Volterra integration operators from Hardy-type tent spaces to Hardy spaces

In this paper, we completely characterize the boundedness and compactness of the Volterra integration operators Jg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$J_{g}$\end{document} acting from the Hardy-type tent spaces HTq,αp(Bn) to the Hardy spaces Ht(Bn) in the unit ball of Cn for all 0<p,q,t<∞\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,t<\infty $\end{document} and α>−n−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha >-n-1$\end{document}. The duality and factorization techniques for tent spaces of sequences play an important role in the proof of the main results.


Introduction
Let B n be the open unit ball in C n , and S n the boundary of B n . Denote by H(B n ) the space of all holomorphic functions on B n . A function g ∈ H(B n ) induces an integration operator (or a Volterra operator) J g given by the formula: where f is holomorphic on B n and Rg is the radial derivative of g, that is, z k ∂g ∂z k (z), z = (z 1 , . . . , z n ) ∈ B n .
The boundedness and compactness of J g have been extensively studied in many spaces of holomorphic functions in the unit ball (see [20] for the corresponding study between Hardy spaces, and [9,19] from Bergman spaces to Hardy spaces, and others [16,23,25] for example).
For 0 < t < ∞, the Hardy space H t (B n ) consists of those holomorphic functions f on B n with where dσ is the surface measure on the unit sphere S n := ∂B n normalized so that σ (S n ) = 1. For 0 < p, q < ∞ and α > -n -1, the weighted tent space T p q,α (B n ) consists of all measurable functions f on B n such that where dv is the volume measure on B n normalized so that v(B n ) = 1, and (ξ ) = {z ∈ B n : |1z, ξ | < (1 -|z| 2 )} is the admissible approach region. In particular, for α = 0, we write T where Q(w) = {z ∈ B n : |1z, w |w| | < 1 -|w| 2 } for w ∈ B n \{0} and Q(0) = B n . For 0 < p, q < ∞ and α > -n -1, the Hardy-type tent space HT p q,α (B n ) consists of holomorphic functions on B n that also belong to T p q,α (B n ), with the same quasinorm, and HT p ∞ (B n ) consists of holomorphic functions on B n that also belong to T p ∞ (B n ). The space CT q,α (B n ) consists of those holomorphic functions that belong to T ∞ q,α (B n ) that is endowed with the same norm. We refer the reader to [21] for more details on Hardy-type tent spaces.
As useful tools, tent spaces play important roles in the study of harmonic analysis and partial differential equations. By the nontangential maximal function characterization of the Hardy space, HT p ∞ (B n ) = H p (B n ) ⊆ HT p q,α (B n ), see [26], and we can consider H p (B n ) as the limit of HT p q,α (B n ) when q → ∞. Hence, we describe the boundedness and compactness of J g : HT p q,α (B n ) → H t (B n ) for all possible ranges 0 < p, q, t < ∞ and α > -n -1. Although only discrete characterizations are described in our theorems, continuous characterizations also can be obtained from subsequent proofs.
Our main results are as follows.
Theorem 1.1 Let 0 < p, q, t < ∞, α > -n -1. Then, the integration operator J g : is bounded if and only if for any r ∈ (0, 1) and an r- satisfies one of the following conditions: Theorem 1.2 Let 0 < p, q, t < ∞, α > -n -1. Then, the integration operator J g : is compact if and only if for any r ∈ (0, 1) and an r- satisfies one of the following conditions: (c) If p = t and q > 2, then This paper is organized as follows: Sect. 2 contains some background materials and the tools used in the proofs. Theorems 1.1 and 1.2 are proved in Sect. 3 and Sect. 4, respectively.
Throughout the paper, constants are used with no attempt to calculate their exact values, and the value of a constant C may change from one occurrence to the next. We also use the notion A B to indicate that there is a constant C > 0 with A ≤ CB. The converse relation A B is defined in an analogous manner, and if A B and A B both hold, we write A B. Given p ∈ [1, ∞], we will denote by p = p/(p -1) its Hölder conjugate, and we agree that 1 = ∞ and ∞ = 1 in this paper.

Preliminaries
In this section, we introduce some basic results that will be used for the proofs of our main theorems.

Area methods and equivalent norms
For ξ ∈ S n and γ > 1, the admissible approach region γ (ξ ) is defined as In this paper we agree that (ξ ) := 2 (ξ ). It is known that for every δ > 1 and γ > 1, there exists γ > 1 so that We will write (ξ ) to indicate this change of aperture. Given z ∈ B n , we can define the set I(z) = S n for z = 0, and I(z) = {ξ ∈ S n : z ∈ (ξ )} ⊂ S n for z = 0. Obviously, 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 We will need the following well-known Calderón's area theorem [8], which will be very important for our arguments, and the variant can be found in [1,20].
Note that Lemma A shows that f ∈ H(B n ) belongs to H t if and only if Rf ∈ HT t 2,1-n . This explains the special role of number 2 in Theorem 1.1 and Theorem 1.2.

Embedding theorems
We need the following embedding theorems for Hardy-type tent spaces, which are the generalizations of Lemma 15 and Lemma 23 in [21]. We prove them by a similar method.
with bounded inclusion.
Proof Let ξ ∈ S n and r > 0. For any z ∈ (ξ ) and f ∈ H(B n ), by the subharmonicity, we have Writing |f | s = |f | q |f | s-q and applying this estimate to the second factor gives Applying this to a fractional differential operator R s, n+1+α 2 and according to [21, Theorem G], we have For any natural number k, Let k be large enough such that 2k > q. Then, by Lemma B, we have We will also need the following Dirichlet-type embedding theorem, which can be found in [7].

Khinchine and Kahane inequalities
Let r k (u) be a sequence of Rademacher functions. We recall first the classical Khinchine's inequality (see [11, Appendix A] for example).
Khinchine's inequality: Let 0 < p < ∞. Then, for any sequence {c k } ∈ l 2 , we have The next result is known as Kahane's inequality, see for instance Lemma 5 of Luecking [18].
Kahane's inequality: Let X be a Banach space, and 0 < p, q < ∞. For any sequence

Separated sequences and lattices
A sequence of points {z j } ⊂ B n is said to be separated if there exists δ > 0 such that β(z i , z j ) ≥ δ for all i and j with i = j, where β(z, w) denotes the Bergman metric on B n . This implies that there is δ > 0 such that the Bergman metric balls D j = {z ∈ B n : β(z, z j ) < δ} are pairwise disjoint. We need a well-known result on decomposition of the unit ball B n . By Theorem 2.23 in [26], there exists a positive integer N such that for any 0 < r < 1 we can find a sequence {a k } in B n with the following properties: (i) B n = k D(a k , r).
(ii) The sets D(a k , r/4) are mutually disjoint. (iii) Each point z ∈ B n belongs to at most N of the sets D(a k , 4r). Any sequence {a k } satisfying the above conditions is called an r-lattice (in the Bergman metric). Obviously any r-lattice is a separated sequence.

Tent spaces of sequences
Let Z = {a k } be an r-lattice. We consider the complex-valued sequences enumerated by this lattice: λ k = f (a k ). For 0 < p, q < ∞, the tent space T p q (Z) consists of those sequences Analogously, the tent space T p ∞ (Z) consists of λ with Another tent space T ∞ q (Z) consists of λ such that We will need the following duality results for the tent spaces of sequences. The proof can be found in [13,14,17].
If 0 < q ≤ 1, then the dual of T p q (Z) is isomorphic to T p ∞ (Z) under the pairing above.
The following result originates from [20], which will be used to construct our test functions.
Lemma F Let 0 < p, q < ∞ and Z = {a k } be an r-lattice. If θ > n max(1, q p , 1 p , 1 q ), then the operator We will also need the following result concerning factorization of sequence tent spaces, which can be found in [19].

Discretization
We will use Khinchine's and Kahane's inequalities throughout the proof of our main results. These tools provide discrete version of the conditions we really need, hence, we need to obtain the continuous characterizations from the discrete ones. The following two results can be found in [19].
Lemma H Let 0 < p, q < ∞ and α > -n -1. There exist r 0 ∈ (0, 1) so that if 0 < r < r 0 and Z = {a k } is an r-lattice, then whenever f is holomorphic on B n and in T p q,α .

whenever f is holomorphic on B n such that the left-hand side is finite.
We also need the following similar result.
Lemma J Let 0 < p < ∞, and α > -n -1, β > 0. There exist r 0 ∈ (0, 1) so that if 0 < r < r 0 and Z = {a k } is an r-lattice, then for any a ∈ B n , we have whenever f is holomorphic on B n such that the left-hand side is finite.
Proof For any a ∈ B n and β > 0, note that By [15,Lemma 2.2], there exist r 0 ∈ (r, 4r), such that for any z ∈ D(a k , r), Thus, we deduce that Since the constants in " " do not depend on r, we can find the desired r 0 , which completes the proof.
3 Proof of Theorem 1.1

Necessity
Suppose that the integration operator J g : HT p q,α (B n ) → H t (B n ) is bounded. We consider first the case p = t, q ≤ 2 or p < t. In this case, for any a ∈ B n and θ > 0, consider the test functions By the standard estimate for H t (B n ) functions, we have Replacing z by a in the inequality above, we obtain In particular, we deduce that sup k |Rg(a k )|(1 -|a k | 2 ) q-(n+1+α) q +n( 1 t -1 p ) < ∞ as desired. Finally, it remains to deal with the other cases. Let Z = {a k } be an r-lattice and r be small enough. Consider the test functions where λ = {λ k } ∈ T p q (Z), r k (x) are the Rademacher functions, and θ is large enough such that Lemma F holds. Then, by Lemma A and Lemma F, we have which is equivalent to Integrating with respect to x from 0 to 1, and using Fubini's theorem, Khinchine's inequality, and Kahane's inequality as in the proof of Theorem 7 in [19], we obtain . Using subharmonicity and bearing in mind z∈ (ξ ) D(z, 4r) ⊂ (ξ ), we obtain S n a k ∈ (ξ ) |λ k | 2 Rg(a k ) 2 Therefore, (a) If p > t and q > 2, for some s large enough such that 2s > 1 and ts > 1, we want to prove Take any v = {v k } ∈ T pts pts-p+t 2qs 2qs-q+2 (Z) and factor it as v k = ρ k · λ 1/s k , where ρ = {ρ k } ∈ T ts ts-1 2s 2s-1 (Z), λ = {λ k } ∈ T p q (Z). Then, by (2) and Hölder's inequalities, we obtain . By the duality of tent spaces of sequences given in Lemma E, we have that u belongs to Note that if q ≤ 2, then 2s-1 2s + 1 qs = 1 δ for some δ ≤ 1. Thus, making some adjustments to the arguments in the proof of (a), we obtain that u belongs to T pt p-t ∞ (Z). (c) If p = t and q > 2, it suffices to prove u 1/s ∈ T ∞ 2qs q-2 (Z) for some s large enough such that 2s > 1 and ts > 1. An appeal to Lemma G gives that Proceeding with the argument as above again, we have that u belongs to T ∞

Sufficiency
To prove the sufficiency of Theorem 1.1, we split it into four cases.
(a) If p > t, q > 2 and u ∈ T pt p-t 2q q-2 (Z), let η = (1n -2α q ) q q-2 . By considering the dilated functions Rg ρ (z) = Rg(ρz) (0 < ρ < 1), an approximation argument (see [21,Lemma 7]) shows that according to Lemma H, we have q-2 ,η (B n ). Then, by Lemma A and Holder's inequalities, we have Next, for the remaining case p < t, there exists some r such that p < r < t and denote that η = ( r p -1)n -1 + r(n+1+α) q . Then, according to Lemma D and Lemma C, we have (1z, a k ) θ+ n+1+α q , z ∈ B n be the bounded operator defined in Lemma F, where Z = {a k } is an r-lattice and r is small enough. Since S Z (E) is a bounded set and J g is compact, the set J g • S Z (E) is relatively compact in H t (B n ). It is well known that a relatively compact set must be a totally bounded set, and then for any ε > 0, there exist a finite number of functions h 1 , . . . , h N , such that Observing that sup i=1,...,N h i H t (B n ) < ∞, for the above ε > 0, there exists ρ 0 ∈ (0, 1) such that Rh i 0 (z) 2 1 -|z| 2 1-n dv(z) t/2 dσ (ξ )