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

In this paper, we completely characterize the compactness of the Volterra type integration operators Jb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_b$$\end{document} acting from weighted Bergman spaces Aαp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^p_{\alpha }$$\end{document} to Hardy spaces Hq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^q$$\end{document} for all 0<p,q<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p,q<\infty $$\end{document}. Furthermore, we give some estimates for the essential norms of Jb:Aαp→Hq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_b:A^p_{\alpha }\rightarrow H^q$$\end{document} in the case 0<p≤q<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p\le q<\infty $$\end{document}. We finally describe the membership in the Schatten(-Herz) class of the Volterra type integration operators.


Introduction
Let B n be the open unit ball of C n and H(B n ) denote the algebra of holomorphic functions on B n . A function b ∈ H(B n ) induces a Volterra type integration operator J b given by the formula: where f ∈ H(B n ) and Rb is the radial derivative of b: Rb(z) = n k=1 z k ∂b ∂z k (z), z = (z 1 , z 2 , . . . , z n ) ∈ B n .
A fundamental property of the operator J b is the following formula involving the radial derivative R: For 0 < p < ∞, the Hardy space H p consists of those holomorphic functions f in B n with where dσ is the surface measure on the unit sphere S n = ∂B n normalized so that σ(S n ) = 1. Given α > −1 and 0 < p < ∞, a function f ∈ H(B n ) belongs to the weighted Bergman space A p α , if Here dv = dv 0 is the Lebesgue measure on B n , normalized so that v(B n ) = 1.
The operator J b has been studied by many authors, see [10,14,16,19] and the references therein. In particular, Wu [19] partially solved the boundedness of J b : A p α → H q in the setting of the unit disk. Recently, Miihkinen et al. [14] 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 J b 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 [14]. Theorem 1.1. Let α > −1, 0 < p, q < ∞ and b ∈ H(B n ). 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   belongs to L pq p−q (S n ).
It is interesting to find out when the compactness of J b : A p α → H q is trivial, i.e., J b is compact if and only if J b = 0. By the above theorem, we can easily obtain the following corollary.
. Then the following hold: (1) In the cases 0 < p ≤ min{2, q} and 2 Let X, Y be (quasi-)Banach spaces and T : X → Y a bounded operator. The essential norm of T , denoted by T e , is its distance from the space of compact operators. It is clear that T is compact if and only if T e = 0. Our next result gives some estimates for the essential norm of J b : A p α → H q in the case 0 < p ≤ q < ∞.
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 Loaiza, López-García and Pérez-Esteva introduced the Schatten-Herz class of Toeplitz operators in [11], which is a generalization of the Schatten class. Recall that, given a positive Borel measure μ on B n , the Toeplitz operator T μ on A 2 α is defined by where K α (z, w) is the reproducing kernel of A 2 α . See [23] for more information about Toeplitz operators. For 0 < p, q < ∞, the Toeplitz operator T μ is said to be in the Schatten-Herz class . Our next result characterizes the Schatten-Herz class of integration operators.
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 [9] and references there for more details. Based on the work concerning Toeplitz operators in [6], our next result (see Theorem 5.4 in Sect. 5) gives a description of the decay of singular values of J b : A 2 α → H 2 . The paper is organized as follows. Some background and preliminary results are given in Sect. 2. In Sect. 3 we consider the compactness of integration operators J b : A p α → H q . In Sect. 4 we estimate the essential norms. Section 5 is devoted to the proof of Theorems 1.4, 1.5 and some descriptions of asymptotic property of singular values of J b : A 2 α → H 2 . We also consider the essential norms and membership in Schatten(-Herz) classes of integration operators from Hardy spaces to Bergman spaces in Sect. 6.
Notation. For 1 < p < ∞, we let p 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.

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

Carleson Measures
For ξ ∈ S n and δ > 0, the non-isotropic metric ball B δ (ξ) is defined by A positive Borel measure μ on B n is said to be a Carleson measure if there is a constant C > 0 such that μ(B δ (ξ)) ≤ Cδ n for all ξ ∈ S n and δ > 0. Obviously every Carleson measure is finite. Hörmander [7] extended to several complex variables the famous Carleson measure embedding theorem [4,5] asserting that, for 0 < p < ∞, the embedding I d : Moreover, with constant depending on t, the supremum of the above integral is comparable to μ CM . A positive Borel measure μ on B n is called a vanishing Carleson measure if uniformly for ξ ∈ S n . Equivalently, one may require that for each (some) t > 0 one has

Separated Sequences and Lattices
We need a well-known result on decomposition of the unit ball B n . By Theorem 2.23 in [21], there exists a positive integer N such that for any 0 < δ < 1 we can find a sequence {a k } in B n with the following properties: The sets D(a k , δ/4) are mutually disjoint; (iii) Each point z ∈ B n 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.

Area Methods and Equivalent Norms
For ξ ∈ S n and γ > 1, recall that the admissible approach region Γ γ (ξ) is defined as In this paper we agree that Γ(ξ) := Γ 2 (ξ). It is known that for every δ > 0 and γ > 1, there exists γ > 1 so that We will write Γ(ξ) to indicate this change of aperture. If I(z) = {ξ ∈ S n : 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 This fact will be used repeatedly throughout the paper. Let us recall the following Littlewood-Paley identity, which can be found in [21].
The next estimate is the Calderón's area theorem [3,13]. The variant we will use can be found in [1] or in [16].
We will also need the following result essentially due to Luecking [12] (see also [16]) 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 B n . Then the identity I d : H p → L s (μ) is bounded if and only if the function defined on S n by

Tent Spaces of Sequences
For 0 < p, q < ∞ and a fixed separated sequence Analogously, the tent space The following theorem is about the dual of tent spaces of sequences. For the proof, see [2,12].
We will also need the following result concerning factorization of sequence tent spaces, which can be found in [14].

Compactness
In this section, we will prove Theorem 1.1. We need the following two lemmas first.

Lemma 3.1. If μ is a vanishing Carleson measure, then
There are two possibilities about the sequence {a k }: as k → ∞. This is contradictory with (3.1). If |a k | 1 − , there are a point a 0 ∈ B n and a subsequence of {a k } converging to a 0 . Without loss of generality, we assume a k → a 0 , then there exists for all z ∈ B n whenever k is large enough, and it is clear that 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.
By the same method as in the proof of [14, Lemma 3], we have It is easy to seer → 1 − as r → 1 − . Combining the above estimate with (3.2), we get 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 [14] 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)| 2 dv 1 (z). Then by Theorem B. For the sake of simplicity, we denote 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 B n . We need to show J b f k H p → 0. For any h ∈ H p/(p−2) , by Hölder's inequality and the fact that μ b is a (vanishing) Carleson measure we have that Since μ b is a vanishing Carleson measure, there exists r 0 (0 < r 0 < 1) such that when k is large enough. Denote by I d k : H p/(p−2) → L 1 (dμ f k ,b ) the embedding operator, and then we get lim k→∞ I d k H p/(p−2) →L 1 (dμ f k ,b ) = 0 by (3.4). Therefore, by Theorem C, we have Thus 3) and the proof of the sufficiency is complete. Next we consider the necessity part. Assume that We see that the measure μ f,b satisfies the conditions of Theorem C for parameters 1 and p p−2 , and this implies I d f : is a bounded sequence converging to zero uniformly on compact subsets of B n . Since Define dν h k ,b (z) = |h k (z)||Rb(z)| 2 dv 1 (z). We now suppose f ∈ B A p α and k > K := max{K 1 , . . . , K m }, then there is j ∈ {1, . . . , m} such that [20, Theorem 54], we have for any δ > 0 and k ≥ 1,

This implies that the embeddings
By subharmonic property, we have It follows from (3.5) and (3.6) that Therefore, we have lim k→∞ Bn This implies that Suppose that J b : A p α → H q is compact, and then it follows from [14, Theorem 1] that V b,0 ∈ L pq/(p−q) (S n ). We want to show that Vol. 76 (2021) Essential Norms and Schatten(-Herz) Classes Page 13 of 33 88 is obvious. We now assume that p − 1 − α ≥ 0. By Lemma 3.2, we only need to prove that for any c ∈ B T p p (Z) . That is, By the same process as in the proof of [14, Theorem 7], we can establish Using the dual and factorization of sequence tent spaces as in the proof of Theorem 7 and Theorem 8 in [14], we can get the desired result. To this end, write x r = {x k,r }, where (3.10) Recall that by (3.8), we want to prove that converges to zero in L p/(p−q) (S n ) as r → 1 − . Namely, we want to prove For every s > 1, this is equivalent to For s large enough, by Theorems D and E, we have and factor it as Then by (2.4) and Hölder's inequality, Combining this with (3.9) and (3.10), we establish whenever r > r 0 . Considering all possible factorizations yields .
We obtain x

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 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(B n ) such that We have the following distance formula for the space B γ .
Proof. If γ = 0, by maximum modulus principle, we have 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, Let δ → 1 − , and we get which completes the lemma.
is a Carleson measure, and define We also define the space CM p γ,0 to be the subspace of CM p γ consisting of b ∈ H(B n ) 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) = B n and for a ∈ B n \{0}, 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, 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 ∈ B n and some absolute constant C > 0, and can be proven by similar methods.
Remark 4.3. The spaces CM p γ are closely related to the so-called Hardy-Carleson type spaces CT q,α studied in [18]. 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 Lemma 4.2 can be obtained for CT q,α . Let 1 ≤ q < ∞, α > −n − 1 and b ∈ CT q,α . Then In the rest part of this section, we agree that for any ζ ∈ S n \S(a) and 0 ≤ r ≤ 1. In fact, if 0 ≤ r ≤ 1 − (1 − |a| 2 ) 2/3 , then for any ζ ∈ S n \S(a) and 0 ≤ r ≤ 1. Therefore, for almost every ζ ∈ S n \S(a), integration by parts yields which implies that b belongs to the Bloch space since 0 < p ≤ q < ∞ and α > −1, and subsequently |b(z)| M log 1 1−|z| for z ∈ B n . Moreover, b = b(0) + J b 1 ∈ H q . Then, noting that s is large enough, we establish Proof of Theorem 1.3. (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, [14, 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 [14, Theorem 4], we have 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 S(a) |Kf a | q dσ → 0 as |a| → 1. Hence we have where the last equality follows from Lemma 4.4. Since K : A p α → H q is an arbitrary compact operator, we obtain

Schatten Class
Recall that A 2 α is a reproducing kernel Hilbert space with the reproducing kernel function given by K z (w) = 1 (1 − w, z ) n+1+α , z,w ∈ B n with norm K z A 2 α = K z (z) 1/2 = (1 − |z| 2 ) −(n+1+α)/2 . The normalized kernel functions are denoted by k z = Kz Kz A 2 α . We also need to introduce some "fractional derivatives" of the kernel functions. For z, w ∈ B n and t ≥ 0, define