The Essential Norm of the Generalized Hankel Operators on the Bergman Space of the Unit Ball in 𝐶^𝑛

Abstract

In 1993, Peloso introduced a kind of operators on the Bergman space 𝐴²(𝐵) of the unit ball that generalizes the classical Hankel operator. In this paper, we estimate the essential norm of the generalized Hankel operators on the Bergman space 𝐴^𝑝(𝐵)(𝑝>1) of the unit ball and give an equivalent form of the essential norm.

1. Introduction

Let 𝐵 be the open unit ball in 𝐶^𝑛, 𝑚 the Lebesgue measure on 𝐶^𝑛 normalized so that 𝑚(𝐵)=1, 𝐻(𝐵) denotes the class of all holomorphic functions on 𝐵. The Bergman space 𝐴²(𝐵) is the Banach space of all holomorphic functions 𝑓 on 𝐵 such that ∫_𝐵|𝑓(𝑧)|²𝑑𝑚(𝑧)<∞. It is easy to show that 𝐴²(𝐵) is a closed subspace of 𝐿²(𝐵,𝑑𝑚).

There is an orthogonal projection of 𝐿²(𝐵,𝑑𝑚) onto 𝐴²(𝐵), denoted by 𝑃 and 𝑃𝑓(𝑧)=_𝐵𝐾(𝑧,𝑤)𝑓(𝑤)𝑑𝑚(𝑤), (1.1) where 𝐾(𝑧,𝑤)=1/(1−⟨𝑧,𝑤⟩)^𝑛+1 is the Bergman kernel on 𝐵.

For a function 𝑓∈𝐻(𝐵), define the Hankel operator 𝐻_𝑓∶𝐴²(𝐵)→𝐴²(𝐵)^⟂ with symbol 𝑓 by 𝐻_𝑓𝑔=(𝐼−𝑃)𝑓𝑔=_𝐵𝑓(𝑧)−𝑓(𝑤)𝐾(𝑧,𝑤)𝑔(𝑤)𝑑𝑚(𝑤), (1.2) where 𝐼 is the identity operator.

Since the Hankel operator 𝐻_𝑓 is connected with the Toeplitz operator, the commutator, the Bloch space, and the Besov space, it has been extensively studied. Important papers in this context are [1, 2] for the case 𝑛=1 and [3–5] for the case 𝑛>1. It is known that 𝐻_𝑓 is bounded on 𝐴²(𝐵) if and only if 𝑓∈𝛽(𝐵) and 𝐻_𝑓 is compact 𝐴²(𝐵) if and only if 𝑓∈𝛽₀(𝐵), where 𝛽(𝐵)=𝑓∈𝐻(𝐵)∶sup_𝑧∈𝐵1−|𝑧|²(𝑧,𝛽𝑅𝑓)<∞₀(𝐵)=𝑓∈𝐻(𝐵)∶1−|𝑧|²𝑅𝑓(𝑧)⟶0,as|𝑧|⟶1. (1.3)

𝑅𝑓 is the radial derivative of 𝑓 defined by 𝑅𝑓(𝑧)=^𝑛_𝑗=1𝑧_𝑗𝜕𝑓(𝑧)𝜕𝑧_𝑗. (1.4)

𝛽(𝐵) is called the Bloch space, and 𝛽₀(𝐵) is called the little Bloch space.

For 𝑛=1, 𝑓∈𝐻(𝐷)  (𝐷 is the open unit disc), 𝐻_𝑓 is in the Schatten class 𝑆_𝑝(1<𝑝<∞) if and only if 𝑓∈𝐵_𝑝(𝐷); 𝐻_𝑓∈𝑆_𝑝(0<𝑝≤1) if and only if 𝑓 is a constant, where 𝐵_𝑝(𝐷)=𝑓∈𝐻(𝐷)∶𝑓^′(𝑧)1−|𝑧|²∈𝐿^𝑝(𝑑𝜆),𝑝>1, (1.5) and 𝑑𝜆(𝑧)=(1−|𝑧|²)⁻²𝑑𝑚(𝑧) is the invariant volume measure on 𝐷, 𝐵_𝑝(𝐷) is called the Besov space on 𝐷. This theorem expresses that there is a cutoff of 𝐻_𝑓 at 𝑝=1.

For 𝑛>1, 𝑓∈𝐻(𝐵), 𝐻_𝑓∈𝑆_𝑝(2𝑛<𝑝<∞) if and only if 𝑓∈𝐵_𝑝(𝐵), 𝐻_𝑓∈𝑆_𝑝(0<𝑝≤2𝑛) if and only if 𝑓 is a constant, where 𝐵_𝑝(𝐵)=𝑓∈𝐻(𝐵)∶1−|𝑧|²𝑅𝑓(𝑧)∈𝐿^𝑝(𝑑𝜆),𝑝>𝑛, (1.6) and 𝑑𝜆(𝑧)=(1−|𝑧|²)^−(𝑛+1)𝑑𝑚(𝑧) is the invariant volume measure on 𝐵. 𝐵_𝑝(𝐵) is called the Besov space on 𝐵. Then, the cutoff phenomenon of 𝐻_𝑓 appears at 𝑝=2𝑛. If 𝑐(𝑛) denotes the value of “cutoff,” then 𝑐(𝑛)=1,𝑛=1,2𝑛,𝑛>1. (1.7) Obviously, 𝑐(𝑛) depends on the dimension 𝑛 of the unit ball.

In 1993, Peloso [3] replaced 𝑓(𝑧)−𝑓(𝑤) with Δ_𝑗𝑓(𝑤,𝑧)=𝑓(𝑤)−_|𝛼|<𝑗𝐷^𝛼𝑓(𝑧)()𝛼!𝑤−𝑧^𝛼 (1.8) to define a kind of generalized Hankel operator: 𝐻_𝑓,𝑗𝑔(𝑧)=_𝐵−Δ_𝑗𝑓()𝐾()𝑔(𝑤)(𝑤),𝐻𝑤,𝑧𝑧,𝑤𝑑𝑚_′𝑓,𝑗𝑔(𝑧)=_𝐵Δ_𝑗𝑓(𝑧,𝑤)𝐾(𝑧,𝑤)𝑔(𝑤)𝑑𝑚(𝑤). (1.9) Here, (𝐷^𝛼𝑓)(𝑧)=(𝜕^|𝛼|𝑓(𝑧))/(𝜕𝑧^𝛼1₁⋯𝜕𝑧^𝛼𝑛_𝑛). Clearly, if 𝑗=1, 𝐻_𝑓,1 and 𝐻^′_𝑓,1 are just the classical Hankel operator 𝐻_𝑓. He proved that 𝐻_𝑓,𝑗 has the same boundedness and compactness properties as 𝐻_𝑓, but the Schatten class property of 𝐻_𝑓,𝑗 is different from that of 𝐻_𝑓. If 𝑛≥2, 𝑓∈𝐻(𝐵), 𝐻_𝑓,𝑗∈𝑆_𝑝((2𝑛/𝑗)<𝑝<∞) if and only if 𝑓∈𝐵_𝑝(𝐵); if 0<𝑝≤(2𝑛/𝑗), 𝐻_𝑓,𝑗∈𝑆_𝑝 if and only if 𝑓 is a polynomial of degree at most 𝑗−1. So the value of “cutoff” of 𝐻_𝑓,𝑗 is 2𝑛/𝑗; this means that the cutoff constant 𝑐(𝑛) depends not only on the dimension but also on the degree of the polynomial _|𝛼|<𝑗𝐷^𝛼𝑓(𝑧)()𝛼!𝑤−𝑧^𝛼, (1.10) and we are able to lower the cutoff constant by increasing 𝑗.

The cutoff phenomenon expressed that the generalized Hankel operator 𝐻_𝑓,𝑗 defined by Peloso and the classical Hankel operator 𝐻_𝑓 are different.

In the present paper, we will consider the generalized Hankel operators 𝐻_𝑓,𝑗 defined by Peloso on the Bergman space 𝐴^𝑝(𝐵) which is the Banach space of all holomorphic functions 𝑓 on 𝐵 such that ∫_𝐵|𝑓(𝑧)|^𝑝𝑑𝑚(𝑧)<∞, for 𝑝>1.

For 𝑓(𝑧)∈𝐻(𝐵), 𝑗 is a positive integer, and we define the generalized Hankel operators 𝐻_𝑓,𝑗 and 𝐻^′_𝑓,𝑗 of order 𝑗 with symbol 𝑓 by 𝐻_𝑓,𝑗𝑔(𝑧)=_𝐵−Δ_𝑗𝑓()𝐾()𝑔(𝑤)(𝑤),𝐻𝑤,𝑧𝑧,𝑤𝑑𝑚_′𝑓,𝑗𝑔(𝑧)=_𝐵Δ_𝑗𝑓(𝑧,𝑤)𝐾(𝑧,𝑤)𝑔(𝑤)𝑑𝑚(𝑤), (1.11) where 𝑔∈𝐴^𝑝(𝐵), Δ_𝑗𝑓(𝑤,𝑧)=𝑓(𝑤)−_|𝛼|<𝑗𝐷^𝛼𝑓(𝑧)()𝛼!𝑤−𝑧^𝛼,Δ_𝑗𝑓(𝑧,𝑤)=𝑓(𝑤)−_|𝛼|<𝑗𝐷^𝛼𝑓(𝑤)()𝛼!𝑧−𝑤^𝛼. (1.12)

Luo and Ji-Huai [6] studied the boundedness, compactness, and the Schatten class property of the generalized Hankel operator 𝐻_𝑓,𝑗 on the Bergman space 𝐴^𝑝(𝐵)(𝑝>1), which extended the known results.

We will study the essential norm of this kind of generalized Hankel operators 𝐻_𝑓,𝑗 and 𝐻^′_𝑓,𝑗. We recall that the essential norm of a bounded linear operator 𝑇 is the distance from 𝑇 to the compact operators; that is, ‖𝑇‖_ess=inf‖𝑇−𝐾‖∶𝐾isacompactoperator. (1.13)

The essential norm of a bounded linear operator 𝑇 is connected with the compactness of the operator 𝑇 and the spectrum of the operator 𝑇.

We know that ‖𝑇‖_ess=0 if and only if 𝑇 is compact, so that estimates on ‖𝑇‖_ess lead to conditions for 𝑇 to be compact. Thus, we will obtain a different proof of the compactness of the generalized Hankel operators 𝐻_𝑓,𝑗 and 𝐻^′_𝑓,𝑗.

Throughout the paper, 𝐶 denotes a positive constant, whose value may change from one occurrence to the next one.

2. Preliminaries

For any fixed point 𝑎∈𝐵−{0}, 𝑧∈𝐵, define the Möbius transformation 𝜑_𝑎 by 𝜑_𝑎(𝑧)=𝑎−𝑃_𝑎(𝑧)−𝑠_𝑎𝑄_𝑎(𝑧)⟨⟩,1−𝑧,𝑎 (2.1) where 𝑠_𝑎=1−|𝑎|² and 𝑃_𝑎 is the orthogonal projection from 𝐶^𝑛 onto the one-dimensional subspace [𝑎] generated by 𝑎, 𝑄_𝑎 is the orthogonal projection from 𝐶^𝑛 onto 𝐶^𝑛![𝑎]. It is clear that 𝑃_𝑎(𝑧)=⟨𝑧,𝑎⟩|𝑎|²𝑎,𝑧∈𝐶^𝑛,𝑄_𝑎(𝑧)⟨⟩=𝑧−𝑧,𝑎|𝑎|²𝑎,𝑧∈𝐵. (2.2)

Lemma 2.1. For every 𝑎∈𝐵, 𝜑_𝑎 has the following properties:(1)𝜑_𝑎(0)=𝑎 and 𝜑_𝑎(𝑎)=0,(2)𝜑_𝑎∘𝜑_𝑎(𝑧)=𝑧,𝑧∈𝐵,(3)1/(1−⟨𝜑_𝑎(𝑧),𝑎⟩)=(1−⟨𝑧,𝑎⟩)/(1−|𝑎|²),𝑧∈𝐵.

Proof. The proofs can be found in [7].

Lemma 2.2. For 𝑠>−1, 𝑡 real, define 𝐼_𝑡(𝑧)=_𝐵1−|𝑤|^2𝑠|1−⟨𝑧,𝑤⟩|^{𝑛+1+𝑠+𝑡}𝑑𝑚(𝑤),𝑧∈𝐵. (2.3) Then,(1)𝑡<0, 𝐼_𝑡(𝑧) is bounded in B,(2)𝑡=0, 𝐼_𝑡(𝑧)∼log(1/(1−|𝑧|²)) as |𝑧|→1⁻,(3)𝑡>0, 𝐼_𝑡(𝑧)∼(1−|𝑧|²)^−𝑡 as |𝑧|→1⁻.

Here, the notation 𝑎(𝑧)∼𝑏(𝑧) means that the ratio 𝑎(𝑧)/𝑏(𝑧) has a positive finite limit as |𝑧|→1⁻.

Proof. This is in [7, Theorem  1.12].

Lemma 2.3. Let 𝑘_𝜉(𝑧)=𝐾(𝑧,𝜉)/‖𝐾_𝜉‖_{𝐿𝑝(𝑑𝑚)}, where 𝐾_𝜉(𝑧)=𝐾(𝑧,𝜉)=1/(1−⟨𝑧,𝜉⟩)^𝑛+1, then 𝑘_𝜉(𝑧) has the following properties:(1)‖𝑘_𝜉‖_{𝐿𝑝(𝑑𝑚)}=1,(2)𝑘_𝜉(𝑧)→0 at every point 𝑧∈𝐵 as |𝜉|→1⁻.

Proof. It is obvious.

Lemma 2.4. Let 𝐾_𝜉(𝑧)=𝐾(𝑧,𝜉). Then, for any positive integer j,(1)𝐻_𝑓,𝑗𝐾_𝜉=−Δ_𝑗𝑓(𝜉,⋅)𝐾_𝜉,(2)𝐻^′_𝑓,𝑗𝐾_𝜉=Δ_𝑗𝑓(⋅,𝜉)𝐾_𝜉.

Proof. The proof is obtained by the definition of 𝐻_𝑓,𝑗 and 𝐻^′_𝑓,𝑗 and the reproducing property of 𝐾(𝑧,𝜉), through the direct computation to get them.

Lemma 2.5. Let 𝑗 be any positive integer, 𝑓∈𝐻(𝐵), and 0<𝑞<∞, then there is a constant 𝐶 independent of 𝑓, such that(1)(1−|𝑧|²)^𝑗|𝑅^𝑗𝑓(𝑧)|≤𝐶{∫_𝐵|Δ_𝑗𝑓(𝜑_𝑧(𝑤),𝑧)|^𝑞𝑑𝑚(𝑤)}^1/𝑞, (2)(1−|𝑧|²)^𝑗|𝑅^𝑗𝑓(𝑧)|≤𝐶{∫_𝐵|Δ_𝑗𝑓(𝑧,𝜑_𝑧(𝑤))|^𝑞𝑑𝑚(𝑤)}^1/𝑞,

where 𝑅^𝑗𝑓 is the 𝑗th order radial derivative of 𝑓, 𝑅^𝑗𝑓(𝑧)=^∞_𝑘=1𝑘^𝑗𝑓_𝑘(𝑧), (2.4) and 𝑓(𝑧)=∑^∞_𝑘=0𝑓_𝑘(𝑧) is the homogeneous expansion.

Proof. This is in [3, Proposition  3.2].

Lemma 2.6. Let 𝑗 be any positive integer, 𝑓∈𝐻(𝐵), and 0<𝜌<1,  𝑝>1, then (1)∫_𝐵|Δ_𝑗𝑓(𝑤,𝑧)|^𝑝((1−|𝑤|²)^−𝜌/|1−⟨𝑧,𝑤⟩|^𝑛+1)𝑑𝑚(𝑤)≤𝐶(1−|𝑧|²)^−𝜌(sup_𝑧∈𝐵(1−|𝑧|²)^𝑗|𝑅^𝑗𝑓(𝑧)|)^𝑝, (2)∫_𝐵|Δ_𝑗𝑓(𝑧,𝑤)|^𝑝((1−|𝑤|²)^−𝜌/|1−⟨𝑧,𝑤⟩|^𝑛+1)𝑑𝑚(𝑤)≤𝐶(1−|𝑧|²)^−𝜌(sup_𝑧∈𝐵(1−|𝑧|²)^𝑗|𝑅^𝑗𝑓(𝑧)|)^𝑝.

Proof. (1) Write 𝐹(𝑤,𝑧) for Δ_𝑗𝑓(𝑤,𝑧). Using the change of variables 𝑤=𝜑_𝑧(𝜉), we obtain _𝐵|𝐹(𝑤,𝑧)|^𝑝1−|𝑤|^2−𝜌|1−⟨𝑧,𝑤⟩|^𝑛+1(𝑤)=𝑑𝑚_𝐵𝐹𝜑_𝑧(𝜉),𝑧^𝑝1−𝜑_𝑧(𝜉)^2−𝜌1−𝑧,𝜑_𝑧(𝜉)^𝑛+11−|𝑧|^2𝑛+11−𝜉,𝑧^2(𝑛+1)(𝜉)=|𝑧|𝑑𝑚1−^2−𝜌_𝐵𝐹𝜑_𝑧(𝜉),𝑧^𝑝1−𝜉^2−𝜌1−𝜉,𝑧^{𝑛+1−2𝜌}𝑑𝑚(𝜉). (∗) Let 1<𝑞^′1<𝑚𝑖𝑛𝜌,𝑛+1𝑛+1−𝜌 (2.5) and set 𝑞=𝑞^′/(𝑞^′−1). Then, applying Hölder's inequality to (*), we obtain _𝐵|𝐹(𝑤,𝑧)|^𝑝1−|𝑤|^2−𝜌|1−⟨𝑧,𝑤⟩|^𝑛+1(𝑤)≤|𝑧|𝑑𝑚1−^2−𝜌_𝐵𝐹𝜑_𝑧(𝜉),𝑧^𝑝𝑞𝑑𝑚(𝜉)^1/𝑞_𝐵1−𝜉^{2−𝜌𝑞′}1−𝜉,𝑧^{(𝑛+1−2𝜌)𝑞′}𝑑𝑚(𝜉)^1/𝑞′. (2.6) Because of our choice of 𝑞^′, it follows that −𝜌𝑞^′>−1 and (𝑛+1−2𝜌)𝑞^′<𝑛+1−𝜌𝑞^′. Now, Lemma 2.2 implies that 1−𝜉_𝐵^{2−𝜌𝑞′}1−𝜉,𝑧^{(𝑛+1−2𝜌)𝑞′}𝑑𝑚(𝜉) (2.7) is bounded by a constant. Therefore, applying [3, Theorem  3.4], we get _𝐵Δ_𝑗𝑓(𝑤,𝑧)^𝑝1−|𝑤|^2−𝜌|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤)≤𝐶1−|𝑧|^2−𝜌sup_𝑧∈𝐵1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧)^𝑝. (2.8)(2) The proof of (2) is similar to that of (1).

3. The Main Result and Its Proof

Theorem 3.1. Let 𝑓∈𝐻(𝐵), 𝑗 any positive integer, 𝑝>1, and the generalized Hankel operators 𝐻_𝑓,𝑗, 𝐻^′_𝑓,𝑗 defined on 𝐴^𝑝(𝐵) by 𝐻_𝑓,𝑗𝑔(𝑧)=_𝐵−Δ_𝑗𝑓()𝐾()𝑔(𝑤)(𝑤),𝐻𝑤,𝑧𝑧,𝑤𝑑𝑚_′𝑓,𝑗𝑔(𝑧)=_𝐵Δ_𝑗𝑓(𝑧,𝑤)𝐾(𝑧,𝑤)𝑔(𝑤)𝑑𝑚(𝑤). (3.1) Suppose that 𝐻_𝑓,𝑗 and 𝐻^′_𝑓,𝑗 are bounded on 𝐴^𝑝(𝐵), then the following quantities are equivalent:(1)‖𝐻_𝑓,𝑗‖_ess and ‖𝐻^′_𝑓,𝑗‖_ess,(2)lim_|𝑧|→1⁻(1−|𝑧|²)^𝑗|𝑅^𝑗𝑓(𝑧)|,(3)lim_|𝑧|→1⁻(1−|𝑧|²)|𝑅𝑓(𝑧)|.

Particularly, 𝐻_𝑓,𝑗 and 𝐻^′_𝑓,𝑗 are compact on 𝐴^𝑝(𝐵) if and only if lim_|𝑧|→1⁻(1−|𝑧|²)|𝑅𝑓(𝑧)|=0.

Proof. First, we will prove that ‖𝐻_𝑓,𝑗‖_ess≥𝐶lim_|𝑧|→1⁻(1−|𝑧|²)^𝑗|𝑅^𝑗𝑓(𝑧)|. By the definition of 𝑘_𝜉(𝑧) of Lemmas 2.3 and 2.4, we have 𝐻_𝑓,𝑗𝑘_𝜉^𝑝_{𝐿𝑝()𝑑𝑚}=_𝐵𝐻_𝑓,𝑗𝑘_𝜉(𝑧)^𝑝(𝑧)=𝑑𝑚_𝐵𝐻_𝑓,𝑗𝐾(𝑧,𝜉)𝐾_{𝜉𝐿𝑝(𝑑𝑚)}^𝑝(𝑧)=1𝑑𝑚𝐾_𝜉^𝑝_{𝐿𝑝()𝑑𝑚}^𝐵𝐻_𝑓,𝑗𝐾(𝑧,𝜉)^𝑝(𝑧)=1𝑑𝑚𝐾_𝜉^𝑝_{𝐿𝑝()𝑑𝑚}^𝐵Δ_𝑗𝑓(𝜉,𝑧)^𝑝𝐾(𝑧,𝜉)^𝑝(𝑧)=1𝑑𝑚𝐾_𝜉^𝑝_{𝐿𝑝()𝑑𝑚}⋅𝐼, (3.2) here 𝐼=∫_𝐵|Δ_𝑗𝑓(𝜉,𝑧)|^𝑝|𝐾(𝑧,𝜉)|^𝑝𝑑𝑚(𝑧).
Use the change of variables 𝑧=𝜑_𝜉(𝜏) in the integral 𝐼, and recall that (𝑧)=𝜉𝑑𝑚1−²1−𝜏,𝜉^2𝑛+1𝑑𝑚(𝜏). (3.3) Thus 𝐼=_𝐵Δ_𝑗𝑓𝜉,𝜑_𝜉(𝜏)^𝑝1−𝜑_𝜉(𝜏),𝜉^𝑝(𝑛+1)⋅1−𝜉²1−𝜏,𝜉^2𝑛+1(𝜏)=1𝑑𝑚1−𝜉^{2(𝑛+1)(𝑝−1)𝐵}Δ_𝑗𝑓𝜉,𝜑_𝜉(𝜏)^𝑝1−𝜏,𝜉^{(2−𝑝)(𝑛+1)}(𝜏)≥1𝑑𝑚1−𝜉^{2(𝑛+1)(𝑝−1)𝐵}Δ_𝑗𝑓𝜉,𝜑_𝜉(𝜏)𝑑𝑚(𝜏)^𝑝×_𝐵11−𝜏,𝜉^{(2−𝑝)(𝑛+1)/(1−𝑝)}𝑑𝑚(𝜏)^1−𝑝≥𝐶1−𝜉^{2(𝑛+1)(𝑝−1)𝐵}Δ_𝑗𝑓𝜉,𝜑_𝜉(𝜏)𝑑𝑚(𝜏)^𝑝≥𝐶1−𝜉^{2(𝑛+1)(𝑝−1)}1−𝜉^2𝑗𝑅^𝑗𝑓(𝜉)^𝑝. (3.4) Here, we have used (3) of Lemma 2.1, Hölder’s inequality for the indexes 𝑝 and 𝑝/(𝑝−1), (1) of Lemma 2.2, and (2) of Lemma 2.5.
Therefore, 𝐻_𝑓,𝑗𝑘_𝜉^𝑝_{𝐿𝑝()𝑑𝑚}≥1𝐾_𝜉^𝑝_{𝐿𝑝()𝑑𝑚}𝐶1−𝜉^{2(𝑛+1)(𝑝−1)}1−𝜉^2𝑗𝑅^𝑗𝑓(𝜉)^𝑝≥𝐶1−𝜉^{2(𝑛+1)(𝑝−1)}11−𝜉^{2(𝑛+1)(𝑝−1)}1−𝜉^2𝑗𝑅^𝑗𝑓(𝜉)^𝑝=𝐶1−𝜉^2𝑗𝑅^𝑗𝑓(𝜉)^𝑝. (3.5) So ‖𝐻_𝑓,𝑗𝑘_𝜉‖_𝐿^𝑝_(𝑑𝑚)≥𝐶(1−|𝜉|²)^𝑗|𝑅^𝑗𝑓(𝜉)|.
For any compact operator 𝑇, by (2) of Lemma 2.3, we have ‖𝑇𝑘_𝜉‖_{𝐿𝑝(𝑑m)}→0 as |𝜉|→1⁻. Then, 𝐻_𝑓,𝑗−𝑇≥lim_𝜉→1−𝐻_𝑓,𝑗−𝑇𝑘_{𝜉𝐿𝑝(𝑑𝑚)}≥lim_𝜉→1−𝐻_𝑓,𝑗𝑘_{𝜉𝐿𝑝(𝑑𝑚)}−𝑇𝑘_{𝜉𝐿𝑝(𝑑𝑚)}=lim_𝜉→1−𝐻_𝑓,𝑗𝑘_{𝜉𝐿𝑝(𝑑𝑚)}≥𝐶lim_𝜉→1−1−𝜉^2𝑗𝑅^𝑗𝑓(𝜉). (3.6) Thus, ‖𝐻_𝑓,𝑗‖_ess≥𝐶lim_|𝑧|→1⁻(1−|𝑧|²)^𝑗|𝑅^𝑗𝑓(𝑧)|.
Now, we will prove that ‖𝐻_𝑓,𝑗‖_ess≤𝐶lim_|𝑧|→1⁻(1−|𝑧|²)^𝑗|𝑅^𝑗𝑓(𝑧)|.
Write 𝐹(𝑧,𝑤) for −Δ_𝑗𝑓(𝑧,𝑤). For 0<𝜌<1 and 𝑔∈𝐴^𝑝(𝐵), let 𝐵(0,𝜌) and 𝐵(0;𝜌,1) denote the ball |𝑧|≤𝜌 and the ring 𝜌<|𝑧|<1, respectively, then we have 𝐻_𝑓,𝑗𝑔(𝑧)=𝜒_𝐵(0,𝜌)(𝑧)_𝐵𝐹(𝑤,𝑧)𝐾(𝑧,𝑤)𝑔(𝑤)𝑑𝑚(𝑤)+𝜒_{𝐵(0;𝜌,1)}(𝑧)_𝐵𝐹(𝑤,𝑧)𝐾(𝑧,𝑤)𝑔(𝑤)𝑑𝑚(𝑤)=𝑇₁𝑔(𝑧)+𝑇₂𝑔(𝑧). (3.7) Here, 𝑇₁𝑔(𝑧)=𝜒_𝐵(0,𝜌)(𝑧)_𝐵𝐹()𝐾()𝑔(𝑤)(𝑤),𝑇𝑤,𝑧𝑧,𝑤𝑑𝑚₂𝑔(𝑧)=𝜒_{𝐵(0;𝜌,1)}(𝑧)_𝐵𝐹(𝑤,𝑧)𝐾(𝑧,𝑤)𝑔(𝑤)𝑑𝑚(𝑤). (3.8)
We first show that 𝑇₁ is compact. Let {𝑔_𝑙} be a sequence weakly converging to 0 and 𝑝^′=𝑝/(𝑝−1), by Hölder’s inequality, then we have 𝑇₁𝑔_𝑙(𝑧)^𝑝=𝜒_𝐵(0,𝜌)(𝑧)_𝐵𝐹(𝑤,𝑧)𝐾(𝑧,𝑤)𝑔_𝑙(𝑤)𝑑𝑚(𝑤)^𝑝≤𝜒_𝐵(0,𝜌)(𝑧)_𝐵|𝐹(𝑔𝑤,𝑧)|^𝑙(𝑤)|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤)^𝑝≤𝜒_𝐵(0,𝜌)(𝑧)_𝐵|𝐹(𝑤,𝑧)|^𝑝′1−|𝑤|^2−1/𝑝|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤)^𝑝/𝑝′×_𝐵𝑔_𝑙(𝑤)^𝑝1−|𝑤|^21/𝑝′|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤). (3.9) By Lemma 2.6, we get 𝑇₁𝑔_𝑙(𝑧)^𝑝≤𝐶𝜒_𝐵(0,𝜌)(𝑧)sup_𝑧∈𝐵1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧)^𝑝×1−|𝑧|^{2−1/𝑝′}_𝐵𝑔_𝑙(𝑤)^𝑝1−|𝑤|^21/𝑝′|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤). (3.10) Thus, 𝑇₁𝑔_𝑙^𝑝_{𝐿𝑝()𝑑𝑚}=_𝐵𝑇₁𝑔_𝑙(𝑧)^𝑝𝑑𝑚(𝑧)≤𝐶_|𝑧|<𝜌sup_𝑧∈𝐵1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧)^𝑝1−|𝑧|^{2−1/𝑝′}×_𝐵𝑔_𝑙(𝑤)^𝑝1−|𝑤|^21/𝑝′|1−⟨𝑧,𝑤⟩|^𝑛+1(𝑤)(𝑧)𝜌𝑑𝑚𝑑𝑚≤𝐶1−^{2−1/𝑝′}sup_𝑧∈𝐵1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧)^𝑝×_𝐵𝑔_𝑙(𝑤)^𝑝1−|𝑤|^21/𝑝′|1−⟨𝑧,𝑤⟩|^𝑛+1(𝑤)(𝑧)𝜌𝑑𝑚𝑑𝑚=𝐶1−^{2−1/𝑝′}sup_𝑧∈𝐵1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧)^𝑝⋅_𝐵𝑔_𝑙(𝑤)^𝑝1−|𝑤|^21/𝑝′_𝐵1|1−⟨𝑧,𝑤⟩|^𝑛+1(𝑧)(𝑤)𝜌𝑑𝑚𝑑𝑚≤𝐶1−^{2−1/𝑝′}sup_𝑧∈𝐵1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧)^𝑝×_𝐵𝑔_𝑙(𝑤)^𝑝1−|𝑤|^21/𝑝′log1−|𝑤|²𝑑𝑚(𝑤)⟶0,as𝑙⟶∞. (3.11)
So, 𝑇₁ is compact.
For 𝑔∈𝐴^𝑝 and 𝑝^′=𝑝/(𝑝−1), by Hölder’s inequality, 𝑇₂𝑔(𝑧)^𝑝=𝜒_{𝐵(0;𝜌,1)}(𝑧)_𝐵𝐹(𝑤,𝑧)𝐾(𝑧,𝑤)𝑔(𝑤)𝑑𝑚(𝑤)^𝑝≤_𝐵𝜒_{𝐵(0;𝜌,1)}(𝑧𝐹(𝑔(𝑤)|𝑤,𝑧)|)|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤)^𝑝≤_𝐵𝜒_{𝐵(0;𝜌,1)}(𝑧)|𝐹(𝑤,𝑧)|^𝑝′1−|𝑤|^2−1/𝑝|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤)^𝑝/𝑝′×_𝐵𝑔(𝑤)^𝑝1−|𝑤|^21/𝑝′|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤). (3.12) So 𝑇₂𝑔^𝑝_{𝐿𝑝(𝑑𝑚)}=_𝐵𝑇₂𝑔(𝑧)^𝑝𝑑𝑚(𝑧)≤_𝐵𝐵𝜒_{𝐵(0;𝜌,1)}(𝑧)|𝐹(𝑤,𝑧)|^𝑝′1−|𝑤|^2−1/𝑝|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤)^𝑝/𝑝′⋅_𝐵𝑔(𝑤)^𝑝1−|𝑤|^21/𝑝′|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤)𝑑𝑚(𝑧). (3.13) Change the variables 𝑤=𝜑_𝑧(𝜉), let 1<𝑞^′<𝑚𝑖𝑛𝑝,𝑛+1,𝑛+1−1/𝑝 (3.14) and set 𝑞=𝑞^′/(𝑞^′−1), by Lemmas 2.1 and 2.2, then we obtain _𝐵𝜒_{𝐵(0;𝜌,1)}(𝑧)|𝐹(𝑤,𝑧)|^𝑝′(1−|𝑤|²)^−1/𝑝|1−⟨𝑧,𝑤⟩|^𝑛+1(𝑤)=𝑑𝑚_𝐵𝜒_{𝐵(0;𝜌,1)}(𝑧)𝐹𝜑_𝑧(𝜉),𝑧^𝑝′1−𝜑_𝑧(𝜉)^2−1/𝑝1−𝑧,𝜑_𝑧(𝜉)^𝑛+11−|𝑧|^2𝑛+11−𝜉,𝑧^2(𝑛+1)(𝜉)=𝑑𝑚_𝐵𝜒_{𝐵(0;𝜌,1)}(𝑧)1−|𝑧|^2−1/𝑝𝐹𝜑_𝑧(𝜉),𝑧^𝑝′1−𝜉^2−1/𝑝1−𝜉,𝑧^{𝑛+1−2/𝑝}(𝜉)≤|𝑧|𝑑𝑚1−^2−1/𝑝_𝐵𝜒_{𝐵(0;𝜌,1)}(𝑧)𝐹𝜑_𝑧(𝜉),𝑧^𝑝′𝑞𝑑𝑚(𝜉)^1/𝑞×_𝐵1−𝜉^{2−𝑞′/𝑝}1−𝜉,𝑧^{(𝑛+1−2/𝑝)𝑞′}𝑑𝑚(𝜉)^1/𝑞′≤𝐶1−|𝑧|^2−1/𝑝_𝐵𝜒_{𝐵(0;𝜌,1)}(𝑧)𝐹𝜑_𝑧(𝜉),𝑧^𝑝′𝑞𝑑𝑚(𝜉)^1/𝑞. (3.15) By the same argument of [3, Theorem  3.4], we know that _𝐵𝜒_{𝐵(0;𝜌,1)}(𝑧)𝐹𝜑_𝑧(𝜉),𝑧^𝑝′𝑞𝑑𝑚(𝜉)^1/𝑞≤𝐶sup_{|𝑧|𝜌<<1}1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧)^𝑝′. (3.16) Applying Fubini’s theorem and Lemma 2.2, we have 𝑇₂𝑔^𝑝_{𝐿𝑝()𝑑𝑚}≤𝐶sup_{|𝑧|𝜌<<1}1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧)^𝑝×_𝐵1−|𝑧|^{2−1/𝑝′}_𝐵𝑔(𝑤)^𝑝1−|𝑤|^21/𝑝′|1−⟨𝑧,𝑤⟩|^𝑛+1𝑑𝑚(𝑤)𝑑𝑚(𝑧)≤𝐶sup_{|𝑧|𝜌<<1}1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧)^𝑝𝑔^𝑝_{𝐿𝑝()𝑑𝑚}. (3.17) So 𝑇₂≤𝐶sup_{|𝑧|𝜌<<1}1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧). (3.18) Thus, by the definition of the essential norm, we have 𝐻_𝑓,𝑗ess≤𝑇₁+𝑇_2ess≤𝑇₂≤𝐶sup_{|𝑧|𝜌<<1}1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧). (3.19) As 𝜌→1, we have 𝐻_𝑓,𝑗ess≤𝐶lim_{|𝑧|→1−}1−|𝑧|^2𝑗𝑅^𝑗𝑓(𝑧). (3.20)
Similarly, we get the equality of ‖𝐻^′_𝑓,𝑗‖_ess and lim_|𝑧|→1⁻(1−|𝑧|²)^𝑗|𝑅^𝑗𝑓(𝑧)|.
By [7, Theorems  3.4 and 3.5], we obtain the equality of lim_|𝑧|→1⁻(1−|𝑧|²)^𝑗|𝑅^𝑗𝑓(𝑧)| and lim_|𝑧|→1⁻(1−|𝑧|²)|𝑅𝑓(𝑧)|.
We complete the proof of Theorem 3.1.

Acknowledgments

The authors would like to thank the referee for the careful reading of the first version of this paper and for the several suggestions made for improvement. The paper was supported by the NNSF of China (no. 10771201) and the Natural Science Foundation of Anhui Province (no. 090416233).