𝐹(𝑝,𝑞,𝑠) Holomorphic Spaces in the Unit Ball of ℂ^𝑛

Abstract

We introduce the 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝜑(𝑝,𝑞,𝑠) vector spaces of holomorphic functions defined in the unit ball of ℂ^𝑛, generalizing previous work like Ouyang et al. (1998), Stroethoff (1989), and Choa et al. (1992). Likewise, we characterize those spaces in terms of harmonic majorants as a generalization of Arellano et al. (2000).

1. Introduction

1.1. Preliminaries in One Complex Variable

Let 𝔻 be the open unit disk in the complex plane ℂ. For 𝑎∈ℂ, let 𝜑_𝑎∶ℂ⧵{1/𝑎}→ℂ be the Möbius transformation defined by 𝜑_𝑎(𝑧)=𝑎−𝑧1−.𝑎𝑧 (1) For 𝑧,𝑎∈𝔻, we denote Green's function of 𝔻 with logarithmic singularity at 𝑎 by 𝑔()1𝑧,𝑎=log𝜑_𝑎(𝑧|)=log1−𝑎𝑧|||.𝑎−𝑧 (2) The Bloch space ℬ is defined as the set of analytic functions 𝑓∶𝔻→ℂ, such that sup_𝑧∈𝔻1−|𝑧|²𝑓^′(𝑧)<+∞. (3)

For 1≤𝑝<∞, Aulaskari and Lappan [1] introduced in 1994 the 𝒬_𝑝 spaces as the family of analytic functions 𝑓∶𝔻→ℂ satisfying sup_{𝑎∈𝔻𝔻}𝑓^′(𝑧)²𝑔^𝑝(𝑧,𝑎)𝑑𝐴(𝑧)<+∞. (4) For 0<𝑝<1, this definition was extended in [2].

Theorem 1 (see [2]). Let 𝑝≥0 and 𝑓∶𝔻→ℂ be an analytic function. Then, 𝑓∈𝒬_𝑝 if and only if sup_{𝑎∈𝔻𝔻}𝑓^′(𝑧)²1−𝜑_𝑎(𝑧)^2𝑝𝑑𝐴(𝑧)<+∞. (5)

With the aim of generalizing and enclosing several weighted function spaces, Zhao in [3] introduced the 𝐹(𝑝,𝑞,𝑠) spaces in the next way.

Let 0<𝑝<∞,−2<𝑞<∞, 0≤𝑠<∞, and 𝑓∶𝔻→ℂ be an analytic function. We will say that 𝑓 belongs to 𝐹(𝑝,𝑞,𝑠) if satisfies the integral condition 𝑓_{𝑝,𝑞,𝑠}∶=sup_{𝑎∈𝔻𝔻}𝑓^′(𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝐴(𝑧)<+∞. (6) So far, Theorem 1 is generalized for functions of 𝐹(𝑝,𝑞,𝑠) by the following.

Theorem 2 (see [3]). Let 0<𝑝<∞, −2<𝑞<∞, 0≤𝑠<∞, and 𝑓∶𝔻→ℂ be an analytic function. Then, 𝑓∈𝐹(𝑝,𝑞,𝑠) if and only if sup_{𝑎∈𝔻𝔻}𝑓^′(𝑧)^𝑝1−|𝑧|^2𝑞1−𝜑_𝑎(𝑧)^2𝑠𝑑𝐴(𝑧)<+∞. (7)

Zhao has shown that for certain intervals of 𝑝, 𝑞, and 𝑠, |𝑓(0)|+{𝑓}_{𝑝,𝑞,𝑠} makes 𝐹(𝑝,𝑞,𝑠) as a Banach space.

In this paper, we present the 𝐹(𝑝,𝑞,𝑠) spaces of holomorphic functions in the unit ball of ℂ^𝑛, that generalize the 𝐹(𝑝,𝑞,𝑠) spaces introduced by Zhao in [3] for analytic functions in the unit disk. At the same time, this work generalizes, mainly, several results of Ouyang et al for 𝒬_𝑝-holomorphic functions appearing in [4]. However, the techniques, the methods, and the structure of our work results are completely different to the quoted reference. At the beginning of Section 2, we introduce the spaces 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝜑(𝑝,𝑞,𝑠), that is, when we use in the integral representation, the invariant Green function, or when we use the biholomorphism 𝜑. We remark that the generalization to several complex variables of 𝐹(𝑝,𝑞,𝑠) requires, for 𝑞 and 𝑠, different intervals to those used in the one-dimensional case. In Theorems 12 and 13, we draw attention to the continuity of the integral expressions defining these spaces to conclude easily the inclusions of the little classes 𝐹_𝐺,0(𝑝,𝑞,𝑠) and 𝐹_𝜑,0(𝑝,𝑞,𝑠) in 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝜑(𝑝,𝑞,𝑠), respectively.

In Section 2.2, we present what is the most natural Bloch space associated to 𝐹_𝐺(𝑝,𝑞,𝑠) and several characterizations that we give in Theorem 22. It is important to compare this statement with the results of [4, Proposition 3.6] and [5, Theorem 2.4]. In Section 3, we present in Corollary 30 the equivalence between 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝜑(𝑝,𝑞,𝑠), generalizing Proposition 3.4 of [4] and Theorem 1 of [3]. Finally in Section 4, we obtain the characterizations of 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝜑(𝑝,𝑞,𝑠) in terms of harmonic majorants. Giving in this way, the holomorphic version of the results of Arellano et al. in [6].

1.2. Preliminaries in Several Variables

Consider the vector space ℂ^𝑛={𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛)∶𝑧_𝑗∈ℂ,𝑗=1,2,…,𝑛} with the Hermitian product defined as ⟨𝑧,𝑤⟩=^𝑛_𝑗=1𝑧_𝑗𝑤_𝑗, (8) and the Euclidean norm on ℂ^𝑛 defined as |𝑧|=⟨𝑧,𝑧⟩^1/2=^𝑛_𝑗=1𝑧_𝑗^21/2. (9)

The open ball with center 𝑎∈ℂ^𝑛 and radius 𝑟>0 denoted by 𝐵_𝑟(𝑎) is defined as 𝐵_𝑟(𝑎)=𝑧∈ℂ^𝑛∶|𝑧−𝑎|<𝑟, (10) and its closure is 𝐵_𝑟(𝑎)=𝑧∈ℂ^𝑛∶|𝑧−𝑎|≤𝑟. (11) We will denote 𝐵₁(0)=𝐵, 𝐵_𝑟(0)=𝐵_𝑟, and 𝐴(𝑟)=𝐵⧵𝐵_𝑟.

The function 𝑓, of class 𝒞²(Ω), is harmonic in Ω if Δ𝑓(𝑥)=0, (12) for all 𝑥∈Ω. Moreover, 𝑓 results harmonic in Ω if and only if ^𝑛_𝑗=1𝜕²𝑓𝜕𝑧_𝑗𝜕𝑧_𝑗=0. (13)

Let 𝑎∈𝐵⧵{0}. The Möbius function 𝜑_𝑎(𝑧) is defined as 𝜑_𝑎(𝑧)=𝑎−𝑃_𝑎(𝑧)−1−|𝑎|^21/2𝑄_𝑎(𝑧)⟨⟩,1−𝑧,𝑎 (14) with 𝑃_𝑎(𝑧)=⟨𝑧,𝑎⟩𝑎/|𝑎|² and 𝑄_𝑎(𝑧)=𝑧−𝑃_𝑎(𝑧), for 𝑧∈𝐵, and define 𝜑₀(𝑧)=−𝑧 (see [7]).

This set of functions is a part of the family of automorphisms on 𝐵 denoted by Aut(𝐵).

Lemma 3 (see [7, 8]). The function 𝜑_𝑎 satisfies the following properties:(i)𝜑_𝑎(𝜑_𝑎(𝑧))=𝑧;(ii)1−|𝜑_𝑎(𝑧)|²=(1−|𝑎|²)(1−|𝑧|²)/|1−⟨𝑧,𝑎⟩|²;(iii)the real Jacobian for 𝜑_𝑎 in 𝑧∈𝐵 is given by 𝐽_ℝ𝜑_𝑎(𝑧)=1−|𝑎|²|1−⟨𝑧,𝑎⟩|^2𝑛+1=1−𝜑_𝑎(𝑧)²1−|𝑧|^2𝑛+1. (15)

Lemma 4. The real Jacobian of 𝜑_𝑎 satisfies 1−|𝑎|1+|𝑎|^𝑛+1≤𝐽_ℝ𝜑_𝑎(𝑧≤|𝑎|)1+1−|𝑎|^𝑛+1. (16)

Proof. It follows directly from 1−|𝑎|≤|1−⟨𝑧,𝑎⟩|≤1+|𝑎|.

Define, for 𝑎∈𝐵 and 0<𝑅<1, 𝐷(𝑎,𝑅)=𝜑_𝑎(𝐵(𝑅))=𝑧∈𝐵∶𝜑_𝑎(𝑧)<𝑅. (17)

1.3. Invariant Gradient and Measure

The collection of all holomorphic functions in 𝐵 is denoted by 𝒪. Let ∇𝑓=(𝜕𝑓/𝜕𝑧₁,…,𝜕𝑓/𝜕𝑧_𝑛) denote the complex gradient of 𝑓 and 𝑅𝑓=∑^𝑛_𝑗=1𝑧_𝑗(𝜕𝑓/𝜕𝑧_𝑗) denote the radial derivative of 𝑓. Let 𝑑𝜆(𝑧)∶=𝑑𝜈(𝑧)/(1−|𝑧|²)^𝑛+1, where 𝑑𝜈 is the normalized volume measure in ℂ^𝑛. Then, 𝑑𝜆 is a Möbius invariant; that is _𝐵𝑓(𝑧)𝑑𝜆(𝑧)=_𝐵𝑓∘𝜓(𝑧)𝑑𝜆(𝑧), (18) for each 𝜆-integrable function 𝑓 and 𝜓∈Aut(𝐵).

Let Δ𝑓(𝑧)=Δ(𝑓∘𝜑_𝑧)(0) denote the invariant Laplacian of 𝑓 [8], and let ∇𝑓(𝑧)=∇(𝑓∘𝜑_𝑧)(0) denote the invariant gradient of 𝑓 [7]. It is well known [7] that, for 𝑓∈𝒪, ∇𝑓(𝑧)²=1−|𝑧|²∇𝑓(𝑧)²−𝑅𝑓(𝑧)²,∇𝑓∘𝜑(𝑧)²=∇𝑓∘𝜑(𝑧)², (19)(20) where 𝜑∈Aut(𝐵). In [7], the invariant Green's function is defined as 𝐺(𝑧,𝑎)=𝑔(𝜑_𝑎(𝑧)), where 𝑔(𝑧)=𝑛+12𝑛¹_|𝑧|1−𝑡^2𝑛−1𝑡^−2𝑛+1𝑑𝑡. (21) As in [7], the 𝑛-dimensional invariant Bloch space ℬ is the set of holomorphic functions 𝑓∶𝐵→ℂ, such that their Bloch invariant norm |𝑓|_ℬ satisfies 𝑓_ℬ=sup_𝑧∈𝐵∇𝑓(𝑧)<+∞; (22) this is equivalent [4, Lemma 3.2] to, say, sup_𝑧∈𝐵1−|𝑧|²∇𝑓(𝑧)<+∞. (23)

Proposition 5 (see [7]). Let 𝑛≥2 be an integer; then there are positive constants 𝐶₁ and 𝐶₂, such that, for all 𝑧∈𝐵⧵{0}, 𝐶₁1−|𝑧|^2𝑛|𝑧|^{−2(𝑛−1)}≤𝑔(𝑧)≤𝐶₂1−|𝑧|^2𝑛|𝑧|^{−2(𝑛−1)}. (24)

Proposition 6 (see [9]). For 𝑓∈𝒪, one has 1−|𝑧|²²∇𝑓(𝑧)²≤∇𝑓(𝑧)²≤1−|𝑧|²∇𝑓(𝑧)²≤∇𝑓(𝑧)². (25)

Theorem 7 (see [7, Theorem 1.12]). Suppose that 𝑐 is real and 𝑡>−1. Then, the integral 𝐽_𝑐,𝑡(𝑧)=_𝐵1−|𝑤|^2𝑡𝑑𝑣(𝑤)1−𝑧,𝜁^{𝑛+1+𝑡+𝑐},𝑧∈𝐵 (26) has the following asymptotic properties.(i)If 𝑐<0, 𝐽_𝑐,𝑡 is bounded in 𝐵.(ii)If 𝑐=0, then 𝐽_𝑐,𝑡(𝑧)1∼ln1−|𝑧|², (27) as |𝑧|→1⁻.(iii)If 𝑐>0, then 𝐽_𝑐,𝑡(𝑧)∼11−|𝑧|^2𝑐, (28) as |𝑧|→1⁻.

Let 0<𝑝<∞,  −(𝑝/2)<𝑞. We say that 𝑝, 𝑞, and 𝑠 satisfy condition (A) if 1−𝑝+2𝑞𝑛2𝑛<𝑠<𝑛−1if0<𝑝+2𝑞𝑛2𝑛<1,0<𝑠<𝑛−1if1≤𝑝+2𝑞2𝑛<∞; (29) and satisfy condition (B) if 1−𝑝+2𝑞2𝑛<𝑠<∞if0<𝑝+2𝑞2𝑛<1,0<𝑠<∞if1≤𝑝+2𝑞2𝑛<∞. (30)

Proposition 8. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). Then, sup_|𝑎|<1𝐵1−|𝑧|^{2(𝑝/2)+𝑞}1−|𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}𝑑𝜆(𝑧)<∞,lim_{|𝑎|→1−𝐵}1−|𝑧|^{2(𝑝/2)+𝑞}1−|𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}𝑑𝜆(𝑧)=0. (31)

Proof. We have the following equalities: 1−|_𝐵𝑧|^{2(𝑝/2)+𝑞}1−|𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}(𝑧)=𝑑𝜆_𝐵1−𝜑_𝑎(𝑤)^{2(𝑝/2)+𝑞}1−|𝑤|^2𝑛𝑠|𝑤|^{2(𝑛−1)𝑠}(𝑤)=𝑑𝜆_𝐵1−|𝑎|^{2(𝑝/2)+𝑞}1−|𝑤|^{2(𝑝/2)+𝑞+𝑛𝑠−𝑛−1}|1−⟨𝑤,𝑎⟩|^𝑝+2𝑞|𝑤|^{2(𝑛−1)𝑠}(𝑤)=|𝑎|𝑑𝜈1−^{2(𝑝/2)+𝑞}_𝐵1−|𝑤|^{2(𝑝/2)+𝑞+𝑛𝑠−𝑛−1}|1−⟨𝑤,𝑎⟩|^𝑝+2𝑞|𝑤|^{2(𝑛−1)𝑠}𝑑𝜈(𝑤). (32) The singularity at 0 is integrable if 𝑠<𝑛/(𝑛−1). Now, if we estimate 1−|_𝐵𝑤|^{2(𝑝/2)+𝑞+𝑛𝑠−𝑛−1}|1−⟨𝑤,𝑎⟩|^𝑝+2𝑞𝑑𝜈(𝑤), (33) as (𝑝/2)+𝑞+𝑛𝑠−𝑛−1>−1 if 1−((𝑝+2𝑞)/2𝑛)<𝑠 and 0<(𝑝/2)+𝑞, 0<𝑠<∞, then the result follows from Theorem 7 for any case of 𝑐=(𝑝/2)+𝑞−𝑛𝑠.

In a similar way, we have the following.

Proposition 9. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (B). Then, sup_|𝑎|<1𝐵1−|𝑧|^{2(𝑝/2)+𝑞}1−𝜑_𝑎(𝑧)^2𝑛𝑠𝑑𝜆(𝑧)<∞,lim_{|𝑎|→1−𝐵}1−|𝑧|^{2(𝑝/2)+𝑞}1−𝜑_𝑎(𝑧)^2𝑛𝑠𝑑𝜆(𝑧)=0. (34)

2. Function Spaces 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝜑(𝑝,𝑞,𝑠)

2.1. Main Definitions

In this section, we define the spaces 𝐹_𝐺(𝑝,𝑞,𝑠), 𝐹_𝜑(𝑝,𝑞,𝑠) and their little associated spaces. We study some properties and relations between them.

Suppose that 𝑝,𝑞, and 𝑠 satisfy condition (A), and let 𝑓∈𝒪. Define 𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎)=_𝐵∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧). (35) Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (B), and let 𝑓∈𝒪. Define 𝐽̃_{𝑝,𝑞,𝑠}𝑓(𝑎)=_𝐵∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞1−𝜑_𝑎(𝑧)^2𝑛𝑠𝑑𝜆(𝑧). (36) It is clear that 𝐽̃_{𝑝,𝑞,𝑠}𝑓(𝑎)≤𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎),foreach𝑎∈𝐵. (37) Associated to these functions, we define the following vector spaces of holomorphic functions. (i)Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). 𝑓∈𝒪 belongs to 𝐹_𝐺(𝑝,𝑞,𝑠) if it satisfies sup_𝑎∈𝐵𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎)<+∞ (38) and belongs to the little space 𝐹_𝐺,0(𝑝,𝑞,𝑠) if it satisfies lim_{|𝑎|→1−}𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎)=0. (39)(ii)Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (B). 𝑓∈𝒪 belongs to 𝐹_𝜑(𝑝,𝑞,𝑠) if it satisfies sup_𝑎∈𝐵𝐽̃_{𝑝,𝑞,𝑠}𝑓(𝑎)<+∞ (40) and belongs to the little space 𝐹_𝜑,0(𝑝,𝑞,𝑠) if it satisfies lim_{|𝑎|→1−}𝐽̃_{𝑝,𝑞,𝑠}𝑓(𝑎)=0. (41)(iii)Let 𝑛<(𝑝+2𝑞)/2. 𝑓∈𝒪 belongs to the Dirichlet space 𝐷_𝑝,𝑞 if ∇𝑓(_𝐵𝑧)^𝑝1−|𝑧|^2𝑞𝑑𝜆(𝑧)<∞. (42)

As ∇(𝑓+ℎ)(𝑧)=∇𝑓(𝑧)+∇ℎ(𝑧), then for 1≤𝑝<∞, the sets 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝜑(𝑝,𝑞,𝑠) are normed vector function spaces, with norms given by𝑓_𝐺∶=𝑓(0)+sup_𝑎∈𝐵𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎)^1/𝑝,𝑓_𝜑∶=𝑓(0)+sup_𝑎∈𝐵𝐽̃_{𝑝,𝑞,𝑠}𝑓(𝑎)^1/𝑝. (43) This is a direct consequence of the usual 𝐿^𝑝(𝐵,𝑑𝜇) theory when we consider the measures 𝑑𝜇(𝑧)=(1−|𝑧|²)^𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧),𝑑𝜇(𝑧)=(1−|𝑧|²)^𝑞(1−|𝜑_𝑎(𝑧)|²)^𝑛𝑠𝑑𝜆(𝑧), respectively. For 0<𝑝<1 and nonnegative 𝑥, 𝑦, we have (𝑥+𝑦)^𝑝≤𝑥^𝑝+𝑦^𝑝; as a consequence, the sets 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝜑(𝑝,𝑞,𝑠) result in metric vector spaces, with the metric given by 𝑑_𝐺𝑓,ℎ=𝑓(0)−ℎ(0)+sup_𝑎∈𝐵𝐼̃_{𝑝,𝑞,𝑠}𝑎),𝑑𝑓−ℎ(_𝜑𝑓,ℎ=𝑓(0)−ℎ(0)+sup_𝑎∈𝐵𝐽̃_{𝑝,𝑞,𝑠}𝑓−ℎ(𝑎), (44) respectively (see [10]). We will prove their completeness in Theorem 26.

Example 10. Let 𝒪 denotes the class of holomorphic functions on a neighborhood of 𝐵. Let 𝑓∈𝒪. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). By (19) we have, for example, the estimation ∇𝑓(_𝐵𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧)≤max_𝑧∈𝐵∇𝑓(𝑧)^𝑝_𝐵1−|𝑧|^{2(𝑝/2)+𝑞}𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧). (45) So, in the definitions, the parameters 𝑝, 𝑞, and 𝑠 were chosen to guarantee that the vector spaces 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝐺,0(𝑝,𝑞,𝑠) include at least this wide class of holomorphic functions, as Propositions 5, 8, and 9 show. Compare this example with Proposition 3.7 and Theorem 3.8 from [4].
Let 𝑓∶𝐵→ℂ, with power series development given by 𝑓(𝑧)=_𝛼𝑎_𝛼𝑧^𝛼, (46) such that ∑^∞_𝑛=0𝑛∑_|𝛼|=𝑛|𝑎_𝛼|<∞. Then, 𝑓 belongs to each one of the previous spaces. By (37), we get the same result for 𝐹_𝜑(𝑝,𝑞,𝑠) and 𝐹_𝜑,0(𝑝,𝑞,𝑠).

The spaces 𝐹_𝐺(𝑝,𝑞,𝑠) are Möbius invariant in the following sense.

Theorem 11. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). If 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠) and 𝜑 is a Möbius transformation, then 𝑓∘𝜑∈𝐹_𝐺(𝑝,𝑞,𝑠).

Proof. Let 𝑎,𝑏∈𝐵,𝑤=𝜑_𝑎(𝑧), and 𝑐=𝜑_𝑎(𝑏). By the change of variables formula and the Möbius invariance of 𝐺(𝑧,𝑎), ∇, and 𝑑𝜆, we have _𝐵∇𝑓∘𝜑_𝑎(𝑤)^𝑝1−|𝑤|^2𝑞𝐺^𝑠()(𝑤)=𝑤,𝑏𝑑𝜆_𝐵∇𝑓∘𝜑_𝑎(𝑤)^𝑝1−|𝑤|^2𝑞𝐺^𝑠(𝑤,𝑏)𝑑𝜆(𝑤)=_𝐵∇𝑓∘𝜑_𝑎𝜑_𝑎(𝑧)^𝑝1−𝜑_𝑎(𝑧)^2𝑞×𝐺^𝑠𝜑_𝑎(𝑧),𝜑_𝑎(𝑐)(𝑧)=𝑑𝜆_𝐵∇𝑓(𝑧)^𝑝1−𝜑_𝑎(𝑧)^2𝑞𝐺^𝑠()(𝑧)≤|𝑎|𝑧,𝑐𝑑𝜆1+1−|𝑎|^|𝑞|_𝐵∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑐)𝑑𝜆(𝑧), (47) where we have used in the last line the estimations of Lemmas 3 and 4.

Observe that if 𝑞=0 and 𝑓∈𝐹_𝐺(𝑝,0,𝑠), then _𝐵∇𝑓∘𝜑_𝑎(𝑤)^𝑝𝐺^𝑠()(𝑤)=𝑤,𝑏𝑑𝜆_𝐵∇𝑓(𝑧)^𝑝𝐺^𝑠(𝑧,𝑐)𝑑𝜆(𝑧), (48) and these spaces are Möbius invariant in the classical way.

Theorem 12. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). Suppose that 𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎)<∞, (49) for all 𝑎∈𝐵. Then, the function 𝐼̃_{𝑝,𝑞,𝑠}𝑓 is continuous on 𝐵.

Proof. Let 𝑎∈𝐵 be fixed and 𝑅, such that |𝑎|<𝑅<1. Consider a neighborhood 𝐵_𝛿(𝑎) with 𝛿=(𝑅−|𝑎|)/2. Let {𝑎_𝑚}_𝑚∈ℕ be a sequence contained in 𝐵_𝛿(𝑎) and convergent to 𝑎. Consider now 𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎_𝑚) and define 𝑑𝜇(𝑧)=|∇𝑓(𝑧)|^𝑝(1−|𝑧|²)^{𝑞+𝑛𝑠}𝑑𝜆(𝑧). Then, 𝐺^𝑠(𝑧,𝑎_𝑚)(1−|𝑧|²)^−𝑛𝑠 is bounded in (𝐵⧵𝐵_𝑅)×𝐵_𝛿(𝑎). Thus, by bounded convergence theorem, _{𝐵⧵𝐵𝑅}𝐺^𝑠𝑧,𝑎_𝑚1−|𝑧|^2𝑛𝑠𝑑𝜇(𝑧) (50) converges to _{𝐵⧵𝐵𝑅}𝐺^𝑠(𝑧,𝑎)1−|𝑧|^2𝑛𝑠𝑑𝜇(𝑧). (51) Now, integrating in 𝐵_𝑅 and using change of variables, ∇𝑓(_𝐵𝑅𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠()(𝑧)=𝑧,𝑎𝑑𝜆_{𝐷(𝑎,𝑅)}∇𝑓𝜑_𝑎(𝑤)^𝑝1−𝜑_𝑎(𝑤)^2𝑞×𝑔^𝑠(𝑤)(𝑤)=𝑑𝜆_{𝐷(𝑎,𝑅)}∇𝑓𝜑_𝑎(𝑤)^𝑝1−|𝑎|^2𝑞|1−⟨𝑤,𝑎⟩|^2𝑞×1−|𝑤|^2𝑞𝑔^𝑠(𝑤)𝑑𝜆(𝑤). (52)
Consider ∇𝑓𝜑_𝑎𝑚(𝑤)^𝑝1−𝑎_𝑚^2𝑞1−𝑤,𝑎_𝑚^2𝑞, (53) and define the measure 𝑑𝜇(𝑤) by 𝑑𝜇(𝑤)=1−|𝑤|^2𝑞𝑔^𝑠(𝑤)𝑑𝜆(𝑤). (54) Now, if we take |𝑤|<1, |𝑎|<𝑅, then 1−𝑅≤|1−⟨𝑤,𝑎_𝑚⟩|. Besides, on 𝐷(𝑎,𝑅), |∇𝑓(𝜑_𝑎𝑚(𝑤))| results are bounded by a constant 𝑀>0, because 𝜑_𝑎𝑚(𝑤)∈𝐵_𝑅. Now, ∇𝑓𝜑_{𝐷(𝑎𝑚,𝑅)𝑎𝑚}(𝑤)^𝑝1−𝑎_𝑚^2𝑞1−𝑤,𝑎_𝑚^2𝑞𝑑𝜇(𝑤)=_𝐵∇𝑓𝜑_𝑎𝑚(𝑤)^𝑝1−𝑎_𝑚^2𝑞1−𝑤,𝑎_𝑚^2𝑞×𝜒_{𝜑𝑎𝑚(𝐵𝑅)}(𝑤)𝑑𝜇(𝑤), (55) where 𝜒_{𝜑𝑎𝑚(𝐵𝑅)} denotes the characteristic function of the set 𝐷(𝑎_𝑚,𝑅). As ∇𝑓𝜑_𝑎𝑚(𝑤)^𝑝1−𝑎_𝑚^2𝑞1−𝑤,𝑎_𝑚^2𝑞𝜒_{𝜑𝑎𝑚(𝐵𝑅)}(𝑤)≤𝑀|1−𝑅|^2𝑞, (56) then by Lebesgue’s dominated convergence theorem, we have that (55) converges to ∇𝑓𝜑_𝐵𝑎(𝑤)^𝑝1−|𝑎|^2𝑞|1−⟨𝑤,𝑎⟩|^2𝑞𝜒_{𝜑𝑎(𝐵𝑅)}(𝑤)𝑑𝜇(𝑤). (57)

Theorem 13. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (B). Suppose that 𝐽̃_{𝑝,𝑞,𝑠}𝑓(𝑎)<∞, (58) for all 𝑎∈𝐵. Then, the function 𝐽̃_{𝑝,𝑞,𝑠}𝑓 is continuous on 𝐵.

Proof. Let 𝑎∈𝐵 and 𝛿>0, such that 𝐵_𝛿(𝑎)⊂𝐵. If 𝐾∶=𝐵×𝐵_𝛿(𝑎), the function 𝑙∶𝐾→ℝ defined by =𝜉𝜂,𝜉⟼𝑙𝜂,𝜉1−^2𝑠1−𝜂,𝜉^2𝑠 (59) results to be uniformly continuous on 𝐾. As 𝑓∈𝐹_𝜑(𝑝,𝑞,𝑠), then ∇𝑓(_𝐵𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑑𝜆(𝑧)=𝐽̃_{𝑝,𝑞,𝑠}𝑓(0)<∞. (60) If 𝑓 is nonconstant, given that 𝜀>0, there exists 𝜌>0, such that |𝜂^′−𝜂|<𝜌 and |𝜉^′−𝜉|<𝜌; then 𝑙𝜂^′,𝜉^′<𝜀−𝑙𝜂,𝜉∫_𝐵∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}(𝑧).𝑑𝜆 (61) Thus, if |𝑎−𝑏|<𝜌, then 𝐽̃_{𝑝,𝑞,𝑠}𝑓(𝑎)−𝐽̃_{𝑝,𝑞,𝑠}𝑓(𝑏≤_𝐵)∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑙𝜂,𝑎−𝑙𝜂,𝑏𝑑𝜆(𝑧)<𝜀. (62)

Remark 14. Last two theorems are the versions in ℂ^𝑛 of Theorem 3.1 of [6].

Some relations between the little spaces 𝐹_𝐺0(𝑝,𝑞,𝑠), 𝐹_𝜑0(𝑝,𝑞,𝑠) and the spaces 𝐹_𝐺(𝑝,𝑞,𝑠), 𝐹_𝜑(𝑝,𝑞,𝑠) are given in the following results.

Proposition 15. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). If 𝑓∈𝐹_𝐺,0(𝑝,𝑞,𝑠), then the function _𝐺ℎ(𝑎)=𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎)|𝑎||𝑎|,𝑖𝑓<1,0,𝑖𝑓=1, (63) is continuous on 𝐵.

Proof. By the Theorem 12, it suffices to prove that if |𝑎₀|=1, given 𝜀>0, there exists 𝛿>0 such that if |𝑎−𝑎₀|<𝛿, then |_𝐺ℎ(𝑎)−_𝐺ℎ(𝑎₀)|<𝜀; that is, |_𝐺ℎ(𝑎)|<𝜀. By hypothesis, lim_|𝑎|→1⁻𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎)=0. So, given 𝜀>0, there exists 𝛿>0, such that if ||𝑎|−1|<𝛿, then |𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎)|<𝜀, or 1−𝛿<|𝑎|<1 implies |_𝐺ℎ(𝑎)|<𝜀. Moreover, as 𝐵 is compact, _𝐺ℎ results uniformly continuous on 𝐵.

Proposition 16. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). Then, 𝐹_𝐺,0(𝑝,𝑞,𝑠) is contained in 𝐹_𝐺(𝑝,𝑞,𝑠).

Proof. In fact if 𝑓∈𝐹_𝐺0(𝑝,𝑞,𝑠), as the function_𝐺ℎ is uniformly continuous on 𝐵, then is bounded and so max_{𝑎∈𝐵𝐵}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧)<∞. (64)

With the same arguments, we get 𝐹_𝜑,0(𝑝,𝑞,𝑠)⊂𝐹_𝜑(𝑝,𝑞,𝑠).

2.2. Characterization as Bloch Spaces

Now, we are going to present several characterizations of Bloch spaces similar to the characterizations obtained by Stroethoff [11, Theorems 4 and 7] and Zhao [3, Theorem 1].

For −(1/2)≤𝛼, the 𝛼-Bloch spaces, ℬ_𝛼, are the families of holomorphic functions 𝑓∶𝐵→ℂ satisfying 𝑓_ℬ𝛼=𝑓(0)+sup_𝑧∈𝐵1−|𝑧|^2𝛼∇𝑓(𝑧)<+∞. (65) It is easy to verify that ‖⋅‖_ℬ𝛼 is a norm on the vector space ℬ_𝛼 and contains at least the holomorphic functions on domains Ω, with Ω⊃𝐵. The respective little 𝛼-Bloch spaces ℬ_𝛼,0 satisfy lim_{|𝑧|→1−}1−|𝑧|^2𝛼∇𝑓(𝑧)=0. (66) It is clear that ℬ_𝛼,0⊂ℬ_𝛼.

Proposition 17. (i) Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). Then, 𝐹_𝐺𝑝,𝑞,𝑠⊂ℬ_𝑞/𝑝, (67) and there exists 𝐶>0, such that 𝑓_ℬ𝑞/𝑝≤𝐶𝑓_𝐺. (68)
(ii) If 𝑛<(𝑝/2)+𝑞. Then, 𝐷_𝑝,𝑞⊂𝐵_𝑞/𝑝,0. (69)

Proof. (i) Let 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠); then by Propositions 5 and 6 and (20) ∞>_𝐵∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠()(𝑧)≥𝑧,𝑎𝑑𝜆_{𝐷(𝑎,𝑅)}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝐶^𝑠₁1−𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}(𝑧)≥𝑑𝜆_{𝐷(𝑎,𝑅)}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝐶^𝑠₁1−𝑅^2𝑛𝑠𝑅^{2(𝑛−1)𝑠}(𝑧)≥𝐶𝑑𝜆_𝑠11−𝑅^2𝑛𝑠𝑅^{2(𝑛−1)𝑠𝐷(𝑎,𝑅)}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞(𝑧)≥𝐶𝑑𝜆_𝑠11−𝑅^2𝑛𝑠𝑅^{2(𝑛−1)𝑠𝐵}_𝑅∇𝑓∘𝜑_𝑎(𝑤)^𝑝×1−𝜑_𝑎(𝑤)^2𝑞(𝑤)≥𝐶𝑑𝜆_𝑠11−𝑅^2𝑛𝑠𝑅^{2(𝑛−1)𝑠𝐵}_𝑅∇𝑓∘𝜑_𝑎(𝑤)^𝑝1−|𝑤|^2𝑝×1−|𝑎|^2𝑞1−|𝑤|^{2𝑞−𝑛−1−𝑝}|1−⟨𝑤,𝑎⟩|^2𝑞(𝑤)≥𝐶𝑑𝜈_𝑠11−𝑅^2𝑛𝑠𝑅^{2(𝑛−1)𝑠}𝐶₂×𝑝,𝑞,𝑛_𝐵𝑅∇𝑓∘𝜑_𝑎(𝑤)^𝑝1−|𝑤|^2𝑝1−|𝑎|^2𝑞(𝑤)|𝑎|𝑑𝜈=𝐶𝑝,𝑞,𝑠,𝑛,𝑅1−^2𝑞_𝐵𝑅∇𝑓∘𝜑_𝑎(𝑤)^𝑝(𝑤)|𝑎|𝑑𝜈≥𝐶𝑝,𝑞,𝑠,𝑛,𝑅1−^2𝑞∇𝑓∘𝜑_𝑎(0)^𝑝=𝐶𝑝,𝑞,𝑠,𝑛,𝑅1−|𝑎|^2𝑞∇𝑓(𝑎)^𝑝=𝐶𝑝,𝑞,𝑠,𝑛,𝑅1−|𝑎|^2𝑞/𝑝∇𝑓(𝑎)^𝑝, (70) where we have used that |∇(𝑓∘𝜑_𝑎)|^𝑝 is a subharmonic function; (i) follows from this estimation.
(ii) Let 𝑓∈𝐷_𝑝,𝑞 and 0<𝑅<1 be fixed. By the absolute continuity of the integral and by Lemma 3, given 𝜀>0, there exists 0<𝑟<1, such that if we choose |𝑎| sufficiently close to 1 then 𝜀>_𝐴(𝑟)∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞(𝑧)≥𝑑𝜆_{𝐷(𝑎,𝑅)}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝑑𝜆(𝑧). (71)
With the same kind of arguments used in (i), we obtain 𝜀≥𝐶𝑝,𝑞,𝑛,𝑅1−|𝑎|^2𝑞/𝑝∇𝑓(𝑎)^𝑝, (72) for all 𝑎∈𝐵, sufficiently close to 1, and we conclude the proof.

The following result says that Dirichlet spaces are the limit case of 𝐹_𝐺,0(𝑝,𝑞,𝑠) and 𝐹_𝜑,0(𝑝,𝑞,𝑠), when 𝑠→0⁺.

Theorem 18. (i) Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). Then, 𝐷_𝑝,𝑞⊂_0<𝑠<3𝐹_𝑔,0𝑝,𝑞,𝑠. (73)
(ii) Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (B). Then, 𝐷_𝑝,𝑞⊂_0<𝑠<∞𝐹_𝜑,0𝑝,𝑞,𝑠. (74)

Proof. We prove the first inclusion. The proof of the second one is similar. Let 𝑓∈𝐷_𝑝,𝑞 and 𝜀>0. By Proposition 17, there exists 0<𝜌<1, such that ∇𝑓(_𝐴(𝜌)𝑧)^𝑝1−|𝑧|^2𝑞𝑑𝜆(𝑧)<𝜀,1−|𝑧|^2𝑞∇𝑓(𝑧)^𝑝<𝜀,∀0<𝜌<|𝑧|<1. (75)(76) Let 0<𝜌<𝑅<𝑅^′<1, such that if 𝑅^′<|𝑎|<1; then 𝐷(𝑎,𝑅)⊂𝐴(𝑅). Thus, 𝐼_{𝑝,𝑞,𝑠}𝑓(𝑎)≤𝐶^𝑠_2𝐵𝑅∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞1−𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}𝑑𝜆(𝑧)+𝐶^𝑠_{2𝐴(𝑅)⧵𝐷(𝑎,𝑅)}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞×1−𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}𝑑𝜆(𝑧)+𝐶^𝑠_{2𝐷(𝑎,𝑅)}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞×1−𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}𝑑𝜆(𝑧). (77) By Proposition 8, the first integral goes to zero as |𝑎|→1⁻. For the second one, consider (75) as following: ∇𝑓(_{𝐴(𝑅)⧵𝐷(𝑎,𝑅)}𝑧)^𝑝1−|𝑧|^2𝑞1−𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}(𝑧)≤1𝑑𝜆𝑅^{2(𝑛−1)𝑠𝐴(𝜌)}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞(𝑧)<𝜀𝑑𝜆𝑅^{2(𝑛−1)𝑠}. (78) Moreover, for the third, using the change of variable 𝜑_𝑎(𝑤)=𝑧, ∇𝑓(_{𝐷(𝑎,𝑅)}𝑧)^𝑝1−|𝑧|^2𝑞1−𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}(𝑧)<𝑑𝜆_{𝐷(𝑎,𝑅)}𝜀1−𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}𝑑𝜆(𝑧)=𝜀_𝐵𝑅1−|𝑤|^2𝑛𝑠|𝑤|^{2(𝑛−1)𝑠}𝑑𝜆(𝑤). (79) and this concludes the proof.

Proposition 19. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A), with 1<𝑠<𝑛/(𝑛−1). Then, ℬ_𝑞/𝑝⊂𝐹_𝐺𝑝,𝑞,𝑠. (80)

Proof. Let 𝑓∈ℬ_𝑞/𝑝. Then, sup_𝑧∈𝐵1−|𝑧|^2𝑞/𝑝∇𝑓(𝑧)≤𝑓_ℬ𝑞/𝑝<∞. (81) Using change of variables and by Proposition 5 we have ∇𝑓(_𝐵𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠()(𝑧)≤𝑧,𝑎𝑑𝜆_𝐵𝑓^𝑝_ℬ𝑞/𝑝1−|𝑧|^2𝑞1−|𝑧|^2𝑞𝐺^𝑠()(𝑧)=𝑓𝑧,𝑎𝑑𝜆_{𝑝ℬ𝑞/𝑝𝐵}𝐺^𝑠()(𝑧)≤𝑓𝑧,𝑎𝑑𝜆_{𝑝ℬ𝑞/𝑝𝐵}𝐶^𝑠₂1−𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}(𝑧)=𝑓𝑑𝜆_{𝑝ℬ𝑞/𝑝}𝐶^𝑠_2𝐵1−|𝑤|^{2𝑛𝑠−𝑛−1}|𝑤|^{2(𝑛−1)𝑠}𝑑𝜈(𝑤), (82) and this integral results convergent because ¹₀(1−𝑟)^{𝑛𝑠−𝑛−1}𝑟^{2(𝑛−𝑛𝑠+𝑠)−1}𝑑𝑟<∞, (83) when 1<𝑠<𝑛/(𝑛−1). Finally, ℬ_𝑞/𝑝⊆𝐹_𝐺(𝑝,𝑞,𝑠) if 1<𝑠<𝑛/(𝑛−1).

Similar results to Propositions 17 and 19 can be obtained for 𝐹_𝜑(𝑝,𝑞,𝑠).

Proposition 20. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (B), with 1<𝑠<∞. Then, 𝐹_𝜑𝑝,𝑞,𝑠⊂ℬ_𝑞/𝑝, (84) and there exists 𝐶>0, such that 𝑓_ℬ𝑞/𝑝≤𝐶𝑓_𝜑. (85)

Proof. Let 𝑓∈𝐹_𝜑(𝑝,𝑞,𝑠); then ∞>_𝐵∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞1−|𝜑_𝑎(𝑧)^2𝑛𝑠(𝑧)=𝑑𝜆_𝐵∇𝑓∘𝜑_𝑎(𝑤)^𝑝1−|𝜑_𝑎(𝑤)^2𝑞×1−|𝑤|^{2𝑛𝑠−𝑛−1}(𝑤)≥𝑑𝜈_𝐵1/2∇𝑓∘𝜑_𝑎(𝑤)^𝑝1−|𝑤|^2𝑝1−|𝑎|^2𝑞1−|𝑤|^2𝑞|1−⟨𝑤,𝑎⟩|^2𝑞×1−|𝑤|^{2𝑛𝑠−𝑛−1+𝑝}𝑑𝜈(𝑤)≥𝐶𝑝,𝑞,𝑠,𝑛_𝐵1/2∇𝑓∘𝜑_𝑎(𝑤)^𝑝1−|𝑤|^2𝑝1−|𝑎|^2𝑞(𝑤)|𝑎|𝑑𝜈≥𝐶𝑝,𝑞,𝑠,𝑛1−^2𝑞_𝐵1/2∇𝑓∘𝜑_𝑎(𝑤)^𝑝𝑑𝜈(𝑤), (86) and we finish the proof as in the Proposition 17.

Proposition 21. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A), with 1<𝑠<∞. Then, ℬ_𝑞/𝑝⊂𝐹_𝜑𝑝,𝑞,𝑠. (87)

Proof. Let 𝑓∈ℬ_𝑞/𝑝. By hypothesis, sup_𝑧∈𝐵1−|𝑧|^2𝑞/𝑝∇𝑓(𝑧)≤𝑓_ℬ𝛼<∞. (88) Thus, ∇𝑓(_𝐵𝑧)^𝑝1−|𝑧|^2𝑞1−𝜑_𝑎(𝑧)^2𝑛𝑠(𝑧)≤𝑑𝜆_𝐵𝑓^𝑝_ℬ𝛼1−|𝑧|^2𝑞1−|𝑧|^2𝑞1−𝜑_𝑎(𝑧)^2𝑛𝑠(𝑧)=𝑓𝑑𝜆_{𝑝ℬ𝛼𝐵}1−|𝑎|^2𝑛𝑠1−|𝑧|^{2𝑛𝑠−𝑛−1}|1−⟨𝑧,𝑎⟩|^2𝑛𝑠(𝑧)=𝑓𝑑𝜈_𝑝ℬ𝛼1−|𝑎|^2𝑛𝑠_𝐵1−|𝑧|^{2𝑛𝑠−𝑛−1}|1−⟨𝑧,𝑎⟩|^2𝑛𝑠𝑑𝜈(𝑧), (89) and by Theorem 7, we obtain that if 𝑠>1, 1−|_𝐵𝑧|^{2𝑛𝑠−𝑛−1}|1−⟨𝑧,𝑎⟩|^2𝑛𝑠(𝑧)≈1𝑑𝜈1−|𝑎|^2𝑛𝑠, (90) and the result follows.

We can synthesize the last propositions in the following.

Theorem 22. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (B). The following statements are equivalent.(i)𝑓∈ℬ_𝑞/𝑝;(ii)𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠) for each 𝑠∈(1,𝑛/(𝑛−1));(iii)𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠) for some 𝑠∈(1,𝑛/(𝑛−1));(iv)𝑓∈𝐹_𝜑(𝑝,𝑞,𝑠) for each 𝑠∈(1,∞);(v)𝑓∈𝐹_𝜑(𝑝,𝑞,𝑠) for some 𝑠∈(1,∞).

Jointly with these properties, there are similar results for the little spaces 𝐹_𝐺,0(𝑝,𝑞,𝑠), 𝐹_𝜑,0(𝑝,𝑞,𝑠), and ℬ_𝑞/𝑝,0.

For these cases, the inclusions 𝐹_𝐺,0(𝑝,𝑞,𝑠),𝐹_𝜑,0(𝑝,𝑞,𝑠)⊆ℬ_𝑞/𝑝,0 follow immediately from the previous estimations. For the other inclusions we have the following.

Proposition 23. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A) and 1<𝑠<𝑛/(𝑛−1). Then, ℬ_𝑞/𝑝,0⊂𝐹_𝐺,0𝑝,𝑞,𝑠. (91)

Proof. Since 1<𝑠<𝑛/(𝑛−1), then 𝐶=∫_𝐵𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧)<∞. Suppose that 𝑓∈ℬ_𝑞/𝑝,0; then lim_{|𝑧|→1−}1−|𝑧|^2𝑞/𝑝∇𝑓(𝑧)=0. (92) Given 𝜀>0, there exists 0<𝑅<1, such that ∇𝑓(𝑧)^𝑝<𝜀𝐶1−|𝑧|^2𝑞 (93) if 𝑅<|𝑧|<1. We split the integral ∇𝑓(_𝐵𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧), (94) as ∇𝑓(_𝐵𝑅𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠()(𝑧)+𝑧,𝑎𝑑𝜆_{𝐵⧵𝐵𝑅}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧). (95) By hypothesis, the second integral satisfies ∇𝑓(_{𝐵⧵𝐵𝑅}𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠()(𝑧)≤𝜀𝑧,𝑎𝑑𝜆𝐶_{𝐵⧵𝐵𝑅}𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧)<𝜀. (96) Now, ∇𝑓(_𝐵𝑅𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧)≤𝐶^𝑠_2𝐵𝑅𝑓_ℬ𝛼1−|𝑧|^2𝑞1−|𝑧|^2𝑞1−𝜑_𝑎(𝑧)^2𝑛𝑠𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}𝑑𝜆(𝑧)≤𝐶^𝑠₂𝑓_ℬ𝛼1−|𝑎|^2𝑛𝑠×_𝐵𝑅1−|𝑧|^{2𝑛𝑠−𝑛−1}|1−⟨𝑧,𝑎⟩|^{2(𝑛−1)𝑠}1𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠}𝑑𝜈(𝑧), (97) and the function ()⟼|𝑧|𝑎,𝑧1−^{2𝑛𝑠−𝑛−1}|1−⟨𝑧,𝑎⟩|^{2(𝑛−1)𝑠}𝜑_𝑎(𝑧)^{2(𝑛−1)𝑠} (98) attains its maximum on the compact set {𝑧∈𝐵∶|𝑧|≤𝑅}×{𝑎∈𝐵∶(𝑅+1)/2≤|𝑎|≤1}. Therefore, if |𝑎|→1⁻, the last estimation tends to be zero.

The proof of ℬ_𝑞/𝑝,0⊂𝐹_𝜑,0𝑝,𝑞,𝑠 (99) is similar.

Lemma 24. Let −(1/2)≤𝛼<∞ and 𝑓∈ℬ_𝛼. Then, 𝑓 is a uniformly continuous function from the metric space 𝐵 equipped with the Bergman metric 𝜌 to the metric space ℂ equipped with the Euclidean metric.

Proof. Let 𝛾∶[0,1]→𝐵 be a piecewise smooth curve. Thus, the lenght of 𝛾 in the Bergman metric 𝜌 of 𝐵 is 𝑙𝛾=¹₀𝑄𝛾(𝑡)𝛾^′(𝑡),𝛾^′(𝑡)𝑑𝑡, (100) where 𝑄(𝑧) is the Bergman matrix of 𝐵 (see [7]). Let 𝑓∶𝐵→ℂ be a holomorphic function. Then, (𝑧(𝑧),∇𝑓)=sup∇𝑓𝑤⟨𝑄(𝑧)𝑤,𝑤⟩∶𝑤∈ℂ^𝑛−{0}. (101) Now, if 𝛾∶[0,1]→𝐵 is a geodesic in the Bergman metric, with 𝛾(0)=𝑧₀, 𝛾₁(1)=𝑧₁,…, and 𝑓∈ℬ_𝛼, we have 𝑓𝑧₁−𝑓𝑧₀≤¹₀𝑓∘𝛾^′(𝑡=)𝑑𝑡₁₀∇𝑓𝛾(𝑡),𝛾^′(𝑡≤)𝑑𝑡₁₀∇𝑓𝛾(𝑡)⟨𝑄𝛾(𝑡)𝛾^′(𝑡),𝛾^′(𝑡≤)𝑑𝑡₁₀𝛾(𝑡)𝛾(𝑡∇𝑓1−)^2𝑞/𝑝1−𝛾(𝑡)^2𝑞/𝑝×⟨𝑄𝛾(𝑡)𝛾^′(𝑡),𝛾^′(𝑡≤𝑓)𝑑𝑡_ℬ𝛼1−sup_𝑡∈[0,1]𝛾(𝑡)^2𝑞/𝑝×¹₀⟨𝑄𝛾(𝑡)𝛾^′(𝑡),𝛾^′(𝑡=𝑓)𝑑𝑡_ℬ𝛼1−sup_𝑡∈[0,1]𝛾(𝑡)^2𝑞/𝑝𝜌𝑧₀,𝑧₁, (102) and the proof is complete.

Theorem 25. Let −(1/2)≤𝛼<∞. The spaces ℬ_𝛼 and ℬ_𝛼,0 are complete metric spaces.

Proof. Let {𝑓_𝑘}⊂ℬ_𝛼 be a Cauchy sequence. Given 𝜀>0, there exists 𝑁∈ℕ, such that ‖𝑓_𝑙−𝑓_𝑘‖_ℬ𝛼<𝜀,   for all   𝑘>𝑙>𝑁. In particular, {𝑓_𝑘(0)} is convergent. Let 0<𝑟<1. We apply the estimation of Lemma 24 to 𝑓_𝑙−𝑓_𝑘,𝑧₀=0, and𝑧₁=𝑧∈𝐵_𝑟. Thus, 𝑓_𝑙(𝑧)−𝑓_𝑘(𝑧)−𝑓_𝑙(0)+𝑓_𝑘(0≤𝑓)^𝑙−𝑓_𝑘ℬ𝛼1−𝑟^2𝑞/𝑝log1+𝑟.1−𝑟 (103)
Then, {𝑓_𝑘} converges uniformly on compact subsets of 𝐵 to some holomorphic function, say 𝑓. Therefore, 𝜕𝑓_𝑘/𝜕𝑧_𝑗 converges uniformly on compact subsets of 𝐵 to 𝜕𝑓/𝜕𝑧_𝑗, for 𝑗=1,…,𝑛. So, we can get |∇(𝑓_𝑙−𝑓_𝑘)|^𝑝 converges to |∇(𝑓−𝑓_𝑘)|^𝑝 uniformly on compact sets of 𝐵 when 𝑙→∞. For 𝑘,𝑙>𝑁, we have 𝑓_𝑘(0)−𝑓_𝑙(0)+1−|𝑧|^2𝛼∇𝑓_𝑘−𝑓_𝑙(𝑧)≤𝑓_𝑘−𝑓_𝑙ℬ𝛼<𝜀, (104) and taking limit when 𝑘→∞, we get 𝑓(0)−𝑓_𝑙(0)+1−|𝑧|^2𝛼∇𝑓−𝑓_𝑙(𝑧)≤𝜀, (105) that is, 𝑓−𝑓_𝑙∈ℬ_𝛼 and hence 𝑓=(𝑓−𝑓_𝑙)+𝑓_𝑙∈ℬ_𝛼.
Let {𝑓_𝑘}⊂ℬ_𝛼,0 be a Cauchy sequence. Then, 𝑓_𝑘 converges to 𝑓∈ℬ_𝛼. Given 𝜀>0, there exists 𝑁∈ℕ, such that ‖𝑓−𝑓_𝑘‖_ℬ𝛼<𝜀/2, for all 𝑘>𝑁. Moreover, there exists 0<𝑟<1, such that 1−|𝑧|^2𝛼∇𝑓_𝑁+1(𝑧<𝜀)2,∀𝑟<|𝑧|<1. (106) Thus, 1−|𝑧|^2𝛼∇𝑓(𝑧≤|𝑧|)1−^2𝛼∇𝑓−𝑓_𝑁+1(𝑧+|𝑧|)1−^2𝛼∇𝑓_𝑁+1(𝑧)<𝜀, (107) for all 𝑟<|𝑧|<1; therefore, ℬ_𝛼,0 is a complete space.

Theorem 26. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). The spaces 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝐺,0(𝑝,𝑞,𝑠) are complete metric spaces.

Proof. We give the proof only in the case 1≤𝑝<∞. Let {𝑓_𝑘}⊂𝐹_𝐺(𝑝,𝑞,𝑠) be a Cauchy sequence. Since {𝑓_𝑘} is too a Cauchy sequence in ℬ_𝑞/𝑝, then converges to some 𝑓∈ℬ_𝑞/𝑝.
We will prove that 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠). Given 𝜀>0, there exists 𝑁∈ℕ, such that 𝐼̃_{𝑝,𝑞,𝑠}𝑓_𝑙−𝑓_𝑘(𝑎)≤𝑓_𝑙−𝑓_𝑘^𝑝_𝐺<𝜀^𝑝,∀𝑙>𝑘>𝑁. (108) By Fatou's Lemma, as 𝑙>𝑁, 𝐼̃_{𝑝,𝑞,𝑠}𝑓_𝑙𝑎)=−𝑓(_𝐵∇𝑓_𝑙−𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧)=_𝐵lim_𝑘→∞∇𝑓_𝑙−𝑓_𝑘(𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧)≤liminf_{𝑘→∞𝐵}∇𝑓_𝑙−𝑓_𝑘(𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧)≤lim_𝑘→∞𝑓_𝑙−𝑓_𝑘^𝑝_𝐺≤𝜀^𝑝. (109) Since {𝑓_𝑘(0)} converges to {𝑓(0)}, if we take supremum on 𝐵 and 𝑝 roots in both sides of the last inequality, we obtain lim_𝑘→∞𝑓_𝑘−𝑓_𝐺=0. (110) Since 𝑓=(𝑓−𝑓_𝑘)+𝑓_𝑘, we get that 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠).
If {𝑓_𝑘} is a Cauchy sequence in 𝐹_𝐺,0(𝑝,𝑞,𝑠), then {𝑓_𝑘} converges to 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠). Given 𝜀>0, there exists 𝑁∈ℕ, such that if 𝑘>𝑁, 𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎)≤2^𝑝𝐼̃_{𝑝,𝑞,𝑠}𝑓−𝑓_𝑘(𝑎)+2^𝑝𝐼̃_{𝑝,𝑞,𝑠}𝑓_𝑘(𝑎)≤2^𝑝𝜀^𝑝+2^𝑝𝐼̃_{𝑝,𝑞,𝑠}𝑓_𝑘(𝑎). (111) Thus, lim_{|𝑎|→1−}𝐼̃_{𝑝,𝑞,𝑠}𝑓(𝑎)≤lim_{|𝑎|→1−}2^𝑝𝜀^𝑝+lim_{|𝑎|→1−}2^𝑝𝐼̃_{𝑝,𝑞,𝑠}𝑓_𝑘(𝑎)=2^𝑝𝜀^𝑝; (112) therefore, 𝑓∈𝐹_𝐺,0(𝑝,𝑞,𝑠).
In the case 0<𝑝<1, the proof is similar.

As a consequence of the previous results and Proposition 17, we have the following.

Corollary 27. Suppose that 1≤𝑝<∞, 𝑞, and 𝑠 satisfy condition (A). The spaces 𝐹_𝐺(𝑝,𝑞,𝑠), 𝐹_𝐺,0(𝑝,𝑞,𝑠), ℬ_𝑞/𝑝, and ℬ_𝑞/𝑝,0 are Banach spaces, and the inclusion operators 𝐼∶𝐹_𝐺(𝑝,𝑞,𝑠)→ℬ_𝑞/𝑝 and 𝐼∶𝐹_𝐺,0(𝑝,𝑞,𝑠)→ℬ_𝑞/𝑝,0 are bounded.

Similar results are valid for the spaces 𝐹_𝜑(𝑝,𝑞,𝑠) and 𝐹_𝜑,0(𝑝,𝑞,𝑠).

3. The Equality 𝐹_𝐺(𝑝,𝑞,𝑠)=𝐹_𝜑(𝑝,𝑞,𝑠)

In this section, we prove the equality 𝐹_𝐺(𝑝,𝑞,𝑠)=𝐹_𝜑(𝑝,𝑞,𝑠).

Theorem 28. Let 0<𝑝<∞, 𝑞∈ℝ, 0<𝑠<𝑛/(𝑛−1), 1/2<𝑅<1, and 𝑓∈𝒪. Then, ∇𝑓(_𝐵1/2𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧)𝑑𝜆(𝑧)≤𝑡_𝐵𝑅∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑑𝜆(𝑧), (113) where 𝑡=𝑡(𝑞,𝑠,𝑛,𝑅).

Proof. Let 1/2<𝑅<1. Define 10<𝜏=𝑞,𝑠,𝑛,𝑅^𝑅_1/21−𝑟^{2𝑞+𝑛𝑠−𝑛−1}𝑟^2𝑛−1𝑑𝑟. (114) For 𝑢∈𝑆^2𝑛−1, as |∇𝑓|^𝑝 is subharmonic on 𝐵, we have 1𝜏𝑞,𝑠,𝑛,𝑅_𝑆^2𝑛−1𝑢∇𝑓2^𝑝=𝑑𝜎_𝑅1/21−𝑟^{2𝑞+𝑛𝑠−𝑛−1}𝑟^2𝑛−1⋅𝑑𝑟_{𝑆2𝑛−1}𝑢∇𝑓2^𝑝𝑑𝜎≤^𝑅_1/21−𝑟^{2𝑞+𝑛𝑠−𝑛−1}𝑟^2𝑛−1⋅𝑑𝑟_{𝑆2𝑛−1}∇𝑓(𝑟𝑢)^𝑝𝑑𝜎=_{𝐵𝑅⧵𝐵1/2}∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}(𝑧)≤𝑑𝜆_𝐵𝑅∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑑𝜆(𝑧). (115) Thus, _{𝑆2𝑛−1}𝑢∇𝑓2^𝑝𝑑𝜎≤𝜏𝑞,𝑠,𝑛,𝑅_𝐵𝑅∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑑𝜆(𝑧). (116) By Proposition 5, ₀^1/21−𝑟^{2𝑞−𝑛−1}𝑟^2𝑛−1𝑔^𝑠(𝑟𝑢)𝑑𝑟≤𝐶^𝑠₂₀^1/21−𝑟^{2𝑞−𝑛−1}𝑟^2𝑛−11−𝑟^2𝑛𝑠𝑟^{2(𝑛−1)𝑠}𝑑𝑟=𝐶^𝑠₂₀^1/21−𝑟^{2𝑞+𝑛𝑠−𝑛−1}𝑟^{2(𝑛−𝑛𝑠+𝑠)−1}𝑑𝑟. (117) This integral results convergent because 0<𝑠<𝑛/(𝑛−1). Denote the last integral by 𝜏^′(𝑞,𝑠,𝑛). Then, ∇𝑓(_𝐵1/2𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧)𝑑𝜆(𝑧)≤𝐶^𝑠₂₀^1/2_{𝑆2𝑛−1}𝑢∇𝑓2^𝑝×1−𝑟^{2𝑞+𝑛𝑠−𝑛−1}𝑟^{2(𝑛−𝑛𝑠+𝑠)−1}𝑑𝜎𝑑𝑟≤𝐶^𝑠₂𝜏^′𝜏×𝑞,𝑠,𝑛𝑞,𝑠,𝑛,𝑅_𝐵𝑅∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑑𝜆(𝑧), (118) and the result follows.

Theorem 29. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). If 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠), then ∇𝑓(_𝐵𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧)𝑑𝜆(𝑧)≤𝑡_𝐵∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑑𝜆(𝑧), (119) where 𝑡=𝑡(𝑝,𝑞,𝑠,𝑛,𝑅) for some 1/2<𝑅<1.

Proof. From Proposition 6 and (113), ∇𝑓(_𝐵1/2𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧)(𝑧)≤𝑑𝜆_𝐵1/2∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧)𝑑𝜆(𝑧)≤𝐶^𝑠₂𝜏𝜏×𝑞,𝑠,𝑛𝑞,𝑠,𝑛,𝑅_𝐵𝑅∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑝1−|𝑧|^2𝑝1−|𝑧|^{2𝑞+𝑛𝑠}(𝑧)≤𝐶𝑑𝜆_𝑠2𝜏𝑞,𝑠,𝑛𝜏𝑞,𝑠,𝑛,𝑅1−𝑅^2𝑝×_𝐵𝑅∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑑𝜆(𝑧). (120)
Now, if 1/2≤|𝑧|<1, ∇𝑓(_{𝐵⧵𝐵1/2}𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧)𝑑𝜆(𝑧)≤𝐶^𝑠_{2𝐵⧵𝐵1/2}∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}|𝑧|^{2(𝑛−1)𝑠}𝑑𝜆(𝑧)≤𝐶^𝑠₂2^{2(𝑛−1)𝑠}_𝐵∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑑𝜆(𝑧). (121)
Finally, ∇𝑓(_𝐵𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧)𝑑𝜆(𝑧)≤𝑡_𝐵∇𝑓(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑑𝜆(𝑧), (122) with 𝑡∶=𝑡(𝑝,𝑞,𝑠,𝑛,𝑅)∶=𝐶^𝑠₂(𝜏^′(𝑞,𝑠,𝑛) 𝜏(𝑞,𝑠,𝑛,𝑅)+2^{2(𝑛−1)𝑠})/(1−𝑅²)^𝑝.

By the Möbius invariance of the space 𝐹_𝐺(𝑝,𝑞,𝑠), if we apply the last result to 𝑓∘𝜑_𝑎, we obtain the following.

Corollary 30. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A) and 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠). Then, for 1/2<𝑅<1, _𝐵∇𝑓∘𝜑_𝑎(𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧)𝑑𝜆(𝑧)≤𝑡𝑝,𝑞,𝑠,𝑛,𝑅_𝐵∇𝑓∘𝜑_𝑎(𝑧)^𝑝1−|𝑧|^{2𝑞+𝑛𝑠}𝑑𝜆(𝑧). (123)

Theorem 31. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A) and 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠). Then, sup_{𝑎∈𝐵𝐵}∇𝑓(𝑤)^𝑝1−|𝑤|^2𝑞𝐺^𝑠(𝑤,𝑎)𝑑𝜆(𝑤)<∞, (124) if and only if sup_{𝑎∈𝐵𝐵}∇𝑓(𝑤)^𝑝1−|𝑤|^2𝑞1−𝜑_𝑎(𝑤)^2𝑛𝑠𝑑𝜆(𝑤)<∞. (125)

Proof. We will abbreviate 𝑡=𝑡(𝑝,𝑞,𝑠,𝑛,𝑅), then from the last corollary and (20) 0≤_𝐵∇𝑓∘𝜑_𝑎(𝑧)^𝑝1−|𝑧|^2𝑞×𝑡1−|𝑧|^2𝑛𝑠−𝑔^𝑠(𝑧(𝑧)=)𝑑𝜆_𝐵∇𝑓𝜑_𝑎(𝑧)^𝑝1−|𝑧|^2𝑞×𝑡1−|𝑧|^2𝑛𝑠−𝑔^𝑠(𝑧)𝑑𝜆(𝑧); (126) using the change of variables 𝑧=𝜑_𝑎(𝑤), we obtain 0≤_𝐵∇𝑓(𝑤)^𝑝1−𝜑_𝑎(𝑤)^2𝑞×𝑡1−𝜑_𝑎(𝑤)^2𝑛𝑠−𝐺^𝑠(𝑤,𝑎)𝑑𝜆(𝑤). (127) Then, 0≤_𝐵∇𝑓(𝑤)^𝑝1−|𝑎|^2𝑞1−|𝑤|^2𝑞|1−⟨𝑤,𝑎⟩|^2𝑞×𝑡1−𝜑_𝑎(𝑤)^2𝑛𝑠−𝐺^𝑠((𝑤)≤(|𝑎𝑤,𝑎)𝑑𝜆1+|)^𝑞(1−|𝑎|)^𝑞_𝐵∇𝑓(𝑤)^𝑝1−|𝑤|^2𝑞×𝑡1−𝜑_𝑎(𝑤)^2𝑛𝑠−𝐺^𝑠(𝑤,𝑎)𝑑𝜆(𝑤), (128) and from here we obtain the result.

Corollary 32. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). Then, 𝐹_𝐺𝑝,𝑞,𝑠=𝐹_𝜑𝑝,𝑞,𝑠,𝐹_𝐺,0𝑝,𝑞,𝑠=𝐹_𝜑,0𝑝,𝑞,𝑠. (129)

4. Harmonic Majorants in 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝜑(𝑝,𝑞,𝑠)

For 𝑧∈𝐵 and 𝑢∈𝑆^2𝑛−1, the function 𝑃 defined by 𝑃()=|𝑧|𝑧,𝑢1−²|𝑧−𝑢|^2𝑛 (130) is the real Poisson kernel for the ball 𝐵⊂ℝ^2𝑛. If ℎ(𝑢) is a real function, integrable on 𝑆^2𝑛−1, we define the Poisson integral formula of ℎ as 𝐻(𝑧)=1𝜎𝑆^{2𝑛−1𝑆2𝑛−1}ℎ(𝑢)𝑃()(𝑢)=1𝑧,𝑢𝑑𝜎𝜎𝑆^{2𝑛−1𝑆2𝑛−1}ℎ(𝑢)1−|𝑧|²|𝑧−𝑢|^2𝑛𝑑𝜎(𝑢), (131) where 𝜎(𝑆^2𝑛−1) is the area measure of 𝑆^2𝑛−1, for 𝑛≥2. This formula gives the classic solution to the Dirichlet problem for ℎ. Moreover, if 0≤𝐻<∞, then 𝐻 is harmonic on 𝐵 and has nontangential limit ℎ(𝑢) a. e. on 𝑆^2𝑛−1 (see [12]).

Theorem 33 ((Harnack) see [13]). Consider 𝐵. Let {𝐻_𝑚} be a monotone sequence of real harmonic functions on 𝐵. Suppose that {𝐻_𝑚(𝑥₀)} is bounded for some 𝑥₀∈𝐵. Then, the sequence {𝐻_𝑚} converges uniformly on compact sets to a harmonic function on 𝐵.

Proposition 34. Let ℎ∶𝐵→ℝ⁺∪{0} be a continuous function on 𝐵. For 0<𝑅<1, let the function ℎ_𝑅∶𝑆^2𝑛−1→ℝ be defined by ℎ_𝑅(𝑢)=max_{0≤𝑟≤𝑅}ℎ(𝑟𝑢). (132) Then, ℎ_𝑅 results in a uniformly continuous function on 𝑆^2𝑛−1. Besides, if 0<𝑅₁<𝑅₂<1, then ℎ_𝑅1(𝑢)≤ℎ_𝑅2(𝑢), (133) for every 𝑢∈𝑆^2𝑛−1.

Proof. The function  ℎ∣_𝐵𝑟is uniformly continuous; that is, given that 𝜀>0, there exists 𝛿>0, such that |ℎ(𝑧)−ℎ(𝑧^′)|<𝜀, for all 𝑧,𝑧^′∈𝐵_𝑅, with |𝑧−𝑧^′|<𝛿. Let 𝑢,𝑣∈𝑆^2𝑛−1. There exist 𝑟_𝑢,𝑟_𝑣∈[0,𝑅], such that ℎ_𝑅(𝑢)=ℎ(𝑟_𝑢) and ℎ_𝑅(𝑣)=ℎ(𝑟_𝑣). Thus, if |𝑢−𝑣|<𝛿, then |ℎ(𝑟_𝑢𝑢)−ℎ(𝑟_𝑢𝑣)|<𝜀 and |ℎ(𝑟_𝑣𝑣)−ℎ(𝑟_𝑣𝑢)|<𝜀. In particular, ℎ_𝑅(𝑢)<𝜀+ℎ(𝑟_𝑢𝑣)≤𝜀+ℎ_𝑅(𝑣) and ℎ_𝑅(𝑣)<𝜀+ℎ(𝑟_𝑣𝑢)≤𝜀+ℎ_𝑅(𝑢). Therefore, if |𝑢−𝑣|<𝛿, then |ℎ_𝑅(𝑢)−ℎ_𝑅(𝑣)|<𝜀.
It follows directly that if 0<𝑅₁<𝑅₂, then max_{0≤𝑟≤𝑅1}ℎ(𝑟𝑢)≤max_{0≤𝑟≤𝑅2}ℎ(𝑟𝑢), (134) so the result follows.

Observe that Poisson's integral formula defines a continuous harmonic function 𝐻_𝑅 on 𝐵, with boundary values given by ℎ_𝑅.

Corollary 35. If ℎ is as in the previous proposition and for every 𝑢∈𝑆^2𝑛−1, sup_0<𝑅<1ℎ_𝑅(𝑢)<∞, define ℎ₁∶𝑆^2𝑛−1→ℝ as ℎ₁(𝑢)=sup_0<𝑅<1ℎ_𝑅(𝑢). (135) Then, one has besides that ℎ₁(𝑢)=lim_𝑅→1−ℎ_𝑅(𝑢). (136)

Theorem 36. If ℎ∶𝐵→[0,∞) is a continuous function and ℎ₁∶𝑆^2𝑛−1→ℝ is the function defined in the previous corollary, let 𝐻₁∶𝐵→ℝ be defined by 𝐻₁(𝑧)=sup_0≤𝑅<1𝐻_𝑅(𝑧), (137) for every 𝑧∈𝐵. Then, 𝐻₁ results in a harmonic function, or 𝐻₁(𝑧)≡∞ on 𝐵. In the first case, 𝐻₁ can be obtained through Poisson's integral formula applied to ℎ₁. Besides, lim_𝑅→1−𝐻_𝑅(𝑢)=ℎ₁(𝑢). (138) If ℎ₁(𝑢)≥0, then the nontangential limits lim_𝑧→𝑢𝐻₁(𝑧)=ℎ₁(𝑢) (139) exist almost everywhere on 𝑆^2𝑛−1.

Proof. Let {𝑅_𝑚} be an increasing sequence of positive numbers in (0,1), such that lim_𝑚→∞𝑅_𝑚=1. For every 𝑚∈ℕ, we define the harmonic continuous function 𝐻_𝑅𝑚(𝑧)=1𝜎𝑆^{2𝑛−1𝑆2𝑛−1}ℎ_𝑅𝑚(𝑢)1−|𝑧|²|𝑧−𝑢|^2𝑛𝑑𝜎(𝑢). (140) Then, if 𝑅₁≤𝑅₂, it follows that 𝐻_𝑅1(𝑧)≤𝐻_𝑅2(𝑧). (141)
Then, the sequence {𝐻_𝑅𝑚} is increasing, so lim_𝑚→∞𝐻_𝑅𝑚(𝑧)=𝐻₁(𝑧), (142) for every 𝑧∈𝐵.
By Harnack's theorem, 𝐻₁≡∞ or 𝐻₁ is a harmonic function on 𝐵, and the limit does not depend on the sequence. As 𝐻_𝑅(𝑢)=ℎ_𝑅(𝑢)≤ℎ₁(𝑢), (143) it follows that 𝐻_𝑅(𝑢)→ℎ₁(𝑢) if 𝑅→1⁻ and lim_𝑧→𝑢𝐻₁(𝑢) there exists almost everywhere.
For every 𝑧∈𝐵 fixed, let 1𝜎𝑆^2𝑛−1ℎ_𝑅𝑚(𝑢)1−|𝑧|²|𝑧−𝑢|^2𝑛, (144) this is a sequence of positive Lebesgue’s integrable functions, such that for 𝑅_𝑚1<𝑅_𝑚2, 10≤𝜎𝑆^2𝑛−1ℎ_𝑅𝑚1(𝑢)1−|𝑧|²|𝑧−𝑢|^2𝑛≤1𝜎𝑆^2𝑛−1ℎ_𝑅𝑚2(𝑢)1−|𝑧|²|𝑧−𝑢|^2𝑛. (145)
Then, by Lebesgue's monotone convergence theorem, 10≤𝜎𝑆^{2𝑛−1𝑆2𝑛−1}ℎ_𝑅𝑚1(𝑢)1−|𝑧|²|𝑧−𝑢|^2𝑛(𝑢)⟶1𝑑𝜎𝜎𝑆^{2𝑛−1𝑆2𝑛−1}ℎ₁(𝑢)1−|𝑧|²|𝑧−𝑢|^2𝑛𝑑𝜎(𝑢), (146) as 𝑚₁→∞.

Let Ω⊂ℂ^𝑛 be an open set and let 𝐺≢−∞. We say that a harmonic function 𝐻∶Ω→ℝ is the least harmonic majorant of 𝐺 if(a)𝐺(𝑧)≤𝐻(𝑧), for every 𝑧∈Ω;(b)𝐼 is any harmonic function on Ω, such that 𝐺(𝑧)≤𝐼(𝑧), for every 𝑧∈Ω, then 𝐻(𝑧)≤𝐼(𝑧).

The least harmonic majorant of a continuous function, bounded and radially increasing is not necessarily the Poisson extension of its boundary values; see, for instance, Example 2.2 of [14].

In a similar way as [6, 14], consider a function 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠). For 0<𝑅<1, let us introduce the family of functions _𝐺ℎ_𝑅∶𝑆^2𝑛−1→ℝ defined by _𝐺ℎ_𝑅(𝑢)∶=max_{𝐺0≤𝑟≤𝑅}ℎ(𝑟𝑢)=max_{𝑎=𝑟𝑢,0≤𝑟≤𝑅𝐵}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧). (147) By Theorem 12 and Proposition 34, this function is well defined for every 0<𝑅<1 and results in a continuous function on 𝑆^2𝑛−1. Moreover, as 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠), for 0<𝑅₁<𝑅₂<1, _𝐺ℎ_𝑅1(𝑢)≤_𝐺ℎ_𝑅2(𝑢)≤_𝐺ℎ₁(𝑢)≤sup_{𝑢∈𝑆2𝑛−1𝐺}ℎ₁(𝑢)=𝑓_{𝑝,𝑞,𝑠}. (148)

Let _𝐺𝐻_𝑅 be the harmonic function associated with _𝐺ℎ_𝑅, that is, the solution of Dirichlet's problem, with boundary values given by _𝐺ℎ_𝑅. It follows from Proposition 34 that sup_{𝐺𝑧∈𝐵}𝐻_𝑅(𝑧)≤sup_{𝐺𝑧∈𝐵}𝐻₁(𝑧)=𝑓_{𝑝,𝑞,𝑠}. (149)

In the same way as we did for _𝐺ℎ_𝑅, we introduce the family of functions _𝜑ℎ_𝑅∶𝑆^2𝑛−1→ℝ defined by _𝜑ℎ_𝑅(𝑢)∶=max_{𝜑0≤𝑟≤𝑅}ℎ(𝑟𝑢)=max_{𝑎=𝑟𝑢,0≤𝑟≤𝑅𝐵}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞×1−|𝜑_𝑎(𝑧)^2𝑛𝑠𝑑𝜆(𝑧). (150) In this case we obtain Theorem 13 and Proposition 34.

Consider sup_{𝜑𝑧∈𝐵}𝐻_𝑅(𝑧)≤sup_{𝜑𝑧∈𝐵}𝐻₁(𝑧)=sup_{𝑢∈𝑆2𝑛−1𝜑}ℎ₁(𝑢)=𝑓_{𝑝,𝑞,𝑠}. (151)

So, we can summarize the previous results in the next theorems.

Theorem 37. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A). Then, 𝑓 belongs to the class 𝐹_𝐺(𝑝,𝑞,𝑠) if and only if the function _𝐺ℎ₁∶𝑆^2𝑛−1→ℝ defined by _𝐺ℎ₁(𝑢)=sup_{𝑎=𝑟𝑢,0≤𝑟<1𝐵}∇𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝐺^𝑠(𝑧,𝑎)𝑑𝜆(𝑧) (152) is bounded. The function _𝐺𝐻₁ is then the Dirichlet solution, with boundary values given by _𝐺ℎ₁. Further, _𝐺𝐻₁ is the least harmonic majorant of the family of harmonic functions {_𝐺𝐻_𝑅}, 0<𝑅<1.

Analogous result is obtained for {_𝜑𝐻_𝑅} and _𝜑𝐻₁.

Now, we can formulate an analogous statement to the Theorem 3.1 in [14] to characterize functions of 𝐹_𝐺(𝑝,𝑞,𝑠) and 𝐹_𝜑(𝑝,𝑞,𝑠) through their harmonic majorant, as in Theorem 3.9 of [6].

Theorem 38. Suppose that 𝑝, 𝑞, and 𝑠 satisfy condition (A) and 𝑓∈𝐹_𝐺(𝑝,𝑞,𝑠). Then, the following conditions are equivalent:(i)𝑓∈𝐹_𝜑(𝑝,𝑞,𝑠);(ii)_𝐺ℎ₁ is bounded and _𝐺𝐻₁ is the classical Dirichlet solution with boundary values given by _𝐺ℎ₁;(iii)_𝜑ℎ₁ is bounded and _𝜑𝐻₁ is the classical Dirichlet solution with boundary values given by _𝜑ℎ₁.

With this development, we have obtained a new version of Theorem 1.3 in [6], in terms of harmonic majorants _𝐺𝐻₁ and _𝜑𝐻₁ and their corresponding boundary values _𝐺ℎ₁ and _𝜑ℎ₁, for the invariant measure.