Maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n$

In this paper, we present maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n.$ The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the radial derivative, the complex gradient, and the invariant gradient. As an application, we obtain new maximal and area integral characterizations of Besov spaces. Moreover, we give an atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces.


Introduction and main results
Let C denote the set of complex numbers. Throughout the paper we fix a positive integer n, and let C n = C × · · · × C denote the Euclidean space of complex dimension n. Addition, scalar multiplication, and conjugation are defined on C n componentwise. For z = (z 1 , · · · , z n ) and w = (w 1 , · · · , w n ) in C n , we write z, w = z 1 w 1 + · · · + z n w n , where w k is the complex conjugate of w k . We also write |z| = |z 1 | 2 + · · · + |z n | 2 .
The open unit ball in C n is the set The boundary of B n will be denoted by S n and is called the unit sphere in C n , i.e., The automorphism group of B n , denoted by Aut(B n ), consists of all biholomorphic mappings of B n . Traditionally, bi-holomorphic mappings are also called automorphisms.
In the case of α = −(n + 1) we denote the resulting measure by and call it the invariant measure on B n , since dτ = dτ • ϕ for any automorphism ϕ of B n .
For α > −1 and p > 0, the (weighted) Bergman space A p α consists of holomorphic functions f in B n with where the weighted Lebesgue measure dv α on B n is defined by dv α (z) = c α (1 − |z| 2 ) α dv(z) and c α = Γ(n + α + 1)/[n!Γ(α + 1)] is a normalizing constant so that dv α is a probability measure on B n . Thus, A p α = H(B n ) ∩ L p (B n , dv α ), where H(B n ) is the space of all holomorphic functions in B n . When α = 0 we simply write A p for A p 0 . These are the usual Bergman spaces. Note that for 1 ≤ p < ∞, A p α is a Banach space under the norm p, α . If 0 < p < 1, the space A p α is a quasi-Banach space with p-norm f p p,α . Recall that D(z, γ) denotes the Bergman metric ball at z D(z, γ) = {w ∈ B n : β(z, w) < γ} with γ > 0, where β is the Bergman metric on B n . It is known that whereafter ϕ z is the bijective holomorphic mapping in B n , which satisfies ϕ z (0) = z, ϕ z (z) = 0 and ϕ z • ϕ z = id.
As is well known, maximal functions play a crucial role in the real-variable theory of Hardy spaces (cf. [11]). In this paper, we first establish a maximalfunction characterization for the Bergman spaces. To this end, we define for each γ > 0 and f ∈ H(B n ) : (1.1) (M γ f )(z) = sup w∈D(z,γ) |f (w)|, ∀z ∈ B n .
We begin with the following simple result.
The norm appearing on the right-hand side of (1.2) can be viewed an analogue of the so-called nontangential maximal function in Hardy spaces. The proof of Theorem 1.1 is fairly elementary (see §2), using some basic facts and estimates on the Bergman balls.
In order to state the real-variable area integral characterizations of the Bergman spaces, we require some more notation. For any f ∈ H(B n ) and z = (z 1 , . . . , z n ) ∈ B n we define and call it the radial derivative of f at z. The complex and invariant gradients of f at z are respectively defined as Now, for fixed 1 < q < ∞ and γ > 0, we define for each f ∈ H(B n ) and z ∈ B n : (1) The radial area function (2) The complex gradient area function (3) The invariant gradient area function We state another main result of this paper as follows.
Then, for any f ∈ H(B n ) the following conditions are equivalent:

Moreover, the quantities
where the comparable constants depend only on q, γ, α, p, and n.
For 0 < p < ∞ and −∞ < α < ∞ we fix a nonnegative integer k with pk+α > −1 and define the so-called Bergman space A p α introduced in [14] as the space of all f ∈ H(B n ) such that (1− |z| 2 ) k R k f ∈ L p (B n , dv α ). One then easily observes that A p α is independent of the choice of k and consistent with the traditional definition when α > −1. Let N be the smallest nonnegative integer such that pN + α > −1 and define Equipped with (1.3), A p α becomes a Banach space when p ≥ 1 and a quasi-Banach space for 0 < p < 1.
It is known that the family of the generalized Bergman spaces A p α covers most of the spaces of holomorphic functions in the unit ball of C n , such as the classical diagonal Besov space B s p and the Sobolev space W p k,β (e.g., [5]), which has been extensively studied before in the literature under different names (e.g., see [14] for an overview). We refer to Arcozzi-Rochberg-Sawyer [3,4], Tchoundja [12] and Volberg-Wick [13] for some recent results on such Besov spaces and more references.
As an application of Theorems 1.1 and 1.2, we obtain new maximal and area integral characterizations of such Besov spaces as follows.
Then for any f ∈ H(B n ), Moreover, where "≈" depends only on q, γ, α, p, k, and n.
To prove Corollaries 1.1 and 1.2, one merely notices that f ∈ A p α if and only if R k f ∈ L p (B n , dv α+pk ) and applies Theorems 1.1 and 1.2 respectively to R k f with the help of Lemma 2.1 below.
There are various characterizations for B s p or W p k,β involving complexvariable quantities in terms of fractional differential operators and in terms of higher order (radical) derivatives and/or complex and invariant gradients, for a review and details see [14] and references therein. However, Corollaries 1.1 and 1.2 can be considered as a unified characterization for such spaces involving real-variable quantities.
In particular, H p s = A p α with α = −2s − 1, where H p s is the Hardy-Sobolev space defined as the set Here, There are several realvariable characterizations of the Hardy-Sobolev spaces obtained by Ahern and Bruna [1] (see also [2]). These characterizations are in terms of maximal and area integral functions on the admissible approach region D α (η) = z ∈ B n : |1 − z, η | < α 2 (1 − |z| 2 ) , η ∈ S n , α > 1. itself. The rest of the paper is organized as follows. In Section 2 we will prove Theorems 1.1 and 1.2 using some elementary facts about Bergman metric and Bergman kernel functions. In particular, we will prove Theorem 1.2 in the case of 0 < p ≤ 1 by using "complex-variable" atom decomposition for Bergman spaces due to Coifman and Rochberg [8]. In Section 3, we will prove an atomic decomposition of A 1 α with respect to Carleson tubes, which is used to give a real-variable proof of Theorem 1.2 in the case p = 1 in Section 4.
In what follows, C always denotes a constant depending (possibly) on n, q, p, γ or α but not on f, which may be different in different places. For two nonnegative (possibly infinite) quantities X and Y, by X Y we mean that there exists a constant C > 0 such that X ≤ CY and by X ≈ Y that X Y and Y X. Any notation and terminology not otherwise explained, are as used in [15] for spaces of holomorphic functions in the unit ball of C n .

Proofs of Theorems 1.1 and 1.2: Complex methods
For the sake of convenience, we collect some elementary facts on the Bergman metric and holomorphic functions in the unit ball of C n as follows.
where N is the constant in Lemma 2.4 depending only on γ and n.
Let 1 ≤ p < ∞ and let E be a complex Banach space. We write L p α (B n , E) for the Banach space of strongly measurable E-valued functions on B n such that It is known that (for example, see [10]) if f is holomorphic in this weak sense, then it is holomorphic in the stronger sense that f is the sum of a power series Then, by merely repeating the proof of the scalar case (e.g., Theorem 3.25 in [15]), we have the following interpolation result.
Moreover, to prove Theorem 1.2 for the case 0 < p ≤ 1, we will use atom decomposition for Bergman spaces due to Coifman and Rochberg [8] (see also [15], Theorem 2.30) as follows.
Proposition 2.1. Suppose p > 0, α > −1, and b > n max{1, 1/p} + (α + 1)/p. Then there exists a sequence {a k } in B n such that A p α consists exactly of functions of the form where {c k } belongs to the sequence space ℓ p and the series converges in the norm topology of A p α . Moreover, where the infimum runs over all the above decompositions.
Also, we need a characterization of Carleson type measures for Bergman spaces as follows, which can be found in [14], Theorem 45.
We are now ready to prove Theorem 1.2. Note that for any f ∈ H(B n ), (e.g., Lemma 2.14 in [15].) We have that (d) implies (c), and (c) implies (b) in Theorem 1.2. Then, it remains to prove that (b) implies (a), and (a) implies (d).
The proof of (a) ⇒ (d) is divided into two steps. At first we prove the case of 0 < p ≤ 1 using the atomic decomposition, then we prove the generic case via complex interpolation.
Proof of (a) ⇒ (d) for 0 < p ≤ 1. To this end, we write An immediate computation yields that and Then we have
Proof of (a) ⇒ (d) for p > 1. Set E = L q (B n , χ D(0,γ) dτ ; C n ). Consider the operator Note that ϕ z (D(0, γ)) = D(z, γ) and the measure dτ is invariant under any automorphism of B n (cf. Proposition 1.13 in [15]), we have . On the other hand, Then, we conclude that T is bounded from B into B(B n , E). Thus, applying Lemma 2.5 to this fact with the case of p = 1 proved above yields that T is bounded from A p α into A p α (B n , E) for any 1 < p < ∞, i.e., where C depends only on q, γ, n, p, and α. The proof is complete.

Atomic decomposition for Bergman spaces
We let d(z, w) = |1 − z, w | 1 2 , z, w ∈ B n . It is known that d satisfies the triangle inequality and the restriction of d to S n is a metric. As usual, d is called the nonisotropic metric.
For any ζ ∈ S n and r > 0, the set is called a Carleson tube with respect to the nonisotropic metric d. We usually write Q = Q r (ζ) in short.
As usual, we define the atoms with respect to the Carleson tube as follows: for 1 < q < ∞, a ∈ L q (B n , dv α ) is said to be a (1, q) α -atom if there is a Carleson tube Q such that (1) a is supported in Q; The constant function 1 is also considered to be a (1, q) α -atom.
Recall that P α is the orthogonal projection from L 2 (B n , dv α ) onto A 2 α , which can be expressed as P α extends to a bounded projection from L p (B n , dv α ) onto A p α (1 < p < ∞). We have the following useful estimates.
Lemma 3.1. For α > −1 and 1 < q < ∞ there exists a constant C q,α > 0 such that To prove Lemma 3.1, we need first to show an inequality for reproducing kernel K α associated with d, which is essentially borrowed from Proposition 2.13 in [12].
Proof of Lemma 3.1. When a is the constant function 1, the result is clear. Thus we may suppose a is a (1, q) α -atom. Let a be supported in a Carleson tuber Q r (ζ) and δr ≤ √ 2, where δ is the constant in Lemma 3.2. Since P α is a bounded operator on L q (B n , dv α ), we have where we have used the fact that v α (Q r ) ≈ r 2(n+1+α) in the third inequality (e.g., Corollary 5.24 in [15]). Thus, we get where C depends only on n and α. Now we turn to the real-variable atomic decomposition of A 1 α (α > −1) with respect to the Carleson tubes. Note that for any (1, q) α -atom a, Then, we define A 1,q α as the space of all f ∈ A 1 α which admits a decomposition where for each i, a i is an (1, q) α -atom and λ i ∈ C so that i |λ i | < ∞. We equip this space with the norm where the infimum is taken over all decompositions of f described above.
It is easy to see that A 1,q α is a Banach space. By Lemma 3.1 we have the contractive inclusion A 1,q α ⊂ A 1 α . We will prove in what follows that these two spaces coincide. That establishes the "real-variable" atomic decomposition of the Bergman space A 1 α . In fact, we will show the remaining inclusion A 1 α ⊂ A 1,q α by duality.
Theorem 3.1. Let 1 < q < ∞ and α > −1. For every f ∈ A 1 α there exist a sequence {a i } of (1, q) α -atoms and a sequence {λ i } of complex numbers such that Moreover, where the infimum is taken over all decompositions of f described above and " ≈ " depends only on α and q.
Recall that the dual space of A 1 α is the Bloch space B (we refer to [15] for details). The Banach dual of A 1 α can be identified with B (with equivalent norms) under the integral pairing (e.g., see Theorem 3.17 in [15].) In order to prove Theorem 3.1, we need the following result, which can be found in [6] (see also Theorem 5.25 in [15]).
for all r > 0 and all ζ ∈ S n , where Moreover, where "≈" depends only on α, p, and n.
As noted above, we will prove Theorem 3.1 via duality. To this end, we first prove the following duality theorem.
Proposition 3.1. For any 1 < q < ∞ and α > −1, we have (A 1,q α ) * = B isometrically. More precisely, (i) Every g ∈ B defines a continuous linear functional ϕ g on A 1,q α by (ii) Conversely, each ϕ ∈ (A 1,q α ) * is given as (3.2) by some g ∈ B. Moreover, we have Proof. Let p be the conjugate index of q, i.e., 1/p + 1/q = 1. We first show For any (1, q) α -atom a, by Lemma 3.3 we have On the other hand, for the constant function 1 we have P α 1 = 1 and so Thus, we deduce that for any finite linear combination f of (1, q) α -atoms. Hence, g defines a continuous linear functional ϕ g on a dense subspace of A 1, q α and ϕ g extends to a continuous linear functional on A 1, q α such that |ϕ g (f )| ≤ C(|g(0)| + g B ) f A 1, q α for all f ∈ A 1, q α .
Next let ϕ be a bounded linear functional on A 1, q α . Note that H q (B n , dv α ) = H(B n ) ∩ L q (B n , dv α ) ⊂ A 1, q α . Then, ϕ is a bounded linear functional on H q (B n , dv α ). By duality there exists g ∈ H p (B n , dv α ) such that Let Q = Q r (ζ) be a Carleson tube. For any f ∈ L q (B n , dv α ) supported in Q, it is easy to check that is a (1, q)-atom. Then, |ϕ(P α a f )| ≤ ϕ and so Hence, for any f ∈ L q (B n , dv α ) we have This concludes that By Lemma 3.3 we have that g ∈ B and g B ≤ C ϕ . Therefore, ϕ is given as (3.2) by g with |g(0)| + g B ≤ C ϕ .
Now we are ready to prove Theorem 3.1.
(1) One would like to expect that when 0 < p < 1, A p α also admits an atomic decomposition in terms of atoms with respect to Carleson tubes. However, the proof of Theorem 3.1 via duality cannot be extended to the case 0 < p < 1. At the time of this writing, this problem is entirely open.
(2) The real-variable atomic decomposition of Bergman spaces should be known to specialists, at least in the case p = 1. Indeed, Coifman and Weiss [9] claimed that the Bergman space A 1 admits an atomic decomposition, based on their theory of harmonic analysis on homogeneous spaces. However, the case 0 < p < 1 seems open as well.

Area integral inequalities: Real-variable methods
In this section, we will prove the area integral inequality for the Bergman space A 1 α via atomic decomposition established in Section 3.
Theorem 4.1. Suppose 1 < q < ∞, γ > 0, and α > −1. Then, This is the assertion that (a) ⇒ (d) in Theorem 1.2 for the case p = 1. The novelty is that the proof we present here utilizes a real-variable method.
The following lemma is elementary.
By the same argument we can estimate I 2 and omit the details.
Next, we estimate I 3 . Note that where the last inequality is achieved by the following estimates , for any w ∈ D(z, γ) and u ∈ Q r (ζ). Thus, by Lemmas 2.1 and 2.3 Hence, as shown above. Similarly, we can estimate I 4 and omit the details. Therefore, combining above estimates we conclude that where C depends only on q, γ, n, and α. Remark 4.1. We remark that whenever A p α have an atomic decomposition in terms of atoms with respect to Carleson tubes for 0 < p < 1, the argument of Theorem 1.2 works as well in this case. However, as noted in Remark 3.1, the problem of the atomic decomposition of A p α with respect to Carleson tubes for 0 < p < 1 is entirely open.
Remark 4.2. The area integral inequality in case 1 < p < ∞ can be also proved through using the method of vector-valued Calderón-Zygmund operators for Bergman spaces. This has been done in [7].