Essential Norms and Schatten(-Herz) Classes of Integration Operators from Bergman Spaces to Hardy Spaces

ESSENTIAL NORMS AND SCHATTEN(-HERZ) CLASSES OF INTEGRATION OPERATORS FROM BERGMAN SPACES TO

HARDY SPACES

JIALE CHEN^∗, JORDI PAU AND MAOFA WANG

Abstract. In this paper, we completely characterize the compactness of the Volterra type integration operators Jb acting from weighted Bergman spaces A^p_α to Hardy spaces H^q for all 0 < p, q < ∞. It is quite surprising that the boundedness and the compactness of Jb : A^p_α → H^q are not equivalent when 0 < q < p ≤ 2, while, due to the well-known results, the compactness and boundedness of Jb : A^p_α → A^q_β (resp. Jb : H^p → H^q) are always equivalent when p > q. Furthermore, we give some estimates for the essential norms of Jb: A^p_α→ H^q in the case 0 < p ≤ q < ∞. We finally describe the membership in the Schatten(-Herz) class of the Volterra type integration operators.

1. Introduction

Let Bnbe the open unit ball of Cⁿand H(Bn) denote the algebra of holomorphic functions on Bn. A function b ∈ H(Bn) induces a Volterra type integration operator J_b given by the formula:

(1.1) J_bf (z) =

Z 1 0

f (tz)Rb(tz)dt

t , z ∈ Bn, where f ∈ H(Bn) and Rb is the radical derivative of b:

Rb(z) =

n

X

k=1

zk

∂b

∂z_k(z), z = (z1, z2, . . . , zn) ∈ Bⁿ.

A fundamental property of the operator J_b is the following formula involving the radical derivative R:

R(J_bf )(z) = f (z)Rb(z), z ∈ Bn.

Date: December 24, 2020.

∗ Corresponding author .

2010 Mathematics Subject Classification. 47G10, 32A35, 32A36.

Key words and phrases. Integration operator, Hardy space, Bergman space, Compactness, Essential norm, Schatten-Herz class.

J. Pau was partially supported by the grants MTM2017-83499-P (Ministerio de Educaci´on y Ciencia) and 2017SGR358 (Generalitat de Catalunya). M. Wang was partially supported by NSFC (No. 11771340) of China.

1

For 0 < p < ∞, the Hardy space H^p consists of those holomorphic functions f in Bn with

kf k^p_Hp = sup

0<r<1

M_p^p(f, r) = sup

0<r<1

Z

Sn

|f (rξ)|^pdσ(ξ) < ∞,

where dσ is the surface measure on the unit sphere Sn = ∂Bn normalized so that σ(Sn) = 1. Given α > −1 and 0 < p < ∞, a function f ∈ H(Bn) belongs to the weighted Bergman space A^p_α, if

kf k^p_Ap α =

Z

Bn

|f (z)|^pdv_α(z) < ∞.

Here dv = dv₀is the Lebesgue measure on Bn, normalized so that v(Bn) = 1. The measure dv_α is given by dv_α(z) = c_n,α(1 − |z|²)^αdv(z) with normalized constant c_n,α so that v_α(Bn) = 1.

The operator J_b has been studied by many authors, see [9, 13, 15, 18] and the references therein. In particular, Wu [18] partially solved the boundedness of J_b : A^p_α → H^q in the setting of the unit disk. Recently, Miihkinen, Pau, Per¨al¨a and Wang [13] completely characterized the boundedness of J_b : A^p_α → H^q for all dimensions n. In this paper, we follow the line of research to completely characterize the compactness of the Volterra type integration operators Jb acting from weighted Bergman spaces A^p_α to Hardy spaces H^q for all 0 < p, q < ∞.

Furthermore, we give some estimates for the essential norms of J_b : A^p_α → H^q in the case 0 < p ≤ q < ∞. We finally describe the membership in Schatten(-Herz) classes of the Volterra type integration operators and give some descriptions of asymptotic property of singular values.

Our first result is the following little version of the main result in [13].

Theorem 1.1. Let α > −1, 0 < p, q < ∞ and b ∈ H(Bn). Then the following hold:

(1) If 0 < p ≤ min{2, q} or 2 < p < q < ∞, then J_b : A^p_α → H^q is compact if and only if

lim

|z|→1⁻Rb(z)(1 − |z|²)^{nq+1−n+1+αp} = 0.

(2) If 2 < p = q < ∞, then J_b : A^p_α → H^q is compact if and only if

|Rb(z)|^p−22p (1 − |z|²)^p−2αp−2 dv(z) is a vanishing Carleson measure.

(3) If p > max{2, q}, then J_b : A^p_α→ H^q is compact if and only if ξ 7→

Z

Γ(ξ)

|Rb(z)|^p−22p (1 − |z|²)^{2−2αp−2 +1−n}dv(z)

^p−2_2p

belongs to L^p−qpq (Sⁿ).

(4) If 0 < q < p ≤ 2, then J_b : A^p_α → H^q is compact if and only if ξ 7→ sup

z∈Γ(ξ)∩{|z|≥r}

|Rb(z)|(1 − |z|²)^p−1−αp

converges to zero in L^p−qpq (Sn) as r → 1⁻.

Two remarks are in order. (1) Comparing the above theorem with [13, Theorem 1], we find that the boundedness and compactness of Jb : A^p_α → H^q are not equivalent when 0 < q < p ≤ 2. This is quite different from the cases J_b : A^p_α → A^q_β and J_b : H^p → H^q, where J_b is compact if and only if it is bounded whenever p > q, see [9, 15].

(2) By Theorem 1.1 (4), we know that if 0 < q < p ≤ 2, Jb : A^p_α → H^q may be not compact even when b is a polynomial. It is quite surprising because the corresponding operators J_b : A^p_α → A^q_β and J_b : H^p → H^q are always compact for any polynomial b if p > q, see [9,15]. See Section 3 for more details.

Let X, Y be (quasi-)Banach spaces and T : X → Y a bounded operator. The essential norm of T , denoted by kT k_e, is its distance from the space of compact operators. It is clear that T is compact if and only if kT k_e = 0. Our next result gives some estimates for the essential norm of Jb : A^p_α → H^q in the case 0 < p ≤ q < ∞.

Theorem 1.2. Let α > −1, 0 < p ≤ q < ∞ and b ∈ H(Bn) such that J_b : A^p_α → H^q is bounded.

(1) If 0 < p ≤ min{2, q} or 2 < p < q < ∞, then kJ_bk_e  lim sup

|a|→1⁻

|Rb(a)|(1 − |a|²)^{nq+1−n+1+αp} . (2) If 2 < p = q < ∞, then

kJ_bk_e. lim sup

|a|→1⁻

Z

Bn

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿ|Rb(z)|^p−22p (1 − |z|²)^p−2αp−2 dv(z)

^p−2_2p .

Recall that if T is a compact operator acting on a separable Hilbert space H, then there exist a nonincreasing sequence {s_k(T )} of nonnegative numbers tending to 0 and orthonormal sets {e_k}, {σ_k} in H such that

T x =X

k

s_k(T )hx, e_kiσ_k

for all x ∈ H. This is the so-called canonical decomposition of the compact operator T . The number s_k(T ) is the kth singular value of T , which is exactly the square root of the kth eigenvalue of the positive operator T^∗T if we rearrange the eigenvalues in nonincreasing order, where T^∗ is the Hilbert adjoint of T . For 0 < p < ∞, the compact operator T belongs to the Schatten class S_p(H) if {s_k(T )} is in the sequence space l^p. If H₁ and H₂ are two separable Hilbert spaces and T : H₁ → H₂ is a compact operator, we say that T is in the Schatten class S_p(H₁, H₂) if T^∗T is in S_p/2(H₁). We refer to [21, Chapter 1] for a brief account on Schatten classes.

Our third result is a complete characterization of the membership in the Schat- ten class S_p(A²_α, H²) of the Volterra type integration operator J_b.

Theorem 1.3. Let α > −1, 0 < p < ∞ and b ∈ H(Bn). Then the following hold:

(1) If p(1 − α)/2 > n, then J_b belongs to S_p(A²_α, H²) if and only if Z

Bⁿ

|Rb(z)|^p(1 − |z|²)^p2(1−α)dλ_n(z) < ∞, where

dλ_n(z) = dv(z) (1 − |z|²)ⁿ⁺¹ is the invariant measure on Bn.

(2) If p(1 − α)/2 ≤ n, then J_b is in S_p(A²_α, H²) if and only if b is constant.

Loaiza, L´opez-Garc´ıa and P´erez-Esteva introduced the Schatten-Herz class of Toeplitz operators in [10], which is a generalization of the Schatten class. Recall that, given a positive Borel measure µ on Bn, the Toeplitz operator T_µ on A²_α is defined by

T_µf (z) = Z

Bn

f (w)K^α(z, w)dµ(w), z ∈ Bn,

where K^α(z, w) is the reproducing kernel of A²_α. See [22] for more information about Toeplitz operators. For 0 < p, q < ∞, the Toeplitz operator T_µ is said to be in the Schatten-Herz class S_p,q(A²_α) if each T_µχk is in S_p(A²_α) and the sequence {kT_µχkk_Sp}_k≥0 is in l^q, where χ_k is the characteristic function of the annulus A_k = {z ∈ Bn : 1 − ₂¹k ≤ |z| < 1 − ₂k+1¹ } for k ≥ 0. Following [7], we say that J_b : A²_α → H² is in the Schatten-Herz class S_p,q(A²_α, H²) if J_b^∗J_b ∈ S^p

2,^q₂(A²_α). Our next result characterizes the Schatten-Herz class of integration operators.

Theorem 1.4. Let α > −1, 0 < p, q < ∞ and b ∈ H(Bⁿ). Then the following hold:

(1) If p(1 − α)/2 > n, then J_b belongs to S_p,q(A²_α, H²) if and only if Z 1

0

M_p^q(Rb, r)(1 − r)^{q2(1−α)−nqp−1}dr < ∞.

(2) If p(1 − α)/2 ≤ n, then J_b is in S_p,q(A²_α, H²) if and only if b is constant.

It is also interesting to consider the speed of s_k(T ) converging to zero if T is a compact operator on a separable Hilbert space. See [8] and references there for more details. Based on the work concerning Toeplitz operators in [5], our next result (see Theorem 5.4 in Section 5) gives a description of the decay of singular values of J_b : A²_α → H².

The paper is organized as follows. Some background and preliminary results are given in Section 2. In Section 3 we consider the compactness of integration operators J_b : A^p_α → H^q. In Section 4 we estimate the essential norms. Section 5 is devoted to the proof of Theorem 1.3, Theorem 1.4 and some descriptions of asymptotic property of singular values of J_b : A²_α → H². We also consider the essential norms and membership in Schatten(-Herz) classes of integration operators from Hardy spaces to Bergman spaces in Section 6.

Notation. For 1 < p < ∞, we let p⁰ denote the conjugate exponent of p. The notation A . B means that A ≤ CB for some inessential constant C > 0. The converse relation A & B is defined in an analogous manner, and if A . B and A & B both hold, we write A  B. In addition, we always let r be a positive number less than 1.

2. Preliminaries

In this section we introduce some well-known results that will be used through- out the paper.

2.1. Carleson measures. For ξ ∈ Sn and δ > 0, the non-isotropic metric ball B_δ(ξ) is defined by

B_δ(ξ) = {z ∈ Bn : |1 − hz, ξi| < δ}.

A positive Borel measure µ on Bn is said to be a Carleson measure if there is a constant C > 0 such that

µ(B_δ(ξ)) ≤ Cδⁿ

for all ξ ∈ Sn and δ > 0. Obviously every Carleson measure is finite. H¨ormander [6] extended to several complex variables the famous Carleson measure embedding theorem [3,4] asserting that, for 0 < p < ∞, the embedding I_d: H^p → L^p(Bn, dµ) is bounded if and only if µ is a Carleson measure. We denote by kµk_CM the infimum of all possible C above. It is well-known (see [19, Theorem 45]) that µ is a Carleson measure if and only if for each (some) t > 0 one has

(2.1) sup

a∈Bn

Z

Bⁿ

(1 − |a|²)^t

|1 − hz, ai|^n+tdµ(z) < ∞.

Moreover, with constant depending on t, the supremum of the above integral is comparable to kµkCM.

A positive Borel measure µ on Bn is called a vanishing Carleson measure if lim

δ→0

µ(B_δ(ξ)) δⁿ = 0

uniformly for ξ ∈ Sn. Equivalently, one may require that for each (some) t > 0 one has

(2.2) lim

|a|→1⁻

Z

Bn

(1 − |a|²)^t

|1 − hz, ai|^n+tdµ(z) = 0,

or for 0 < p < ∞, the embedding I_d : H^p → L^p(Bn, dµ) is compact.

2.2. Separated sequences and lattices. A sequence of points {z_j} ⊂ Bnis said to be separated if there exists δ₀ > 0 such that β(z_i, z_j) ≥ δ₀ for all i and j with i 6= j, where β(z, w) denotes the Bergman metric on Bn. This implies that there is δ > 0 such that the Bergman metric balls D(z_j, δ) = {z ∈ Bn : β(z, z_j) < δ}

are pairwise disjoint.

We need a well-known result on decomposition of the unit ball Bn. By Theorem 2.23 in [20], there exists a positive integer N such that for any 0 < δ < 1 we can find a sequence {a_k} in Bn with the following properties:

(i) Bn =S

kD(a_k, δ);

(ii) The sets D(a_k, δ/4) are mutually disjoint;

(iii) Each point z ∈ Bn belongs to at most N of the sets D(a_k, 4δ).

Any sequence {a_k} satisfying the above conditions is called a δ-lattice (in the Bergman metric). Obviously any δ-lattice is a separated sequence.

2.3. Area methods and equivalent norms. For ξ ∈ Sⁿ and γ > 1, recall that the admissible approach region Γγ(ξ) is defined as

Γ_γ(ξ) =



z ∈ Bn: |1 − hz, ξi| < γ

2(1 − |z|²)

 .

In this paper we agree that Γ(ξ) := Γ₂(ξ). It is known that for every δ > 0 and γ > 1, there exists γ⁰ > 1 so that

(2.3) [

z∈Γγ(ξ)

D(z, δ) ⊂ Γ_γ⁰(ξ).

We will write eΓ(ξ) to indicate this change of aperture. If I(z) = {ξ ∈ Sn : z ∈ Γ(ξ)}, then σ(I(z))  (1 − |z|²)ⁿ, and it follows from Fubini’s theorem that, for a positive measurable function ϕ, and a finite positive measure ν, one has (2.4)

Z

Bn

ϕ(z)dν(z)  Z

Sn

Z

Γ(ξ)

ϕ(z) dν(z) (1 − |z|²)ⁿ

 dσ(ξ).

This fact will be used repeatedly throughout the paper.

Let us recall the following Littlewood-Paley identity, which can be found in [20].

Theorem A. Suppose 0 < p < ∞. Then kf − f (0)k^p_Hp = p²

2n Z

Bn

|Rf (z)|²|f (z) − f (0)|^p−2|z|⁻²ⁿlog 1

|z|dv(z) for all f ∈ H^p. In particular, if f (0) = 0,

kf k^p_Hp  Z

Bn

|Rf (z)|²|f (z)|^p−2(1 − |z|²)dv(z).

The next estimate is the Calder´on’s area theorem [2, 12]. The variant we will use can be found in [1] or in [15].

Theorem B. Let 0 < p < ∞. If f ∈ H(Bn) and f (0) = 0, then kf k^p_Hp 

Z

Sn

Z

Γ(ξ)

|Rf (z)|²(1 − |z|²)¹⁻ⁿdv(z)

p/2

dσ(ξ).

We will also need the following result essentially due to Luecking [11] (see also [15]) describing those positive Borel measures for which the embedding from H^p into L^s(µ) is bounded when s < p.

Theorem C. Let 0 < s < p < ∞ and let µ be a positive Borel measure on Bn. Then the identity I_d : H^p → L^s(µ) is bounded if and only if the function defined on Sn by

µ(ξ) =e Z

Γ(ξ)

(1 − |z|²)⁻ⁿdµ(z)

belongs to L^p/(p−s)(Sn). Moreover, one has kI_dk_H^p→L^s(µ) kµke ^1/s_Lp/(p−s)

(Sn). 3. Compactness

In this section, we will prove Theorem1.1. We need the following two lemmas first.

Lemma 3.1. If µ is a vanishing Carleson measure, then lim

r→1⁻kχ_(rBn)^cµk_CM = 0,

where (rBⁿ)^c = Bⁿ \ rBⁿ, and χ_(rBn)^c is the characteristic function of the set (rBⁿ)^c.

Proof. Since kχ_(rBn)^cµk_CM  sup_a∈BnR

(rBn)^c

(1−|a|²)ⁿ

|1−hz,ai|²ⁿdµ(z), it is sufficient to show that

lim

r→1⁻

sup

a∈Bn

Z

(rBn)^c

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿdµ(z) = 0.

We complete the proof by contradiction. Suppose that lim

r→1⁻

sup

a∈Bn

Z

(rBn)^c

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿdµ(z) 6= 0,

and then there exist ₀ > 0, an increasing sequence of positive numbers {r_k}, r_k→ 1⁻, and {a_k} ⊂ Bn, such that

(3.1)

Z

Bn

(1 − |ak|²)ⁿχ_(rkBn)^c(z)

|1 − hak, zi|²ⁿ dµ(z) ≥ ₀, ∀k ≥ 1.

There are two possibilities about the sequence {a_k}: |a_k| → 1⁻ and |a_k| 9 1⁻. If |a_k| → 1⁻, then by (2.2) we have

Z

Bn

(1 − |a_k|²)ⁿχ_(rkBn)^c(z)

|1 − ha_k, zi|²ⁿ dµ(z) ≤ Z

Bn

(1 − |a_k|²)ⁿ

|1 − ha_k, zi|²ⁿdµ(z) → 0

as k → ∞. This is contradictory with (3.1). If |a_k| 9 1⁻, there are a point a₀ ∈ Bn and a subsequence of {a_k} converging to a₀. Without loss of generality, we assume a_k→ a₀, then there exists δ₀ > 0 such that B(a₀, δ₀) = {z : |z − a₀| ≤ δ₀} ⊂ Bn and a_k ∈ B(a₀, δ₀) if k is large enough. Therefore, we have

(1 − |a_k|²)ⁿχ_(rkBn)^c(z)

|1 − ha_k, zi|²ⁿ ≤ 1

[1 − (|a₀| + δ₀)]²ⁿ

for all z ∈ Bnwhenever k is large enough, and it is clear that ^(1−|ak|

2)ⁿχ_(rkBn)c(z)

|1−ha_k,zi|²ⁿ → 0 for all z ∈ Bⁿ. Then we get

lim

k→∞

Z

Bn

(1 − |a_k|²)ⁿχ_(rkBn)^c(z)

|1 − ha_k, zi|²ⁿ dµ(z) = 0

by dominated convergence theorem since µ is a vanishing Carleson measure (in particular, µ is finite). This, again, is contradictory with (3.1). Thus the proof

is finished. 

Lemma 3.2. Let 0 < p < ∞, β ≥ 0 and Z = {a_k} be a δ-lattice. Suppose f ∈ H(Bn) and R

Snsup_z∈Γ(ξ)|f (z)|^p(1 − |z|²)^βdσ(ξ) < ∞. If

(3.2) lim

r→1⁻

Z

Sⁿ

sup

ak∈Γ(ξ)∩{|w|≥r}

|f (a_k)|^p(1 − |a_k|²)^βdσ(ξ) = 0 whenever δ > 0 is small enough, then

lim

r→1⁻

Z

Sn

sup

z∈Γ(ξ)∩{|w|≥r}

|f (z)|^p(1 − |z|²)^βdσ(ξ) = 0.

Proof. Given  > 0, choose δ > 0 small enough so that δ^p < . Fix this δ, and let

˜

r = inf{|a_k| : a_k ∈ Z, a_k ∈ D(z, δ) for some |z| ≥ r}.

By the same method as in the proof of [13, Lemma 3], we have sup

z∈Γ(ξ)∩{|w|≥r}

|f (z)|^p(1 − |z|²)^β . δ^p sup

z∈eeΓ(ξ)

|f (z)|^p(1 − |z|²)^β + sup

ak∈eΓ(ξ)∩{|w|≥˜r}

|f (ak)|^p(1 − |ak|²)^β. Integrating over Sⁿ yields

Z

Sⁿ

sup

z∈Γ(ξ)∩{|w|≥r}

|f (z)|^p(1 − |z|²)^βdσ(ξ)

. δ^p Z

Sn

sup

z∈eΓ(ξ)e

|f (z)|^p(1 − |z|²)^βdσ(ξ) + Z

Sn

sup

ak∈eΓ(ξ)∩{|w|≥˜r}

|f (a_k)|^p(1 − |a_k|²)^βdσ(ξ)

.  + Z

Sn

sup

ak∈Γ(ξ)∩{|w|≥˜r}

|f (a_k)|^p(1 − |a_k|²)^βdσ(ξ).

It is easy to see ˜r → 1⁻ as r → 1⁻. Combining the above estimate with (3.2), we get

lim sup

r→1⁻

Z

Sn

sup

z∈Γ(ξ)∩{|w|≥r}

|f (z)|^p(1 − |z|²)^βdσ(ξ) . .

The proof is finished since  > 0 is arbitrary.  Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. We only prove (2) and the necessary part of (4). The rest parts can be proved by standard modifications of the corresponding parts in [13]

and so are omitted. Fix 0 <  < 1.

(2) For any f ∈ A^p_α, consider the measure dµ_f,b(z) = |f (z)|²|Rb(z)|²dv₁(z).

Then

(3.3) kJ_bf k^p_Hp 

Z

Sn

µg_f,b(ξ)^p/2dσ(ξ) by TheoremB. For the sake of simplicity, we denote

dµ_b(z) = |Rb(z)|^p−22p (1 − |z|²)^p−2αp−2 dv(z).

We first consider the sufficiency part. Suppose µ_b is a vanishing Carleson measure and {f_k} is a bounded sequence in A^p_α converging to 0 uniformly on compact subsets of Bn. We need to show kJ_bf_kk_H^p → 0. For any h ∈ H^p/(p−2), by H¨older’s inequality and the fact that µ_b is a (vanishing) Carleson measure we have that

Z

Bn

|h(z)|dµ_fk,b(z)

=

Z

rBn

+ Z

(rBn)^c



|h(z)||fk(z)|²|Rb(z)|²dv1(z)

. khk_Hp−2^p ·

Z

rBn

|f_k(z)|^pdv_α(z)

²_p

+ khk

H

p

p−2 · kχ_(rBn)^cµ_bk

p−2 p

CM. Since µ_b is a vanishing Carleson measure, there exists r₀ (0 < r₀ < 1) such that

kχ_(r0Bn)^cµ_bk

p−2 p

CM <  by Lemma3.1. Fix this r₀, we have

 R

r0Bn|f_k(z)|^pdv_α(z)

2/p

<  when k is large enough since f_k → 0 uniformly on r₀Bn. Thus

(3.4)

Z

Bⁿ

|h(z)|dµ_fk,b(z) . khk_Hp−2^p

when k is large enough. Denote by I_dk : H^p/(p−2) → L¹(dµ_fk,b) the embedding operator, and then we get lim_k→∞kI_dkk_Hp/(p−2)→L¹(dµ_fk,b) = 0 by (3.4). Therefore, by TheoremC, we have

Z

Sn

µg_fk,b(ξ)^p/2dσ(ξ)  kI_dkk^p/2_{Hp/(p−2)→L1(dµ}

fk,b) → 0.

Thus

kJbfkk^p_Hp  Z

Sn

µgf_k,b(ξ)^p/2dσ(ξ) → 0 by (3.3) and the proof of the sufficiency is complete.

Next we consider the necessity part. Assume that J_b : A^p_α → H^p is compact.

Then for any f ∈ A^p_α, by (3.3), we have Z

Sn

gµ_f,b(ξ)^p/2dσ(ξ)  kJ_bf k^p_Hp < ∞.

We see that the measure µ_f,bsatisfies the conditions of TheoremCfor parameters 1 and _p−2^p , and this implies I_df : H^p/(p−2) → L¹(dµ_f,b) is compact. Moreover, for any h ∈ H^p/(p−2),

Z

Bn

|h(z)|dµ_f,b(z) . kJbf k²_Hp· khk

H

p p−2.

Let B_A^p_α = {f ∈ A^p_α : kf k_A^p_α ≤ 1} be the closed unit ball of A^p_α, then J_b(B_A^p_α) is relatively compact in H^p. We can find f₁, · · · , f_m ∈ B_A^p_α, such that for any f ∈ B_A^p_α, there are some j ∈ {1, 2, · · · , m} with

kJbf − JbfjkH^p < .

Suppose {h_k} ⊂ H^p/(p−2) is a bounded sequence converging to zero uniformly on compact subsets of Bn. Since I_df : H^p/(p−2) → L¹(dµ_f,b) is compact for any f ∈ A^p_α, then for every j ∈ {1, · · · , m}, there is K_j > 0 such that k > K_j implies

Z

Bn

|h_k(z)|dµ_fj,b(z) < ².

Define dν_hk,b(z) = |h_k(z)||Rb(z)|²dv₁(z). We now suppose f ∈ B_A^p_α and k > K :=

max{K₁, · · · , K_m}, then there is j ∈ {1, · · · , m} such that

Z

Bⁿ

|f (z)|²dν_hk,b(z)

1/2

≤

Z

Bn

|f (z) − f_j(z)|²dν_hk,b(z)

1/2

+

Z

Bn

|f_j(z)|²dν_hk,b(z)

1/2

=

Z

Bn

|h_k(z)|dµ_{f −fj,b}(z)

1/2

+

Z

Bn

|h_k(z)|dµ_fj,b(z)

1/2

. kJb(f − f_j)k_H^p· kh_kk^1/2_Hp/(p−2)+

Z

Bn

|h_k(z)|dµ_fj,b(z)

1/2

. .

This implies that the embeddings I_dk : A^p_α → L²(dν_hk,b) are bounded and

k→∞lim kI_dkk_A^p_α→L²_(dν

hk,b)= 0.

Therefore, since p > 2, by [19, Theorem 54], we have for any δ > 0 and k ≥ 1, ˆ

νh_k,b(z) := ν_hk,b(D(z, δ))

(1 − |z|²)^n+1+α ∈ L^p/(p−2)(Bⁿ, dvα) and

(3.5) kˆν_hk,bk_Lp/(p−2)(dvα) . kIdkk²_A^p

α→L²(dν_hk,b). By subharmonic property, we have

(3.6) ν_hk,b(D(z, δ)) & |hk(z)||Rb(z)|²(1 − |z|²)ⁿ⁺².

It follows from (3.5) and (3.6) that Z

Bⁿ

|h_k(z)|^p−2p |Rb(z)|^p−22p (1 − |z|²)^p−2αp−2 dv(z) . kIdkk

2p p−2

A^pα→L²(dν_hk,b). Therefore, we have

k→∞lim Z

Bn

|h_k(z)|^p−2p dµ_b(z) = 0.

This implies that I_d: H^p/(p−2) → L^p/(p−2)(dµ_b) is compact. Consequently, µ_b is a vanishing Carleson measure.

(4) Let

V_b,r(ξ) = sup

z∈Γ(ξ)∩{|z|≥r}

|Rb(z)|(1 − |z|²)^p−1−αp .

Suppose that Jb : A^p_α → H^q is compact, and then it follows from [13, Theorem 1] that V_b,0 ∈ L^pq/(p−q)(Sn). We want to show that

(3.7) lim

r→1⁻kV_b,rk_Lpq/(p−q)(Sn)= 0.

If p−1−α < 0, then Vb,0 ∈ L^pq/(p−q)(Sⁿ) implies b is constant. So (3.7) is obvious.

We now assume that p − 1 − α ≥ 0. By Lemma 3.2, we only need to prove that ξ 7→ sup

ak∈Γ(ξ)∩{|z|≥r}

|Rb(a_k)|^q(1 − |a_k|²)^{qp(p−1−α)}

converges to zero in L^p/(p−q)(Sn) as r → 1⁻, where Z = {a_k} is a δ-lattice with δ small enough.

Let B_Tp^p_(Z) = {c ∈ T_p^p(Z) : kck_Tp^p_(Z) ≤ 1} be the closed unit ball of T_p^p(Z). For any c = {c_k} ∈ B_Tp^p_(Z), define

S(c)(z) =X

k

(1 − |a_k|²)^n/pc_kg_k(z), z ∈ Bn, where

g_k(z) = (1 − |a_k|²)b−(n+1+α)/p

(1 − hz, a_ki)^b

with b > n max{1, 1/p} + (α + 1)/p. Then we have S(c) ∈ A^p_α with kS(c)k_A^p_α . 1 for any c ∈ B_Tp^p_(Z) by [20, Theorem 2.30].

Since J_b : A^p_α → H^q is compact, the set J_b ◦ S(B_Tp^p_(Z)) is relatively compact in H^q. We can choose 0 < r0 < 1 such that

Z

Sn

Z

Γ(ξ)∩{|z|≥r0}

|Rb(z)|²|S(c)(z)|²(1 − |z|²)¹⁻ⁿdv(z)

q/2

dσ(ξ) . ^q for any c ∈ B_Tp^p_(Z). That is,

Z

Sn

Z

Γ(ξ)∩{|z|≥r0}

Rb(z)X

k

(1 − |a_k|²)^n/pc_kg_k(z)    

2

dv_1−n(z)

q/2

dσ(ξ) . ^q.

By a same process as in the proof of [13, Theorem 7], we can establish Z

Sn



X

ak∈Γ(ξ)∩{|z|≥r⁰₀}

|c_k|²|Rb(a_k)|²(1 − |a_k|²)^{2−2(1+α)/p}

q/2

dσ(ξ) . ^q,

where r₀⁰ = inf{|a_k| : D(a_k, δ) ⊂ {|z| ≥ r₀}}. Thus, for any c ∈ T_p^p(Z), we have Z

Sn



X

ak∈Γ(ξ)∩{|z|≥r₀⁰}

|c_k|²|Rb(a_k)|²(1 − |a_k|²)^{2−2(1+α)/p}

q/2

dσ(ξ) . ^qkck^q_T^p

p(Z). (3.8)

Using the dual and factorization of sequence tent spaces (see the proof of Theorem 7 and Theorem 8 in [13]), by (3.8), we get

ξ 7→ sup

ak∈Γ(ξ)∩{|z|≥r}

|Rb(a_k)|^q(1 − |a_k|²)^{qp(p−1−α)}

converges to zero in L^p/(p−q)(Sn) as r → 1⁻ and then the necessity part holds.

The proof of Theorem 1.1 is now finished. 

As mentioned in Introduction, here we give an example indicating that the integration operator Jb : A^p_α → H^q may be not compact for some polynomials b and some parameters p, q and α. For simplicity, we consider the case n = 1. Let b(z) = z and

e_k(z) =

r(k + 1)(k + 2) 2 z^k. Then kekk_A²

1 = 1 and ek→ 0 weakly in A²₁. By an elementary calculation, we get Jbek(z) =

Z z 0

ek(ζ)b⁰(ζ)dζ =

s k + 2

2(k + 1)z^k+1. Thus

kJ_be_kk_H^q = sup

0<r<1

Z

S1

|J_be_k(rξ)|^qdσ(ξ)

1/q

= s

k + 2

2(k + 1) → 1

√2

as k → ∞. Consequently, J_b : A²₁ → H^q is not compact for any 0 < q < ∞.

However, if 0 < q < 2, it is easy to see that b(z) = z satisfies the integrable condition in [13, Theorem 1(4)]. That is, J_b : A²₁ → H^q is bounded if 0 < q < 2.

Therefore, the boundedness and compactness of J_b are not equivalent for 0 < q <

p ≤ 2. This is in sharp contrast with the classical case where the boundedness and the compactness of Jb : H^p → H^q (or Jb : A^p_α → A^q_β) are always equivalent for any 0 < q < p < ∞.

4. Essential norms

In order to estimate the essential norm of J_b : A^p_α → H^q, we need some auxiliary results.

For γ ≥ 0, let B_γ be the γ-Bloch space, that is, the space of holomorphic functions b ∈ H(Bn) such that

kbkB_γ := sup

z∈Bn

|Rb(z)|(1 − |z|²)^γ < ∞.

It becomes a Banach space provided that we identify functions that differ by a constant. Let B_γ,0 be the closed subspace of B_γ consisting of functions b ∈ H(Bn) such that

lim

|z|→1⁻|Rb(z)|(1 − |z|²)^γ = 0.

We have the following distance formula for the space B_γ. Lemma 4.1. Let γ ≥ 0 and b ∈ B_γ. Then

dist(b, B_γ,0)  lim sup

|z|→1⁻

|Rb(z)|(1 − |z|²)^γ. Proof. If γ = 0, by maximum modulus principle, we have

dist(b, B_0,0) = kbkB₀ = sup

z∈Bn

|Rb(z)| = lim sup

|z|→1⁻

|Rb(z)|.

We now assume γ > 0. The lower estimate can be easily deduced by triangle inequality. We consider the upper estimate. It is clear that b_r ∈ B_γ,0 for any 0 < r < 1. Here, b_r(z) = b(rz). Hence, for any 0 < δ < 1,

dist(b, B_γ,0) ≤ lim sup

r→1⁻

kb − b_rkBγ

≤ lim sup

r→1⁻

sup

|z|≤δ

|Rb(z) − Rb(rz)|(1 − |z|²)^γ+ lim sup

r→1⁻

sup

|z|>δ

|Rb(z) − Rb(rz)|(1 − |z|²)^γ

≤ sup

|z|>δ

|Rb(z)|(1 − |z|²)^γ+ lim sup

r→1⁻

sup

|z|>δ

|Rb(rz)|(1 − |z|²)^γ. Let δ → 1⁻, and we get

dist(b, B_γ,0) . lim sup

|z|→1⁻

|Rb(z)|(1 − |z|²)^γ,

which completes the lemma. 

Given 1 ≤ p < ∞ and γ > −1, let CM^p_γ be the space of holomorphic functions b ∈ H(Bn) such that

dµ_b,p,γ(z) = |Rb(z)|^p(1 − |z|²)^γdv(z) is a Carleson measure, and define

kbk_CM^p_γ = kµb,p,γk

1 p

CM.

We also define the space CM^p_γ,0to be the subspace of CM^p_γconsisting of b ∈ H(Bn) such that µ_b,p,γ is a vanishing Carleson measure. We have the following distance formulas for the space CM^p_γ. Here we denote Q(0) = Bn and for a ∈ Bn\{0},

Q(a) = {z ∈ Bn : |1 − hz, a

|a|i| < 1 − |a|²}.

Lemma 4.2. Let 1 ≤ p < ∞, γ > −1 and b ∈ CM^p_γ. Then dist(b, CM^p_γ,0)  lim sup

|a|→1⁻

Z

Bⁿ

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿ|Rb(z)|^p(1 − |z|²)^γdv(z)

_p¹

 lim sup

|a|→1⁻

 1

(1 − |a|²)ⁿ Z

Q(a)

|Rb(z)|^p(1 − |z|²)^γdv(z)

¹_p . Proof. The lower estimate follows from triangle inequality. We deduce the upper estimate. For any 0 < r < 1, it is easy to see b_r ∈ CM^p_γ,0. Moreover, for any 0 < δ < 1,

sup

|a|≤δ

Z

Bn

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿdµ_b−br,p,γ(z)

≤ 1

(1 − δ)²ⁿ Z

Bn

|Rb(z) − Rb(rz)|^p(1 − |z|²)^γdv(z) → 0 as r → 1⁻. Thus we have

dist(b, CM^p_γ,0) ≤ lim sup

r→1⁻

kb − b_rk

1 p

CM^pγ

. lim sup

r→1⁻

sup

|a|>δ

Z

Bn

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿ|Rb(z) − Rb(rz)|^p(1 − |z|²)^γdv(z)

¹_p

≤ sup

|a|>δ

Z

Bn

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿdµ_b,p,γ(z)

_p¹ +

lim sup

r→1⁻

sup

|a|>δ

Z

Bn

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿdµbr,p,γ(z)

¹_p . Let δ → 1⁻, and we get

dist(b, CM^p_γ,0) . lim sup

|a|→1⁻

Z

Bn

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿdµ_b,p,γ(z)

¹_p , which establishes the estimate

dist(b, CM^p_γ,0)  lim sup

|a|→1⁻

Z

Bn

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿdµb,p,γ(z)

¹_p .

Noting that a positive Borel measure µ is a Carleson measure if and only if µ(Q(a)) ≤ C(1 − |a|²)ⁿ for all a ∈ Bn and some absolute constant C > 0, and

kµk_CM = sup

a∈Bn

µ(Q(a)) (1 − |a|²)ⁿ,

the equivalent relation

dist(b, CM^p_γ,0)  lim sup

|a|→1⁻

 µ_b,p,γ(Q(a)) (1 − |a|²)ⁿ

¹_p

can be proven by similar methods. 

Remark 4.3. The spaces CM^p_γare closely related to the so-called Hardy-Carleson type spaces CTq,α studied in [17]. In fact, we have b ∈ CM^p_γ if and only if Rb ∈ CT_p,γ−n, and this relation also holds for the little versions of these spaces.

Therefore, the distance formulas similar to Lemma4.2 can be obtained for CT_q,α. Let 1 ≤ q < ∞, α > −n − 1 and b ∈ CT_q,α. Then

dist(b, CT⁰_q,α)  lim sup

|a|→1⁻

Z

Bn

(1 − |a|²)ⁿ

|1 − hz, ai|²ⁿ|b(z)|^q(1 − |z|²)^α+ndv(z)

¹_q

 lim sup

|a|→1⁻

 1

(1 − |a|²)ⁿ Z

Q(a)

|b(z)|^q(1 − |z|²)^α+ndv(z)

¹_q . In the rest part of this section, we agree that

f_a(z) = (1 − |a|²)s−(n+1+α)/p

(1 − hz, ai)^s

for sufficiently large s and a ∈ Bn. It is obvious that kf_ak_A^p_α  1 and f_a → 0 uniformly on compact subsets of Bn as |a| → 1⁻. For any a ∈ Bn, let

S(a) = {ζ ∈ Sn : |1 − hζ, ai| < (1 − |a|²)^1/3}.

Lemma 4.4. Let 0 < p ≤ q < ∞, α > −1 and b ∈ H(Bn) such that J_b : A^p_α → H^q is bounded. Then

|a|→1lim Z

Sn\S(a)

|Jbfa|^qdσ = 0.

Proof. Let √

63/8 < |a| < 1. We claim that

|1 − hrζ, ai| > 1

4(1 − |a|²)^2/3

for any ζ ∈ Sn\ S(a) and 0 ≤ r ≤ 1. In fact, if 0 ≤ r ≤ 1 − (1 − |a|²)^2/3, then

|1 − hrζ, ai| ≥ 1 − r|hζ, ai| ≥ (1 − |a|²)^2/3 > 1

4(1 − |a|²)^2/3. If 1 − (1 − |a|²)^2/3< r ≤ 1, then

|1 − hrζ, ai| ≥ r|1 − hζ, ai| − (1 − r)

≥ r(1 − |a|²)^1/3− (1 − |a|²)^2/3

> (1 − |a|²)^1/3 1

2 − (1 − |a|²)^1/3



> 1

4(1 − |a|²)^2/3.

Consequently,

|fa(ζ)| = (1 − |a|²)s−(n+1+α)/p

|1 − hζ, ai|^s ≤ 4^s(1 − |a|²)^s3−n+1+αp and

|Rf_a(rζ)| = s|hrζ, ai|(1 − |a|²)s−(n+1+α)/p

|1 − hrζ, ai|^s+1 ≤ 4^ssr(1 − |a|²)^{s−23 −n+1+αp}

for any ζ ∈ Sn\ S(a) and 0 ≤ r ≤ 1. Therefore, for almost every ζ ∈ Sn\ S(a), integration by parts yields

|J_bf_a(ζ)|^q . |fa(ζ)b(ζ)|^q+

Z 1 0

|Rf_a(tζ)||b(tζ)|dt t

q

. (1 − |a|²)^{q(s−2)3 −(n+1+α)qp}



|b(ζ)|^q+

Z 1 0

|b(tζ)|dt

q . Since J_b : A^p_α→ H^q is bounded, we have

M := sup

z∈Bn

|Rb(z)|(1 − |z|²)^{nq+1−n+1+αp} < ∞,

which implies that b belongs to the Bloch space since 0 < p ≤ q < ∞ and α > −1, and subsequently |b(z)| . M log_1−|z|¹ for z ∈ Bn. Moreover, b = b(0) + J_b1 ∈ H^q. Then, noting that s is large enough, we establish

Z

Sn\S(a)

|Jbfa|^qdσ . (1 − |a|²)

q(s−2)

3 −^(n+1+α)q_p Z

Sn

|b|^qdσ + M^q

Z 1 0

log 1 1 − tdt

q

= (1 − |a|²)^{q(s−2)3 −(n+1+α)qp} (kbk^q_Hq + M^q) → 0

as |a| → 1 . 

Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2. (1) When n/q + 1 − (n + 1 + α)/p < 0, the boundedness of J_b : A^p_α → H^q implies that b is a constant, and then there is nothing to prove.

Suppose now γ = n/q + 1 − (n + 1 + α)/p ≥ 0.

The upper estimate can be deduced easily by Theorem 1.1, [13, Theorem 4]

and Lemma 4.1. In fact, for any g ∈ B_γ,0, by Theorem 1.1, J_g : A^p_α → H^q is compact. Therefore, by the norm estimate for J_b in [13, Theorem 4], we have

kJ_bk_e≤ kJ_b− J_gk  kb − gkB_γ.

Since g ∈ B_γ,0 is arbitrary, the upper estimate follows from Lemma 4.1.

We now take care of the lower estimate. Suppose K : A^p_α → H^q is a compact operator. It is easy to see R

S(a)|Kf_a|^qdσ → 0 as |a| → 1. Hence we have kJ_b− Kk & lim sup

|a|→1⁻

k(J_b − K)f_ak_H^q

≥ lim sup

|a|→1⁻



min{1, 2^1−1q}

Z

S(a)

|J_bf_a|^qdσ

1/q

−

Z

S(a)

|Kf_a|dσ

1/q