# OnL2-Boundedness of h-Pseudodifferential Operators

Authors: Jie Yang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

LetTah be the h-pseudodifferential operators with symbol a. When a∈Sρ,1m and m=nρ−1/2, it is well known that Tah is not always bounded in L2ℝn. In this paper, under the condition ax,ξ∈L∞Sρnρ−1/2ω, we show that Tah is bounded on L2.


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## Body

## 1. Introduction and Main Results

Letx,ξ∈ℝn×ℝn and Sρ,δm be the sets of all the symbol ax,ξ satisfying, for m∈ℝ, 0≤ρ,δ≤1, and α,β∈ℕn,
(1)∂xβ∂ξαax,ξ≲ξm−ρ∣α∣+δβ.The classical pseudodifferential operatorTa associated with the symbol ax,ξ∈Sρ,δm is defined by
(2)Taf=12πn∫ℝneixξax,ξf^ξdξ,wheref^ξ is the Fourier transform of f.The generalized form of pseudodifferential operator ish-pseudodifferential operator, which is introduced by Helffer in [1]. Helffer studied this operator to discuss the operator −h2Δ+Vx associated with the parameter h, where Δ is the Laplace operator and Vx denotes some potential function for any x∈ℝn. Moveover, the h-pseudodifferential operator also provides a rigorous way to establish relationship each other for quantum physics and classical mechanics; see, for example, [2, 3]. Furthermore, by using h-pseudodifferential operator, the Cauchy problem of semiclassical elliptic partial differential equations is studied in [3–5]. Because of these operators importance, many scholars have studied one; see, for example, [6–12] and the references given there.Now, we consider theh-pseudodifferential operator as follows:
(3)Tahf=12πn∫ℝndξ∫ℝnax,hξeix−yξfydy=12πhn∫ℝndξ∫ℝnax,ξei/hx−yξfydy=12πhn∫ℝnei/hxξax,ξf^ξdξ,whereh∈0,h0 and
(4)f^ξ,h=∫ℝne−i/hyξfydy.First of all, we reviewL2 regularity theory for the classical pseudodifferential operators Ta. In [13], Hörmander shown that Ta is bounded in L2ℝn, when a∈Sρ,δm,δ<1 and m≤nρ−δ/2. For a∈S1,10, Ching [14] proved that Ta is not bounded in L2ℝn. Moveover, for a∈Sρ,1m, Rodino [15] shown that Ta is bounded in L2ℝn if and only if m<nρ−1/2. However, the operator Ta is not always L2-boundedness for a∈Sρ,1nρ−1/2; see, for example, [13–15]. The necessary and sufficient conditions of L2-boundedness of Ta are obtained by Higuchi [16] as m=nρ−1/2. Recently, Aitemrar and Senoussaoui [6] studied boundedness and compactness on L2 for h-Fourier integral operators, where the symbol a∈Γρμ. Moveover, Aitemrar and Senoussaoui [9] also discussed regularity on Bessel potential spaces. Furthermore, Aitemrar and Senoussaoui [7] considered the symbol for h-pseudodifferential operator, which belongs to the class L∞Sρm, namely,
(5)∂ξαa·,ξL∞ℝn≤Cαξm−ρα.By using this condition and the compact setSuppξax,ξ, they obtained the global Lp-boundedness for h-Fourier integral operators. The noncompact case for Suppξax,ξ, they also proved Lp-boundedness for m<nρ−1/2. Motivated by this noncompact case, for δ=1,m=nρ−1/2, we consider the L2-boundedness of h-pseudodifferential operators. In what follows, we mainly concentrate on the L2-boundedness for h-pseudodifferential operator Tah with the specific symbol a. Our main result could be stated as follows:Theorem 1.
LetTah be the h-pseudodifferential operators given by (3) with symbol ax,ξ∈L∞Sρnρ−1/2ω, where L∞Sρnρ−1/2ω is as in Definition 3. Then, for 0≤ρ≤1, there exists a constant C>0 such that
(6)TahuL2≤CuL2.Remark 2.
For the classical pseudodifferential operator, in [16, 17], they have constructed an example that the operator is not bounded in L2ℝn as a∈Sρ,1nρ−1/2. Moveover, here, we remark that the symbol L∞Sρnρ−1/2ω is different from the symbol a∈Γρμ in [6].


## 2. Definitions, Notations, and Preliminaries

Letℝn be an n-dimensional Euclidean space, x=x1,⋯,xn∈ℝn, ℕ=1,2,⋯, ℤ+=ℕ∪0, and ℤ+n=ℤ+n. For any multi-index α=α1,⋯,αn and β=β1,⋯,βn∈ℤ+n, we let
(7)α=∑j=1nαj,α+β=α1+β1,⋯,αn+βn,∂xα=∂α∂x1α1⋯∂xnαn,and ∇ξ=∂ξ1,⋯,∂ξn. Also, we also define ∣ξ∣ and ξ as follows:
(8)ξ=∑j=1nξj21/2,ξ=1+ξ21/2.Throughout this article, we denote byC a positive constant which is independent of the main parameters, but it may vary from line to line. We sometimes write A≲B as shorthand for A≤CB.We give the classL∞Sρmω defined below, which will play significant role in our investigations.Definition 3.
Letm be a real number. A function ax,ξ which is smooth in the frequency variable ξ and bounded measurable in the spatial variable x, belongs to the symbol class L∞Sρmω, if it satisfies, for all multi-indices α,
(9)∂ξαax,ξL∞ℝn≤Cαξm−ραωξ,where ωt is a nonnegative and decreasing function on 1,∞ and satisfies
(10)∫1∞ωt2tdt<∞.Remark 4.
Ifωt satisfies (10), then ∑j=0∞‍ω22j<∞.To prove the main theorem, we need the following lemma.Lemma 5.
LetTah be the h-pseudodifferential operators given by (3) with symbol ax,ξ∈L∞Sρnρ−1/2 and Suppξax,ξ⊂ξ:∣ξ∣≤R. Then, for 0≤ρ≤1, there exists a constant C>0 such that
(11)TahuL2≤CuL2.The proof method of lemma is similar to Corollary 3.2 in [18], the details being omitted.


## 3. Proof of Main Result

In this section, we shall prove the main result Theorem1.First we need a dyadic partition of unity, letA be the annulus A=ξ∈ℝn;1/2≤ξ≤2,(12)χ0ξ+∑j=1∞χjξ=1forallξ∈ℝn,where χ0ξ∈C0∞B0,2 and χjξ=χ2−jξ. When j≥1 with χξ∈C0∞A, we decompose the operator Tah as
(13)Tah=Tχ0D+∑j=1∞TχjD=T0D+∑j=1∞TjD.The first term in (13) is bounded on L2ℝn from Lemma 5. After a change of variables, we have
(14)TjD=12πhn∫ℝnei/hxξχjξax,ξu^ξdξ=2jρn2πhn∫ℝnei/h·2jρxξχj2jρξax,2jρξu^2jρξdξ=2jρn2πhn∫ℝnei/h·2jρxξχj2jρξax,2jρξ∫ℝne−i/h2jρξ·yuydydξ=2jρn2πhn∫ℝn×ℝnei/h·2jρxξ−y·ξχj2jρξax,2jρξuydξdy.The kernel of the operatorTjD is given by
(15)Tjx,y=2jρn2πhn∫ℝnei/h·2jρxξ−y·ξχj2jρξax,2jρξdξ.Let(16)ajx,ξ=χ2jρ−1ξax,2jρξ.Then(17)Aj=Suppξaj⊂ξ;2−12j1−ρ<ξ<2·2j1−ρ,and satisfying
(18)∂ξαajx,ξ≤Cα2jnρ−1/2.Now, we deal with the high frequency componentTj of Tφa. Let Sj=TjTj∗. Then
(19)Sjux=12πhn∫ℝn×ℝnei/hxξ−yξχj2ξax,ξay,ξ¯uydydξ=2jρn2πhn∫ℝn×ℝnei/h2jρxξ−yξχj22jρξax,2jρξay,2jρξ¯uydξdy.The kernel of the operatorSj=TjTj∗ reads
(20)Sjx,y=2jρn2πhn∫ℝnei/h2jρxξ−yξχj22jρξax,2jρξay,2jρξ¯dξ.We define(21)bjx,y,ξ=χj22jρξax,2jρξay,2jρξ¯.By this, we have(22)Suppbj⊂ξ:2j1−ρ2<∣ξ∣<2·2j1−ρ.Now, we claim that(23)∂ξαbjx,y,ξ≤Cα2jnρ−1ω22j.In fact, by using (18), we have
(24)∂ξαbjx,y,ξ=∂ξαχj22jρξax,2jρξay,2jρξ¯=∑α1+α2=α∂ξα1ax,2jρξay,2jρξ¯∂ξα2χ22−j1−ρξ≲∑α1+α2=α2jρα1∂ξα1a·a¯x,2jρξω22j·2−j1−ρα2∂ξα2χ2−j1−ρξ≲∑α1+α2=α2jρα12jρξnρ−1−ρα12−j1−ρα2ω22j≲∑α1+α2=α2jρ∣α1∣2jnρ−1−ρ∣α1∣2−j1−ρ∣α2∣ω22j=∑α1+α2=α2jnρ−1−j1−ρ∣α2∣ω22j=2jnρ−1∑α2‍2−j1−ρα2ω22j≲2jnρ−1ω22j.Next, we consider the following differential operators, forj∈ℕ,(25)Ljx,y,D=h∇ξi2jρx−y.So(26)LjNx,y,Dei/h2jρx−yξ=ei/h2jρx−yξ,(27)Lj∗x,y,D=−h∇ξi2jρx−y.From this and (25), it follows that
(28)Sjx,y=2jρn2πhn∫ℝnei/h2jρxξ−yξχj22jρξax,2jρξay,2jρξ¯dξ=2jρn2πhn∫ℝnLjNei/h2jρx−yξbjx,y.ξdξ=2jρn2πhn∫ℝnei/h2jρx−yξLj∗Nbjdξ.Moreover, by (27) and (24), we further obtain
(29)Lj∗Nbj≲12jρN·hNx−yN2jnρ−1ω22j=hN2jρx−yN2jnρ−1ω22j.Integration by parts yields(30)Sjx,y=2jρn2πhn∫ℝnei/h2jρx−yξLj∗Nbjdξ≲hN2jρn2jρ∣x−y∣N2jnρ−1ω22j2j1−ρn≲2jρn2jρ∣x−y∣Nω22j.Thus,(31)Sjx,y∑l=0N2jρx−yl≲2jρnω22j,which implies that
(32)Sjx,y≲2jρn1+2jρx−yNω22j,and hence
(33)supx∫ℝnSjx,ydy≲2jρnω22j∫ℝn11+2jρx−yNdy≲ω22j∫ℝn11+zNdz≲ω22j.By Young’s inequality, we obtain(34)SjuxL2≲ω22juxL2.Therefore, we have(35)Tj∗uL22=Tj∗u,Tj∗u=u,TjTj∗u≤uL2SjuL2≤Cω22juL22.Namely(36)TjuL2≤Cω2juL2.Next, we need the Littlewood-Paley decomposition. Letψ0:ℝn→ℝ be a smooth radial function which equals to one on the unit ball centric at the origin and supported on its concentric double. Set ψξ=ψ0ξ−ψ02ξ and ψkξ=ψ2−kξ. Then, for any ξ∈ℝn,
(37)ψ0ξ+∑k=1∞ψkξ=1,andsuppψkξ⊂ξ:2k−1≤ξ≤2k+1 for k≥1. Thus
(38)u^ξ=∑k=0∞u^ξψkξ.Furthermore, we have(39)Tahu=∑j=0∞Tju=∑k=0∞∑j=1∞∫ℝnei/hxξajx,ξu^kξdξ,where ∣j−k∣≤2; in order to simplify, let
(40)∑k=0∞∑j=1∞∫ℝnei/hxξajx,ξu^kξdξ=∑j=1∞Tjuj.And hence(41)TahuL2=∑j=0∞TjujL2≲∑j=0∞ω2jujL2≲∑j=0∞ω22j12∑j=0∞∥uj∥L2212≲uL2.We use Remark4 here. This finishes the proof of Theorem 1.



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*Source: 6690963-2021-02-20.xml*