Maximal and Area Integral Characterizations of Bergman Spaces in the Unit Ball of ℂ^𝑛

Abstract

We present maximal and area integral characterizations of Bergman spaces in the unit ball of 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.

1. 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 (1) denote the Euclidean space of complex dimension 𝑛. Addition, scalar multiplication, and conjugation are defined on ℂ^𝑛 component wise. For 𝑧=(𝑧₁,…,𝑧_𝑛) and 𝑤=(𝑤₁,…,𝑤_𝑛) in ℂ^𝑛, we write ⟨𝑧,𝑤⟩=𝑧₁𝑤₁+⋯+𝑧_𝑛𝑤_𝑛, (2) where 𝑤_𝑘 is the complex conjugate of 𝑤_𝑘. We also write |𝑧|=𝑧₁²+⋯+𝑧_𝑛². (3) The open unit ball in ℂ^𝑛 is the set 𝔹_𝑛=𝑧∈ℂ^𝑛∶|𝑧|<1. (4) The boundary of 𝔹_𝑛 will be denoted by 𝕊_𝑛 and is called the unit sphere in ℂ^𝑛, that is, 𝕊_𝑛=𝑧∈ℂ^𝑛∶|𝑧|=1. (5) Also, we denote by 𝔹_𝑛 the closed unit ball, that is, 𝔹_𝑛=𝑧∈ℂ^𝑛∶|𝑧|≤1=𝔹_𝑛∪𝕊_𝑛. (6) 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.

For 𝛼∈ℝ, the weighted Lebesgue measure 𝑑𝑣_𝛼 on 𝔹_𝑛 is defined by 𝑑𝑣_𝛼(𝑧)=𝑐_𝛼1−|𝑧|^2𝛼𝑑𝑣(𝑧), (7) where 𝑐_𝛼=1 for 𝛼≤−1 and 𝑐_𝛼=Γ(𝑛+𝛼+1)/[𝑛!Γ(𝛼+1)] if 𝛼>−1, which is a normalizing constant so that 𝑑𝑣_𝛼 is a probability measure on 𝔹_𝑛. In the case of 𝛼=−(𝑛+1), we denote the resulting measure by 𝑑𝜏(𝑧)=𝑑𝑣1−|𝑧|^2𝑛+1 (8) 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 𝑓_𝑝,𝛼=_𝔹𝑛𝑓(𝑧)^𝑝𝑑𝑣_𝛼(𝑧)^1/𝑝<∞, (9) where the weighted Lebesgue measure 𝑑𝑣_𝛼 on 𝔹_𝑛 is defined by 𝑑𝑣_𝛼(𝑧)=𝑐_𝛼1−|𝑧|^2𝛼𝑑𝑣(𝑧), (10) 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=ℋ𝔹_𝑛∩𝐿^𝑝𝔹_𝑛,𝑑𝑣_𝛼, (11) where ℋ(𝔹_𝑛) is the space of all holomorphic functions in 𝔹_𝑛. When 𝛼=0, we simply write 𝒜^𝑝 for 𝒜^𝑝₀. 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 𝑧𝐷𝑧,𝛾=𝑤∈𝔹_𝑛∶𝛽(𝑧,𝑤)<𝛾, (12) with 𝛾>0, where 𝛽 is the Bergman metric on 𝔹_𝑛. It is known that 𝛽()=1𝑧,𝑤2𝜑log1+_𝑧(𝑤)1−𝜑_𝑧(𝑤),𝑧,𝑤∈𝔹_𝑛, (13) 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. [1]). In this paper, we first establish a maximal-function characterization for the Bergman spaces. To this end, we define for each 𝛾>0 and 𝑓∈ℋ(𝔹_𝑛): 𝑀_𝛾𝑓(𝑧)=sup_{𝑤∈𝐷𝑧,𝛾}𝑓(𝑤),∀𝑧∈𝔹_𝑛. (14) Then we have the following result.

Theorem 1. Suppose 𝛾>0 and 𝛼>−1. Let 0<𝑝<∞. Then for any 𝑓∈ℋ(𝔹_𝑛),𝑓∈𝒜^𝑝_𝛼 if and only if 𝑀_𝛾𝑓∈𝐿^𝑝(𝔹_𝑛,𝑑𝑣_𝛼). Moreover, 𝑓_𝑝,𝛼≈𝑀_𝛾𝑓_𝑝,𝛼, (15) where “≈” depends only on 𝛾,𝛼,𝑝, and 𝑛.

The norm appearing on the right-hand side of (15) can be viewed as an analogue of the so-called nontangential maximal function in Hardy spaces. The proof of Theorem 1 is fairly elementary (see Section 2), using some basic facts and estimates on the Bergman balls.

In order to state the area integral characterizations of the Bergman spaces, we require some more notation. For any 𝑓∈ℋ(𝔹_𝑛) and 𝑧=(𝑧₁,…,𝑧_𝑛)∈𝔹_𝑛, we define ℛ𝑓(𝑧)=^𝑛_𝑘=1𝑧_𝑘𝜕𝑓(𝑧)𝜕𝑧_𝑘 (16) and call it the radial derivative of 𝑓 at 𝑧. The complex and invariant gradients of 𝑓 at 𝑧 are, respectively, defined as (𝑧)=(𝑧)∇𝑓𝜕𝑓𝜕𝑧₁(𝑧),…,𝜕𝑓𝜕𝑧_𝑛,∇𝑓(𝑧)=∇𝑓∘𝜑_𝑧(0). (17)

Now, for fixed 1<𝑞<∞ and 𝛾>0, we define for each 𝑓∈ℋ(𝔹_𝑛) and 𝑧∈𝔹_𝑛:(1)the radial area integral function 𝐴_ℛ^𝛾,𝑞𝑓(𝑧)=_{𝐷(𝑧,𝛾)}|(1−|𝑤|²)ℛ𝑓(𝑤)|^𝑞𝑑𝜏(𝑤)^1/𝑞, (18)(2)the complex gradient area integral function 𝐴_∇^𝛾,𝑞𝑓(𝑧)=_{𝐷(𝑧,𝛾)}|(1−|𝑤|²)∇𝑓(𝑤)|^𝑞𝑑𝜏(𝑤)^1/𝑞, (19)(3)the invariant gradient area integral function 𝐴^𝛾,𝑞∇𝑓(𝑧)=_{𝐷(𝑧,𝛾)}|∇𝑓(𝑤)|^𝑞1𝑑𝜏(𝑤)𝑞. (20)

We state the second main result of this paper as follows.

Theorem 2. Suppose 1<𝑞<∞,𝛾>0, and 𝛼>−1. Let 0<𝑝<∞. Then, for any 𝑓∈ℋ(𝔹_𝑛), the following conditions are equivalent:(a)𝑓∈𝒜^𝑝_𝛼,(b)𝐴_ℛ^𝛾,𝑞(𝑓) is in 𝐿^𝑝(𝔹_𝑛,𝑑𝑣_𝛼),(c)𝐴_∇^𝛾,𝑞(𝑓) is in 𝐿^𝑝(𝔹_𝑛,𝑑𝑣_𝛼),(d)𝐴^𝛾,𝑞∇(𝑓) is in 𝐿^𝑝(𝔹_𝑛,𝑑𝑣_𝛼).
Moreover, the quantities 𝐴_ℛ^𝛾,𝑞(𝑓)_𝑝,𝛼,𝐴_∇^𝛾,𝑞(𝑓)_𝑝,𝛼,𝐴^𝛾,𝑞∇(𝑓)_𝑝,𝛼 (21) are all comparable to ‖𝑓−𝑓(0)‖_𝑝,𝛼, 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 generalized Bergman space 𝒜^𝑝_𝛼 as introduced in [2] to be the space of all 𝑓∈ℋ(𝔹_𝑛) such that (1−|𝑧|²)^𝑘ℛ^𝑘𝑓∈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. Put 𝑓_𝑝,𝛼=𝑓(0)+_𝔹𝑛1−|𝑧|^2𝑝𝑁ℛ^𝑁𝑓(𝑧)^𝑝𝑑𝑣_𝛼(𝑧)^1/𝑝,𝑓∈𝒜^𝑝_𝛼. (22) Equipped with (22), 𝒜^𝑝_𝛼 becomes a Banach space when 𝑝≥1 and a quasi-Banach space for 0<𝑝<1.

clized 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, which has been extensively studied before in the literature under different names (see e.g., [2] for an overview).

There are various characterizations for 𝐵^𝑠_𝑝 or 𝑊^𝑝_𝑘,𝛽 involving complex-variable quantities in terms of radical derivatives, complex and invariant gradients, and fractional differential operators (for a review and details see [2] and references therein). However, as an application of Theorems 1 and 2, we obtain new maximal and area integral characterizations of the Besov spaces as follows, which can be considered as a unified characterization for such spaces involving real-variable quantities.

Corollary 3. Suppose 𝛾>0 and 𝛼∈ℝ. Let 0<𝑝<∞ and 𝑘 a positive integer such that 𝑝𝑘+𝛼>−1. Then for any 𝑓∈ℋ(𝔹_𝑛),𝑓∈𝒜^𝑝_𝛼 if and only if 𝑀_𝛾^(𝑘)(𝑓)∈𝐿^𝑝(𝔹_𝑛,𝑑𝑣_𝛼), where 𝑀_𝛾^(𝑘)𝑓(𝑧)=sup_{𝑤∈𝐷𝑧,𝛾}1−|𝑤|^2𝑘ℛ^𝑘𝑓(𝑤),𝑧∈𝔹_𝑛. (23) Moreover, 𝑓−𝑓(0)_𝑝,𝛼≈𝑀_𝛾(ℛ^𝑘𝑓)_𝑝,𝛼, (24) where “≈” depends only on 𝛾,𝛼,𝑝,𝑘, and 𝑛.

Corollary 4. Suppose 1<𝑞<∞,𝛾>0, and 𝛼∈ℝ. Let 0<𝑝<∞ and 𝑘 a nonnegative integer such that 𝑝𝑘+𝛼>−1. Then for any 𝑓∈ℋ(𝔹_𝑛),𝑓∈𝒜^𝑝_𝛼 if and only if 𝐴_ℛ^𝛾,𝑞_𝑘+1(𝑓) is in 𝐿^𝑝(𝔹_𝑛,𝑑𝑣_𝛼), where 𝐴_ℛ^𝛾,𝑞_𝑘+1𝑓(𝑧)=_{𝐷(𝑧,𝛾)}1−|𝑤|^2𝑘+1ℛ^𝑘+1𝑓(𝑤)^𝑞𝑑𝜏(𝑤)^1/𝑞. (25) Moreover, 𝑓−𝑓(0)_𝑝,𝛼≈𝐴_ℛ^𝛾,𝑞_𝑘+1(𝑓)_𝑝,𝛼, (26) where “≈” depends only on 𝑞,𝛾,𝛼,𝑝,𝑘, and 𝑛.

To prove Corollaries 3 and 4, one merely notices that 𝑓∈𝒜^𝑝_𝛼 if and only if ℛ^𝑘𝑓∈𝐿^𝑝(𝔹_𝑛,𝑑𝑣_{𝛼+𝑝𝑘}) and applies Theorems 1 and 2, respectively, to ℛ^𝑘𝑓 with the help of Lemma 5 below.

The paper is organized as follows. In Section 2 we will prove Theorems 1 and 2. An atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces will be presented in Section 3 via duality method. Finally, in Section 4, we will prove Theorem 2 through using the real-variable atomic decomposition of Bergman spaces established in the preceding section.

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 [3] for spaces of holomorphic functions in the unit ball of ℂ^𝑛.

2. Proofs of Theorems 1 and 2

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.

Lemma 5 (cf. [3, Lemma 2.20]). For each 𝛾>0, 1−|𝑎|²≈1−|𝑧|²≈|1−⟨𝑎,𝑧⟩|, (27) for all 𝑎 and 𝑧 in 𝔹_𝑛 with 𝛽(𝑎,𝑧)<𝛾.

Lemma 6 (cf. [3, Lemma 2.24]). Suppose 𝛾>0,𝑝>0, and 𝛼>−1. Then there exists a constant 𝐶>0 such that for any 𝑓∈ℋ(𝔹_𝑛), 𝑓(𝑧)^𝑝≤𝐶1−|𝑧|^{2𝑛+1+𝛼𝐷(𝑧,𝛾)}𝑓(𝑤)^𝑝𝑑𝑣_𝛼(𝑤),∀𝑧∈𝔹_𝑛. (28)

Lemma 7 (cf. [3, Lemma 2.27]). For each 𝛾>0, |1−⟨𝑧,𝑢⟩|≈|1−⟨𝑧,𝑣⟩|, (29) for all 𝑧 in 𝔹_𝑛 and 𝑢,𝑣 in 𝔹_𝑛 with 𝛽(𝑢,𝑣)<𝛾.

2.1. Proof of Theorem 1

One needs the following result (cf. [4, Lemma 5]).

Lemma 8. For fixed 𝛾>0, there exist a positive integer 𝑁 and a sequence {𝑎_𝑘} in 𝔹_𝑛 such that(1)𝔹_𝑛=⋃_𝑘𝐷(𝑎_𝑘,𝛾), and(2)each 𝑧∈𝔹_𝑛 belongs to at most 𝑁 of the sets 𝐷(𝑎_𝑘,3𝛾).

Proof of Theorem 1. Let 𝑝>0. By Lemmas 8, 6, and 5, one has _𝔹𝑛𝑀_𝛾(𝑓)(𝑧)^𝑝𝑑𝑣_𝛼(𝑧)≤_{𝑘𝐷(𝑎𝑘,𝛾)}𝑀_𝛾(𝑓)(𝑧)^𝑝𝑑𝑣_𝛼(𝑧)=_{𝑘𝐷(𝑎𝑘,𝛾)}sup_{𝑤∈𝐷(𝑧,𝛾)}𝑓(𝑤)^𝑝𝑑𝑣_𝛼(𝑧)≲_{𝑘𝐷(𝑎𝑘,𝛾)}sup_{𝑤∈𝐷𝑧,𝛾}11−|𝑤|^2𝑛+1+𝛼×_{𝐷(𝑤,𝛾)}𝑓(𝑢)^𝑝𝑑𝑣_𝛼(𝑢)𝑑𝑣_𝛼(𝑧)≲_{𝑘𝐷(𝑎𝑘,𝛾)}11−𝑎_𝑘^2𝑛+1+𝛼×_{𝐷(𝑎𝑘,3𝛾)}𝑓(𝑢)^𝑝𝑑𝑣_𝛼(𝑢)𝑑𝑣_𝛼(𝑧)≲_{𝑘𝐷(𝑎𝑘,3𝛾)}𝑓(𝑢)^𝑝𝑑𝑣_𝛼(𝑢)≲𝑁_𝔹𝑛𝑓(𝑢)^𝑝𝑑𝑣_𝛼(𝑢), (30) where 𝑁 is the constant in Lemma 8 depending only on 𝛾 and 𝑛.

2.2. Proof of Theorem 2

Recall that ℬ(𝔹_𝑛) is defined as the space of all 𝑓∈ℋ(𝔹_𝑛) so that 𝑓_ℬ=sup_{𝑧∈𝔹𝑛}∇𝑓(𝑧)<∞. (31)ℬ(𝔹_𝑛) with the norm ‖𝑓‖=|𝑓(0)|+‖𝑓‖_ℬ is a Banach space and called the Bloch space. Then, the following interpolation result holds.

Lemma 9 (cf. [3, Theorem 3.25]). Let 1<𝑝<∞. Suppose 𝛼>−1 and 1𝑝=1−𝜃𝑝^′, (32) for 0<𝜃<1 and 1≤𝑝^′<∞. Then 𝒜^𝑝_𝛼𝔹_𝑛=𝒜^𝑝′_𝛼𝔹_𝑛,ℬ𝔹_𝑛𝜃, (33) with equivalent norms.

Moreover, to prove Theorem 2 for the case 0<𝑝≤1, one will use atom decomposition for Bergman spaces due to Coifman and Rochberg [5] (see also [3, Theorem 2.30]) as follows.

Proposition 10. Suppose 𝑝>0,𝛼>−1, and 𝑏>𝑛max{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 𝑓(𝑧)=^∞_𝑘=1𝑐_𝑘1−𝑎_𝑘^{2(𝑝𝑏−𝑛−1−𝛼)/𝑝}1−⟨𝑧,𝑎_𝑘⟩^𝑏,𝑧∈𝔹_𝑛, (34) where {𝑐_𝑘} belongs to the sequence space ℓ^𝑝 and the series converges in the norm topology of 𝒜^𝑝_𝛼. Moreover, _𝔹𝑛𝑓(𝑧)^𝑝𝑑𝑣_𝛼(𝑧)≈inf_𝑘𝑐_𝑘^𝑝, (35) 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 [2, Theorem 45].

Proposition 11. Suppose 𝑛+1+𝛼>0 and 𝜇 is a positive Borel measure on 𝔹_𝑛. Then, there exists a constant 𝐶>0 such that 𝜇𝑄_𝑟(𝜁)≤𝐶𝑟^{2(𝑛+1+𝛼)},∀𝜁∈𝕊_𝑛,𝑟>0, (36) if and only if for each 𝑠>0, there exists a constant 𝐶>0 such that 1−|_𝔹𝑛𝑧|^2𝑠|1−⟨𝑧,𝑤⟩|^{𝑛+1+𝛼+𝑠}𝑑𝜇(𝑤)≤𝐶, (37) for all 𝑧∈𝔹_𝑛.

We are now ready to prove Theorem 2. Note that for any 𝑓∈ℋ(𝔹_𝑛), 1−|𝑧|²ℛ𝑓(𝑧)≤1−|𝑧|²(𝑧≤(𝑧∇𝑓)∇𝑓),∀𝑧∈𝔹_𝑛 (38) (cf. [3, Lemma 2.14]). We have that (d) implies (c) and (c) implies (b) in Theorem 2. Then, it remains to prove that (b) implies (a) and (a) implies (d).

Proof of (𝑏)⇒(𝑎). Since ℛ𝑓(𝑧) is holomorphic, by Lemma 6 we have ℛ𝑓(𝑧)^𝑞≤𝐶1−|𝑧|^{2𝑛+1𝐷(𝑧,𝛾)}ℛ𝑓(𝑤)^𝑞𝑑𝑣(𝑤)≤𝐶_{𝛾𝐷(𝑧,𝛾)}ℛ𝑓(𝑤)^𝑞𝑑𝜏(𝑤). (39) Then, 1−|𝑧|²(𝑧|𝑧|ℛ𝑓)≤𝐶1−²_{𝐷(𝑧,𝛾)}ℛ𝑓(𝑤)^𝑞𝑑𝜏(𝑤)^1/𝑞≤𝐶_{𝛾𝐷(𝑧,𝛾)}(1−|𝑤|²)ℛ𝑓(𝑤)^𝑞𝑑𝜏(𝑤)^1/𝑞=𝐶_𝛾𝐴_ℛ^𝛾,𝑞𝑓(𝑧). (40) Hence, for any 𝑝>0, if 𝐴_ℛ^𝛾,𝑞(𝑓)∈𝐿^𝑝(𝔹_𝑛,𝑑𝑣_𝛼) then (1−|𝑧|²)|ℛ𝑓(𝑧)| is in 𝐿^𝑝(𝔹_𝑛,𝑑𝑣_𝛼), which implies that 𝑓∈𝒜^𝑝_𝛼 (cf. [3, Theorem 2.16]).

The proof of (a)⇒(d) is divided into two steps. We first prove the case 0<𝑝≤1 using the atomic decomposition and then the remaining case via complex interpolation.

Proof of (𝑎)⇒(𝑑)𝑓𝑜𝑟0<𝑝≤1. To this end, we write 𝑓_𝑘(𝑧)=1−𝑎_𝑘^{2(𝑝𝑏−𝑛−1−𝛼)/𝑝}1−⟨𝑧,𝑎_𝑘⟩^𝑏. (41) An immediate computation yields that ∇𝑓_𝑘(𝑧)=𝑏𝑎_𝑘1−𝑎_𝑘^{2(𝑝𝑏−𝑛−1−𝛼)/𝑝}1−⟨𝑧,𝑎_𝑘⟩^𝑏+1,ℛ𝑓_𝑘(𝑧)=𝑏⟨𝑧,𝑎_𝑘⟩1−𝑎_𝑘^{2(𝑝𝑏−𝑛−1−𝛼)/𝑝}1−⟨𝑧,𝑎_𝑘⟩^𝑏+1. (42) Then we have ∇𝑓_𝑘(𝑧)²=1−|𝑧|²∇𝑓_𝑘(𝑧)²−ℛ𝑓_𝑘(𝑧)²=𝑏²1−|𝑧|²1−𝑎_𝑘^{22(𝑝𝑏−𝑛−1−𝛼)/𝑝}𝑎_𝑘²−⟨𝑧,𝑎_𝑘⟩²1−⟨𝑧,𝑎_𝑘⟩^2(𝑏+1). (43) By Lemmas 5 and 7, one has 𝐴^𝛾,𝑞∇𝑓_𝑘(𝑧)=_{𝐷(𝑧,𝛾)}∇𝑓_𝑘(𝑤)^𝑞𝑑𝜏(𝑤)^1/𝑞≤𝑏1−𝑎_𝑘^{2(𝑝𝑏−𝑛−1−𝛼)/𝑝}×_{𝐷(𝑧,𝛾)}11−𝑤,𝑎_𝑘^𝑞𝑏𝑑𝜏(𝑤)^1/𝑞≤𝐶_𝛾𝑏1−𝑎_𝑘^{2(𝑝𝑏−𝑛−1−𝛼)/𝑝}1−⟨𝑧,𝑎_𝑘⟩^𝑏, (44) where we have used the fact 𝑣(𝐷(𝑧,𝛾))≈(1−|𝑧|²)^𝑛+1. Note that 𝑣_𝛼(𝑄_𝑟)≈𝑟^{2(𝑛+1+𝛼)} (cf. [3, Corollary 5.24]); by Proposition 11 we have ∫_𝔹𝑛𝐴^𝛾,𝑞∇𝑓_𝑘(𝑧)^𝑝𝑑𝑣_𝛼(𝑧)≤𝐶𝑏^𝑝∫_𝔹𝑛1−𝑎_𝑘^{2(𝑝𝑏−𝑛−1−𝛼)}1−𝑧,𝑎_𝑘^𝑝𝑏𝑑𝑣_𝛼(𝑧)≤𝐶_𝑝,𝛼. (45) Hence, for 0<𝑝≤1, we have for 𝑓=∑^∞_𝑘=1𝑐_𝑘𝑓_𝑘 with ∑_𝑘|𝑐_𝑘|^𝑝<∞, _𝔹𝑛𝐴^𝛾,𝑞∇𝑓(𝑧)^𝑝𝑑𝑣_𝛼≤^∞_𝑘=1𝑐_𝑘^𝑝_𝔹𝑛𝐴^𝛾,𝑞∇𝑓_𝑘(𝑧)^𝑝𝑑𝑣_𝛼≤𝐶^∞_{𝑝,𝛼𝑘=1}𝑐_𝑘^𝑝. (46) This concludes that _𝔹𝑛𝐴^𝛾,𝑞∇𝑓(𝑧)^𝑝𝑑𝑣_𝛼≤𝐶_𝑝,𝛼inf^∞_𝑘=1𝑐_𝑘^𝑝≤𝐶_{𝑝,𝛼𝔹𝑛}𝑓(𝑧)^𝑝𝑑𝑣_𝛼(𝑧). (47) The proof is complete.

Proof of (𝑎)⇒(𝑑) for 𝑝>1. Set 𝐸=𝐿^𝑞(𝔹_𝑛,𝜒_𝐷(0,𝛾)𝑑𝜏;ℂ^𝑛). Consider the operator 𝑇𝑓(𝑧,𝑤)=∇𝑓𝜑_𝑧(𝑤),𝑓∈ℋ𝔹_𝑛. (48) Note that 𝜑_𝑧(𝐷(0,𝛾))=𝐷(𝑧,𝛾) and the measure 𝑑𝜏 is invariant under any automorphism of 𝔹_𝑛 (cf. [3, Proposition 1.13]); we have 𝑇𝑓(𝑧)_𝐸=_𝔹𝑛∇𝑓𝜑_𝑧(𝑤)^𝑞𝜒_𝐷(0,𝛾)(𝑤)𝑑𝜏(𝑤)^1/𝑞=_𝔹𝑛∇𝑓(𝑤)^𝑞𝜒_{𝐷(𝑧,𝛾)}(𝑤)𝑑𝜏(𝑤)^1/𝑞=𝐴^𝛾,𝑞∇𝑓(𝑧). (49) On the other hand, 𝐴^𝛾,𝑞∇𝑓(𝑧)≤𝐶_𝛾1−|𝑧|^2−𝑛−1𝑣(𝐷(𝑧,𝛾))^1/2𝑓_ℬ≤𝐶𝑓_ℬ. (50) This follows that 𝑇 is bounded from ℬ into 𝐿^∞_𝛼(𝔹_𝑛,𝐸). Notice that we have proved that 𝑇 is bounded from 𝒜¹_𝛼 to 𝐿¹_𝛼(𝔹_𝑛,𝐸). Thus, by Lemma 9 and the well known fact that 𝐿_𝑝𝔹_𝑛,𝐸=𝐿¹_𝛼(𝔹_𝑛,𝐸),𝐿^∞_𝛼𝔹_𝑛,𝐸_𝜃with1𝜃=1−𝑝, (51) we conclude that 𝑇 is bounded from 𝒜^𝑝_𝛼 into 𝐿^𝑝_𝛼(𝔹_𝑛,𝐸) for any 1<𝑝<∞; that is, 𝐴^𝛾,𝑞∇(𝑓)_𝑝,𝛼≤𝐶𝑓_𝑝,𝛼,∀𝑓∈𝒜^𝑝_𝛼, (52) where 𝐶 depends only on 𝑞, 𝛾, 𝑛, 𝑝, and 𝛼. The proof is complete.

Remark 12. From the proofs of that (b)⇒(a) and that (a)⇒(d) for 𝑝>1, we find that Theorem 2 still holds true for the Bloch space. That is, for any 𝑓∈ℋ(𝔹_𝑛),𝑓∈ℬ if and only if one (or equivalently, all) of 𝐴_ℛ^𝛾,𝑞(𝑓),𝐴_∇^𝛾,𝑞(𝑓), and 𝐴^𝛾,𝑞∇(𝑓) is (or, are) in 𝐿^∞(𝔹_𝑛). Moreover, 𝑓_ℬ≈𝐴_ℛ^𝛾,𝑞𝑓_{𝐿∞(𝔹𝑛)}≈𝐴_∇^𝛾,𝑞𝑓_{𝐿∞(𝔹𝑛)}≈𝐴^𝛾,𝑞∇(𝑓)_{𝐿∞(𝔹𝑛)}, (53) where “≈” depends only on 𝑞,𝛾, and 𝑛.

3. Atomic Decomposition for Bergman Spaces

We let 𝑑(𝑧,𝑤)=|1−⟨𝑧,𝑤⟩|^1/2,𝑧,𝑤∈𝔹_𝑛. (54) 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 𝑄_𝑟(𝜁)=𝑧∈𝔹_𝑛∶𝑑(𝑧,𝜁)<𝑟 (55) 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;(2)‖𝑎‖_𝐿^𝑞_{(𝔹𝑛,𝑑𝑣𝛼)}≤𝑣_𝛼(𝑄)^(1/𝑞)−1; (3)∫_𝔹𝑛𝑎(𝑧)𝑑𝑣_𝛼(𝑧)=0.

The constant function (1) is also considered to be a (1,𝑞)_𝛼-atom. Note that for any (1,𝑞)_𝛼-atom 𝑎, ‖𝑎‖_1,𝛼=_𝑄|𝑎|𝑑𝑣_𝛼≤𝑣_𝛼(𝑄)^1−1/𝑞‖𝑎‖_𝑞,𝛼≤1. (56)

Recall that 𝑃_𝛼 is the orthogonal projection from 𝐿²(𝔹_𝑛,𝑑𝑣_𝛼) onto 𝒜²_𝛼, which can be expressed as 𝑃_𝛼𝑓(𝑧)=_𝔹𝑛𝐾^𝛼(𝑧,𝑤)𝑓(𝑤)𝑑𝑣_𝛼(𝑤),∀𝑓∈𝐿¹𝔹_𝑛,𝑑𝑣_𝛼,𝛼>−1, (57) where 𝐾^𝛼()=1𝑧,𝑤(1−⟨𝑧,𝑤⟩)^𝑛+1+𝛼,𝑧,𝑤∈𝔹_𝑛. (58)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 13. For 𝛼>−1 and 1<𝑞<∞, there exists a constant 𝐶_{𝑞,𝛼,𝑛}>0 such that 𝑃_𝛼(𝑎)_1,𝛼≤𝐶_{𝑞,𝛼,𝑛}, (59) for any (1,𝑞)_𝛼-atom 𝑎.

To prove Lemma 13, we need first to show an inequality for reproducing kernel 𝐾^𝛼 associated with 𝑑, which is essentially borrowed from [6, Proposition 2.13].

Lemma 14. For 𝛼>−1, there exists a constant 𝛿>0 such that for all 𝑧,𝑤∈𝔹_𝑛,𝜁∈𝕊_𝑛 satisfying 𝑑(𝑧,𝜁)>𝛿𝑑(𝑤,𝜁), we have 𝐾^𝛼(𝑧,𝑤)−𝐾^𝛼(𝑧,𝜁)≤𝐶_𝛼,𝑛𝑑(𝑤,𝜁)𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+1}. (60)

Proof. Note that 𝐾^𝛼(𝑧,𝑤)−𝐾^𝛼()=𝑧,𝜁₁₀𝑑1𝑑𝑡(1−⟨𝑧,𝜁⟩−𝑡⟨𝑧,𝑤−𝜁⟩)^𝑛+1+𝛼𝑑𝑡. (61) We have 𝐾^𝛼(𝑧,𝑤)−𝐾^𝛼(≤𝑧,𝜁)₁₀(𝑛+1+𝛼)𝑧,𝑤−𝜁1−𝑧,𝜁−𝑡𝑧,𝑤−𝜁^𝑛+2+𝛼𝑑𝑡. (62)
Write 𝑧=𝑧₁+𝑧₂ and 𝑤=𝑤₁+𝑤₂, where 𝑧₁ and 𝑤₁ are parallel to 𝜁, while 𝑧₂ and 𝑤₂ are perpendicular to 𝜁. Then ⟨𝑧,𝑤⟩−⟨𝑧,𝜁⟩=⟨𝑧₂,𝑤₂⟩−⟨𝑧₁,𝑤₁−𝜁⟩ (63) and so ⟨𝑧,𝑤⟩−𝑧,𝜁≤𝑧₂𝑤₂+𝑤₁−𝜁. (64) Since |𝑤₁−𝜁|=|1−⟨𝑤,𝜁⟩|, 𝑧₂²=|𝑧|²−𝑧₁²<1−𝑧₁²<1+𝑧₁1+𝑧₁≤1−𝑧₁,𝜁=21−𝑧,𝜁, (65) and similarly 𝑤₂²≤21−𝑤,𝜁, (66) we have ⟨𝑧,𝑤⟩−𝑧,𝜁≤21−𝑧,𝜁^1/21−𝑤,𝜁^1/2+(𝑑()(11−𝑤,𝜁=2𝑑𝑤,𝜁)𝑧,𝜁+𝑑𝑤,𝜁)≤21+𝛿1𝛿𝑑²(𝑧,𝜁). (67) This concludes that there is 𝛿>1 such that <1𝑧,𝑤−𝜁21−𝑧,𝜁,∀𝑧,𝑤∈𝔹_𝑛,𝜁∈𝕊_𝑛, (68) whenever 𝑑(𝑧,𝜁)>𝛿𝑑(𝑤,𝜁). Then, we have >>11−𝑧,𝜁−𝑡𝑧,𝑤−𝜁1−𝑧,𝜁−𝑡𝑧,𝜁−𝑤21−𝑧,𝜁. (69) Therefore, 𝐾^𝛼(𝑧,𝑤)−𝐾^𝛼(≤2𝑧,𝜁)_𝑛+3+𝛼(𝑛+1+𝛼)(1+1/𝛿)𝑑(𝑤,𝜁)𝑑(𝑧,𝜁)1−⟨𝑧,𝜁⟩^𝑛+2+𝛼≤𝐶_𝛼,𝑛𝑑(𝑤,𝜁)𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+1}, (70) and the lemma is proved.

Proof of Lemma 13. When 𝑎 is the constant function 1, the result is clear. Thus we may suppose that 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 14. Since 𝑃_𝛼 is a bounded operator on 𝐿^𝑞(𝔹_𝑛,𝑑𝑣_𝛼), we have _𝑄𝛿𝑟𝑃_𝛼(𝑎)𝑑𝑣_𝛼(𝑧)≤𝑣_𝛼𝑄_𝛿𝑟^1−(1/𝑞)𝑃_𝛼(𝑎)_𝑞,𝛼≤𝑃_𝛼𝐿𝑞𝑣_𝛼𝑄_𝛿𝑟^1−(1/𝑞)‖𝑎‖_𝑞,𝛼≤𝑃_𝛼𝐿𝑞. (71) Next, if 𝑑(𝑧,𝜁)>𝛿𝑟 then _𝔹𝑛𝑎(𝑤)(1−⟨𝑧,𝑤⟩)^𝑛+1+𝛼𝑑𝑣_𝛼(𝑤=_{𝑄𝑟(𝜁)})𝑎(𝑤1)(1−⟨𝑧,𝑤⟩)^𝑛+1+𝛼−11−𝑧,𝜁^𝑛+1+𝛼𝑑𝑣_𝛼(𝑤)≤𝐶_{𝑄𝑟(𝜁)}|𝑎(𝑤𝑑())|𝑤,𝜁𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+1}𝑑𝑣_𝛼(𝑤)≤𝐶𝑟_{𝑄𝑟(𝜁)}|𝑎(𝑤)|𝑑𝑣_𝛼(𝑤)1𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+1}≤𝐶𝑟𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+1}. (72) Then _{𝑑()𝑧,𝜁>𝛿𝑟}𝑃_𝛼(𝑎)𝑑𝑣_𝛼(𝑧)≤𝐶𝑟_{𝑑(𝑧,𝜁)>𝛿𝑟}1𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+1}𝑑𝑣_𝛼(𝑧)=𝐶𝑟_{𝑘≥02𝑘𝛿𝑟<𝑑(𝑧,𝜁)≤2𝑘+1𝛿𝑟}1𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+1}𝑑𝑣_𝛼(𝑧)≤𝐶𝑟_𝑘≥0𝑣_𝛼𝑄₂^𝑘+1_𝛿𝑟2^𝑘𝛿𝑟^{2(𝑛+1+𝛼)+1}≤𝐶𝑟^∞_𝑘=02^𝑘+1𝛿𝑟^{2(𝑛+1+𝛼)}2^𝑘𝛿𝑟^{2(𝑛+1+𝛼)+1}≤𝐶, (73) where we have used the fact that 𝑣_𝛼(𝑄_𝑟)≈𝑟^{2(𝑛+1+𝛼)} in the third inequality (cf. [3, Corollary 5.24]). Thus, we get _𝔹𝑛𝑃_𝛼(𝑎)𝑑𝑣_𝛼(𝑧)=_𝑄𝛿𝑟𝑃_𝛼(𝑎)𝑑𝑣_𝛼(𝑧)+_{𝑑(𝑧,𝜁)>𝛿𝑟}𝑃_𝛼(ℎ)𝑑𝑣_𝛼(𝑧)≤𝐶, (74) where 𝐶 depends only on 𝑞,𝑛, and 𝛼.

Now we turn to the atomic decomposition of 𝒜¹_𝛼(𝛼>−1) with respect to the Carleson tubes. Recall that ‖𝑎‖_1,𝛼≤1 for any (1,𝑞)_𝛼-atom 𝑎. Then, we define 𝒜_𝛼^1,𝑞 as the space of all 𝑓∈𝒜¹_𝛼 which admits a decomposition 𝑓=_𝑖𝜆_𝑖𝑃_𝛼𝑎_𝑖,_𝑖𝜆_𝑖≤𝐶_𝑞𝑓_1,𝛼, (75) 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 𝑓_𝒜𝛼^1,𝑞=inf_𝑖𝜆_𝑖∶𝑓=_𝑖𝜆_𝑖𝑃_𝛼𝑎_𝑖, (76) where the infimum is taken over all decompositions of 𝑓 described above.

It is easy to see that 𝒜_𝛼^1,𝑞 is a Banach space. By Lemma 13, we have the contractive inclusion 𝒜_𝛼^1,𝑞⊂𝒜¹_𝛼. We will prove in what follows that these two spaces coincide. That establishes the “real-variable” atomic decomposition of the Bergman space 𝒜¹_𝛼. In fact, we will show the remaining inclusion 𝒜¹_𝛼⊂𝒜_𝛼^1,𝑞 by duality.

Theorem 15. Let 1<𝑞<∞ and 𝛼>−1. For every 𝑓∈𝒜¹_𝛼 there exist a sequence {𝑎_𝑖} of (1,𝑞)_𝛼-atoms and a sequence {𝜆_𝑖} of complex numbers such that 𝑓=_𝑖𝜆_𝑖𝑃_𝛼𝑎_𝑖,_𝑖𝜆_𝑖≤𝐶_𝑞𝑓_1,𝛼. (77) Moreover, 𝑓_1,𝛼≈inf_𝑖𝜆_𝑖, (78) where the infimum is taken over all decompositions of 𝑓 described above and “≈” depends only on 𝛼,𝑛, and 𝑞.

Recall that the dual space of 𝒜¹_𝛼 is the Bloch space ℬ (we refer to [3] for details). The Banach dual of 𝒜¹_𝛼 can be identified with ℬ (with equivalent norms) under the integral pairing 𝑓,𝑔_𝛼=lim_{𝑟→1−𝔹𝑛}𝑓(𝑟𝑧)𝑔(𝑧)𝑑𝑣_𝛼(𝑧),𝑓∈𝒜¹_𝛼,𝑔∈ℬ. (79) (cf. [3, Theorem 3.17].)

In order to prove Theorem 15, we need the following result, which can be found in [4] (see also [3, Theorem 5.25]).

Lemma 16. Suppose 𝛼>−1 and 1≤𝑝<∞. Then, for any 𝑓∈ℋ(𝔹_𝑛), 𝑓 is in ℬ if and only if there exists a constant 𝐶>0 depending only on 𝛼 and 𝑝 such that 1𝑣_𝛼𝑄_𝑟(𝜁)_{𝑄𝑟(𝜁)}𝑓−𝑓_{𝛼,𝑄𝑟(𝜁)}^𝑝𝑑𝑣_𝛼≤𝐶, (80) for all 𝑟>0 and all 𝜁∈𝕊_𝑛, where 𝑓_{𝛼,𝑄𝑟(𝜁)}=1𝑄_𝑟(𝜁)_{𝑄𝑟(𝜁)}𝑓(𝑧)𝑑𝑣_𝛼(𝑧). (81) Moreover, 𝑓_ℬ≈sup_{𝑟>0,𝜁∈𝕊}1𝑣_𝛼(𝑄_𝑟(𝜁))_{𝑄𝑟(𝜁)}𝑓−𝑓_{𝛼,𝑄𝑟(𝜁)}^𝑝𝑑𝑣_𝛼^1/𝑝, (82) where “≈” depends only on 𝛼,𝑝, and 𝑛.

As noted above, we will prove Theorem 15 via duality. To this end, we first prove the following duality theorem.

Proposition 17. For any 1<𝑞<∞ and 𝛼>−1, one has (𝒜_𝛼^1,𝑞)^∗=ℬ isometrically. More precisely,(i)every 𝑔∈ℬ defines a continuous linear functional 𝜑_𝑔 on 𝒜_𝛼^1,𝑞 by𝜑_𝑔𝑓=lim_{𝑟→1−𝔹𝑛}𝑓(𝑟𝑧)𝑔(𝑧)𝑑𝑣_𝛼(𝑧),∀𝑓∈𝒜_𝛼^1,𝑞, (83)(ii)conversely, each 𝜑∈(𝒜_𝛼^1,𝑞)^∗is given as (83) by some 𝑔∈ℬ.
Moreover, we have 𝜑_𝑔≈𝑔(0)+𝑔_ℬ,∀𝑔∈ℬ. (84)

Proof. Let 𝑝 be the conjugate index of 𝑞, that is, 1/𝑝+1/𝑞=1. We first show ℬ⊂(𝒜_𝛼^1,𝑞)^∗. Let 𝑔∈ℬ. For any (1,𝑞)_𝛼-atom 𝑎, by Lemma 16 one has _𝔹𝑛𝑃_𝛼𝑎(𝑧)𝑔(𝑧)𝑑𝑣_𝛼(𝑧=𝑃_𝛼)𝑎_𝑗,𝑔_𝛼=_𝔹𝑛𝑎𝑔𝑑𝑣_𝛼=_𝔹𝑛𝑎𝑔−𝑔_𝑄𝑑𝑣_𝛼≤_𝑄|𝑎|^𝑞𝑑𝑣_𝛼^1/𝑞_𝑄𝑔−𝑔_𝑄^𝑝𝑑𝑣_𝛼^1/𝑝≤1𝑣_𝛼(𝑄)_𝑄𝑔−𝑔_𝑄^𝑝𝑑𝑣_𝛼^1/𝑝≤𝐶𝑔_ℬ. (85) On the other hand, for the constant function (1), we have 𝑃_𝛼1=1 and so _𝔹𝑛𝑃_𝛼1(𝑧)𝑔(𝑧)𝑑𝑣_𝛼(𝑧)=_𝔹𝑛𝑔(𝑧)𝑑𝑣_𝛼(𝑧)=𝑔(0). (86) Thus, we deduce that _𝔹𝑛𝑓𝑔𝑑𝑣_𝛼≤𝐶𝑓_𝒜𝛼^1,𝑞𝑔(0)+𝑔_ℬ, (87) 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_𝛼^1,𝑞, and 𝜑_𝑔 extends to a continuous linear functional on 𝒜_𝛼^1,𝑞 such that 𝜑_𝑔𝑓≤𝐶𝑔(0)+𝑔_ℬ𝑓_𝒜𝛼^1,𝑞, (88) for all 𝑓∈𝒜_𝛼^1,𝑞.
Next let 𝜑 be a bounded linear functional on 𝒜_𝛼^1,𝑞. Note that ℋ^𝑞𝔹_𝑛,𝑑𝑣_𝛼=ℋ𝔹_𝑛∩𝐿^𝑞𝔹_𝑛,𝑑𝑣_𝛼⊂𝒜_𝛼^1,𝑞. (89) 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 𝜑𝑓=_𝔹𝑛𝑓𝑔𝑑𝑣_𝛼,∀𝑓∈ℋ^𝑞𝔹_𝑛,𝑑𝑣_𝛼. (90) 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 𝑎_𝑓=𝑓−𝑓_𝑄𝜒_𝑄𝑓_𝐿𝑞𝑣_𝛼(𝑄)^1/𝑝 (91) is a (1,𝑞)-atom. Then, |𝜑(𝑃_𝛼𝑎_𝑓)|≤‖𝜑‖ and so 𝜑𝑃_𝛼𝑓−𝑓_𝑄𝜒_𝑄≤𝜑𝑓_𝐿𝑞𝑣_𝛼(𝑄)^1/𝑝. (92) Hence, for any 𝑓∈𝐿^𝑞(𝔹_𝑛,𝑑𝑣_𝛼), we have _𝑄𝑓𝑔−𝑔_𝑄𝑑𝑣_𝛼=_𝑄𝑓−𝑓_𝑄𝑔𝑑𝑣_𝛼=_𝔹𝑛𝑓−𝑓_𝑄𝜒_𝑄𝑔𝑑𝑣_𝛼=_𝔹𝑛𝑃_𝛼𝑓−𝑓_𝑄𝜒_𝑄𝑔𝑑𝑣_𝛼=𝜑𝑃_𝛼𝑓−𝑓_𝑄𝜒_𝑄≤𝜑𝑓−𝑓_𝑄𝜒_{𝑄𝐿𝑞(𝔹𝑛,𝑑𝑣𝛼)}𝑣_𝛼(𝑄)^1/𝑝≤2𝜑𝑓_{𝐿𝑞(𝑄,𝑑𝑣𝛼)}𝑣_𝛼(𝑄)^1/𝑝. (93) This concludes that 1𝑣_𝛼(𝑄)_𝑄𝑔−𝑔_𝑄^𝑝𝑑𝑣_𝛼^1/𝑝≤2𝜑. (94) By Lemma 16, we have that 𝑔∈ℬ and ‖𝑔‖_ℬ≤𝐶‖𝜑‖. Therefore, 𝜑 is given as (83) by 𝑔 with |𝑔(0)|+‖𝑔‖_ℬ≤𝐶‖𝜑‖.

Now we are ready to prove Theorem 15.

Proof of Theorem 15. By Lemma 13 we know that 𝒜_𝛼^1,𝑞⊂𝒜¹_𝛼. On the other hand, by Proposition 17 we have (𝒜¹_𝛼)^∗=(𝒜_𝛼^1,𝑞)^∗. Hence, by duality we have ‖𝑓‖_1,𝑞≈‖𝑓‖_𝒜𝛼^1,𝑞.

Remark 18. (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 15 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 in the case 𝑝=1. Indeed, based on their theory of harmonic analysis on homogeneous spaces, Coifman and Weiss [7] claimed that the Bergman space 𝒜¹ admits an atomic decomposition in terms of atoms with respect to (𝔹_𝑛,𝜚,𝑑𝑣), where 𝜚()=𝑧|−|𝑤+1𝑧,𝑤||||1−|𝑧𝑤|⟨,||𝑧,𝑤⟩if𝑧,𝑤∈𝔹_𝑛⧵{0},|𝑧|+|𝑤|,otherwise. (95) This also applies to 𝒜¹_𝛼 because (𝔹_𝑛,𝜚,𝑑𝑣_𝛼) is a homogeneous space for 𝛼>−1 (see e.g. [6]). However, the approach of Coifman and Weiss is again based on duality and therefore not constructive and cannot be applied to the case 0<𝑝<1. Recently, the present authors [8] extend this result to the case 0<𝑝<1 through using a constructive method.

4. Area Integral Inequalities: Real-Variable Methods

In this section, we will prove the area integral inequality for the Bergman space 𝒜¹_𝛼 via atomic decomposition established in Section 3.

Theorem 19. Suppose 1<𝑞<∞,𝛾>0, and 𝛼>−1. Then, 𝐴^𝛾,𝑞∇(𝑓)_1,𝛼≲𝑓_1,𝛼,∀𝑓∈ℋ𝔹_𝑛. (96)

This is the assertion (a)⇒(d) of Theorem 2 in the case 𝑝=1. The novelty of the proof here is to involve a real-variable method.

The following lemma is elementary.

Lemma 20. Suppose 1<𝑞<∞,𝛾>0 and 𝛼>−1. If 𝑓∈𝒜^𝑞_𝛼, then _𝔹𝑛𝐴^𝛾,𝑞∇(𝑓)(𝑧)^𝑞𝑑𝑣_𝛼≈_𝔹𝑛𝑓(𝑧)−𝑓(0)^𝑞𝑑𝑣_𝛼, (97) where “≈” depends only on 𝑞,𝛾,𝛼, and 𝑛.

Proof. Note that 𝑣_𝛼(𝐷(𝑧,𝛾))≈(1−|𝑧|²)^𝑛+1+𝛼. Then _𝔹𝑛𝐴^𝛾,𝑞∇𝑓(𝑧)^𝑞𝑑𝑣_𝛼=_{𝔹𝑛𝐷(𝑧,𝛾)}1−|𝑤|^2−1−𝑛∇𝑓(𝑤)^𝑞𝑑𝑣(𝑤)𝑑𝑣_𝛼(𝑧)=_𝔹𝑛𝑣_𝛼𝐷𝑤,𝛾1−|𝑤|^2−1−𝑛∇𝑓(𝑤)^𝑞𝑑𝑣(𝑤)≈_𝔹𝑛∇𝑓(𝑤)^𝑞𝑑𝑣_𝛼(𝑤)≈_𝔹𝑛𝑓(𝑤)−𝑓(0)^𝑞𝑑𝑣_𝛼(𝑤). (98) In the last step, we have used [3, Theorem 2.16(b)].

Proof of (96). By Theorem 15, it suffices to show that for 1<𝑞<∞,𝛾>0, and 𝛼>−1, there exists 𝐶>0 such that 𝐴^𝛾,𝑞∇(𝑃_𝛼𝑎)_1,𝛼≤𝐶, (99) for all (1,𝑞)_𝛼-atoms 𝑎. Given an (1,𝑞)_𝛼-atom 𝑎 supported in 𝑄=𝑄_𝑟(𝜁), by Lemma 20, we have _2𝑄𝐴^𝛾,𝑞∇𝑃_𝛼𝑎𝑑𝑣_𝛼≤𝑣_𝛼(2𝑄)^1−(1/𝑞)_2𝑄𝐴^𝛾,𝑞∇𝑃_𝛼𝑎^𝑞𝑑𝑣_𝛼^1/𝑞≤𝐶𝑣_𝛼(𝑄)^1−(1/𝑞)_𝔹𝑛𝑃_𝛼𝑎(𝑧)−𝑃_𝛼𝑎(0)^𝑞𝑑𝑣_𝛼^1/𝑞≤𝐶𝑣_𝛼(𝑄)^1−(1/𝑞)‖𝑎‖_𝑞,𝛼≤𝐶, (100) where 2𝑄=𝑄_2𝑟(𝜁). On the other hand,_()2𝑄𝑐𝐴^𝛾,𝑞∇𝑃_𝛼𝑎𝑑𝑣_𝛼=_{(2𝑄)𝑐𝐷(𝑧,𝛾)}∇𝑃_𝛼𝑎(𝑤)^𝑞𝑑𝜏(𝑤)^1/𝑞𝑑𝑣_𝛼(𝑧)=_{(2𝑄)𝑐𝐷(𝑧,𝛾)𝑄}∇_𝑤[𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(𝑤,𝜁)]𝑎(𝑢)𝑑𝑣_𝛼(𝑢)^𝑞𝑑𝜏(𝑤)^1/𝑞𝑑𝑣_𝛼(𝑧)≤‖𝑎‖_{𝑞,𝛼(2𝑄)𝑐𝐷(𝑧,𝛾)𝑄}∇_𝑤𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(𝑤,𝜁)^{𝑞/(𝑞−1)}𝑑𝑣_𝛼(𝑢)^𝑞−1𝑑𝜏(𝑤)^1/𝑞𝑑𝑣_𝛼(𝑧)≤_{(2𝑄)𝑐𝐷(𝑧,𝛾)}sup_𝑢∈𝑄∇_𝑤𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(𝑤,𝜁)^𝑞𝑑𝜏(𝑤)^1/𝑞𝑑𝑣_𝛼(𝑧), (101)where (2𝑄)^𝑐=𝔹_𝑛⧵2𝑄.
An immediate computation yields that ∇_𝑤𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(=(𝑤,𝜁)𝑛+1+𝛼)𝑢(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼−𝜁(1−⟨𝑤,𝜁⟩)^𝑛+2+𝛼=(𝑛+1+𝛼)𝑢1−𝑤,𝜁^𝑛+2+𝛼−𝜁(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼1−𝑤,𝜁^𝑛+2+𝛼,ℛ_𝑤𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(=(⟨⟩𝑤,𝜁)𝑛+1+𝛼)𝑤,𝑢(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼−𝑤,𝜁(1−⟨𝑤,𝜁⟩)^𝑛+2+𝛼=()×()𝑛+1+𝛼⟨𝑤,𝑢⟩1−⟨𝑤,𝜁⟩^𝑛+2+𝛼−⟨𝑤,𝜁⟩(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼(1−⟨𝑤,𝜁⟩)^𝑛+2+𝛼. (102) Moreover,∇_𝑤[𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(𝑤,𝜁)]²=(𝑛+1+𝛼)²×|𝑢|²1−⟨𝑤,𝜁⟩^{2(𝑛+2+𝛼)}+|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}1−𝑤,𝜁^{2(𝑛+2+𝛼)}−(1−⟨𝑤,𝜁⟩)^𝑛+2+𝛼(1−⟨𝑢,𝑤⟩)^𝑛+2+𝛼⟨𝜁,𝑢⟩|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}1−𝑤,𝜁^{2(𝑛+2+𝛼)}−(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼(1−⟨𝜁,𝑤⟩)^𝑛+2+𝛼⟨𝑢,𝜁⟩|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}1−𝑤,𝜁^{2(𝑛+2+𝛼)},ℛ_𝑤𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(𝑤,𝜁)²=(𝑛+1+𝛼)²×|⟨𝑤,𝑢⟩|²1−𝑤,𝜁^{2(𝑛+2+𝛼)}+𝑤,𝜁²|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}1−𝑤,𝜁^{2(𝑛+2+𝛼)}−⟨𝑤,𝑢⟩⟨𝜁,𝑤⟩(1−⟨𝑤,𝜁⟩)^𝑛+2+𝛼(1−⟨𝑢,𝑤⟩)^𝑛+2+𝛼|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}1−𝑤,𝜁^{2(𝑛+2+𝛼)}−⟨𝑤,𝜁⟩⟨𝑢,𝑤⟩(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼(1−⟨𝜁,𝑤⟩)^𝑛+2+𝛼|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}1−𝑤,𝜁^{2(𝑛+2+𝛼)}. (103)Then, we have ∇_𝑤𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(𝑤,𝜁)²−ℛ_𝑤𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(𝑤,𝜁)²=(𝑛+1+𝛼)²|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}1−𝑤,𝜁^{2(𝑛+2+𝛼)}×|𝑢|²−|⟨𝑤,𝑢⟩|²1−⟨𝑤,𝜁⟩^{2(𝑛+2+𝛼)}+1−𝑤,𝜁²|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}+()×()⟨𝑤,𝑢⟩⟨𝜁,𝑤⟩−⟨𝜁,𝑢⟩1−⟨𝑤,𝜁⟩^𝑛+2+𝛼(1−⟨𝑢,𝑤⟩)^𝑛+2+𝛼+()×()⟨𝑤,𝜁⟩⟨𝑢,𝑤⟩−⟨𝑢,𝜁⟩1−⟨𝑤,𝑢⟩^𝑛+2+𝛼(1−⟨𝜁,𝑤⟩)^𝑛+2+𝛼. (104) Note that for any 𝑓∈ℋ(𝔹_𝑛), ∇𝑓(𝑧)²=1−|𝑧|²∇𝑓(𝑧)²−ℛ𝑓(𝑧)²,𝑧∈𝔹_𝑛 (105) (cf. [3, Lemma 2.13]). It is concluded that ∇_𝑤𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(≤()|𝑤|𝑤,𝜁)𝑛+1+𝛼1−^21/2|1−⟨𝑤,𝑢⟩|^𝑛+2+𝛼1−𝑤,𝜁^𝑛+2+𝛼×1−|⟨𝑤,𝑢⟩|²1−𝑤,𝜁^{2(𝑛+2+𝛼)}+1−𝑤,𝜁²|1−⟨𝑤,𝑢⟩|^{2(𝑛+2+𝛼)}+𝑤,𝑢−𝜁𝜁,𝑤+𝑤,𝜁²+×()−11−𝜁,𝑢1−⟨𝑤,𝜁⟩^𝑛+2+𝛼(1−⟨𝑢,𝑤⟩)^𝑛+2+𝛼+⟨𝑤,𝜁−𝑢⟩⟨𝑢,𝑤⟩+|⟨𝑤,𝑢⟩|²+×()−11−𝑢,𝜁1−⟨𝑤,𝑢⟩^𝑛+2+𝛼(1−⟨𝜁,𝑤⟩)^{𝑛+2+𝛼1/2}≤(𝑛+1+𝛼)1−|𝑤|^21/2𝑀₁+𝑀₂+𝑀₃+𝑀₄^1/2|1−⟨𝑤,𝑢⟩|^𝑛+2+𝛼1−⟨𝑤,𝜁⟩^𝑛+2+𝛼, (106) where 𝑀₁=1−𝑤,𝜁^𝑛+2+𝛼|1−⟨𝑢,𝑤⟩|^𝑛+2+𝛼×(,𝑀⟨𝑤,𝑢−𝜁⟩⟨𝜁,𝑤⟩+1−⟨𝜁,𝑢⟩)₂=|1−⟨𝑤,𝑢⟩|^𝑛+2+𝛼1−𝜁,𝑤^𝑛+2+𝛼×,𝑀⟨𝑤,𝜁−𝑢⟩⟨𝑢,𝑤⟩+1−𝑢,𝜁₃=1−|⟨𝑤,𝑢⟩|²1−𝜁,𝑤^𝑛+2+𝛼×1−𝑤,𝜁^𝑛+2+𝛼−(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼,𝑀₄=1−𝑤,𝜁²|1−⟨𝑢,𝑤⟩|^𝑛+2+𝛼×(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼−(1−⟨𝑤,𝜁⟩)^𝑛+2+𝛼, (107) for 𝑤∈𝐷(𝑧,𝛾),𝑢∈𝑄_𝑟(𝜁), and 𝑧∈𝔹_𝑛,𝜁∈𝕊_𝑛.
Hence, _(2𝑄)𝑐𝐴^𝛾,𝑞∇𝑃_𝛼𝑎𝑑𝑣_𝛼≤_{(2𝑄)𝑐𝐷(𝑧,𝛾)}sup_𝑢∈𝑄∇_𝑤𝐾^𝛼(𝑤,𝑢)−𝐾^𝛼(𝑤,𝜁)^𝑞𝑑𝜏(𝑤)^1/𝑞𝑑𝑣_𝛼(𝑧)≤(𝑛+1+𝛼)_(2𝑄)𝑐𝐼₁+𝐼₂+𝐼₃+𝐼₄𝑑𝑣_𝛼(𝑧), (108)where𝐼₁=_{𝐷(𝑧,𝛾)}sup_𝑢∈𝑄1−|𝑤|^2𝑞/2𝑀₁^𝑞/2|1−⟨𝑤,𝑢⟩|^{𝑞(𝑛+2+𝛼)}1−⟨𝑤,𝜁⟩^{𝑞(𝑛+2+𝛼)}𝑑𝜏(𝑤)^1/𝑞,𝐼₂=_{𝐷(𝑧,𝛾)}sup_𝑢∈𝑄1−|𝑤|^2𝑞/2𝑀₂^𝑞/2|1−⟨𝑤,𝑢⟩|^{𝑞(𝑛+2+𝛼)}1−𝑤,𝜁^{𝑞(𝑛+2+𝛼)}𝑑𝜏(𝑤)^1/𝑞,𝐼₃=_{𝐷(𝑧,𝛾)}sup_𝑢∈𝑄1−|𝑤|^2𝑞/2𝑀₃^𝑞/2|1−⟨𝑤,𝑢⟩|^{𝑞(𝑛+2+𝛼)}1−⟨𝑤,𝜁⟩^{𝑞(𝑛+2+𝛼)}𝑑𝜏(𝑤)^1/𝑞,𝐼₄=_{𝐷(𝑧,𝛾)}sup_𝑢∈𝑄1−|𝑤|^2𝑞/2𝑀₄^𝑞/2|1−⟨𝑤,𝑢⟩|^{𝑞(𝑛+2+𝛼)}1−𝑤,𝜁^{𝑞(𝑛+2+𝛼)}𝑑𝜏(𝑤)^1/𝑞. (109)
We first estimate 𝐼₁. Note that 𝑀₁≤+×𝑤,𝑢−𝜁1−𝜁,𝑢1−⟨𝑤,𝜁⟩^𝑛+2+𝛼|1−⟨𝑤,𝑢⟩|^𝑛+2+𝛼≤21−𝑢,𝜁^1/21−𝑤,𝜁^1/2+1−𝑢,𝜁^1/2+×1−𝜁,𝑢1−𝑤,𝜁^𝑛+2+𝛼|1−⟨𝑤,𝑢⟩|^𝑛+2+𝛼≤21−𝑢,𝜁^1/2𝐶_𝛾1−𝑧,𝜁^1/2+121−𝑧,𝜁^1/2+×1−𝜁,𝑢1−𝑤,𝜁^𝑛+2+𝛼|1−⟨𝑤,𝑢⟩|^𝑛+2+𝛼≤𝐶_𝛾𝑟1−𝑧,𝜁^1/2+𝑟²×1−𝑤,𝜁^𝑛+2+𝛼|1−⟨𝑤,𝑢⟩|^𝑛+2+𝛼, (110) where the second inequality is the consequence of the following fact which has appeared in the proof of Lemma 14: 𝑤,𝑢−𝜁≤21−⟨𝑢,𝜁⟩^1/21−⟨𝑤,𝜁⟩^1/2+1−⟨𝑢,𝜁⟩^1/2; (111) the third inequality is obtained by Lemma 7, and the fact 1−⟨𝑢,𝜁⟩^1/21<𝑟<21−⟨𝑧,𝜁⟩^1/2, (112) for 𝑢∈𝑄 and 𝑧∈(2𝑄)^𝑐. Since |1−⟨𝑧,𝑢⟩|^1/2≥1−⟨𝑧,𝜁⟩^1/2−1−⟨𝑢,𝜁⟩^1/2≥1−𝑧,𝜁^1/2−121−𝑧,𝜁^1/2≥121−𝑧,𝜁^1/2, (113) by Lemmas 5 and 7, one has𝐼₁≤_{𝐷(𝑧,𝛾)}sup_𝑢∈𝑄1−|𝑤|^2𝑞/2𝐶𝑟1−⟨𝑧,𝜁⟩^1/2+𝑟^2𝑞/2|1−⟨𝑤,𝑢⟩|^{(𝑞/2)(𝑛+2+𝛼)}1−𝑤,𝜁^{(𝑞/2)(𝑛+2+𝛼)}𝑑𝜏(𝑤)^1/𝑞≤𝐶_{𝛾𝐷(𝑧,𝛾)}sup_𝑢∈𝑄1−|𝑧|^2𝑞/2𝐶𝑟1−⟨𝑧,𝜁⟩^1/2+𝑟^2𝑞/2|1−⟨𝑧,𝑢⟩|^{(𝑞/2)(𝑛+2+𝛼)}1−𝑧,𝜁^{(𝑞/2)(𝑛+2+𝛼)}𝑑𝜏(𝑤)^1/𝑞≤𝐶_𝛾1−|𝑧|^2𝑞/2𝑟1−⟨𝑧,𝜁⟩^1/2+𝑟^2𝑞/21−⟨𝑧,𝜁⟩^{𝑞(𝑛+2+𝛼)1/𝑞}≤𝐶_𝛾𝑟1−⟨𝑧,𝜁⟩^1/2+𝑟^21/21−⟨𝑧,𝜁⟩^{𝑛+(3/2)+𝛼}≤𝐶_𝛾𝑟^1/2𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+(1/2)}+𝑟𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+1}. (114)Hence, _(2𝑄)𝑐𝐼₁𝑑𝑣_𝛼(𝑧)≲_(2𝑄)𝑐𝑟^1/2𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+(1/2)}𝑑𝑣_𝛼(𝑧)+_(2𝑄)𝑐𝑟𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+1}𝑑𝑣_𝛼(𝑧). (115) The second term on the right hand side has been estimated in the proof of Lemma 13. The first term can be estimated as follows: _{𝑑()𝑧,𝜁>2𝑟}𝑟^1/2𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+(1/2)}𝑑𝑣_𝛼(𝑧)=𝑟^1/2_{𝑘≥02𝑘𝑟<𝑑(𝑧,𝜁)≤2𝑘+1𝑟}1𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+(1/2)}𝑑𝑣_𝛼(𝑧)≤𝑟^1/2_𝑘≥0𝑣_𝛼𝑄₂^𝑘+1_𝑟2^𝑘𝑟^{2(𝑛+1+𝛼)+(1/2)}≤𝐶𝑟^∞1/2_𝑘=02^𝑘+1𝑟^{2(𝑛+1+𝛼)}2^𝑘𝑟^{2(𝑛+1+𝛼)+(1/2)}≤𝐶, (116) where we have used the fact that 𝑣_𝛼(𝑄_𝑟)≈𝑟^{2(𝑛+1+𝛼)} in the third inequality (cf. [3, Corollary 5.24]).
By the same argument we can estimate 𝐼₂ and omit the details.
Next, we estimate 𝐼₃. Note that 𝑀₃≤1−|⟨𝑤,𝑢⟩|²1−⟨𝑤,𝜁⟩^𝑛+2+𝛼×1−𝑤,𝜁^𝑛+2+𝛼−(1−⟨𝑤,𝑢⟩)^𝑛+2+𝛼≤2|1−⟨𝑤,𝑢⟩|1−⟨𝑤,𝜁⟩^𝑛+2+𝛼×¹₀𝑑()𝑢𝑑𝑡1−𝑤,𝑡𝜁+1−𝑡^𝑛+2+𝛼(𝑑𝑡=2𝑛+2+𝛼)|1−⟨𝑤,𝑢⟩|1−⟨𝑤,𝜁⟩^𝑛+2+𝛼×⟨𝑤,𝜁−𝑢⟩¹₀1−𝑤,𝑡𝜁+(1−𝑡)𝑢^𝑛+1+𝛼𝑑𝑡≤𝐶_𝛾|1−⟨𝑤,𝑢⟩|1−⟨𝑤,𝜁⟩^𝑛+2+𝛼𝑟1−⟨𝑧,𝜁⟩^{𝑛+3/2+𝛼}, (117) where the last inequality is achieved by the following estimates: 1−𝑤,𝑡𝜁+(1−𝑡)𝑢≤𝐶_𝛾1−𝑧,𝑡𝜁+(1−𝑡)𝑢≤𝐶_𝛾|1−⟨𝑧,𝑢⟩|+𝑧,𝜁−𝑢≤𝐶_𝛾1−𝑧,𝜁,𝑤,𝜁−𝑢≤𝐶_𝛾𝑟1−⟨𝑧,𝜁⟩^1/2, (118) for any 𝑤∈𝐷(𝑧,𝛾) and 𝑢∈𝑄_𝑟(𝜁). Thus, by Lemmas 5 and 7𝐼₃≤𝐶_{𝛾𝐷(𝑧,𝛾)}sup_𝑢∈𝑄1−|𝑤|^2𝑞/2𝑟^𝑞/21−⟨𝑧,𝜁⟩^{(𝑞/2)(𝑛+(3/2)+𝛼)}|1−⟨𝑤,𝑢⟩|^{𝑞(𝑛+1+𝛼)+(𝑞/2)}1−𝑤,𝜁^{(𝑞/2)(𝑛+2+𝛼)}𝑑𝜏(𝑤)^1/𝑞≤𝐶_{𝛾𝐷(𝑧,𝛾)}sup_𝑢∈𝑄(1−|𝑧|²)^𝑞/2𝑟^𝑞/2|1−⟨𝑧,𝜁⟩|^{(𝑞/2)(𝑛+(3/2)+𝛼)}|1−⟨𝑧,𝑢⟩|^{𝑞(𝑛+1+𝛼)+(𝑞/2)}1−𝑧,𝜁^{(𝑞/2)(𝑛+2+𝛼)}𝑑𝜏(𝑤)^1/𝑞≤𝐶_𝛾(1−|𝑧|²)^𝑞/2𝑟^𝑞/21−⟨𝑧,𝜁⟩^{𝑞(𝑛+1+𝛼)+(3/4)𝑞1/𝑞}≤𝐶_𝛾𝑟^1/2𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+(1/2)}. (119)Hence, _(2𝑄)𝑐𝐼₃𝑑𝑣_𝛼(𝑧)≤𝐶_{𝛾(2𝑄)𝑐}𝑟^1/2𝑑(𝑧,𝜁)^{2(𝑛+1+𝛼)+(1/2)}𝑑𝑣_𝛼(𝑧)≤𝐶_𝛾, (120) as shown above.
Similarly, we can estimate 𝐼₄ and omit the details. Therefore, combining the above estimates we conclude that _(2𝑄)𝑐𝐴_𝛾∇𝑃_𝛼𝑎𝑑𝑣_𝛼≤𝐶, (121) where 𝐶 depends only on 𝑞,𝛾,𝑛, and 𝛼.

Remark 21. We remark that whenever 𝒜^𝑝_𝛼 has an atomic decomposition in terms of atoms with respect to Carleson tubes for 0<𝑝<1, the argument of Theorem 2 works as well in this case. However, as noted in Remark 18(1), the problem of the atomic decomposition of 𝒜^𝑝_𝛼 with respect to Carleson tubes in 0<𝑝<1 is entirely open.

Remark 22. 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 [9].

Acknowledgment

This research was supported in part by the NSFC under Grant no. 11171338.