# Toeplitz Operators withIMOs Symbols between Generalized Fock Spaces

Authors: Yiyuan Zhang; Guangfu Cao; Li He
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we study the mapping properties of Toeplitz operatorsTf associated with IMOs symbols f acting between two generalized Fock spaces Fφp, where 1<s≤∞. We characterize bounded or compact Toeplitz operators Tf from one generalized Fock space Fφp to another Fφq, respectively, in four cases.


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## Body

## 1. Introduction

Letℂn be the n-dimensional complex Euclidean space and ω0=ddc·2 be the Euclidean Kähler form on ℂn, where d denotes the usual exterior derivative and dc=−1/4∂¯−∂. In what follows, φ∈C2ℂn is assumed to be real-valued such that
(1)c1ω0≤ddcφ≤c2ω0,for two positive constants c1 and c2.Given1≤p<∞, the space Lφp consists of all measurable functions f on ℂn such that
(2)fp,φ=∫ℂnfze−φzpdVz1/p<∞,where dV is the usual Lebesgue measure on ℂn.LetHℂn be the class of all entire functions on ℂn. For 1≤p<∞; the generalized Fock space is defined by
(3)Fφp=Hℂn⋂Lφp,and
(4)Fφ∞=f∈Hℂn:∥f∥∞,φ=supz∈ℂn∣fz∣e−φz<∞.It is well known thatFφp is a Banach space under the norm ∥·∥p,φ when 1≤p≤∞. The generalized Fock spaces Fφp have been studied by many mathematics researchers (refer to [1–4] and the references therein). For a particular choice of φ, Fφp corresponds to some spaces that we are familiar with. For example, when φ=α/2z2 with α>0, it becomes the classical Fock space studied in [5]. When φz=1/2z2−mln∣z∣, it becomes the Fock-Sobolev space introduced in [6].It is clear thatFφ2 is a Hilbert space with inner product
(5)f,gFφ2=∫ℂnfzgz¯e−2φzdVz.Forz,w∈ℂn, the reproducing kernel of Fφ2 is given by Kzw=Kw,z and the projection P from Lφ2 to Fφ2 is given by
(6)Pfz=∫ℂnfwKz,we−2φwdVw.Forz∈ℂn, we denote kz=Kz/∥Kz∥2,φ by the normalized reproducing kernel for Fφ2. By Proposition 4.2 in [4], we see that kz⟶0 weakly in Fφp as z⟶∞ for 1≤p≤∞.Givenz∈ℂn and r>0, write Bz,r=w∈ℂn:w−z<r. For a locally Lebesgue integrable function f on ℂn (written as f∈Lloc1), the averaging function f^rz on ℂn is defined by
(7)f^rz=1VBz,r∫Bz,rfwdVw,and the mean oscillation of f at z is given by
(8)MOrfz=1VBz,r∫Bz,rfw−f^rzdVw.For1<s≤∞ and r>0, the space IMOrs, s-th integrable mean oscillation, is defined to be the class of all f∈Lloc1 such that
(9)fIMOrs=MOrfLs<∞,where Ls=Lsℂn,dV.Forr>0, denote BMOr by the space of functions on ℂn with bounded mean oscillation, consisting of those functions f∈Lloc1 such that
(10)fBMOr=supz∈ℂnMOrfz<∞.The spacesIMOrs and BMOr are both independent of r (see Remark 5 below and Lemma 3.1 of [2]); thus, we will respectively write them IMOs and BMO for simplicity. From the definition, it is easy to see that IMO∞=BMO. Hence, IMOs spaces are generalizations of BMO spaces.Givenf∈IMOs, the Toeplitz operator Tf on Fφp is defined by
(11)Tfgz=∫ℂnfwgwKz,we−2φwdVw,z∈ℂn.For anyf on ℂn with fkz2∈Lφ1 and z∈ℂn, the Berezin transform of f is given by
(12)f~z=∫ℂnfwkzw2e−2φwdVw.It is easy to check thatf~z=Tfkz,kz.For1<s≤∞ and r>0, let IArs be the space of all functions f∈Lloc1 with
(13)fIArs=∣f∣^rLs<∞.Lemma 2.3 of [1] shows that the space IArs is independent of r, and we write it IAs for simplicity. Moreover,
(14)f∈IAsifandonlyif∣f∣~∈Ls.Given a continuous functionf on ℂn, the oscillation of f at z is defined to be
(15)ωrfz=supfw−fz:w−z<r.For1<s≤∞ and r>0, let IOrs denote the space of continuous functions on ℂn such that
(16)fIOrs=ωrfLs<∞.It follows from Lemma 2.1 in [7] that the space IOrs is independent of the choice of r, we simply write IOs instead of IOrs.TheBMO space in the Bergman metric was first introduced by Zhu in [8] and further studied by Berger et al. in [9]. Over the past few decades, a great number of research work have been done on the boundedness and compactness of Toeplitz operators induced by BMO symbols on Bergman spaces or Fock spaces (see [10–14] and the references therein). The IMOs space was firstly introduced and studied by Hu et al. in [2]. Recently, Wang [12] studied the the boundedness and compactness of Toeplitz operators induced by IMOs symbols between two classical Fock spaces. Inspired by his work, we proceed to extend the results to the generalized Fock spaces. Moreover, the largest generalized Fock spaces Fφ∞ are taken into account and some interesting results are obtained.In this paper, we aim to answer when the Toeplitz operatorsTf with f∈IMOs between two generalized Fock spaces are bounded or compact, where 1<s≤∞. Specifically, we characterize the boundedness and compactness of Toeplitz operators Tf from Fφp to Fφq in four cases, i.e., 1<p≤q<∞, 1<q<p<∞, 1<p<∞ and q=∞, and p=∞ and 1<q<∞. As we see, the proofs of the main results in [12] rely heavily on the explicit formula of the reproducing kernel. Unfortunately, we do not know whether there is a similar explicit formula on the generalized Fock space setting. Therefore, some different techniques and methods are needed to show our main conclusions on the generalized Fock space.Throughout this paper, for each1<p<∞, we write p′ for its conjugate exponent, that is, 1/p+1/p′=1. We will use the same letter C to denote various constants which may change at different occurrences. We denote f≲g whenever there exists a positive number C, independent of the essential variables, such that f≤Cg, and f≃g if both f≲g and g≲f hold.


## 2. Preliminaries

We begin with some estimates for the reproducing kernelK·,· of Fφ2; results were proved in [4].Lemma 1.
The reproducing kernelK·,· for Fφ2 satisfies:
(1)
Forz,w∈ℂn, there exists some θ>0 such that(17)Kz,we−φze−φw≲e−θ∣z−w∣.(2)
Forz∈ℂn, there exists some r>0 such that(18)Kz,we−φze−φw≃1,forw∈Bz,r.(3)
Forz∈ℂn and 1≤p≤∞,(19)K·,zp,φ≃eφz≃Kz,z.The following lemma from Lemma 2.4 in [1] is called the atomic decomposition for Fφp.Lemma 2.
For1≤p≤∞, let ajj=1∞ be an r-lattice and λjj=1∞∈lp (the p-summable sequence space) and set
(20)fz=∑j=1∞λjkajz,z∈ℂn.
Then,f∈Fφp and there exists a constant C>0 such that
(21)fp,φ≤Cλjj=1∞lp.Lemma 3.
For anyz∈ℂn and w∈Bz,r, where r is taken as in (18), there exists some constant C>0 depending only on r such that
(22)1VBz,r≤Ckzw2e−2φw.Proof.
By estimate (18), for w∈Bz,r, we have
(23)kzw2e−2φw≃1.
SinceVBz,r≃r2n, we have
(24)1VBz,r≲max1r2n,1≃max1r2n,1kzw2e−2φw.
This completes the proof.☐☐Proposition 4.
Supposef∈Lloc1 and 1<s≤∞. Then, the following statements are equivalent:
(1)
f∈IMOrs for some (or any) r>0.(2)
f∈IMOs.(3)
f=f1+f2, where f1∈IOs and f2∈IAs.(4)
f acting on ℂn with fkz2∈Lφ1 satisfies(25)∫ℂnfw−f~zkzw2e−2φwdVw∈Ls.Proof.
The equivalences of1⇔2⇔3 follow from Theorem 2.2 of [7].
3⇒4. For f∈IAs, by the triangle inequality, we obtain
(26)∫ℂnfw−f~zkzw2e−2φwdVw≤∫ℂnfwkzw2e−2φwdVw+∫ℂnf~zkzw2e−2φwdVw≤2∣f∣~z.
Sincef∈IAs if and only if ∣f∣~∈Ls by (14), we obtain (25) whenever f∈IAs.
Meanwhile, forf∈IOs, by using Lemma 2.1, the changing of variables, Lemma 4.2 in [7], and Fubini’s theorem, we deduce that
(27)∫ℂnfw−f~zkzw2e−2φwdVw≲∫ℂnfw−f~ze−2θz−wdVw=∫ℂnfz−w−f~ze−2θwdVw=∫ℂnfz−w−∫ℂnfξkzξ2e−2φξdVξe−2θwdVw≤∫ℂn∫ℂnfz−w−fξkzξ2e−2φξdVξe−2θwdVw≲∫ℂn∫ℂnfz−w−fξe−2θz−ξdVξe−2θwdVw=∫ℂn∫ℂnfz−w−fz−ξe−2θξdVξe−2θwdVw≲∫ℂn∫ℂn1+w−ξ∫01ωfz−w+tw−ξdte−2θwdVwe−2θξdVξ=∫ℂn∫ℂn∫011+w−ξωfz−w+tw−ξdte−2θ∣w∣dVwe−2θ∣ξ∣dVξ.
LetdVt,w,ξ≔dte−2θ∣w∣dVwe−2θ∣ξ∣dVξ and apply Hölder’s inequality; then, the last integral above is less than or equal to C times
(28)∫ℂn×ℂn×0,11+∣w−ξ∣sωfsz−w+tw−ξdVt,w,ξ1/s.
For∣w−ξ∣≥1, the integral above is less than or equal to C times:
(29)∫ℂn×ℂn×0,1ws+ξsωfsz−w+tw−ξdVt,w,ξ1/s.
It follows by Fubini’s theorem that(30)∫ℂn∫ℂnfw−f~zkzw2e−2φwdVwsdVz≲∫ℂn∫ℂn×ℂn×0,1ws+ξsωfsz−w+tw−ξdVt,w,ξdVz=∫ℂn×ℂn×0,1ws+ξsdVt,w,ξ∫ℂnωfsz−w+tw−ξdVz≲∥f∥IOss.
For∣w−ξ∣<1, the integral in (28) is less than or equal to C times:
(31)∫ℂn×ℂn×0,1ωfsz−w+tw−ξdVt,w,ξ1/s.
Thus, we obtain(32)∫ℂn∫ℂnfw−f~zkzw2e−2φwdVwsdVz≲∫ℂn∫ℂn×ℂn×0,1ωfsz−w+tw−ξdVt,w,ξdVz=∫ℂn×ℂn×0,1dVt,w,ξ∫ℂnωfsz−w+tw−ξdVz≲∥f∥IOss.
Hence,3⇒4 holds.
4⇒1. Suppose f on ℂn with fkz2∈Lφ1 satisfies condition (25), we are to show f∈IMOs.
Lemma3 implies that
(33)MOrfz=1VBz,r∫Bz,rfw−f^rzdVw≤1VBz,r∫Bz,rfw−f~zdVw+f^rz−f~z≤2VBz,r∫Bz,rfw−f~zdVw≲∫ℂnfw−f~zkzw2e−2φwdVw,and this shows that MOrfz∈Ls; hence, f∈IMOrs, which gives 4⇒1. The proof ends here.☐☐Remark 5.
It follows from Proposition4 that the spaces IMOrs are independent of r. Although the equivalences of 1⇔2⇔3 hold for all 0<s≤∞, it is unknown whether they are equivalent to 4 in the case of 0<s≤1. We tried to prove it, but there were two difficulties. Firstly, the techniques of Hölder’s inequality or Jensen’s inequality are not available when 0<s≤1. Secondly, the function ωf in Proposition 4 is not subharmonic, which makes the method of lattice invalid in this case.Corollary 6.
Let1<s≤∞. If f∈IMOs, then ∣f∣~z−∣f~z∣∈Ls for all z∈ℂn.Proof.
Forz∈ℂn, by the triangle inequality, we get
(34)f~z−f~z=∫ℂnfw−f~zkzw2e−2φwdVw≤∫ℂnfw−f~zkzw2e−2φwdVw.
Taking into account Proposition4, we obtain
(35)∫ℂnfw−f~zkzw2e−2φwdVw∈Ls,thus, we have ∣f∣~z−∣f~z∣∈Ls for all z∈ℂn. This completes the proof.☐☐Lemma 7 (see [1], Lemma 3.2).
Let1≤p<∞. A subset E⊂Fφp is relatively compact if and only if for any ε>0, there exists some R>0 such that
(36)supf∈E∫∣z∣≥Rfze−φzpdVz<ε.Lemma 8 (see [4], Proposition 2.3).
Suppose1≤p<∞ and r>0. Then, for f∈Hℂn and z∈ℂn, we have
(37)fze−φzp≲1r2n∫Bz,rfwe−φwpdVw.Lemma 9.
Supposef∈IMOs with 1<s≤∞. Then, for any g∈Hℂn, there exists some constant C>0 such that
(38)∫ℂngze−φzpfzdVz≤C∫ℂngze−φzpf^rzdVz,
forr>0 and 1≤p<∞.Proof.
By Lemma8 and Fubini’s theorem, we have
(39)∫ℂngze−φzp∣fz∣dVz≲1r2n∫ℂn∫Bz,rgwe−φwpdVw∣fz∣dVz=∫ℂngwe−φwp1r2n∫Bw,r∣fz∣dVzdVw≲∫ℂngwe−φwp∣f∣^rwdVw,which completes the proof.☐☐Lemma 10.
Let1≤p≤q<∞. Suppose E is relatively compact in Fφp, then E is relatively compact in Fφq.Proof.
SupposeE is relatively compact in Fφp. Given any ε>0, by Lemma 7, there exists some R>0 such that
(40)supf∈E∫∣z∣≥Rfze−φzpdVz<ε.
Lemma8 gives
(41)fz≲∥f∥p,φeφz,forf∈E.
Thus,(42)∫z≥Rfze−φzqdVz=∫z≥Rfzpfzq−pe−qφzdVz≤∥f∥p,φq−p∫z≥Rfze−φzpdVz<∥f∥p,φq−pε,for f∈E. Note that by the proof of Lemma 2.7 in [1], the fact that E is relatively compact in Fφp implies that E is bounded in Fφp. It follows that there is a constant C>0 such that
(43)supf∈E∫∣z∣≥Rfze−φzqdVz<Cε.
An application of Lemma7 implies that E is relatively compact in Fφq, and the lemma is proved.☐☐


## 3. Toeplitz Operators withIMOs Symbols

In this section, we will derive necessary and sufficient conditions for which Toeplitz operatorsTf with IMOs symbols are bounded or compact from Fφp to Fφq, respectively, in four cases, where 1<s≤∞.Case 1.
1<p≤q<∞.
The following theorems are the main results in the case of1<p≤q<∞.Theorem 11.
Let1<p≤q<∞ and f∈IMO∞. Then, Tf:Fφp⟶Fφq is bounded if and only if f~∈L∞.Proof.
Assume thatf~∈L∞, then Tf is bounded on Fφp by Theorem 3.2 in [1]. That is, for any function g∈Fφp, we have Tfg∈Fφp with
(44)∥Tfg∥p,φ≲∥g∥p,φ.
Observe thatFφp⊂Fφq when 1<p≤q<∞. Thus,
(45)∥Tfg∥q,φ≲∥Tfg∥p,φ≲∥g∥p,φ.
This implies thatTf:Fφp⟶Fφq is bounded.
Conversely, ifTf:Fφp⟶Fφq is bounded for 1<p≤q<∞. Note that ∥kz∥p,φ≃1 by estimate (19). Applying Hölder’s inequality, we get
(46)∣f~z∣=Tfkz,kz≤∥Tfkz∥q,φ∥kz∥q′,φ≲∥Tf∥Fφp⟶Fφq∥kz∥p,φ≃∥Tf∥Fφp⟶Fφq.
It follows thatf~∈L∞.
This completes the proof.☐☐Furthermore, we equivalently characterize the compact Toeplitz operatorsTf with f∈IMO∞.Theorem 12.
Let1<p≤q<∞ and f∈IMO∞. Then, Tf:Fφp⟶Fφq is compact if and only if
(47)limz⟶∞supw∈Bz,rTfkz,kwFφ2=0,foranyr>0.Proof.
For1<p≤q<∞, since kz⟶0 weakly in Fφp as z⟶∞ and Tf:Fφp⟶Fφq is compact by assumption, we deduce that
(48)limz⟶∞Tfkzq,φ=0.
Thus, by Hölder’s inequality, we have(49)supw∈Bz,rTfkz,kwFφ2≤supw∈Bz,r∥Tfkz∥q,φ∥kw∥q′,φ≲∥Tfkz∥q,φ⟶0,for any r>0 as z⟶∞.
Conversely, suppose(50)limz⟶∞supw∈Bz,rTfkz,kwFφ2=0,
for anyr>0, then by Theorem 3.2 in [1], we see that Tf is compact on Fφp, namely, Tf maps any bounded set in Fφp to a relatively compact set in Fφp. Lemma 10 implies that a relatively compact set in Fφp is also relatively compact in Fφq for 1<p≤q<∞. Therefore, Tf maps any bounded set in Fφp to a relatively compact set in Fφq. This shows Tf is compact from Fφp to Fφq.
This completes the proof.☐☐Case 2.
1<q<p<∞.
Before we show the main result in the case of1<q<p<∞, we need to introduce some notations firstly. Recall that the Rademacher functions are defined by
(51)r0t=1,0≤t−t<12,−1,12≤t−t<1,rnt=r02nt,n≥1,where t denotes the largest integer less than or equal to t.
The Khinchine’s inequality [15] is given as follows.
For0<p<∞, there exist positive constants C1 and C2 depending only on p such that, for all natural numbers m and all complex numbers b1,b2,⋯,bm,
(52)C1∑j=1mbj2p/2≤∫01∑j=1mbjrjtpdt≤C2∑j=1mbj2p/2.
Given somer>0, a sequence ajj=1∞ in ℂn is called an r-lattice if the balls Baj,rj=1∞ cover ℂn and Baj,r/4j=1∞ are pairwise disjoint. For any δ>0, it is elementary to prove that there exists some positive integer N (depending only on r and δ) such that each z∈ℂn belongs to at most N balls of Baj,δj=1∞. Moreover, given r>0, it is easy to choose aj such that ajj=1∞ forms an r-lattice.Theorem 13.
Let1<q<p<∞ and set s=pq/p−q. Then, for f∈IMOs, the following statements are equivalent:
(1)
Tf:Fφp⟶Fφq is bounded.(2)
Tf:Fφp⟶Fφq is compact.(3)
f~∈Ls.Proof.
The implication2⇒1 is trivial, it suffices to prove the implications 1⇒3 and 3⇒2.
1⇒3. Given any r>0, let ajj=1∞ be an r-lattice. For z∈ℂn and 1<p<∞, let λjj=1∞∈lp, by Lemma 2, we see that the function
(53)gtz=∑j=1∞λjrjtkajz
belongs toFφp and ∥gt∥p,φ≲λjj=1∞lp. By assumption, Tf:Fφp⟶Fφq is bounded; it follows that
(54)Tfgtq,φ≤TfFφp⟶Fφqgtp,φ≲TfFφp⟶Fφqλjj=1∞lp.
Applying Fubini’s theorem, Khinchine’s inequality, and Lemma8, we obtain
(55)∫01Tfgtq,φqdt=∫01∫ℂn∑j=1∞λjrjtTfkajzqe−qφzdVzdt=∫ℂn∫01∑j=1∞λjrjtTfkajzqdte−qφzdVz≳∫ℂn∑j=1∞λj2Tfkajz2q/2e−qφzdVz≳∑i=1∞∫Bai,r∑j=1∞λj2Tfkajz2q/2e−qφzdVz≥∑i=1∞λiq∫Bai,rTfkaizqe−qφzdVz≳∑i=1∞λiqTfkaiaiqe−qφai.
By the reproducing property and estimate (19), we have
(56)Tfkaiaiqe−qφai=Tfkai,KaiFφ2qe−qφai=Tfkai,kaiFφ2qe−qφai∥Kai∥p,φq≃f~aiq.
This together with (54) and (55) gives
(57)∑i=1∞λiqf~aiq≲TfFφp⟶Fφqqλjj=1∞lpq=TfFφp⟶Fφqqλjqj=1∞lp/q.
Sincep/q>1, the classical duality lp/q∗≃lp/p−q now implies
(58)∑i=1∞f~ais≲TfFφp⟶Fφqs.
Choose a pointwi∈Bai,r¯ such that
(59)f~wi=supz∈Bai,rf~z.
Then, we obtain(60)∫ℂnf~zsdVz≤∑i=1∞∫Bai,rf~zsdVz≲∑i=1∞f~wis.
Note that the setwii is a finite union of δ-lattices and (58) holds for any r-lattice, this along with (58) implies that f~∈Ls.
3⇒2. Assume that f~∈Ls. Since s=pq/p−q>1, by Corollary 6, we have ∣f∣~∈Ls. Notice that
(61)0≤Ref+,Ref−,Imf+,Imf−≤f,we get Ref+~, Ref−~, Imf+~, and Imf−~ are all in Ls. By Theorem 3.6 in [1], Toeplitz operators induced by these nonnegative symbols Ref+, Ref−, Imf+, and Imf− are all compact from Fφp to Fφq. Consequently, Tf is compact from Fφp to Fφq, since
(62)f=Ref+iImf=Ref+−Ref−+iImf+−Imf−.
This completes the proof.☐☐Case 3.
1<p<∞,q=∞.
We now state the main result in the case of1<p<∞,q=∞.Theorem 14.
Let1<p<∞ and f∈IMO∞. Then, the following statements are equivalent:
(1)
Tf:Fφp⟶Fφ∞ is bounded.(2)
Tf:Fφp⟶Fφ∞ is compact.(3)
f~∈L∞.Proof.
The implication2⇒1 is trivial; we are to prove the implications 1⇒3 and 3⇒2.
1⇒3. Assume that Tf:Fφp⟶Fφ∞ is bounded. For any z∈ℂn, since ∥kz∥p,φ≃1 by the estimate (19), we have
(63)f~z=Tfkz,kzFφ2≤∫ℂnTfkzwkzwe−2φwdVw≤∥Tfkz∥∞,φ∫ℂnkzwe−φwdVw≲∥Tf∥Fφp⟶Fφ∞∥kz∥p,φ∥kz∥1,φ≃∥Tf∥Fφp⟶Fφ∞.
Therefore,f~∈L∞.
3⇒2. Suppose f~∈L∞, we have ∣f∣~∈L∞ by Corollary 6, equivalently, ∣f∣^r∈L∞ by (14). Assume that the sequence gj⟶0 weakly in Fφp as j⟶∞, we just need to show that
(64)limj⟶∞Tfgj∞,φ⟶0,since Fφp is reflexive for 1<p<∞.
For anyε>0, there exists an R>0 large enough such that
(65)∫∣w∣>Rkzwp′e−p′φwdVw1/p′<ε,since kz∈Fφp′.
In views of Proposition 4.1 in [4], our assumption on gjj implies that gjp,φ≲1 and gj⟶0 uniformly on compact subsets of ℂn as j⟶∞. This shows
(66)∫∣w∣≤Rgjwkzwe−2φwdVw<ε,for arbitrary ε>0 and fixed R above.
Combing the estimate (19), Lemma 9 (to the weight 2φ), Hölder’s inequality, (65), and (66), we get
(67)∣Tfgjz∣e−φz≤∫ℂnfwgjwKz,we−2φwdVwe−φz≃∫ℂnfwgjwkzwe−2φwdVw≲∫ℂngjwkzwe−2φw∣f∣^rwdVw≲f^rL∞∫ℂngjwkzwe−2φwdVw≲∫∣w∣≤R∣gjwkzw∣e−2φwdVw+∫∣w∣>Rgjwkzwe−2φwdVw≲ε+∥gj∥p,φ∫∣w∣>Rkzwp′e−p′φwdVw1/p′≲ε
asj⟶∞, which yields (64). Hence, Tf:Fφp⟶Fφ∞ is compact.
This finishes the proof.☐☐Remark 15.
It is interesting and surprising that the boundedness and compactness ofTf from Fφp to Fφ∞ are equivalent when 1<p<∞, which is quite different from that in the case of 1<p≤q<∞.We will end this section by proving the last case.Case 4.
p=∞,1<q<∞.
The following theorem is the main result in the case ofp=∞,1<q<∞.Theorem 16.
Let1<q<∞ and f∈IMOq. Then, the following statements are equivalent:
(1)
Tf:Fφ∞⟶Fφq is bounded.(2)
Tf:Fφ∞⟶Fφq is compact.(3)
f~∈Lq.Proof.
It is trivial that2⇒1, it remains to prove the implications 1⇒3 and 3⇒2. The proof is similar to that of Theorem 13, but we include it here for the sake of completeness.
1⇒3. Let λjj=1∞ be any bounded sequence on ℂn and ajj=1∞ be an r-lattice. For z∈ℂn, set
(68)gtz=∑j=1∞rjtλjkajz.
By Lemma2, we have gt∈Fφ∞ with
(69)gt∞,φ≲λjj=1∞l∞.
SinceTf:Fφ∞⟶Fφq is bounded by hypothesis, we have Tfgt∈Fφq. Applying Khinchine’s inequality and Fubini’s theorem, we deduce that
(70)∫ℂn∑j=1∞λjTfkajz2q/2e−qφzdVz≲∫ℂn∫01∑j=1∞rjtλjTfkajzqdte−qφzdVz=∫01Tf∑j=1∞rjtλjkajq,φqdt≲∫01∥Tf∥Fφ∞⟶Fφqqgt∞,φqdt≲∥Tf∥Fφ∞⟶Fφqq∥λjj=1∞∥l∞q.
By Lemma8 and the estimate (56), we get
(71)∫ℂn∑j=1∞λjTfkajz2q/2e−qφzdVz∑i=1∞∫Bai,r∑j=1∞λjTfkajz2q/2e−qφzdVz≥∑i=1∞∫Bai,rλiTfkaizqe−qφzdVz∑i=1∞λiqTfkaiaiqe−qφai≃∑i=1∞λiqf~aiq.
This along with the estimate (70) yields
(72)∑i=1∞λiqf~aiq≲∥Tf∥Fφ∞⟶Fφqq∥λii=1∞∥l∞q.
Takebi=λiq for each i. Then, bii=1∞∈l∞ and
(73)∑i=1∞bif~aiq≲∥Tf∥Fφ∞⟶Fφqq∥bii=1∞∥l∞.
It follows that(74)∑i=1∞f~aiq≲∥Tf∥Fφ∞⟶Fφqq.
Choose one pointwi∈Bai,r¯ such that
(75)f~wi=supz∈Bai,rf~z.
Then, we obtain(76)∫ℂnf~zqdVz≤∑i=1∞∫Bai,rf~zqdVz≲∑i=1∞f~wiq.
Note that the setwii is finite union of some δ-lattices and (74) holds for any r-lattice, this along with (74) implies that f~∈Lq.
3⇒2. Assume that f~∈Lq, by Corollary 6, we have ∣f∣~∈Lq. Since
(77)0≤Ref+,Ref−,Imf+,Imf−≤f,we see that Ref+~, Ref−~, Imf+~, and Imf−~ are all in Lq. By Theorem 2.5 in [3], Toeplitz operators induced by these nonnegative symbols Ref+, Ref−, Imf+, and Imf− are all compact from Fφ∞ to Fφq. Notice that
(78)f=Ref+iImf=Ref+−Ref−+iImf+−Imf−,we conclude that Tf is compact from Fφ∞ to Fφq.
This finishes the proof.☐☐



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*Source: 8044854-2021-07-31.xml*