# Boundedness and Compactness of Hankel Operators on Large Fock Space

Authors: Xiaofeng Wang; Zhicheng Zeng
Journal: Journal of Function Spaces
 (2022)

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## Abstract

We introduce the BMO spaces and use them to characterize complex-valued functionsf such that the big Hankel operators Hf and Hf¯ are both bounded or compact from a weighted large Fock space Fpϕ into a weighted Lebesgue space Lpϕ when 1≤p<∞.


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## Body

## 1. Introduction

Letℂ denote the set of complex numbers. For a fixed positive integer n, let ℂn be the n-dimensional Euclidean space. Given z∈ℂn and r>0, let Bz,r denote the Euclidean ball defined as
(1)Bz,r=w∈ℂn:∣w−z∣<r.LetΔ be the Laplace operator and dv denote the Lebesgue volume measure on ℂn. Suppose ϕ:ℂn⟶ℝ is a C2-plurisubharmonic function. 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)
Δϕ satisfies the reverse-Ho¨lder’s inequality
(3)ΔϕL∞Bz,r≤Cr−2n∫Bz,rΔϕdv,∀z∈ℂn,r>0,for some 0<C<∞(C)
The eigenvalues ofHϕ are comparable; i.e., there exists δ0>0 such that
(4)Hϕzu,u≥δ0Δϕzu2,∀z,u∈ℂn,where
(5)Hϕ=∂2ϕ∂zj∂z¯kj,k.Letϕ∈W. For 0<p<∞, the weighted Lebesgue space Lpϕ is the set of those measurable functions f on ℂn such that
(6)fLpϕ≔∫ℂnfzpe−pϕzdvz1/p<∞.Forp=∞, the space L∞ϕ consists of all measurable functions f on ℂn such that
(7)fL∞ϕ≔supz∈ℂn∣fz∣e−ϕz<∞.The weighted Fock spaceFpϕ consists of all analytic functions on ℂn such that fLpϕ<∞.The weight classW is first introduced by Dall’Ara in [1] where the pointwise estimation of the Bergman kernel is given. Recently, the research on the weighted Fock space has been gradually improved. Lv [2] obtained characterizations on the boundedness of the Bergman projection. Arroussi and Tong [3] gave the Lpϕ-norm estimation of the Bergman kernel and studied the boundedness, compactness, and Schatten class membership of the weighted composition operators. Liu and Wang [4] consider the duality of large Fock spaces and obtained equivalent conditions for compactness of weakly localized operators. The weight class W contains abundant functions. For example, when ϕz is replaced by z2/2, F2ϕ is the classical Fock space (see [5]). The weight function ϕ satisfying ddcϕ≃ddcz2 in [6, 7] is the member of W. It is interesting that the weight class W also contains some nonradial weights. An example of nonradial weight is given in [3].For1≤p<∞, Fpϕ is a Banach space. In particular, F2ϕ is a Hilbert space. Let Kϕw,z be the Bergman kernel of F2ϕ and denote Kzw=Kϕw,z for simplicity. We know that Kϕw,z¯=Kϕz,w. The Bergman projection P can be represented as
(8)Pgz=∫ℂnKϕz,wgwe−2ϕwdvw,z∈ℂn.Moreover,P is bounded from Lpϕ to Fpϕ and Pg=g for any g∈Fpϕ when 1≤p<∞. See [2, 4] for more details. From [4], we know that
(9)Γ=∑j=1NajKϕ·,zj:N∈ℕ,aj∈ℂ,zj∈ℂn,∀1≤j≤N,is dense in Fpϕ. To let Hf make sense on Γ, we naturally consider those measurable functions f in the symbol class S defined as
(10)S=f:fg∈Lpϕ,∀g∈Γ.It is obvious that the spaceL∞ℂn of essentially bounded measurable functions on ℂn is contained in S. Given some symbol function f∈S, one defines the big Hankel operator as
(11)Hfg=fg−Pfg.Using the property of reproducing kernel, it has the form of(12)Hfgz=∫ℂnfz−fwgwKϕz,we−2ϕwdvw,z∈ℂn.Hankel operators on Fock spaces and Bergman spaces have been studied by many mathematicians during the past few decades; see [5, 8] for instance. The main problem we consider in this paper is to characterize the simultaneous boundedness and compactness of Hankel operators Hf and Hf¯ from Fpϕ to Lpϕ with 1≤p<∞ in terms of suitable defined BMO spaces and VMO spaces. There are exclusive references for this issue both in the setting of Bergman spaces and Fock spaces. See [9–12] for Hankel operators that are simultaneously bounded or compact between the same exponents and [13–16] between different exponents.This paper is organized as follows. In Section2, we will give some basic useful results including the Lpϕ-norm as well as pointwise estimation of the reproducing kernel. The space of BMOrp will be established in Section 3. The decomposition theorem of BMOrp is given. We will use the BMOrp space to characterize simultaneously bounded Hankel operators Hf and Hf¯ in Section 4. The space VMOrp is introduced in Section 5, and it has similar results of the space BMOrp. Compact Hankel operators are characterized in Section 6.Throughout this paper, we will useC to denote a positive constant whose value may change from line to line, but does not depend on functions being considered. We denote A≲B if there is a positive constant C such that A≤CB. A≃B stands for A≲B and B≲A.


## 2. Preliminaries

In this section, we are going to give some basic definitions and results, which will be frequently used. Forz∈ℂn, set
(13)ρz≔supr>0:supw∈Bz,rΔϕw≤r−2.Givenr>0, write Brz=Bz,rρz. We will call ρz a radius function.Lemma 1.
Letϕ∈W. Then, the function ρ satisfies the following properties:
(A)
ρ is bounded on ℂn, that is
(14)ρz≤supz∈ℂnρz≕M,∀z∈ℂn.(B)
For0<r<1, and w∈Brz, there holds
(15)1−rρz≤ρw≤1+rρz.Proof.
See [3].☐Letkp,z be the normalized reproducing kernel in Fpϕ, that is
(16)kp,zw=KzwKzLpϕ.We are going to give some estimations of the reproducing kernel. One can find the proof in [3].Lemma 2.
Letϕ∈W, then one has
(A)
For0<p≤∞(17)KzLpϕ≃eϕzρz2n1−p/p,z∈ℂn.(B)
There is someα∈0,1 such that(18)kp,zwpe−pϕw≃ρz−2n,w∈Bαz.(C)
The setkp,z:z∈ℂn is bounded in Fpϕ and kp,z⟶0 uniformly on every compact subsets of ℂn when ∣z∣⟶∞.In the following argument, we will always assumeα to be a constant as in (B) of Lemma 2. Notice that α∈0,1, so inequality (15) always holds when 0<r<α. Before giving the pointwise estimation of the Bergman kernel, we need to introduce a distance function associated with the radius function ρ. In fact, we are interested with the Riemannian distance. To describe it explicitly, given z,w∈ℂn, we define
(19)dz,w=infγ∫01∣γ′t∣ργtdt,where the infimum is taken as γ:0,1⟶ℂn which varies over the collection of piecewise C1-curves with γ0=z and γ1=w. We define the ball of d-metric as
(20)Bρz,r=w∈ℂn:dz,w<r.The following lemma gives the pointwise estimation ofKϕz,w.Lemma 3.
There is a constantσ>0 such that
(21)∣Kϕz,w∣≲eϕw+ϕzexp−σdz,wρznρwn,∀z∈ℂn.Proof.
See Theorem 20 of [1].
In what follows, we will always assumeσ to be a constant as in Lemma 3.☐


## 3. The Space BMO

For1≤p<∞, let Llocpℂn be the set of all p-th locally Lebesgue integrable functions on ℂn. Suppose f∈Lloc1ℂn and r>0, the averaging function f^r is defined as
(22)f^rz=1∣Brz∣∫Brzfwdvw,where ∣Brz∣ is the volume of Brz. For f∈Llocpℂn, the p-th mean oscillation of f at z is given by
(23)MOrpfz=1Brz∫Brzfw−f^rzpdvw1/p.Supposer>0 and 1≤p<∞, the space BMOrp consists of those functions f∈Llocpℂn such that
(24)fBMOrp=supz∈ℂnMOrpfz<∞.The following proposition reveals that the functionfr^z in (23) can be replaced by any complex-valued function in ℂn.Proposition 4.
Let1≤p<∞, f∈Llocpℂn, and r>0. Then, f∈BMOrp if and only if there exists a complex-valued function czonℂn such that
(25)1∣Brz∣∫Brzfw−czpdvw≤C,z∈ℂn,where C is a constant.Proof.
Iff∈BMOrp, then (25) holds with cz=f^rz and C=fBMOrpp.
Conversely, if (25) is satisfied, by the triangle inequality for the Lp-norm, we obtain
(26)MOrpfz=1Brz∫Brzfw−fr^zpdvw1/p≤1Brz∫Brzfw−czpdvw1/p+1Brz∫Brzfr^z−czpdvw1/p=1Brz∫Brzfw−czpdvw1/p+fr^z−cz.
For1<p<∞, by the Hölder’s inequality, one can get
(27)f^rz−cz=1Brz∫Brzfw−czdvw≤1Brz∫Brzfw−czpdvw1/p.
Forp=1, it is obvious that
(28)∣f^rz−cz∣≤1∣Brz∣∫Brz∣fw−cz∣dvw.
So we have(29)MOrpfz≤21Brz∫Brzfw−czpdvw1/p,which implies
(30)fBMOrpp≤2pC.
This completes the proof.☐LetHBrz be the set of all analytic functions on Brz.Lemma 5.
Let1≤p<∞, f∈Llocpℂn, and r>0. If for any z∈ℂn, there exist functions h1,h2∈HBrz and constants C1,C2>0 such that
(31)1∣Brz∣∫Brzfw−h1wpdvw≤C1,1∣Brz∣∫Brzf¯w−h2wpdvw≤C2,then f∈BMOrp.Proof.
The approach is similar to [13]. It is well known that, if v:B0,1⟶ℝ is pluriharmonic, there exists some pluriharmonic function u:B0,1⟶ℝ such that u+iv∈HB0,1. Theorem 1 from [17] tells us there is some constant C>0 such that
(32)u−u0LpB0,1,dv≤CvLpB0,1,dv.
Hence, for anyr>0, by change of variables, we know that, if v:Brz⟶ℝ is pluriharmonic, there exists a pluriharmonic function u such that u+iv∈HBrz and
(33)u−uzLpBrz,dv≤CvLpBrz,dv.
Forf∈Llocpℂn, set fp,z=1/∣Brz∣∫Brzfpdv1/p. By the triangle inequality, we have
(34)f+f¯2−h1+h22p,z≤f−h12p,z+f¯−h22p,z≤12C11/p+C21/p.
Sincef+f¯ is real-valued, we get
(35)Imh1+h22p,z≤12C11/p+C21/p.
Notice thath1+h2∈HBrz, by (33), one has
(36)Reh1+h22−Reh1+h22zp,z≤CImh1+h22p,z≤C2C11/p+C21/p.
By the triangle inequality, there holds(37)f+f¯2−Reh1+h22zp,z≤f+f¯2−Reh1+h22p,z+Reh1+h22−Reh1+h22zp,z≤f+f¯2−h1+h22p,z+Reh1+h22−Reh1+h22zp,z≤1+C2C11/p+C21/p.
Similarly, there holds(38)Imf−f¯2−Imh1−h22zp,z≲C11/p+C21/p.
Setcz=Reh1+h2/2z+i·Imh1−h2/2z. By the triangle inequality again, we have
(39)f−czp,z≲C11/p+C21/p.
From this and Proposition4, the desired result follows immediately.☐To arrive at the decomposition theorem, we need to introduce two subspaces ofBMOrp: one is BOr, the other is BArp.For a continuous functionf on ℂn and radius r>0, let
(40)ωrfz=supfz−fw:w∈Brz.The functionωrfz is called the oscillation of f at z. For r>0, let BOr denote the space of all continuous functions f on ℂn such that
(41)fBOr=supz∈ℂnωrfz<∞.Before we give a characterization of functionsf∈BOr, we need the following lemma.Lemma 6.
There holds(42)Bρz,r1+r⊆Brz⊆Bρz,r1−r,for z∈ℂn and r∈0,1.Proof.
Ifw∈Brz, take the curve γt=z+tz−w. For t∈0,1, γt∈Brz. By (15), there holds
(43)ργt≥1−rρz,∀t∈0,1.
It follows that(44)dz,w≤∫01∣z−w∣ργtdt≤∣z−w∣1−rρz≤rρz1−rρz=r1−r,which makes Brz⊆Bρz,r/1−r.
Ifw∉Brz, take any piecewise C1-curve γ:0,1⟶ℂn connecting z and w, and let T0 be the minimum time such that ∣z−γT0∣=rρz. By (15), there holds
(45)ργt≤1+rρz,∀t∈0,T0.
This implies that(46)∫01∣γ′t∣ργtdt≥∫0T0∣γ′t∣ργtdt≥∫0T0∣γ′t∣1+rρzdt.
With the fact that∫0T0∣γ′t∣dt≥rρz, one has dz,w≥r/1+r. It follows that Bρz,r/1+r⊆Brz. The proof is completed.☐Proposition 7.
Supposer∈0,1. Then, the space BOr is independent of r. Moreover, a continuous function f on ℂn belongs to BOr if and only if for any z,w∈ℂn, there exists a constant Cf,r>0 such that
(47)fz−fw≤Cf,rdz,w+1,where Cf,r≃fBOr.Proof.
Supposef∈BOr. Set Mr=r/1+2r<r/1+r. According to Lemma 6, we get w∈Brz and z∈Brw whenever dz,w≤Mr.
Ifdz,w≤Mr, then for any z∈ℂn, there holds
(48)∣fz−fw∣≤ωrfz≤fBOrdz,w+1.
TakeCf,r=fBOr, then the inequality (47) holds in the case of dz,w≤Mr.
Ifdz,w>Mr, we replace points z0=z,z1,⋯,zN=w on the geodesic in the d-metric from z to w in such a way: dzi,zi−1=Mr for i=1,2,⋯,N−1 and dzN,zN−1≤Mr. Note that it is possible to avoid using the completeness of the d-metric. More specifically, we do not have to use geodesics. Instead, we can simply use a curve from z and w whose length in the d-metric differs slightly from dz,w. Since dz,w=N−1Mr+dzN−1,w, one can get
(49)Mr+dz,w=NMr+dzN−1,w≥NMr.
It follows that(50)N≤1+Mr−1dz,w≤2+1rdz,w+1.
By the triangle inequality, there holds(51)∣fz−fw∣≤∑i=1N∣fzi−fzi−1∣≤∑i=1Nωrfzi≤NfBOr.
TakeCf,r=2+1/rfBOr, then the inequality (47) comes from combining (50) and (51).
Conversely, suppose (47) holds. Set Cr=2r/1−r. According to Lemma 6, we have dz,w≤Cr whenever w∈Brz. By (47), there holds
(52)fz−fw≤Cr+1Cf,r,∀z∈ℂn,w∈Brz.
It follows that(53)fBOr≲Cf,r.
This completes the proof.☐From the above proposition, we have proved thatBOr is independent of the radius r for any r∈0,1. So one can simply write BO for short.For1≤p<∞ and r>0, let BArp denote the space of all locally p-th integrable functions f on ℂn such that
(54)fBArp=supz∈ℂnfp^rz1/p<∞.Given a Borel measureμ, a radius r>0, and 1≤t<∞, we define the averaging function with respect to μ as follows:
(55)μ^rz=1Brz∫Brzdμw,∀z∈ℂn.Supposeμ is a Borel measure. Let r>0 and 1≤t<∞; we define the t-Berezin transform with respect to μ as follows:
(56)μ~tz=∫ℂnkt,zwte−tϕwdμw.In particular, for a functionf on ℂn, set dμ=fdv, then
(57)f~tz=∫ℂnfwkt,zwte−tϕwdvw.For1≤p<∞, we call a positive Borel measure μ on ℂn a Carleson measure if there is a positive constant C such that
(58)∫ℂngwe−ϕwpdμw1/p≤CgLpϕ,for every function g∈Fpϕ. This definition means that the identity mapping
(59)Iμ:Fpϕ⟶Lpℂn,e−pϕdμ,is bounded. The following lemma characterizes the Carleson measure, and one can find it from Theorem 3.2 in [3] and its proof.Lemma 8.
Letμ be a finite positive Borel measure on ℂn. Assume 0<p<∞. Then, the following conditions are all equivalent:
(A)
The measureμ is a Carleson measure for Fpϕ(B)
The functionμ~t∈L∞ℂn,dv for t>0(C)
The functionμ^r∈L∞ℂn,dv for 0<r<α. Moreover(60)Iμp≲μ~tL∞ℂn,dv≃μ^rL∞ℂn,dv.Setdμ=fpdv. Then, we have μ^rz=fp^rz and μ~tz=fp~tz. The following lemma comes immediately.Lemma 9.
Let1≤p<∞ and f∈Llocpℂn. Then, the following conditions are all equivalent:
(A)
The measuredμ=fpdv is a Carleson measure for Fpϕ(B)
The functionfp~t∈L∞ℂn,dv for t>0(C)
The functionfp^r∈L∞ℂn,dv for 0<r<α, i.e., f∈BArp. Moreover(61)Iμp≲fp~tL∞ℂn,dv≃fBArpp.From the previous lemma, one can see that the spaceBArp is independent of the choice of r∈0,α. So we will write BAp for short.Lemma 10.
Letp>0,k∈ℝ, then there holds
(62)∫ℂnρwke−pdz,wdvw≲ρzk+2n.Proof.
See [2].☐Theorem 11.
Let0<r<α, 1≤p<∞, and 0<t,l<∞. Then, the following conditions are all equivalent:
(A)
f∈BMOrp(B)
f admits a decomposition f=f1+f2 where f1∈BOr and f2∈BArp(C)
The function∫ℂnfw−f~tzp·kl,zwle−lϕwdvw is bounded. Moreover(63)f1BOr+f2BArp≃fBMOrp.Proof.
A⇒B: suppose f∈BMOrp. For w∈Br/4z, we have Br/4w⊆Brz. In fact, let ξ∈Br/4w. Since 0<r<α≤1, using inequality (15), we know that
(64)1−r4ρz≤ρw≤1+r4ρz.
Applying the triangle inequality, we get(65)ξ−z≤∣ξ−w∣+∣w−z∣<r4ρw+r4ρz≤1+r4r4ρz+r4ρz<rρz.
Setf1=f^r/4 and f2=f−f1. For 1<p<∞ and w∈Br/4z, by the triangle inequality and Hölder’s inequality, one can get
(66)f1z−f1w≤f1z−f^rz+f^rz−f1w≤1∣Br/4z∣∫Br/4z∣fu−f^rz∣dvu+1∣Br/4w∣∫Br/4w∣fu−f^rz∣dvu≲1∣Brz∣∫Brzfu−f^rzpdvu1/p.
Forp=1 and w∈Br/4z, there also holds
(67)∣f1z−f1w∣≲1∣Brz∣∫Brz∣fu−f^rz∣dvu.
Thusf1∈BOr/4. By Proposition 7, we have f1∈BOr. Moreover,
(68)f1BOr≲fBMOrp.
Next, we prove the part with regard tof2. By the triangle inequality, one can get
(69)MOr/4pfz≤1∣Br/4z∣∫Br/4zfu−f^rzpdvu1/p+f^rz−f^r/4z.
For1<p<∞, using the Hölder’s inequality, there holds
(70)∣f^rz−f^r/4z∣=1∣Br/4z∣∫Br/4zfu−f^rzdvu≤1∣Br/4z∣∫Br/4zfu−f^rzpdvu1/p.
Forp=1, the above inequality is obvious. So we conclude that MOr/4pfz≲MOrpfz for all z∈ℂn.
(71)f2p^r/4z1/p=1∣Br/4z∣∫Br/4zfu−f1updvu1/p≤1∣Br/4z∣∫Br/4zfu−f^r/4zpdvu1/p+1∣Br/4z∣∫Br/4zf1u−f1zpdvu1/p≤MOr/4pfz+ωr/4f1z≲MOrpfz+ωrf1z.
Thus,f2∈BAr/4p. By Lemma 9, we have f2∈BArp. Moreover,
(72)f2BArp≲fBMOrp.
(B)⇒ (C): set f=f1+f2 where f1∈BOr and f2∈BArp. By the triangle inequality, there holds
(73)∫ℂnf1w−f1~tzp·kl,zwle−lϕwdvw1/p≤∫ℂnf1w−f1zp·kl,zwle−lϕwdvw1/p+∫ℂnf1z−f1~tzp·kl,zwle−lϕwdvw1/p=∫ℂnf1w−f1zp·kl,zwle−lϕwdvw1/p+f1z−f1~tz.
By (17) and Lemma 3, we have
(74)kl,zwle−lϕw≃ρznl−2nρw−nle−σldz,w.
Sincex+1ae−bx⟶0 as x⟶+∞ for any a,b>0, there exists a constant C such that
(75)supz,w∈ℂndz,w+1pe−1/2σldz,w≤C.
By Proposition7, Lemma 10, (74), and (75), one can get
(76)∫ℂnf1w−f1zp·kl,zwle−lϕwdvw≲f1BOrp∫ℂndz,w+1pρznl−2nρw−nle−σldz,wdvw≲f1BOrp∫ℂnρznl−2nρw−nle−1/2σldz,wdvw≲f1BOrp.
For1<p<∞, by Hölder’s inequality and (76), we obtain
(77)∣f1z−f1~tz∣=∫ℂnf1z−f1w·kt,zwte−tϕwdvw≤∫ℂnf1w−f1zp·kt,zwte−tϕwdvw1/p≲f1BOr.
The same is true for (77) when p=1. By (73), (76), and (77), there holds
(78)∫ℂnf1w−f1~tzp·kl,zwle−lϕwdvw1/p≲f1BOr.
Next, we prove the part with regard tof2. For 1<p<∞, applying Hölder’s inequality, we obtain
(79)∣f2~tz∣≲f2p~tz1/p.
Forp=1, inequality (79) is obvious. By the triangle inequality, (79), and Lemma 9, there holds
(80)∫ℂnf2w−f2~tzp·kl,zwle−lϕwdvw1/p≤∫ℂnf2wp·kl,zwle−lϕwdvw1/p+∫ℂnf2~tzp·kl,zwle−lϕwdvw1/p=f2p~lz1/p+f2~tz≤f2p~lz1/p+f2p~tz1/p≲f2BArp.
By (78), (80), and the triangle inequality, one can get
(81)∫ℂnfw−f~tzp·kl,zwle−lϕwdvw1/p≤f1BOr+f2BArp.
(C)⇒ (A): by (18) and ∣Brz∣≃ρz2n, we get
(82)1∣Brz∣≃kl,zwle−lϕw,w∈Brz.
Hence, there holds(83)1∣Brz∣∫Brzfw−f~tzpdvw≃∫Brzfw−f~tzp·kl,zwle−lϕwdvw≤∫ℂnfw−f~tzp·kl,zwle−lϕwdvw.
By Proposition4 with cz=f~tz, we have f∈BMOrp.
The estimation (63) comes from (68), (72), (81), and (83). The proof is finished.☐


## 4. Bounded Hankel Operators

In this section, we are devoted to characterizing complex-valued functionsf in S such that the big Hankel operators Hf and Hf¯ are simultaneously bounded from Fpϕ into Lpϕ with 1≤p<∞.Theorem 12.
Let1≤p<∞, 0<r<α, ϕ∈W, and f∈S. If f∈BOr, then Hf:Fpϕ⟶Lpϕ is bounded. Moreover,
(84)Hf≲fBOr.Proof.
By Lemma3 and Proposition 7, we have
(85)∣Hfgz∣e−ϕz=∣I−Pfgz∣e−ϕz≤e−ϕz∫ℂn∣fz−fw∣·∣gw∣·∣Kϕz,w∣e−2ϕwdvw≲fBOr∫ℂndz,w+1ρz−nρw−ne−σdz,w∣gw∣e−ϕwdvw.
By Lemma10, one can get
(86)∫ℂnρz−nρw−ne−1/2σdz,w≤C.
By (75) and (86), there holds
(87)Hfgze−ϕz≲fBOrgL∞ϕ,for every z∈ℂn. It follows that Hf is bounded from F∞ϕ to L∞ϕ and Hf≲fBOr.
Forp=1, applying Fubini’s theorem, (75), and (86), we have
(88)∫ℂn∣Hfgz∣e−ϕzdvz≲fBOr∫ℂn∣gw∣e−ϕwdvw∫ℂndz,w+1ρz−nρw−ne−σdz,wdvz≲fBOrgL1ϕ.
Thus, we see thatHf:F1ϕ⟶L1ϕ is bounded and Hf≲fBOr.
By the Riesz-Thorin interpolation theorem, we conclude thatHf:Fpϕ⟶Lpϕ is bounded for all 1≤p<∞ and Hf≲fBOr.☐Lemma 13.
Let1≤p<∞, 0<r<α, ϕ∈W, and f∈S. If f∈BArp, then Hf is bounded from Fpϕ to Lpϕ. Moreover,
(89)Hf≲fBArp.Proof.
By Lemma9, f∈BArp if and only if fpdv is a Carleson measure for Fpϕ. Moreover,
(90)∫ℂnfwgwe−ϕwpdvw1/p≲fBArpgLpϕ.for any g∈Fpϕ. Since P is bounded on Lpϕ, there holds
(91)HfgLpϕ≤I−P·fgLpϕ≲fBArpgLpϕ.
It follows thatHf is bounded and Hf≲fBArp.☐Theorem 14.
Let1≤p<∞, 0<r<α, ϕ∈W, and f∈S. Then, Hf and Hf¯ are both bounded from Fpϕ to Lpϕ if and only if f∈BMOrp. Moreover,
(92)Hf¯+Hf≃fBMOrp.Proof.
By (18), for any z∈ℂn, we have
(93)Hfp≥Hfkp,zLpϕp=∫ℂnfwkp,zw−Pfkp,zwpe−pϕwdvw≥∫Brzfw−kp,z−1wPfkp,zwpkp,zwpe−pϕwdvw≃1∣Brz∣∫Brzfw−kp,z−1wPfkp,zwpdvw.
Similarly, there holds(94)Hf¯p≳1∣Brz∣∫Brzf¯w−kp,z−1wPf¯kp,zwpdvw.
By (18), for 0<r<α and w∈Brz, we have kp,zw≠0. Thus,
(95)kp,zw−1Pfkp,zw∈HBrz,(96)kp,zw−1Pf¯kp,zw∈HBrz.
Therefore, by Lemma5, (93), and (94), we have f∈BMOrp. Furthermore, following the proof of Lemma 5, we will get f−czp,z≲Hf+Hf¯. Then, checking the proof of Proposition 4, we obtain
(97)fBMOrp≲Hf+Hf¯.
Conversely, supposef∈BMOrp. By Theorem 11, f has a decomposition f=f1+f2 where f1∈BOr and f2∈BArp. Then, by Lemma 12 and Lemma 13, Hf is bounded from Fpϕ to Lpϕ. Moreover,
(98)Hf≲f1BOr+f2BArp≃fBMOrp.
Sincef∈BMOrp if and only if f¯∈BMOrp, it follows that Hf¯ is bounded from Fpϕ to Lpϕ. Moreover,
(99)Hf¯≲f1BOr+f2BArp≃fBMOrp.
Then, the norm estimation follows from these two inequalities above and (97). This completes the proof.☐We immediately have the following corollaries.Corollary 15.
Let1≤p<∞, 0<r<α, ϕ∈W, and f∈S. If f is real-valued on ℂn, then Hf is bounded from Fpϕ to Lpϕ if and only if f∈BMOrp.Corollary 16.
Let1≤p<∞, 0<r<α, ϕ∈W, and f∈S. If f is analytic on ℂn, then Hf¯ is bounded from Fpϕ to Lpϕ if and only if f∈BMOrp.


## 5. The Space VMO

Supposer>0. We say that f belongs to VMOrp if f∈BMOrp and
(100)lim∣z∣⟶∞MOrpfz=0.As in the space BMO, we have the following two lemmas. The proof is similar to the corresponding case of BMO. One can prove them with easy modifications, so we omit details here.Lemma 17.
Let1≤p<∞, f∈Llocpℂn, and r>0. Then, f∈VMOrp if and only if there exists a complex-valued function cz on ℂnsuch that
(101)lim∣z∣⟶∞1∣Brz∣∫Brzfw−czpdvw=0.Lemma 18.
Let1≤p<∞, f∈Llocpℂn, and r>0. If for any z∈ℂn, there exist functions h1,h2∈HBrz such that
(102)lim∣z∣⟶∞1∣Brz∣∫Brzfw−h1wpdvw=0,lim∣z∣⟶∞1∣Brz∣∫Brzf¯w−h2wpdvw=0,then f∈VMOrp.Letr>0. We define VOr as the space of complex-valued functions f in BOr satisfying
(103)lim∣z∣⟶∞ωrfz=0.DefineVArp as the space of complex-valued functions f in BArp satisfying
(104)lim∣z∣⟶∞fp^rz1/p=0.The next result shows thatVOr is independent of the radius r.Lemma 19.
Let0<r1,r2<α. Then, f∈VOr1 if and only if f∈VOr2.Proof.
Without loss of generality, we may assumer1<r2. Since ωr1fz≤ωr2fz, we have f∈VOr1 if f∈VOr2.
Conversely, supposef∈VOr1. For any z∈ℂn, since f is continuous on ℂn, there holds
(105)ωr2fz=sup∣fz−fu∣:u∈Br2z¯.
Hence, there exists somew∈Br2z¯ such that
(106)ωr2fz=fz−fw.
For a sufficiently smallε>0, 1+εr2<1, we can apply Lemma 6 with r=1+εr2 and obtain
(107)Br2z¯⊆B1+εr2z⊆Bρz,1+εr21−1+εr2.
Therefore,(108)dz,w<1+εr21−1+εr2.
Letε⟶0, we get dz,w≤r2/1−r2. We then follow a similar argument as Proposition 7. We replace points
(109)z0=z,z1,⋯,zN=w,on the geodesic in the d-metric from z to w in such a way: dzi,zi−1=Mr1 for i=1,2,⋯,N−1 and dzN,zN−1≤Mr1 where Mr1=r1/1+2r1 is an appropriate constant such that zi∈Br1zi−1. We will get a similar inequality as (50):
(110)N≤2+1r1dz,w+1≤2+1r1r21−r2+1≕N0.
Sincef∈VOr1, for any ε>0, there exists some positive number M1 such that
(111)ωr1fz<εN0,∣z∣>M1.
By the triangle inequality, we get(112)∣z−zi∣≤∣z−z1∣+⋯+∣zi−zi−1∣≤r1ρz+⋯+r1ρzi−1≤N0r1supz∈ℂnρz≕M2.
For∣z∣≥M1+M2, we clearly have ∣zi∣≥∣z∣−∣zi−z∣≥M1. Thus, if we set M0=M1+M2, then for ∣z∣>M0, there holds
(113)ωr2fz=∣fz−fw∣≤∑i=1N∣fzi−fzi−1∣≤∑i=1Nωr1fzi≤NεN0≤ε.
The proof is finished.☐A positive Borel measureμ on ℂn is called a vanishing Carleson measure if
(114)limj⟶∞∫ℂngjwe−ϕwpdμw=0,whenevergjj=1∞ is a bounded sequence in Fpϕ and converges to 0 uniformly on any compact subset of ℂn as j⟶∞.The following lemma characterizes the vanishing Carleson measure, and one can find it from Theorem 3.9 in [3].Lemma 20.
Letμ be a finite positive Borel measure on ℂn. Assume 0<p<∞. Then, the following conditions are all equivalent:
(A)
The measureμ is a vanishing Carleson measure(B)
The functionμ~t⟶0 as ∣z∣⟶∞ for t>0(C)
The functionμ^r⟶0 as ∣z∣⟶∞ for 0<r<αSetdμ=fpdv. Then, we have μ^rz=fp^rz and μ~tz=fp~tz. The following lemma comes immediately.Lemma 21.
Let1≤p<∞ and f∈Llocpℂn. Then, the following conditions are all equivalent:
(A)
The measuredμ=fpdv is a vanishing Carleson measure(B)
The functionfp~t⟶0 as ∣z∣⟶∞ for t>0(C)
The functionfp^r⟶0 as ∣z∣⟶∞ for 0<r<α, i.e., f∈VArpFrom the previous lemma, one can see that the spaceVArp is independent of the choice of r∈0,α. So we will write VAp for short.By the following decomposition theorem, we know that the spaceVMOrp is independent of the choice of r∈0,α.Theorem 22.
Let0<r<α, 1≤p<∞, and 0<t,l<∞. Then, the following conditions are all equivalent:
(A)
f∈VMOrp(B)
f admits a decomposition f=f1+f2 where f1∈VOr and f2∈VArp(C)
The function∫ℂnfw−f~tzp·kl,zwle−lϕdvw⟶0 as ∣z∣⟶∞Proof.
The proof is similar to Theorem11. Notice that kl,z converges uniformly to 0 as ∣z∣⟶∞ on any compact subset of ℂn. One can conduct the same argument to prove that (B) implies (C).☐


## 6. Compact Hankel Operators

Theorem 23.
Let1≤p<∞, 0<r<α, ϕ∈W, and f∈S. Then, Hf and Hf¯ are both compact from Fpϕ to Lpϕ if and only if f∈VMOrp.Proof.
Notice thatkp,z converges to 0 on any compact subset of ℂn. Similar to the proof of Theorem 14, we have
(115)lim∣z∣⟶∞1∣Brz∣∫Brzfw−kp,z−1wPfkp,zwpdvw=0,lim∣z∣⟶∞1∣Brz∣∫Brzf¯w−kp,z−1wPf¯kp,zwpdvw=0.
By (95), (96), and Lemma 18, we get f∈VMOrp.
Conversely, supposef∈VMOrp. Let fR≔fχB0,R where χB0,R is the characteristic function of B0,R. Since f∈VMOrp, for any ε>0, there is some R>0 such that MOrpfz<ε when ∣z∣>R/2. Without loss of generality, we may suppose r<R/4M where M=supz∈ℂnρz. Then, we have MOrpf−fRz=MOrpfz<ε for ∣z∣>2R. Therefore, f−fRBMOrp⟶0 as R⟶∞. Applying Theorem 14 to f−fR, there holds
(116)Hf−fR≲f−fRBMOrp.
By the fact that the space of compact operators is norm closed, we only need to prove thatHfR:Fpϕ⟶Lpϕ is compact.
FixR>0. Let gjj=1∞ be a bounded sequence in Fpϕ that converges to 0 uniformly on any compact subset of ℂn. From Theorem 11, we know that fR^r belongs to BOr, since fR∈BMOrp. It is clear that fR^r has compact support. For z in some compact subset of ℂn, using Proposition 7, we obtain
(117)∣fR^rz−fR^r0∣≲dz,0+1<∞.
Hence,fR^r is bounded. From Lemma 3.3 of [3], we know that
(118)∫ℂngjzpe−pϕzfRpzdvz≤C∫ℂngjzpe−pϕzfRp^rzdvz.
Therefore,(119)fRgjLpϕ=∫ℂnfRzgjzpe−pϕzdvz1/p≲∫ℂngjzpe−pϕzfRp^rzdvz1/p=∫Egjzpe−pϕz1Brz∫BrzfRwpdvwdvz1/p,
whereE is the support of fRp^r. We also have
(120)fRw−fR^rz+fR^rzp≲fRw−fR^rzp+fR^rzp.
For everyε>0, there is some j0 such that
(121)supz∈Egjz<ε,for j≥j0. Consequently,
(122)fRgjLpϕ≲∫Egjzpe−pϕz1Brz∫BrzfRw−fR^rzp+fR^rzpdvwdvz1/p=∫Egjzpe−pϕzMOrpfRzp+fR^rzpdvz1/p≲εfRBMOrp+εfR^rL∞ℂn,dv,
forj≥j0. This means that limj⟶∞fRgjLpϕ=0. Since fRgj∈Lpϕ and P is bounded on Lpϕ, there holds
(123)HfRgjLpϕ≤I−P·fRgjLpϕ.
Thus,HfR is compact. Since f∈VMOrp if and only if f¯∈VMOrp, Hf¯ is also compact.☐Corollary 24.
Let1≤p<∞, 0<r<α, ϕ∈W, and f∈S. If f∈VOr, then Hf:Fpϕ⟶Lpϕ is compact.Corollary 25.
Let1≤p<∞, 0<r<α, ϕ∈W, and f∈S. If f∈VArp, then Hf:Fpϕ⟶Lpϕ is compact.Corollary 26.
Let1≤p<∞, 0<r<α, ϕ∈W, and f∈S. If f is real-valued on ℂn, then Hf is compact from Fpϕ to Lpϕ if and only if f∈VMOrp.Corollary 27.
Let1≤p<∞, 0<r<α, ϕ∈W, and f∈S. If f is analytic on ℂn, then Hf¯ is compact from Fpϕ to Lpϕ if and only if f∈VMOrp.



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