# Duality of Large Fock Spaces in Several Complex Variables and Compact Localization Operators

Authors: Youqi Liu; Xiaofeng Wang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, dual spaces of large Fock spacesFϕp with 0<p<∞ are characterized. Also, algebraic properties and equivalent conditions for compactness of weakly localized operators are obtained on Fϕp0<p<∞.


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## Body

## 1. Introduction

Letℂn be the n-dimensional complex Euclidean space. Let dv denote the Lebesgue volume measure on ℂn. For any two points z=z1,⋯,zn and w=w1,⋯,wn in ℂn, we write z,w=z1w1¯+⋯+znwn¯ and z=z12+⋯+zn2.For eachz∈ℂn and r >0,(1)Bz,r=w∈ℂn:w−z<rdenotes the Euclidean ball centered at z with radius r.Let ∆ denote the Laplacian operator. Supposeϕ:ℂn⟶ℝ is a C2 plurisubharmonic function (see [1]). We say that ϕ belongs to the weight class W if ϕ satisfies the following statements:(A)
There existsc>0 such that for z∈ℂn(2)infz∈ℂnsupw∈Bz,cΔϕw>0;(B)
For anyz∈ℂn and r>0, ∆ϕ satisfies the reverse-Hölder inequality(3)∆ϕL∞Bz,r≤Cr−2n∫Bz,r∆ϕdvfor some 0<C<∞;(C)
The eigenvalues ofHϕ are comparable, i.e., for every z,u∈ℂn, there exists δ0>0 such that(4)Hϕzu,u≥δ0∆ϕzu2, where(5)Hϕ=∂2ϕ∂zj∂zk¯j,k.Suppose0<p<∞, ϕ∈W. The space Lϕp consists of all Lebesgue measurable functions f on ℂn for which(6)fp,ϕ=∫ℂnfzpe−pϕzdvz1p<∞.Lϕ∞is the set of all Lebesgue measurable functions f on ℂn with(7)f∞,ϕ=supz∈ℂnfze−ϕz<∞.LetHℂn be the family of all entire functions on ℂn. The large Fock space is defined as(8)Fϕp=Lϕp∩Hℂn.Fϕpis a Banach space under ·p,ϕ if p≥1, and Fϕp is a quasi-Banach space with distance df,g=f−gp,ϕp if 0<p<1. Assume that ϕz=z2/2, then Fϕp is the classical Fock space which has been studied in [2–4] for example. Also, the weight function ϕ on ℂn with the restriction that ddcϕ⋍ddcz2 belongs to W, where d=∂¯+∂ and dc=−1/4∂¯−∂. See [5, 6] for more details.Particularly,Fϕ2 is a reproducing kernel Hilbert space. That is, for any f∈Fϕ2, there exists a unique function Kz∈Fϕ2 so that fz=f,KzFϕ2, where (9)f,gFϕ2=∫ℂnfzgz¯e−2ϕzdvz,f,g∈Fϕ2.We say that the functionKz· is the reproducing kernel of Fϕ2. It is well known that the orthogonal projection P:Lϕ2⟶Fϕ2 is given by(10)Pfz=∫ℂnfwKz,we−2ϕwdvw,f∈Lϕ2.As we know if1≤p<∞ and q is the conjugate exponents of p, then the dual space of Lϕp can be identified with Lϕq by the integral pairing ,Fϕ2 defined by (9). In general, for 1≤p<∞, no less important than the Hahn-Banach theorem is the Bergman projection to explore the dual spaces of Fϕp. However, there are some differences for these quasi-Banach spaces Fϕp0<p<1. To do this, we will mainly apply Hörmander’s solution of the ∂¯ equation and the Lebesgue dominated convergence theorem to consider the duality of Fϕp0<p<1.The “weakly localized” operators were introduced for the first time in [7], and the authors studied the compactness of these operators on the Bergman space Ap and weighted the Bargmann-Fock space Fφp with 1<p<∞. In fact, this kind of operators is interesting since these weakly localized operators contain Toeplitz operators which are induced by bounded symbols. Indeed, Toeplitz operators are a kind of significant operators, and these Toeplitz operators induced by diverse functions enjoy abundant properties, see more in [8, 9]. As a further research, Hu, Lv, and Wick characterized the compactness of these weakly localized operators on generalized Fock spaces Fφp with 0<p≤1, see [5]. Besides, in generalized Bergman space setting [10], there are two questions: whether Toeplitz operators induced by bounded symbols are weakly localized operators? Would these weakly localized operators form an algebra?This paper is devoted to consider the compactness of these weakly localized operators on large Fock spacesFϕp with 0<p<∞. To ensure the validity of these fascinating operators, we show these localization operators contain Toeplitz operators induced by bounded symbols on Fϕp, see Theorem 16. Meanwhile, we also give affirmative answer about the second question on our Fock spaces, see Theorem 15.Notice that although in the one-dimensional case, the diverse weight function gives another Bergman metric, and the resulting Bergman disk will be changed. Furthermore, there is no inclusion relation betweenFϕp and Fϕq if p≠q. The above properties are much different from [5], so we have to apply more techniques to discuss the compactness of weakly localized operators in case 0<p≤1. For case 1<p<∞, the ideas to study compact weakly localized operators in [7] are not entirely applicable to the situation we are discussing. Hence, we finally combine the skills in [5, 7] to consider the compactness of these operators on Fϕp1<p<∞. Eventually, when p>1, we bring new consequences even if Fϕp is the generalized Fock space in [5].This paper is organized as follows. In Section2, we give some lemmas which will play key roles in our proofs. In Section 3, we show some properties of projection and dual spaces of large Fock spaces Fϕp when 0<p<∞. In Section 4, we conclude the algebraic properties and boundedness of localization operators. Finally, in Section 5, we consider the compactness of weakly localized operators on our Fock spaces.Throughout this paper, we writeA≲B for two quantities A and B if there is a constant C>0 such that A≤CB. Furthermore, A⋍B means that both A≲B and B≲A are satisfied.


## 2. Preliminaries and Basic Estimates

In this section, we will give some useful estimates for our proofs. Forz∈ℂn, set(11)ρϕz=supr>0:supw∈Bz,rΔϕw≤r−2.In the following, we writeρz instead of ρϕz for short.By [11] (see also [12]), we have the following consequences.Lemma 1.
Letϕ be as defined in (2). Then, the function ρ satisfies the following properties:(A)
There existsM>0 such that(12)supz∈ℂnρz≤M.(B)
The functionρ is Lipschitz, that is(13)ρz−ρw≤z−w,z,w∈ℂn.(C)
Forr∈0,1andw∈B(z,rρz), there holds(14)1−rρz≤ρw≤1+rρz.(D)
There exista,b>0 such that(15)z−a≲ρz≲zb,forz>1.Letr>0, we writeBrz=Bz,rρz and Bz=B1z. In fact, it is easily obtained from estimate (14) that there is some constant cr such that cr−1ρz≤ρw≤crρz, where cr=1−r−1 for any r∈0,1. That is for every r∈0,1, we have ρw⋍ρz whenever w∈Brz. Besides, (14) and the triangle inequality give m1 and m2 so that(16)Bz⊂Bm1wandBw⊂Bm2z,wheneverw∈Bz.Givenr>0, there is a sequence akk=1∞ in ℂn such that Brakk covers ℂn, and the balls Br/5akk are pairwise disjoint. We say the sequence akk is an r-lattice. For the r-lattice akk and m>0, there exists some integer N such that any z in ℂn belongs to at most N balls of Bmrakk. That is, for every z∈ℂn,(17)∑k=1∞χBmrakz≤N.Now, we are going to state the properties of the reproducing kernelKz. Let ϕ∈W, and it follows from [11–13] that(A)
Forz,w∈ℂn, there are constants ϵ,α>0 such that(18)Kzw≲eϕz+ϕwρznρwne−ϵw−zρzα.(B)
Forz∈ℂn, there exists β∈0,1 such that(19)Kzw⋍eϕz+ϕwρz2n,w∈Bβz.(C)
For0<p≤∞, there holds(20)Kzp,ϕ⋍eϕzρz2n1/p−1,z∈ℂn.With the help of Lemmas1 and 2 in [12], we get the following lemma.Lemma 2.
Givenp,α>0 and k∈ℝ, there exists C>0 such that for z∈ℂn(21)∫ℂnρwke−pw−zρzαdvw≤Cρz2n+k.Forr>0 and z∈ℂn, we write Brzc=ℂn∖Brz. Let dσ=ρ−2ndv. It is directly from ([14], Lemma 2.7) that we have the next estimate.Lemma 3.
For anyα>0, p>0, k≥0, r≥1, and z∈ℂn, there is a constant Cα,p,kr>0 such that(22)∫Brzcw−zke−pw−zρzαdσw≤Cα,p,krρzk,
andCα,p,kr⟶0 whenever r⟶∞.We will writekp,zw=Kzw/Kzp,ϕ for the normalized reproducing kernel at z∈ℂn, where 0<p<∞ and w∈ℂn.Lemma 4.
Let0<p≤2. Then, for every z∈ℂn, we have(23)∫Brzckp,zwe−ϕwpdvw→0asr⟶∞.Proof.
By joining (18) and (20), we have(24)∫Brzckp,zwe−ϕwpdvw≃∫BrzcKzweϕz+ϕwpρz2np−2ndvw≲ρznp−2n∫Brzcρw2n−pne−pw−zρzαdσw≲ρznp−2n∫Brzce−pw−zρzαdσw.Here, the last step is from the estimate (12). Thus, the assertion follows from Lemma 3 with k=0 for any fixed z∈ℂn.The next lemma is immediately from ([12], Lemma 4) (see also ([11], Lemma 2) for any r>0.Lemma 5.
For 0<p<∞, there is a constant C>0 such that for each r>0,f∈Hℂn and z∈ℂn, we have(25)fze−ϕz≤crρz2n/p∫Brzfwe−ϕwpdvw1/p.Forr>0 and some domain Ω⊂ℂn, write Ωr+=⋃z∈ΩBrz. Let d·,· be the Euclidean distance, and we have the following lemma.Lemma 6.
For0<p≤1, 0<r<1, and α∈ℝ, there is some constant C (depending on p, n, and r) such that for any domain Ω⊂ℂn and f∈Hℂn,(26)∫Ω∣fwe−ϕwρwα∣dvwp≤C∫Ωr+fwe−ϕwρwαpρw2p−2ndvw.Proof.
Consider ther-lattice L=z1,z2,⋯,zj,⋯ in ℂn. For 0<r<1 and z∈ℂn, we get ρw≃ρz whenever w∈Brz. By letting w∈Brzj, it follows from (16) that(27)Brzj⊂Bmrw⊂Bm2rzj,
wherem=mr>1. Also notice that a+bp≤ap+bp for positive a,b and 0<p≤1. Let r be sufficiently small so that m2r<1. Thus, the above inequality, (17) and (25) show(28)∫Ωfwe−ϕwρwαdvwp≤∑zj∈L∫Ω∩Brzj∣fwe−ϕwρwα∣dvwp≲C∑zj∈L,dzj,Ω<rρzjsup∣w−zj∣≤rρzjfwe−ϕwpρw2n+αp≲C∑zj∈L,dzj,Ω<rρzj∫Bm2rzjfze−ϕzpρzj2n+αp−2ndvz≃C∑zj∈L,dzj,Ω<rρzj∫Bm2rzjfze−ϕzpρz2n+αp−2ndvz≤C∫Ωr+∑zj∈L,dzj,Ω<rρzjχBm2rzjwfwe−ϕwρwαpρw2p−2ndvw≤CN∫Ωr+fwe−ϕwρwαpρw2p−2ndvw,which completes the proof.


## 3. Bergman Projection and Duality

The paper [12] points out that the Bergman projection P is bounded on Fϕp for 0<p<∞. And there is no answer to whether Pf=f on Fϕp. In what follows, we use the classical Ho¨rmander theorem to prove that the projection P is an identity operator on Fϕp0<p<∞.Theorem 7 ([15], Theorem 4.2.6).
LetX be a pseudo-convex open set in ℂn, φ  a plurisubharmonic function inX, and a>0. Ifψis inL0,12 locally in X and ∂¯ψ=0, then the equation ∂¯u=ψ has a solution u∈Lloc2X such that(29)a∫Xuz2e−φz1+z2−advz≤∫Xψz2e−φz1+z22−advz.
For1<p<∞, we let q be the conjugate exponent of p such that 1/p+1/q=1.Theorem 8.
If0<p<∞ and f∈Fϕp, then  Pf=f.Proof.
Suppose thath0z∈C∞ℂn satisfying h0z=1 if ∣z∣≤RR>1, 0<h0z<1 if R<∣z∣<R+1, h0z=0 if ∣z∣≥R+1 and(30)∂¯h0z2≲h0z.SetΩj=z:∣z∣≤jR where j=1,2,⋯. It follows that for any z∈Ωj,(31)h0,jz≔h0zj=1.Because of (15), there are a,b>0 so that z−a≲ρz≲zb whenever ∣z∣>1. Indeed, by choosing r>0 sufficiently small, we obtain Ωjcr+⊂w:∣w∣>jR−1 when j is large enough.If0<p≤1, then by Lemma 6, we have(32)Pfz−Pfh0,jzp≤∫w>jRfwKz,we−2ϕwdvwp≲∫Ωjcr+fwKz,wpe−2pϕwρw2p−2ndvw.This together with (18), Lemma 2 and Fubini’s theorem give(33)Pf−Pfh0,jp,ϕp≲∫ℂn∫Ωjcr+fwKz,wpe−2pϕwρw2p−2ne−pϕzdvwdvz=∫Ωjcr+fwpe−2pϕwρw2p−2n∫ℂnKz,wpe−pϕzdvzdvw≲∫Ωjcr+fwpe−pϕwdvw≤∫w>jR−1fwpe−pϕwdvw.We now let1<p<∞. Notice that estimate (18) and Lemma 2 indicate(34)∫ℂn∣Kz,w∣e−ϕz−ϕwdvw≲∫ℂnρz−nρw−ne−ϵ∣w−z∣ρzαdvw<∞.So,Ho¨lder’s inequality and Fubini’s theorem show(35)Pf−Pfh0,jp,ϕp≲∫ℂn∫Ωjcfwpe−pϕw∣Kz,w∣e−ϕz−ϕwdvw×∫Ωjc∣Kz,w∣e−ϕz−ϕwdvwpqdvz≲∫Ωjcfwpe−pϕw∫ℂn∣Kz,w∣e−ϕz−ϕwdvzdvw≲∫Ωjcfwpe−pϕwdvw≤∫∣w∣>jR−1fwpe−pϕwdvw.And then, for0<p<∞, we get(36)limj⟶∞Pf−Pfh0,jp,ϕp≤Climj⟶∞∫∣w∣>jR−1fwpe−pϕwdvw=0.This combined with (25) means that(37)Pfz−Pfh0,jz≤Ceϕzρz−2np∥Pf−Pfh0,j∥p,ϕ⟶0,asj⟶∞.On the other hand, applying Theorem 7 witha=2 to the solution of ∂¯u=ψ in Lϕ2, we have(38)∫Xuw2e−2ϕw1+w2−2dvw≤∫Xψw2e−2ϕwdvw.Hence, forz∈Ωj and let j be sufficiently large, it follows immediately from esitmates (25), (30), and Lemma 2 that(39)fz−Pfh0,jz2e−2ϕzρz4np=fh0,jz−Pfh0,jz2e−2ϕzρz4np≲ρz2n2p−1∫Brzfh0,jw−Pfh0,jw21+w2−2e−2ϕwe−∣z−w∣ρzαdvw≲ρz2n2p−1∫ℂnfh0,jw−Pfh0,jw21+w2−2e−2ϕwe−∣z−w∣ρzαdvw≲ρz2n2p−1∫ℂn1jfw∂¯h0wj2e−2ϕwe−∣z−w∣ρzαdvw≲1j2ρz2n2p−1∥f∥p,ϕ2∫jR<∣w∣<jR+1ρw−4npe−∣z−w∣ρzαdvw≲1j2∥f∥p,ϕ2.By combining the above estimate and (37), we finally obtain(40)∣Pfz−fz∣≤∣Pfz−Pfh0,jz∣+∣Pfh0,jz−fz∣⟶0,asj⟶∞. This ends the proof.We now proceed to identify the dual space ofFϕp when 0<p<∞. Arguing as in [16], we let(41)Fs,ϕ∞=f∈Hℂn:∥f∥∞,s,ϕ=supz∈ℂn∣fz∣ρzse−ϕz<∞.Theorem 9.
Suppose0<p≤1. Then, Fϕp∗=F2n−2n/p,ϕ∞.Proof.
For anyf∈Fϕp, consider Lg·=·,g where g∈F2n−2n/p,ϕ∞. Then, (25) says(42)Lgf≤∫ℂn∣fzgz∣e−2ϕzdvz≤supz∈ℂn∣fz∣e−ϕzρz2np1−p∫ℂn∣gz∣ρz2n−2npe−ϕzfze−ϕzpdvz≤C∥g∥∞,2n−2np,ϕ∥f∥p,ϕ.The above inqueality shows thatLg is a bounded linear functional on Fϕp and ∥Lg∥≤C∥g∥∞,2n−2n/p,ϕ.Forw∈ℂn, define gw=LK·,w¯ where L is a bounded linear functional on Fϕp. Pick an r>0 such that w+Δw∈Brw. For some m>0 and every z∈ℂn, using Cauchy’s estimates, we have(43)Kw+Δw,z−Kw,zΔwp≤sup0≤t≤1∂K∂ww+tΔw,zp≤1mrρwpsupξ−w=mrρwKξ,zp≲1r2n+pρw2n+p∫Bm2rwKu,zpdvu.We note that for any fixedw∈ℂn, the function ∫Bm2rwKu,·pdvu1/p is in Lϕp. Fix w and z, and we get(44)limΔw⟶0Kw+Δw,z−Kw,zΔw=∂K∂ww,z.Thus Lebesgue dominated convergence theorem indicates(45)limΔw⟶0Kw+Δw,·−Kw,·Δw−∂K∂ww,·p,ϕ=0.Hence, for anyL∈Fϕp∗, we obtain gw∈Hℂn since(46)g′w=limΔw⟶0LK·,w+Δw¯−LK·,w¯Δw=L∂K∂ww,·¯¯,and∣gw∣≤LK·,wp,ϕ≤CLeϕwρw2n1/p−1. The result is(47)g∈F2n−2np,ϕ∞andg∞,2n−2np,ϕ≤CL.To complete the proof, it only remains to show that(48)Lf=∫ℂnfwgw¯e−2ϕwdvw.Letann be an r-lattice. For 0<R<∞, we consider(49)Sr,Rfz=∑nKz,an∫Bran\⋃j<nBraj⋂B0,Rfwe−2ϕwdvw.SinceB0,R¯ is compact, there exists k>0 so that ⋃n=1kBran⊃B0,R. Moreover, we see that Sr,Rf∈Hℂn because it is actually a finite sum of analytic functions. And there is R′>R such that ⋃n=1kBran⊂B0,R′. It follows from (43) that(50)Sr,Rfz−PfχB0,Rz=∣∑n∫Bran\⋃j<nBraj⋂B0,RKz,an−Kz,wfwe−2ϕwdvw∣≤supw∈B0,R∣fwe−2ϕw∣∑n∫Bran\⋃j<nBraj⋂B0,RKz,an−Kz,wdvw≲supw∈B0,Rfwe−2ϕw∑Bran⋂B0,Rsup∣ξ∣<2R′Kξ,zrρan2n≲CRr2nsup∣ξ∣<2R′∣Kξ,z∣goes to 0 by lettingr⟶0, where χB0,R denotes the characteristic function for the ball B0,R. This means that(51)limr⟶0Sr,Rf=PfχB0,R.It is clear that, for each fixedz∈ℂn, sup∣an∣≤2R′∣Kan,z∣ is in Lϕp. Hence, by the following estimate,(52)Sr,Rfz≤supw∈B0,Rfwe−2ϕw∑Bran\⋃j<nBraj⋂B0,R∫BranKan,zdvw≲CRsupan≤2R′Kan,z,and the Lebesgue dominated convergence theorem we deduce(53)limr⟶0∥Sr,Rf−PfχB0,R∥p,ϕ=0.Furthermore, we claim that(54)∣LSr,Rf·−∫w≤RfwLK·,we−2ϕwdvw∣≤supw∈Bran⋂B0,R∣LK·,an−K·,w∣∫w≤R∣fw∣e−2ϕwdvw≤CRLsupw∈Bran⋂B0,RK·,an−K·,wp,ϕ⟶0,asr⟶0. Here, the last assertion follows from fact that K·,an−K·,wp,ϕ⟶0 whenever r⟶0. To see this, by letting r⟶0 and w∈Bran, we then get Kw⟶Kan. Indeed, (18) gives us a dominating function, and it is from Lemma 2 that the function is in Lϕp since eϕw+ϕz/ρwnρzne−ϵz−w/ρwαp,ϕ≲eϕwρw2n/p−2n for any fixed w. Then, the desired assertion holds by Lebesgue dominated convergence theorem again. So, we have(55)limr⟶0LSr,Rf=∫∣w∣≤RfwLK·,we−2ϕwdvw.Therefore, we have by Theorem8, (51) and (55) that(56)Lf=limR⟶∞LPfχB0,R=limR⟶∞limr⟶0LSr,Rf=limR⟶∞∫∣w∣≤RfwLK·,we−2ϕwdvw=∫ℂnfwLK·,we−2ϕwdvw.This finishes the proof.Theorem 10.
Suppose1<p<∞. Then, Fϕp∗=Fϕq under the pairing(57)f,g=∫ℂnfzgz¯e−2ϕzdvz.Proof.
Ifg∈Fϕq, define(58)Lg·=·,g.For anyf∈Fϕp, Ho¨lder’s inequality gives(59)∣Lgf∣≤∫ℂn∣fzgz¯∣e−2ϕzdvz≤∥g∥q,ϕ∥f∥p,ϕ.This means thatLg is a bounded linear functional on Fϕp and ∥Lg∥≤∥g∥q,ϕ.On the other hand, letL:Fϕp⟶ℂ be a bounded linear functional. The Hahn-Banach extension theorem implies that L can be extended to a bounded linear functional L~ on Lϕp. It follows from the duality theory of Lϕp that there is a function G∈Lϕq such that ∥G∥q,ϕ≤∥L~∥=∥Lg∥ and(60)L~f=f,G,f∈Lϕp.Setg=PG, then ∥g∥q,ϕ≤∥P∥∥G∥q,ϕ since P is bounded. Also, note that Theorem 8 shows Pf=f for f∈Fϕp. So, (60) indicates(61)L~f=f,G=Pf,G=f,PG=f,g,f∈Fϕp.This completes the proof.Corollary 11.
Suppose0<p<∞. Then, the linear span E of all reproducing kernel functions Kz· is dense in Fϕp.Proof.
Let0<p≤1. It is immediately from Theorem 8 and the proof of Theorem 9, for any f∈Fϕp, that(62)limR⟶∞limr⟶0Sr,Rf−fp,ϕ≤limR⟶∞limr⟶0Sr,Rf−PfχB0,Rp,ϕ+PfχB0,R−fp,ϕ=0.Next, we assume thatp>1. By Theorem 10 and the Hahn-Banach theorem, it suffices to show that for any g∈E, we have f=0 if f∈Fϕq satisfies f,g=0. This follows from the fact that fz=Pfz=f,Kz=0 for every z∈ℂn.


## 4. Localization Operators

In this section, we will explore some properties of weakly localized operators on our Fock spaces. In particular, we will show the algebraic properties of these localization operators.Before stating weakly localized operators, we consider firstly the following proposition.Proposition 12.
Suppose0<p≤1. Then,(63)∥K·,w∥2n−2np,ϕ≃eϕwρw−2np.Proof.
It is from (11) that there exists some r0>0 such that ρu>r0 for each u∈ℂn. Fix w∈ℂn, and we have(64)ρw2np−nρun−2npe−ϵ∣u−w∣ρwα<r0n−2npρw2np−ne−ϵ∣u−w∣ρwα.For everyw∈ℂn, let r>0 be sufficiently large and let ∣u−w∣≥r, and it follows that estimate (18) together with (64) gives (65)∣Ku,w∣e−ϕuρu2n−2np<eϕwρw−2npr0n−2npρw2np−ne−ϵrρwαis a dimensionless. For fixedw∈ℂn, we get r≃βρw, where β∈0,1. Hence, Theorem 9 together with (19) shows that(66)Ku,w2n−2np,ϕ=supz∈ℂn∣Ku,w∣e−ϕuρu2n−2np≃supu∈Bw,r∣Ku,w∣e−ϕuρu2n−2np≃eϕwρw−2np,which is the desired estimate.Now, with the above preparations, we are ready for the definition of weakly localized operators.Definition 13.
Let0<p≤1. A linear operator T  on  Fϕp is called weakly localized for Fϕp if(67)supz∈ℂn∫ℂnTkp,z,k2n−2np,wFϕ2pρw−2ndvw<∞,supz∈ℂn∫ℂnT∗k2n−2np,z,kp,wFϕ2pρw−2ndvw<∞,(68)limr⟶∞supz∈ℂn∫BrzcTkp,z,k2n−2np,wFϕ2pρw−2ndvw=0,limr⟶∞supz∈ℂn∫BrzcT∗k2n−2np,z,kp,wFϕ2pρw−2ndvw=0,where(69)k2n−2np,⋅u≃Ku,⋅eϕ⋅ρ⋅−2n/p,u∈ℂn.Recall that, for1<p<∞, q is the conjugate exponent of p so that 1/p+1/q=1.Definition 14.
Suppose1<p<∞. A linear operator T on Fϕp is called weakly localized for Fϕp if(70)supz∈ℂn∫ℂn∣Tkp,z,kq,wFϕ2∣ρw−2ndvw<∞,supz∈ℂn∫ℂn∣T∗kq,z,kp,wFϕ2∣ρw−2ndvw<∞,(71)limr⟶∞supz∈ℂn∫Brzc∣Tkp,z,kq,wFϕ2∣ρw−2ndvw=0,limr⟶∞supz∈ℂn∫Brzc∣T∗kq,z,kp,wFϕ2∣ρw−2ndvw=0.Next, we are going to answer the questions raised at the beginning of the paper in our Fock spaces. In fact, each set of these weakly localized operators onFϕp is an algebra.Theorem 15.
Suppose0<p<∞. Then, weakly localized operators on Fϕp form an algebra.Proof.
Suppose operatorsT and S are weakly localized. So, it remains to show that TS is a weakly localized operator because the linear combination of two weakly localized operators is also a weakly localized operator.We let0<p≤1. It follows from (68) that there is some r>0 such that(72)∫BrazcTkp,z,k2n−2np,wFϕ2pρw−2ndvw<ε,wherea=1+r1/2+1 and any ε>0 (when 1<p<∞, estimate (71) gives an analogous representation). For z,x∈ℂn, if x∈Br/az then ρx≤1+r/aρz and by the triangle inequality, we have Br/ax⊂Brz. That is, Brzc⊂Br/axc whenever x∈Br/az.By joining Lemma6 and Fubini’s theorem, we get(73)∫BrzcTSkp,z,k2n−2np,wFϕ2pρw−2ndvw=∫BrzcSkp,z,T∗k2n−2np,wFϕ2pρw−2ndvw≃∫BrzcSKz,T∗KwFϕ2pe−pϕz−pϕwρz2np−2ndvw=∫Brzc∫ℂnSKz,KxFϕ2Kx,T∗KwFϕ2e−2ϕxdvxpe−pϕz−pϕwρz2np−2ndvw≃∫Brzc∫ℂnSkp,z,k2n−2np,xFϕ2kp,x,T∗k2n−2np,wFϕ2ρx−2ndvxpρw−2ndvw≲∫Brzc∫ℂnSkp,z,k2n−2np,xFϕ2pkp,x,T∗k2n−2np,wFϕ2pρx−2ndvxρw−2ndvw=∫ℂnSkp,z,k2n−2np,xFϕ2pρx−2n∫BrzcTkp,x,k2n−2np,wFϕ2pρw−2ndvwdvx.SinceT and S are weakly localized, hence(74)I1≔∫BrazSkp,z,k2n−2np,xFϕ2pρx−2n∫BrzcTkp,x,k2n−2np,wFϕ2pρw−2ndvwdvx≤∫BrazSkp,z,k2n−2np,xFϕ2pρx−2n∫BraxcTkp,x,k2n−2np,wFϕ2pρw−2ndvwdvx<ε∫BrazSkp,z,k2n−2np,xFϕ2pρx−2ndvx,I2≔∫BrazcSkp,z,k2n−2np,xFϕ2pρx−2n∫BrzcTkp,x,k2n−2np,wFϕ2pρw−2ndvwdvx≤∫BrazcSkp,z,k2n−2np,xFϕ2pρx−2n∫ℂnTkp,x,k2n−2np,wFϕ2pρw−2ndvwdvx≤supx∈ℂn∫ℂnTkp,x,k2n−2np,wFϕ2pρw−2ndvw∫BrazcSkp,z,k2n−2np,xFϕ2pρx−2ndvx<εsupx∈ℂn∫ℂnTkp,x,k2n−2np,wFϕ2pρw−2ndvw.Therefore, by combiningI1 and I2, we get(75)∫BrzcTSkp,z,k2n−2np,wFϕ2pρw−2ndvw<Cε,where the constantC does not depend on ε. This means(76)supz∈ℂn∫BrzcTSkp,z,k2n−2np,wFϕ2pρw−2ndvw⟶0,whenr⟶∞. Meanwhile, we also get (77)supz∈ℂn∫BrzcTS∗k2n−2np,z,kp,wFϕ2pρw−2ndvw⟶0,wheneverr⟶∞.On the other hand, let now1<p<∞, by Fubini’s theorem, and we have(78)∫Brzc∣TSkp,z,kq,wFϕ2∣ρw−2ndvw≃∫Brzc∣∫ℂnSkp,z,kq,xFϕ2kp,x,T∗kq,wFϕ2ρx−2ndvx∣ρw−2ndvw≤∫ℂn∣Skp,z,kq,xFϕ2∣ρx−2n∫Brzc∣Tkp,x,kq,wFϕ2∣ρw−2ndvwdvx.We split again the above integral onℂn into the corresponding integrals on Br/az and Br/azc, then(79)supz∈ℂn∫Brzc∣TSkp,z,kq,wFϕ2∣ρw−2ndvw,supz∈ℂn∫Brzc∣TS∗kq,z,kp,wFϕ2∣ρw−2ndvwall go to 0 asr⟶∞. This ends the proof since others are obvious.LetWLpϕ denote the algebra generated by weakly localized operators for Fϕp. Let Tf be a Toeplitz operator (see [8]) on Fϕp, where f is called a symbol function. Then, each WLpϕ contains some special Toeplitz operators.Theorem 16.
Suppose0<p<∞ and f∈L∞. Then, Toeplitz operator Tf∈WLpϕ.Proof.
We first suppose0<p≤1. Clearly, it suffices to prove that(80)supz∈ℂn∫BrzcTfkp,z,k2n−2np,wFϕ2pρw−2ndvw
converges to 0 asr⟶∞.SinceTfkp,z=Pfkp,z for any fixed z, hence Lemma 4 gives that(81)∫BrzcTfkp,z,k2n−2np,wFϕ2pρw−2ndvw≲fL∞p∫Brzckp,z,k2n−2np,wFϕ2pρw−2ndvw≃fL∞p∫Brzckp,zwpe−pϕwdvwgoes to 0 wheneverr⟶∞.Now, assume that1<p<∞. It is easily obtained from (18), (20), and Lemma 2 that(82)∫ℂn∣Tfkp,z,kq,wFϕ2∣ρw−2ndvw≃∫ℂnfwKzwe−ϕz−ϕwρz2n−2npρw−2nqdvw≲∥f∥L∞∫ℂnρzn−2npρw−n−2nqe−ϵ∣w−z∣ρzαdvw≲∥f∥L∞.Thus, we only need to show(83)supz∈ℂn∫Brzc∣Tfkp,z,kq,wFϕ2∣ρw−2ndvw⟶0asr⟶∞. In fact, ∣w−z∣≥rρz if w∈Brzc. This together with (82) indicates(84)∫Brzc∣Tfkp,z,kq,wFϕ2∣ρw−2ndvw≲∥f∥L∞∫Brzcρzn−2npρw−n−2nqe−ϵ2∣w−z∣ρzαe−ϵ2rαdvw≲e−ϵ2rα∥f∥L∞.Therefore, the desired conclusion follows whenr⟶∞. This ends the proof.Remark. Moreover, Theorem16 indicates that the identity operator is also in WLpϕ. Namely, each algebra WLpϕ possesses an unit.We next consider the boundedness of operatorT∈WLpϕ for 0<p<∞.Theorem 17.
If0<p<∞ and T∈WLpϕ, then T is bounded on Fϕp.Proof.
First, we see that(85)Tfz=Tf,KzFϕ2=f,T∗KzFϕ2=∫ℂnfwKw,T∗KzFϕ2e−2ϕwdvw.Let0<p≤1 and let(86)M1=supw∈ℂn∫ℂnTkp,w,k2n−2np,zFϕ2pρz−2ndvz.Estimate (20) combined with Lemma 6 yields(87)Tfze−ϕzp≤∫ℂn∣fwKw,T∗KzFϕ2∣e−2ϕwe−ϕzdvwp≃∫ℂn∣fwKw,T∗k2n−2np,zFϕ2∣e−2ϕwρz−2npdvwp≃ρz−2n∫ℂnfwkp,w,T∗k2n−2np,zFϕ2e−ϕwρw2np−2ndvwp≲ρz−2n∫ℂnfwkp,w,T∗k2n−2np,zFϕ2pe−pϕwdvw.So, we conclude by Fubini’s theorem that(88)∥Tf∥p,ϕp≲∫ℂnfwpe−pϕw∫ℂnTkp,w,k2n−2np,zFϕ2pρz−2ndvzdvw≤M1∫ℂnfwpe−pϕwdvw=M1∥f∥p,ϕp.We now assume thatp>1. Set(89)M2=maxsupw∈ℂn∫ℂn∣Tkp,w,kq,zFϕ2∣ρz−2ndvz,supz∈ℂn∫ℂnT∗kq,z,kp,wFϕ2ρw−2ndvw.By Fubini’s theorem andHo¨lder’s inequality, we obtain(90)∥Tf∥p,ϕp≤∫ℂn∫ℂn∣fwTKw,KzFϕ2∣e−2ϕwdvwpe−pϕzdvz≃∫ℂn∫ℂn∣fw∣e−ϕw∣Tkp,w,kq,zFϕ2∣ρw2np−2nρz2nq−2ndvwpdvz≤∫ℂn∫ℂnfwpe−pϕw∣Tkp,w,kq,zFϕ2∣ρz−2ndvw·∫ℂn∣T∗kq,z,kp,wFϕ2∣ρw−2ndvwpqdvz≲M2pq∫ℂnfwpe−pϕw∫ℂn∣Tkp,w,kq,zFϕ2∣ρz−2ndvzdvw≲∥f∥p,ϕp,which completes the proof.Now, it follows from Theorem17 that each WLpϕ0<p≤1 is analogous to a Banach algebra.Theorem 18.
Suppose0<p≤1 and T,S∈WLpϕ. Then, ∥TS∥p,ϕ≲∥T∥p,ϕ∥S∥p,ϕ.Proof.
Suppose0<p≤1. For every z∈ℂn, by the proof of Theorem 15, we see that(91)TSkp,zp,ϕp=∫ℂnTSkp,zwpe−pϕwdvw≃∫ℂnTSkp,z,k2n−2np,wFϕ2pρw−2ndvw≲∫ℂnSkp,z,k2n−2np,xFϕ2pρx−2n∫ℂnTkp,x,k2n−2np,wFϕ2pρw−2ndvwdvx≃∫ℂnSkp,zxpe−pϕx∫ℂnTkp,xwpe−pϕwdvwdvx=∫ℂnTkp,xp,ϕpSkp,zxpe−pϕxdvx.SinceT,S∈WLpϕ, then Theorem 17 says T and S are bounded on Fϕp. Thus, the above estimate implies TSkp,zp,ϕ≲Tp,ϕSp,ϕ. This completes the proof since the supremum of TSkp,zp,ϕ is no more than C times Tp,ϕSp,ϕ.Theorem 19.
If0<p≤1, then WLpϕ is closed under the operator norm on Fϕp.Proof. See Lemma 2.6 of [5].
We omit the details.


## 5. Equivalent Conditions for Compactness

For this section, we use the ideas in [5, 7] to characterize compactness of weakly localized operators on large Fock spaces. Indeed, it is more complex than [5] because Bergman metric works in a different way than in Euclidean metric.We begin with the following preparations. Recall that, for fixedr>0, there is an r-lattice zjj such that Brzjj covers ℂn. Let Fj=Brzj\⋃i<jBrzi. It follows that Fjj is also a covering of ℂn and Fj∩Fk=∅j≠k. We write Fjr+=⋃x∈FjBrx, and it is from estimate (16) that we consider(92)Gj=y:dy,Fj≤m4rρzj,m=mr>1.In what follows, we always defineFj and Gj as above. Also, there is some constant N such that(93)∑j=1∞χFjw≤∑j=1∞χFjr+w≤Nforanyw∈ℂn.Lemma 20.
If0<p≤1 and T∈WLpϕ, then for every ε>0, there exists sufficiently large r>0 such that for the covering Fjj (associated to r), we obtain(94)T−P∑j=1∞MχFjTPMχGjFϕp⟶Fϕp<ε.Proof.
SinceT∈WLpϕ, then there is some r>0 sufficiently large such that(95)supu∈ℂn∫BrucTkp,u,k2n−2np,wFϕ2pρw−2ndvw<ε.DefineS=TP−∑j=1∞MχFjTPMχGj. Then, Lemma 6 indicates that(96)PSfzp≤∫ℂnSfwKz,we−2ϕwdvwp=∫ℂn∑j=1∞MχFjTPMχGjcfwKz,we−2ϕwdvwp≤∑j=1∞∫ℂnMχFjTPMχGjcfwKz,we−2ϕwdvwp≤∑j=1∞∫FjTPMχGjcfwKz,we−2ϕwdvwp≲∑j=1∞∫Fjt+TPMχGjcfwpKz,wpe−2pϕwρw2np−2ndvw.Notice that(97)∣TPMχGjcfw∣=∣TPMχGjcf,KwFϕ2∣=∣MχGjcf,T∗KwFϕ2∣=∣∫ℂnMχGjcfT∗Kw¯e−2ϕudvu∣≤∫Gjc∣fuKu,T∗KwFϕ2∣e−2ϕudvu.This together with, for somet,s>0, estimate (20), Lemma 6, and Proposition 12 shows(98)PSfzp≲∑j=1∞∫Fjt+∫GjcfuKu,T∗KwFϕ2e−2ϕudvupKz,wpe−2pϕwρw2np−2ndvw≃∑j=1∞∫Fjt+Kz,wpe−pϕwρw2np−2n×∫Gjcfue−ϕuTkp,u,k2n−2np,wFϕ2ρw−2npρu2np−2ndvupdvw≲∑j=1∞∫Fjt+Kz,wpe−pϕwρw2np−4n∫Gjcs+fue−ϕuTkp,u,k2n−2np,wFϕ2pdvudvw.We claim first thatFjt+⊂Bruc if w∈Fjt+ and u∈Gjcs+. Furthermore, applying (18), (93), Lemma 6, and Fubini’s theorem, we obtain(99)∥PSfz∥p,ϕp≲∑j=1∞∫Gjcs+fue−ϕup∫Fjt+χBrucwTkp,u,k2n−2np,wFϕ2pe−pϕwρw2np−4n×∫ℂnKz,wpe−pϕzdvzdvwdvu≲N∫ℂnfue−ϕup∫BrucTkp,u,k2n−2np,wFϕ2pρw−2ndvwdvu≲∫ℂnfue−ϕupsupu∈ℂn∫BrucTkp,u,k2n−2np,wFϕ2pρw−2ndvwdvu<ε∥f∥p,ϕp.SincePTP=T on Fϕp, thus PS=T−P∑j=1∞MχFjTPMχGj is well defined.Thus it only remains to prove thatFjt+⊂Bruc.In fact, (16) gives Brzj⊂Bm2rzj⊂Bm4rzj where m=mr>1. We then choose a t satisfying w∈Fjt+⊂Bm2rzj.Note that for anyu∈ℂn, there exists M>0 so that ρu≤M by estimate (12). For fixed r>0, it follows that ∣u∣−rρu>0 whenever ∣u∣ is sufficiently large. So, for any j>0, there is some R>0 such that(100)u−rρu>m2rρzj+zj,whenever∣u∣>R. Thus, the triangle inequality implies ∣u−zj∣≥∣u∣−∣zj∣>rρu+m2rρzj. Further, the above inequality concludes that Fjt+∩Bru=∅.Next, we assume that∣u∣≤R. For any fixed j, there is a r0 satisfying u∈Br0zj. Since ρ is a Lipschitz function, then ρzj≥1/cr0ρu where cr0>1. Notice that m=mr>1, thus it allows us to let m>cr0. It follows that mrρzj>rρu. For u∈Gjcs+ and ∣u∣≤R, we can pick an appropriate s so that sρu<mrρzj−rρu. Hence,(101)∣u−zj∣>m4+1rρzj−sρu>m4−m+1rρzj+rρu>m2rρzj+rρushowsFjt+∩Bru=∅.Therefore, the desired assertion holds, and the proof is finished.Lemma 21.
If1<p<∞ and T∈WLpϕ, then for every ε>0, there exists sufficiently large r>0 such that for the covering Fjj (associated to r), we obtain(102)TP−∑j=1∞MχFjTPMχGjFϕp⟶Fϕp<ε.Proof.
By (71), for any ε>0 and w∈ℂn, there is some r>0 such that(103)∫Brwc∣T∗kq,w,kp,uFϕ2∣ρu−2ndvu<ε,and (70) shows(104)∫ℂn∣Tkp,u,kq,wFϕ2∣ρw−2ndvw<M<∞.We also considerS=TP−∑j=1∞MχFjTPMχGj. For any fixed w∈ℂn, we assume that w∈Fj0⊂Brzj0. Note first that if u∈Gj0c, then ∣u−zj0∣>m4+1rρzj0 where m=mr>1. For w∈Fj0, we have ∣w−zj0∣<rρzj0 and ρzj0≥cr−1ρw since ρ is a Lipschitz function. Suppose m4>cr, so ∣u−w∣>m4rρzj0>rρw, that is Gj0c⊂Brwc. It follows from Lemma 20 that(105)∣Sfw∣=∣∑j=1∞MχFjTPMχGjcfw∣≤∑j=1∞MχFjw∫Gjc∣fuKu,T∗KwFϕ2∣e−2ϕudvu≤∫Brwc∣fuKu,T∗KwFϕ2∣e−2ϕudvu.This together withHo¨lder’s inequality and Fubini’s theorem implies(106)∥Sf∥p,ϕp≤∫ℂn∫Brwc∣fuTKu,KwFϕ2∣e−2ϕudvupe−pϕwdvw≃∫ℂn∫Brwc∣fu∣e−ϕu∣Tkp,u,kq,wFϕ2∣ρu2np−2nρw2nq−2ndvupdvw≲∫ℂn∫Brwcfupe−pϕu∣Tkp,u,kq,wFϕ2∣ρw−2ndvu×∫Brwc∣T∗kq,w,kp,uFϕ2∣ρu−2ndvupqdvw<εpq∫ℂnfupe−pϕu∫ℂn∣Tkp,u,kq,wFϕ2∣ρw−2ndvudvw<Mεpq∥f∥p,ϕp,which proves the desired result.Lemma 22.
Suppose0<p<∞. For any bounded linear operator Ton  Fϕp, there is some constant C such that(107)limsupm⟶∞∥PTm∥Fϕp⟶Fϕp≤Climsupm⟶∞supw∈⋃j>mGjr+∥Tkp,w∥p,ϕ,
whereTm=∑j>mMχFjTPMχGj.Proof.
First, let0<p≤1. Suppose f∈Fϕp and f≠0, define(108)gj=PχGjf∥χGjr+f∥p,ϕ.SincePf=f on Fϕp, then by (85) we get(109)∣Tgjz∣=∣∫ℂngjwTKwze−2ϕwdvw∣=∣∫ℂnχGjfw∥χGjr+f∥p,ϕTKwze−2ϕwdvw∣≤∫GjfwTKwze−2ϕw∥χGjr+f∥p,ϕdvw.By combining the above estimate and Lemma6, we obtain(110)PMχFjTgjup=∫FjTgjzKu,ze−2ϕzdvzp≲∫Fjr+TgjzpKu,zpe−2pϕzρz2np−2ndvz≤∫Fjr+Ku,zpe−2pϕz∫GjfwTKwze−2ϕwχGjr+fp,ϕdvwpρz2np−2ndvz≲∫Fjr+Ku,zpe−2pϕzρz2np−2n×∫Gjr+fwpTKwzpe−2pϕwχGjr+fp,ϕpρw2np−2ndvwdvz.Notice that there is ar′∈0,1 such that Fjr+⊂Br′zj (we can let r be small enough), then for z∈Br′zj, we get ρz≃ρzj by (14). Now, by (18), (25), Lemma 2, and Fubini’s theorem, we have(111)∥PMχFjTgj∥p,ϕp≲∫Gjr+fwp∥χGjr+f∥p,ϕpe−2pϕwρw2np−2n×∫Fjr+TKwzpe−2pϕzρz2np−2n∫ℂnKu,zpe−pϕudvudvwdvz≃∫Gjr+fwp∥χGjr+f∥p,ϕpe−pϕw∫Fjr+Tkp,wzpe−pϕzdvzdvw≲supw∈Gjr+∥Tkp,w∥p,ϕp∫Br′zjρz−2ndvz∫Gjr+fwp∥χGjr+f∥p,ϕpe−pϕwdvw≲supw∈Gjr+∥Tkp,w∥p,ϕp.Therefore, we deduce(112)∥PTmf∥p,ϕp≤∑j>m∥PMχFjTPMχGjf∥p,ϕp=∑j>m∥PMχFjTgj∥p,ϕp∥χGjr+f∥p,ϕp≲∑j>msupw∈Gjr+∥Tkp,w∥p,ϕp∥χGjr+f∥p,ϕp≤Nsupw∈⋃j>mGjr+∥Tkp,w∥p,ϕp∥f∥p,ϕp.Now,1<p<∞. It comes from the above proof that(113)PTmfup=P∑j>mMχFjTPMχGjfup≤∑j>m∫Fj∣Ku,z∣e−2ϕz∫GjfwTKw,KzFϕ2e−2ϕwdvwdvzp.For fixedj and r>0, let x∈Fj\zj. Note that Fj⊂Brzj, we then get Fj⊂Bbrx and Gj⊂Bb5rx by (16), where b=br>1. Furthermore, there exists some r′>0 such that Bb5rx⊂Gjr′+. It follows from estimate (14) we have ρz≃ρx if z∈Bbrx and ρw≃ρx if w∈Bb5rx (here we also assume r is small enough so that b5r<1). Hence, by estimates (18), (20), (25), (93), Minkowski’s inequality, and Lemma 6, we obtain(114)∥PTmf∥p,ϕ≤N∫ℂn∫Bb5rx∣fw∣e−2ϕw×∫Bbrx∣TKwz∣e−ϕz∣Ku,z∣e−ϕz−ϕudvzdvwpdvu1p≲∫Bb5rx∣fw∣e−2ϕw∫Bbrx∣TKwz∣e−ϕz×∫ℂnKu,zpe−pϕz−pϕudvu1pdvzdvw≃∫Bb5rx∣fw∣e−2ϕw∫Bbrx∣TKwz∣e−ϕzρz2np−2ndvzdvw≃∫Bb5rx∣fw∣e−ϕwρw2np−2n∫Bbrx∣Tkp,wz∣e−ϕzρz2np−2ndvzdvw≃∫Bb5rx∣fw∣e−ϕwρw2np−2n∥Tkp,w∥p,ϕ∫Bbrxρz−2ndvzdvw≲supw∈Bb5rx∥Tkp,w∥p,ϕ∥f∥p,ϕρx−2nvBbrx∫Bb5rxρx−2ndvw≲Cb,rsupw∈⋃j>mGjr′+∥Tkp,w∥p,ϕ∥f∥p,ϕ.This completes the proof.Theorem 23.
Suppose0<p<∞ and T∈WLpϕ. Then, the following conditions are equivalent (q≔2n−2n/pif0<p≤1):(A)
limz⟶∞∥Tkp,z∥p,ϕ=0;(B)
limz⟶∞supw∈Brz∣Tkp,z,kq,wFϕ2∣=0 for any r>0;(C)
limz⟶∞supw∈ℂn∣Tkp,z,kq,wFϕ2∣=0.Proof.
Suppose condition (A) holds. If0<p≤1, then (25) gives(115)Tkp,z,k2n−2np,wFϕ2p≃Tkp,zwe−ϕwρw2npp≤C∫BrwTkp,zupe−pϕudvu≲∥Tkp,z∥p,ϕp.Similarly, when1<p<∞, we obtain(116)Tkp,z,kq,wp≃Tkp,zwe−ϕwρw2n−2nqp≲∫BrwTkp,zupe−pϕudvu≲∥Tkp,z∥p,ϕp.Then (A) implies (C).BecauseC⇒B is clear, so it remains to prove that the implication B⇒A. If T∈WLpϕ and 0<p≤1, then, by (68), there is some r>0 such that(117)∫BrzcTkp,z,k2n−2np,wFϕ2pρw−2ndvw<ε.Note that by the definition of functionρ, we get ρ≥α>0. Then, (B) shows(118)∥Tkp,z∥p,ϕp=∫ℂnTkp,z,KwFϕ2pe−pϕwdvw≃∫Brzc⋃BrzTkp,z,k2n−2np,wFϕ2pρw−2ndvw<ε+∫BrzTkp,z,k2n−2np,wFϕ2pρw−2ndvw≤ε+supw∈BrzTkp,z,k2n−2np,wFϕ2p∫Brzρw−2ndvw≲ε,whenever∣z∣ is large enough. By hypothesis, A holds when 0<p≤1.Suppose1<p<∞. It follows from (20), (25), and Theorem 17 that(119)∣Tkp,z,kq,wFϕ2∣≃∣Tkp,z,KwFϕ2∣e−ϕwρw2n−2nq≲∥Tkp,z∥p,ϕ≲1.Therefore, joining conditionB and (71), we deduce(120)∥Tkp,z∥p,ϕp≃∫ℂnTkp,z,kq,wFϕ2pρw−2ndvw≲∫Brzc∣Tkp,z,kq,wFϕ2∣ρw−2ndvw+∫Brz∣Tkp,z,kq,wFϕ2∣ρw−2ndvw<ε+∫Brz∣Tkp,z,kq,wFϕ2∣ρw−2ndvw≲ε+supw∈Brz∣Tkp,z,kq,wFϕ2∣∫Brzρw−2ndvw≲εfor sufficiently largez. This completes the proof.It is similar to ([17], Lemma 3.2) that we have the following assertion about relatively compact. That is, for every ε>0, there is some R>0 such that(121)supf∈E∫∣z∣≥Rfze−ϕzpdvz<εif and only if a bounded subsetE⊂Fϕp is relatively compact. In what follows, we call KFϕp the set of compact operators on Fϕp.Theorem 24.
If0<p<∞ and T∈KFϕp, then(122)limR⟶∞∥PMχB0,RT−T∥Fϕp⟶Fϕp=0.Proof.
We omit the details for0<p≤1, see ([5], Lemma 2.11).If1<p<∞, then Ho¨lder’s inequality, Fubini’s theorem, (18), and Lemma 2 imply(123)PMχB0,RT−Tfp,ϕp≤∫ℂn∫∣w∣≥R∣Tfw∣e−ϕw∣Kz,w∣e−ϕw−ϕzdvwpdvz≤∫ℂn∫∣w∣≥RTfwpe−pϕw∣Kz,w∣e−ϕw−ϕzdvw×∫∣w∣≥R∣Kz,w∣e−ϕw−ϕzdvwpqdvz≲∫∣w∣≥RTfwpe−pϕw∫ℂn∣Kz,w∣e−ϕw−ϕzdvzdvw≲∫∣w∣≥RTfwpe−pϕwdvwconverges to 0 as R⟶∞. This finishes the proof.Due toPMχGj that can be viewed as a Toeplitz operator induced by χGj, then PMχGj is compact (the reason is similar to ([18], Lemma 3.1)).Theorem 25.
Suppose1<p<∞. Then, there exists r such that(124)∥T∥e≤Climsupz⟶∞supw∈BrzTkp,z,kq,wFϕ2where ∥T∥e means the essential norm of a bounded operator T on Fϕp.Proof.
SincePf=f for f∈Fϕp, then ∥T∥e=∥TP∥e, and we always assume that ∥TP∥e>0. Thus, Lemma 21 shows there is some r>0 satisfying(125)TP−∑j=1∞MχFjTPMχGjFϕp⟶Fϕp<12TPe.Because of∑j≤mMχFjTPMχGj is a compact operator where m∈ℕ, then(126)TPe≤TP−∑j=1mMχFjTPMχGjFϕp⟶Fϕp≤TP−∑j=1∞MχFjTPMχGjFϕp⟶Fϕp+TmFϕp⟶Fϕp<12TPe+TmFϕp⟶Fϕp,where Tm=∑j>mMχFjTPMχGj. For the rest of the task, we show(127)limsupm⟶∞TmFϕp≲Climsupz⟶∞supw∈Brz∣Tkp,z,kq,wFϕ2∣+14∥TP∥e.Letf∈Fϕp and ∥f∥Fϕp≤1. Note that Fj∩Fk=∅j≠k, hence(128)TmfFϕpp=∑j>mMχFjTPMχGjfFϕpp=∑j>mMχFjTljFϕppMχGjfFϕpp≤Nsupj>mMχFjTljFϕpp∥f∥Fϕpp,wherelj≔PMχGjf/∥MχGjf∥Fϕp. It follows that(129)∥Tm∥Fϕp≲supj>msup∥f∥Fϕp≤1MχFjTljFϕp:lj≔PMχGjf∥MχGjf∥Fϕp.Furthermore, we have(130)limsupm⟶∞∥Tm∥Fϕp≲limsupj⟶∞sup∥f∥Fϕp≤1MχFjTljFϕp:lj≔PMχGjf∥MχGjf∥Fϕp.Letfj be a sequence with ∥fj∥Fϕp≤1 such that(131)limsupj⟶∞sup∥f∥Fϕp≤1MχFjTljFϕp:lj≔PMχGjf∥MχGjf∥Fϕp−14TPe≤limsupj⟶∞MχFjTgjFϕp,wheregj≔PMχGjfj/∥MχGjfj∥Fϕp. Fix j, for z∈Fj⊂Gj, and there is a b=br>0 such that Gj⊂Bbrz by (16). By joining the proof of Lemma 22 and Ho¨lder’s inequality, we deduce(132)limsupj⟶∞MχFjTgjFϕpp≤limsupj⟶∞∫Fj∫GjfwTKwze−2ϕwχGjfp,ϕdvwpe−pϕzdvz≃limsupj⟶∞∫Fj∫Gjfwe−ϕwχGjfp,ϕTkp,w,kq,zFϕ2ρz2nq−2nρw2np−2ndvwpdvz≲limsupj⟶∞∫Fj∫Gjfwpe−pϕwχGjfp,ϕpTkp,w,kq,zFϕ2ρz−2ndvw∫Gjρw−2ndvwpqdvz≲limsupj⟶∞∫Fj∫Gjfwpe−pϕwχGjfp,ϕpTkp,w,kq,zFϕ2ρz−2ndvwdvz≲limsupz⟶∞supw∈BbrzTkp,z,kq,wFϕ2psupj∫Gjρz−2n∫Gjfwpe−pϕwχGjfp,ϕpdvwdvz≤Crlimsupz⟶∞supw∈BbrzTkp,z,kq,wFϕ2p,whereCr is independent of j. This finishes the proof.Theorem 26.
Suppose0<p<∞ and T∈WLpϕ. Then, T∈KFϕp if and only if limz⟶∞∥Tkp,z∥p,ϕ=0.Proof.
(⇐) For any ε>0, by Lemma 20, we get(133)T−P∑j=1∞MχFjTPMχGjFϕp⟶Fϕp<ε.ConsiderTm=∑j>mMχFjTPMχGj, then P∑j=1mMχFjTPMχGj is a compact operator on Fϕp, where m is any positive integer. Hence,(134)∥T∥ep≤T−P∑j=1mMχFjTPMχGjFϕp⟶Fϕpp≤T−P∑j=1∞MχFjTPMχGjFϕp⟶Fϕp+PTmFϕp⟶Fϕpp<ε+PTmFϕp⟶Fϕpp.With our assumption, there is anR>0 such that ∥Tkp,z∥p,ϕ<ε for ∣z∣>R. Since ⋃j>mGjr+⊂B0,Rc whenever m is large enough, then Lemma 22 indicates ∥T∥e=0 for 0<p≤1. Also, when p>1, ∥T∥e=0 follows immediately from Theorems 23 and 25.(⇒) The case 1<p<∞ is similar to the following discussion of 0<p≤1.Consider0<p≤1 and r>0. With the help of Theorem 23, we will finish the proof if supw∈BrzTkp,z,k2n−2n/p,wFϕ2⟶0 as z⟶∞. Recall first that, for f∈Fϕp, by (25), we have(135)fz≤Ceϕzρz−2npfp,ϕ.SinceTkp,z∈Fϕp, estimate (18), Kwu=Kuw, and Proposition 12, then (136)PMχB0,RTkp,z,k2n−2np,wFϕ2=∫B0,RTkp,zuk2n−2np,wu¯e−2ϕudvu≤∫B0,RTkp,zuk2n−2np,wu¯e−2ϕudvu≲Tkp,zp,ϕ∫B0,Rk2n−2np,wu¯e−ϕuρu−2npdvu≲Tkp,zp,ϕ∫B0,RKwue−ϕu−ϕwρw2npρu−2npdvu≲Tkp,zp,ϕρw2np−n∫B0,Rρu−n−2npe−ϵ∣w∣−∣u∣ρuαdvu≲R2nTkp,zp,ϕρw2np−ne−gwϵgoes to 0 asw⟶∞, where gw=∣w∣−R/maxρu:u∈B0,R¯.On the other hand, Theorems17 and 24 conclude that(137)PMχB0,RcTkp,z,k2n−2np,wFϕ2≃T−PMχB0,RTkp,z,KwFϕ2e−ϕwρw2np≲T−PMχB0,RTkp,zp,ϕ≤T−PMχB0,RTp,ϕ⟶0asR⟶∞. Altogether gives that(138)Tkp,z,k2n−2np,wFϕ2≤PMχB0,RTkp,z,k2n−2np,wFϕ2+PMχB0,RcTkp,z,k2n−2np,wFϕ2,which ends the proof since supw∈BrzTkp,z,k2n−2n/p,wFϕ2⟶0 as z⟶∞.



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*Source: 6619459-2021-02-08.xml*