always denote a ball, and tBt>0 denotes the ball B dilated by t.


## 2. Notations and Auxiliary Lemma

In this section, we first present an auxiliary lemma 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 3.1 in [9]; see also [5]).Lemma 1.
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(6)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(7)supα+β=MDxαDyβKx,y≤CM1x−yM+m+n/ρ,x≠y.Moreover, for any multi-indexα,β∈ℕn and N∈ℕ,(8)supx−y≥1/2x−yNDxαDyβKx,y≤Cα,β,N.The following usefulLωp bound for the pseudodifferential operator T is obtained by Michalowski et al. [7].Lemma 2.
LetT∈ℒρ,δm, 0≤δ<1,0<ρ≤1, and m<−n1−ρ. Then, for each p∈1,∞ and ω∈Ap, there exists a constant C>0 such that(9)TfLωpℝn≤CfLωpℝn.Remark 1.
Obviously, theLω2 bounds of pseudodifferential operators T are established automatically.Remark 2.
m
≤
−
n
+
1
1
−
ρ satisfies the condition of m in Lemma 2.
The following usefulL2 bound of the pseudodifferential operator T∈ℒρ,δm is obtained by Alvarez and Hounie [5].Lemma 3.
LetT∈ℒρ,δm, 0≤δ<1,0<ρ≤1, and m<min0,nρ−δ/2. Then, the operator T is bounded on L2ℝn.Remark 3.
The range ofm,p here is m<−n1−ρ1/p−1/2+min0,nρ−δ/2 and 1<p<∞, respectively.Remark 4.
Obviously,m≤−n+11−ρ satisfies the condition of m in Lemma 3.
Letp∈1,∞. A nonnegative locally integrable function ω belongs to Muckenhoupt class Ap, if there exists a constant C>0, such that for all balls B⊂ℝn,(10)1B∫Bωxdx1B∫Bωx−1/p−1dxp−1≤Cif1<p<∞,1B∫Bωxdx≤Cessinfx∈Bωxifp=1.
We denoteA∞=∪p≥1Ap.
It is well known thatω∈Ap implies ω∈Aq for all q>p. Also, if ω∈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 [13]).Lemma 4.
Letω∈Ap and p≥1. Then, there exists a constant C>0 such that(11)CEBp≤ωEωB,for all balls B and measurable subsets E⊂B.Given a weight functionω on ℝn, we denote by Lωpℝn the weighted Lebesgue space of all functions f satisfying(12)fLωpℝn=∫ℝnfxpωxdx1/p<∞.Whenp=∞, Lω∞ℝn is L∞ℝn. Analogous to the classical Hardy space, the weighted Hardy space Hω1ℝn can be defined in terms of maximal functions.Definition 1.
Letω∈A∞. The weighted Hardy space Hω1ℝn is defined by(13)Hω1ℝn=f∈S′ℝn:ϕ∗fx=supt>0ϕt∗fx∈Lω1ℝn,where ϕ∈Sℝn is a fixed function with ∫ϕdx≠0 and ϕty=1/tnϕy/t for any t>0. Moreover, we define fHω1ℝn=ϕ∗fLω1ℝn.Remark 5.
Definition1 is independent of the choice of ϕ (see [14]).Definition 2.
Letω be a weight with the critical index qω. An 1,∞‐atom with respect to ω is a function a satisfying(14)suppa⊂BaL∞≤ωB−1,and ∫axxαdx=0 for every multi-index α with α≤nqω−1.
The Hardy spaceHω1ℝn is a linear space spanned by all of 1,∞‐atoms with respect to ω. Namely, f∈Hω1ℝn if and only if f can be written as (see [13])(15)f=∑jλjaj,in the sense of S′, where each aj is an 1,∞-atom with respect to ω and λj satisfies(16)∑jλj<∞.
Moreover,fHω1ℝn=inf∑j=1∞λj:f=∑j=1∞λjaj.Definition 3.
LetT be a pseudodifferential operator in ℒρ,δm. We say T∗1=0 if ∫ℝnTaxdx=0 for all a∈L∞ℝn with compact support and ∫ℝnaxdx=0.


## 3. The Proof of Theorem 1

In this section, we prove that the pseudodifferential operatorsT in ℒρ,δm are bounded from Hω1ℝn into Lω1ℝn.Proposition 1.
Letω∈Ap, p∈1,∞ and ε=min1,1+m+n/ρ. Assume pseudodifferential operator T∈ℒρ,δm with 0<ρ≤1, 0≤δ<1, and −n+1<m≤−n+11−ρ. Then, there exists a constant C>0 such that(17)TaLω12k+1B\2kB≤C2−kε+n1−p,holds for all 1,∞-atoms a with respect to ω, where suppa⊂B=Bx0,r.Proof.
Inspired by the proof of Lemma 3.2 in [15], we consider two cases about the radius r.Case 1.
Whenr≥1. For every x∈2k+1B\2kB and y∈Bx0,r, we have(18)x−y≥x−x0−y−x0≥2kr−r≥1.
Hence, by (8) and properties of 1,∞-atoms with respect to ω, we have(19)Tax=∫ℝnKx,yaydy≤∫BKx,yaydy≤C∫B1x−yn+1aydy≤C1x−x0n+1∫Baydy≤C2−kB2kBωB−1,for all x∈2k+1B\2kB. Thus,(20)TaLω12k+1B/2kB≤C2−kB2kBω2k+1BωB≤C2−kε+n1−p.Case 2.
When0<r<1. For every x∈2k+1B/2kB and y∈Bx0,r, by moments condition, we have(21)Tax=∫ℝnKx,yaydy≤∫BKx,y−Kx,x0aydy.
By the mean value theorem,1+m+n>0, and (7), we have(22)Tax≤C∫By−x0x−x01+m+n/ρay≤Cr2kr1+m+n/ρBωB,where we take M=1 and use the fact that x−ξ∼x−x0 if ξ∈Bx0,r. Let us now consider two subcases.Subcase 1.
If2k−1r≥1, then, for any y∈Bx0,r and x∈2k+1B\2kB,(23)x−y≥x−x0−y−x0≥2k−1r≥1.
Similar to the caser≥1, we get(24)Tax≤CB2kBr2kr1+m+n/ρωB−1.
Since0<r<1 and 2kr>1, we have(25)r2kr1+m+n/ρ≤2−k1+m+n/ρ,1+m+nρ≤1;2−k,1+m+nρ≥1.
Notingε=min1,1+m+n/ρ, it easy to see Tax≤C2−εkB/2kBωB−1. This implies (17).Subcase 2.
If2k−1r<1. Since m≤−n+11−ρ, (22) yields(26)TaLω12k+1B\2kB≤Cr2krB1+m+n/ρω2k+1BωB≤C2−kε+n1−p.
In view of (20) and (26), we finish the proof of Proposition 1.Proof.
The proof of Theorem1 is motivated by the atomic decomposition for Hω1ℝn. Let f∈Hω1ℝn. We obtain an atomic decomposition of f satisfying (15) and (16). So, to prove that the pseudodifferential operators T are bounded from Hω1ℝn into Lω1ℝn, it suffices to show that for each 1,∞-atom a with respect to ω, we have TaLω1ℝn≤C. Recall that an 1,∞-atom a with respect to ω is a function satisfying(27)suppa⊂B,aL∞≤ωB−1,∫axdx=0,for some ball B=Bx0,r.
Now, leta be such an atom and write(28)∫Taω=∫2BTaω+∫2BcTaω=I1+I2.
It is easy to estimate the termI1. Using Hölder inequality and Lω2-boundedness for the pseudodifferential operator T (see Remark 1), we get(29)I1≤∫2BTa2ω1/2∫2Bω1/2≤CTaLω2ℝnω2B1/2≤CaLω2ℝnω2B1/2≤CωB−1/2ωB1/2≤C,where C is independent of a.
For the termI2, we write(30)I2=∫2BcTaω≤∑k=1∞∫2k+1B\2kBTaω=∑k=1∞TaLω12k+1B\2kB.
By Proposition 1, we get(31)I2≤∑k=1∞C2−kε+n1−p≤C,since 1≤p<1+ε/n. Combing (29) and (31), we finish the proof of Theorem 1.


## 4. The Proof of Theorem 2

In this section, we establish the weighted norm inequality on weighted Hardy spacesHω1ℝn for pseudodifferential operators T in ℒρ,δm.Proof.
Without loss of generality, we assume1≤p<1+μ/n, where μ=1+m+n/ρ−n. Fix ϕ∈Sℝn and ∫ℝnϕxdx≠0. By (15), it is sufficient to show that for each 1,∞-atom a with respect to ω, Ta∗Lω1ℝn≤C with C independent of a. In order to do this, one can suppose suppa⊂B=Bx0,r and write(32)Ta∗Lω1ℝn=∫x−x0≤4rTa∗xωxdx+∫x−x0>4rTa∗xωxdx=I1+I2.
For the termI1, by Hölder inequality, Lω2-boundedness of the maximal function Ta∗, Lω2-boundedness of the pseudodifferential operator T, and (14), we get(33)I1≤∫x−x0≤4rTa∗x2ωxdx1/2∫x−x0≤4rωxdx1/2≤CTaLω2ℝnωBx0,4r1/2≤CaLω2ℝnωBx0,4r1/2≤C,where C is independent of a.
To estimateI2, we first estimate Ta∗x for x−x0>4r. For any t>0, since T∗1=0(see Definition 3), we have(34)Ta∗ϕtx=∫ℝnTay1tnϕx−ytdy=∫ℝnTay1tnϕx−yt−ϕx−x0tdy≤1tn∫y−x0<2rTayϕx−yt−ϕx−x0tdy+1tn∫2r≤y−x0<x−x0/2⋯+1tn∫y−x0≥x−x0/2⋯=E1+E2+E3.
For the termE1, by the mean value theorem and Hölder’s inequality, we have(35)E1≤1tn+1TaL2ℝn×∫y−x0<2r∇ϕx−x0−γy−x0t2y−x02dy1/2≤Crn+1x−x0n+1ωB,where γ∈0,1 depends on x,y, and x0, and ∇=∂/∂x1,…,∂/∂xn. Here, we use the inequalities(36)x−x0−γy−x0n+1tn+1∇ϕx−x0−γy−x0t≤C,and x−x0−γy−x0≥x−x0−γy−x0≥x−x0−y−x0≥x−x0/2, and L2-boundedness of the pseudodifferential operator T (see Lemma 3).
To estimateE2 and E3, we first estimate Tay when y−x0>2r and consider two cases about r.Case 3.
Ifr>1, then for every y∈ℝn\Bx0,2r and z∈B, we have y−z≥y−x0−z−x0≥r≥1. Hence, by (8), we have(37)Tay=∫ℝnKy,zazdz≤∫BKy,zazdz≤C∫B1y−zN+n+1aL∞dz≤C1y−x0n+1aL∞B≤Crn+1y−x0n+1ωB.Case 4.
In the case of0<r<1, we have ∫Bazdz=0. Thus, for every y∈ℝn\Bx0,2r, from 1+n+m>nρ>0, (7) yields(38)Tay=∫ℝnKy,zazdz≤∫BKy,z−Ky,x0azdz≤C∫Bz−x0y−x01+n+m/ρaL∞dz≤Cry−x01+n+m/ρaL∞B≤Crn+1y−x01+n+m/ρωB,where we use the fact that y−ξ∼y−x0 if ξ∈Bx0,r and y−x0>2r.
Let us now continue to estimateE2. When r≥1, using the mean value theorem and (37), we have(39)E2=1tn+1∫2r≤y−x0<x−x0/2∫B2r≤y−x0<x−x02×∇ϕx−x0−γy−x0ty−x0dy≤Crn+1x−x0n+1ωB∫2r≤y−x0<x−x0/21y−x0ndy≤Crn+1x−x0n+1ωBlnx−x04r.
Here, we use the fact thatx−x0−γy−x0∼x−x0 under the condition of 2r≤y−x0<x−x0/2.
Similarly, in the case of0<r<1, by the moments condition for a, the mean value condition, and (38), we get(40)E2=1tn+1∫2r≤y−x0<x−x0/2∫BKy,z−ky,x0azdz×∇ϕx−x0−γy−x0ty−x0dy≤Crn+1x−x0n+1ωB∫2r≤y−x0<x−x0/21y−x01+n+m/ρ−1dy≤Crn+1x−x0n+1ωBlnx−x04r,1+m+nρ=n+1;Crn+μx−x0n+μωB,1+m+nρ<n+1.
For the termE3, we have(41)E3≤1tn∫y−x0≥x−x0/2Tayϕx−yt+ϕx−x0tdy.
Sincey−x0≥x−x0/2, we have x−y≥x−x0/2. Thus,(42)1tnϕx−yt≤Cx−yn≤Cx−x0n.
Meanwhile,(43)1tnϕx−x0t≤Cx−x0n.
So, in the case ofr≥1, by (37), (42), and (43), we have(44)E3≤C∫y−x0≥x−x0/2rn+1y−x0n+1ωB1x−x0ndy=Crn+1x−x0nωB∫y−x0≥x−x0/21y−x0n+1dy=Crn+1x−x0n+1ωB.
In the case of0<r<1, by (38), (42), and (43), we have(45)E3≤C∫y−x0≥x−x0/2rn+1y−x01+n+m/ρωB1x−x0ndy=Crn+1x−x0nωB∫y−x0≥x−x0/21y−x01+n+m/ρdy=Crn+1x−x0n+μωB=Crn+μx−x0n+μωB,since ρn<1+n+m≤ρn+1.
Letx∉Bx0,4r. In view of (35), (39), (40), (44), and (45), we shall unify these formulas. Firstly, ρn−n+1<m≤ρ−1n+1 implies μ∈0,1. Secondly, x∉Bx0,4r implies r/x−x0<1. Therefore,(46)rn+1x−x0n+1ωB≤rn+μx−x0n+μωBholds for any μ∈0,1. Finally, since lnx≤1/αxα holds for x≥1 and α∈0,1, we have(47)rn+1x−x0n+1ωBlnx−x04r≤Crn+1x−x0n+1ωBx−x0r1−μ=Crn+μx−x0n+μωB.
Using these three facts, we have(48)Ta∗x≤Crn+μx−x0n+μωB.
Note thatp≤1+μ/n. Then, in all cases, we have n+μ>np and(49)I2≤C∫x−x0>4rrn+μx−x0n+μωBωxdx=Crn+μωB∑l=0∞∫2l+2r≤x−x0<2l+3rωxx−x0n+μ≤C∑l=0∞12ln+μ−np≤C.
This concludes the proof of Theorem2.



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