# F

Authors: Augusto Guadalupe Miss Paredes; Lino Feliciano Reséndis Ocampo; Luis Manuel Tovar Sánchez
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We introduce theF~G(p,q,s) and F~φ(p,q,s) vector spaces of holomorphic functions defined in the unit ball of ℂn, 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).


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## Body

## 1. Introduction

### 1.1. Preliminaries in One Complex Variable

Let𝔻 be the open unit disk in the complex plane ℂ. For a∈ℂ, let φa:ℂ∖{1/a¯}→ℂ be the Möbius transformation defined by
(1)φa(z)=a-z1-a¯z.
For z,a∈𝔻, we denote Green's function of 𝔻 with logarithmic singularity at a by
(2)g(z,a)=log1|φa(z)|=log|1-a¯z||a-z|.
The Bloch space ℬ is defined as the set of analytic functions f:𝔻→ℂ, such that
(3)supz∈𝔻(1-|z|2)|f′(z)|<+∞.For1≤p<∞, Aulaskari and Lappan [1] introduced in 1994 the 𝒬p spaces as the family of analytic functions f:𝔻→ℂ satisfying
(4)supa∈𝔻∫𝔻‍|f′(z)|2gp(z,a)dA(z)<+∞.
For 0<p<1, this definition was extended in [2].Theorem 1 (see [2]).
Letp≥0 and f:𝔻→ℂ be an analytic function. Then, f∈𝒬p if and only if
(5)supa∈𝔻∫𝔻‍|f′(z)|2(1-|φa(z)|2)pdA(z)<+∞.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.Let0<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
(6){f}p,q,s:=supa∈𝔻∫𝔻‍|f′(z)|p(1-|z|2)qgs(z,a)dA(z)<+∞.
So far, Theorem 1 is generalized for functions of F(p,q,s) by the following.Theorem 2 (see [3]).
Let0<p<∞, -2<q<∞, 0≤s<∞, and f:𝔻→ℂ be an analytic function. Then, f∈F(p,q,s) if and only if
(7)supa∈𝔻∫𝔻‍|f′(z)|p(1-|z|2)q(1-|φa(z)|2)sdA(z)<+∞.Zhao has shown that for certain intervals ofp, q, and s, |f(0)|+{f}p,q,s makes F(p,q,s) as a Banach space.In this paper, we present theF~(p,q,s) spaces of holomorphic functions in the unit ball of ℂn, 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 𝒬p-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 Section2.2, 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 spaceℂn={z=(z1,z2,…,zn):zj∈ℂ,j=1,2,…,n} with the Hermitian product defined as
(8)〈z,w〉=∑‍j=1nzjwj¯,
and the Euclidean norm on ℂn defined as
(9)|z|=〈z,z〉1/2=(∑‍j=1n|zj|2)1/2.The open ball with centera∈ℂn and radius r>0 denoted by Br(a) is defined as
(10)Br(a)={z∈ℂn:|z-a|<r},
and its closure is
(11)Br(a)¯={z∈ℂn:|z-a|≤r}.
We will denote B1(0)=B, Br(0)=Br, and A(r)=B∖Br.The functionf, of class 𝒞2(Ω), is harmonic in Ω if
(12)Δf(x)=0,
for all x∈Ω. Moreover, f results harmonic in Ω if and only if
(13)∑‍j=1n∂2f∂zj∂z¯j=0.Leta∈B∖{0}. The Möbius function φa(z) is defined as
(14)φa(z)=a-Pa(z)-(1-|a|2)1/2Qa(z)1-〈z,a〉,
with Pa(z)=〈z,a〉a/|a|2 and Qa(z)=z-Pa(z), for z∈B, and define φ0(z)=-z (see [7]).This set of functions is a part of the family of automorphisms onB denoted by Aut(B).Lemma 3 (see [7, 8]).
The functionφa satisfies the following properties:(i)
φa(φa(z))=z;(ii)
1-|φa(z)|2=(1-|a|2)(1-|z|2)/|1-〈z,a〉|2;(iii)
the real Jacobian forφa in z∈B is given by
(15)Jℝφa(z)=(1-|a|2|1-〈z,a〉|2)n+1=(1-|φa(z)|21-|z|2)n+1.Lemma 4.
The real Jacobian ofφa satisfies
(16)(1-|a|1+|a|)n+1≤|Jℝφa(z)|≤(1+|a|1-|a|)n+1.Proof.
It follows directly from1-|a|≤|1-〈z,a〉|≤1+|a|.Define, fora∈B and 0<R<1,
(17)D(a,R)=φa(B(R))={z∈B:|φa(z)|<R}.


### 1.3. Invariant Gradient and Measure

The collection of all holomorphic functions inB is denoted by 𝒪. Let ∇f=(∂f/∂z1,…,∂f/∂zn) denote the complex gradient of f and Rf=∑j=1n‍zj(∂f/∂zj) denote the radial derivative of f. Let dλ(z):=dν(z)/(1-|z|2)n+1, where dν is the normalized volume measure in ℂn. Then, dλ is a Möbius invariant; that is
(18)∫B‍f(z)dλ(z)=∫B‍f∘ψ(z)dλ(z),
for each λ-integrable function f and ψ∈Aut(B).LetΔ~f(z)=Δ(f∘φz)(0) denote the invariant Laplacian of f [8], and let ∇~f(z)=∇(f∘φz)(0) denote the invariant gradient of f [7]. It is well known [7] that, for f∈𝒪,
(19)|∇~f(z)|2=(1-|z|2)(|∇f(z)|2-|Rf(z)|2),(20)|∇~(f∘φ)(z)|2=|(∇~f)∘φ(z)|2,
where φ∈Aut(B). In [7], the invariant Green's function is defined as G(z,a)=g(φa(z)), where
(21)g(z)=n+12n∫|z|1‍(1-t2)n-1t-2n+1dt.
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
(22)|f|ℬ=supz∈B|∇~f(z)|<+∞;
this is equivalent [4, Lemma 3.2] to, say,
(23)supz∈B(1-|z|2)|∇f(z)|<+∞.Proposition 5 (see [7]).
Letn≥2 be an integer; then there are positive constants C1 and C2, such that, for all z∈B∖{0},
(24)C1(1-|z|2)n|z|-2(n-1)≤g(z)≤C2(1-|z|2)n|z|-2(n-1).Proposition 6 (see [9]).
Forf∈𝒪, one has
(25)(1-|z|2)2|∇f(z)|2≤|∇~f(z)|2≤(1-|z|2)|∇f(z)|2≤|∇f(z)|2.Theorem 7 (see [7, Theorem 1.12]).
Suppose thatc is real and t>-1. Then, the integral
(26)Jc,t(z)=∫B‍(1-|w|2)tdv(w)|1-〈z,ζ〉|n+1+t+c,z∈B
has the following asymptotic properties.(i)
Ifc<0, Jc,t is bounded in B.(ii)
Ifc=0, then
(27)Jc,t(z)~ln11-|z|2,
as |z|→1-.(iii)
Ifc>0, then
(28)Jc,t(z)~1(1-|z|2)c,
as |z|→1-.Let0<p<∞,  -(p/2)<q. We say that p, q, and s satisfy condition (A) if
(29)1-p+2q2n<s<nn-1if0<p+2q2n<1,0<s<nn-1if1≤p+2q2n<∞;
and satisfy condition (B) if
(30)1-p+2q2n<s<∞if0<p+2q2n<1,0<s<∞if1≤p+2q2n<∞.Proposition 8.
Suppose thatp, q, and s satisfy condition (A). Then,
(31)sup|a|<1∫B‍(1-|z|2)(p/2)+q(|1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)<∞,lim|a|→1-∫B‍(1-|z|2)(p/2)+q(|1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)=0.Proof.
We have the following equalities:(32)∫B‍(1-|z|2)(p/2)+q(|1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)=∫B‍(1-|φa(w)|2)(p/2)+q(1-|w|2)ns|w|2(n-1)sdλ(w)=∫B‍(1-|a|2)(p/2)+q(1-|w|2)(p/2)+q+ns-n-1|1-〈w,a〉|p+2q|w|2(n-1)sdν(w)=(1-|a|2)(p/2)+q∫B‍(1-|w|2)(p/2)+q+ns-n-1|1-〈w,a〉|p+2q|w|2(n-1)sdν(w).
The singularity at 0 is integrable if s<n/(n-1). Now, if we estimate
(33)∫B‍(1-|w|2)(p/2)+q+ns-n-1|1-〈w,a〉|p+2qdν(w),
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 thatp, q, and s satisfy condition (B). Then,
(34)sup|a|<1∫B‍(1-|z|2)(p/2)+q(1-|φa(z)|2)nsdλ(z)<∞,lim|a|→1-∫B‍(1-|z|2)(p/2)+q(1-|φa(z)|2)nsdλ(z)=0.


## 1.1. Preliminaries in One Complex Variable

Let𝔻 be the open unit disk in the complex plane ℂ. For a∈ℂ, let φa:ℂ∖{1/a¯}→ℂ be the Möbius transformation defined by
(1)φa(z)=a-z1-a¯z.
For z,a∈𝔻, we denote Green's function of 𝔻 with logarithmic singularity at a by
(2)g(z,a)=log1|φa(z)|=log|1-a¯z||a-z|.
The Bloch space ℬ is defined as the set of analytic functions f:𝔻→ℂ, such that
(3)supz∈𝔻(1-|z|2)|f′(z)|<+∞.For1≤p<∞, Aulaskari and Lappan [1] introduced in 1994 the 𝒬p spaces as the family of analytic functions f:𝔻→ℂ satisfying
(4)supa∈𝔻∫𝔻‍|f′(z)|2gp(z,a)dA(z)<+∞.
For 0<p<1, this definition was extended in [2].Theorem 1 (see [2]).
Letp≥0 and f:𝔻→ℂ be an analytic function. Then, f∈𝒬p if and only if
(5)supa∈𝔻∫𝔻‍|f′(z)|2(1-|φa(z)|2)pdA(z)<+∞.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.Let0<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
(6){f}p,q,s:=supa∈𝔻∫𝔻‍|f′(z)|p(1-|z|2)qgs(z,a)dA(z)<+∞.
So far, Theorem 1 is generalized for functions of F(p,q,s) by the following.Theorem 2 (see [3]).
Let0<p<∞, -2<q<∞, 0≤s<∞, and f:𝔻→ℂ be an analytic function. Then, f∈F(p,q,s) if and only if
(7)supa∈𝔻∫𝔻‍|f′(z)|p(1-|z|2)q(1-|φa(z)|2)sdA(z)<+∞.Zhao has shown that for certain intervals ofp, q, and s, |f(0)|+{f}p,q,s makes F(p,q,s) as a Banach space.In this paper, we present theF~(p,q,s) spaces of holomorphic functions in the unit ball of ℂn, 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 𝒬p-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 Section2.2, 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 spaceℂn={z=(z1,z2,…,zn):zj∈ℂ,j=1,2,…,n} with the Hermitian product defined as
(8)〈z,w〉=∑‍j=1nzjwj¯,
and the Euclidean norm on ℂn defined as
(9)|z|=〈z,z〉1/2=(∑‍j=1n|zj|2)1/2.The open ball with centera∈ℂn and radius r>0 denoted by Br(a) is defined as
(10)Br(a)={z∈ℂn:|z-a|<r},
and its closure is
(11)Br(a)¯={z∈ℂn:|z-a|≤r}.
We will denote B1(0)=B, Br(0)=Br, and A(r)=B∖Br.The functionf, of class 𝒞2(Ω), is harmonic in Ω if
(12)Δf(x)=0,
for all x∈Ω. Moreover, f results harmonic in Ω if and only if
(13)∑‍j=1n∂2f∂zj∂z¯j=0.Leta∈B∖{0}. The Möbius function φa(z) is defined as
(14)φa(z)=a-Pa(z)-(1-|a|2)1/2Qa(z)1-〈z,a〉,
with Pa(z)=〈z,a〉a/|a|2 and Qa(z)=z-Pa(z), for z∈B, and define φ0(z)=-z (see [7]).This set of functions is a part of the family of automorphisms onB denoted by Aut(B).Lemma 3 (see [7, 8]).
The functionφa satisfies the following properties:(i)
φa(φa(z))=z;(ii)
1-|φa(z)|2=(1-|a|2)(1-|z|2)/|1-〈z,a〉|2;(iii)
the real Jacobian forφa in z∈B is given by
(15)Jℝφa(z)=(1-|a|2|1-〈z,a〉|2)n+1=(1-|φa(z)|21-|z|2)n+1.Lemma 4.
The real Jacobian ofφa satisfies
(16)(1-|a|1+|a|)n+1≤|Jℝφa(z)|≤(1+|a|1-|a|)n+1.Proof.
It follows directly from1-|a|≤|1-〈z,a〉|≤1+|a|.Define, fora∈B and 0<R<1,
(17)D(a,R)=φa(B(R))={z∈B:|φa(z)|<R}.


## 1.3. Invariant Gradient and Measure

The collection of all holomorphic functions inB is denoted by 𝒪. Let ∇f=(∂f/∂z1,…,∂f/∂zn) denote the complex gradient of f and Rf=∑j=1n‍zj(∂f/∂zj) denote the radial derivative of f. Let dλ(z):=dν(z)/(1-|z|2)n+1, where dν is the normalized volume measure in ℂn. Then, dλ is a Möbius invariant; that is
(18)∫B‍f(z)dλ(z)=∫B‍f∘ψ(z)dλ(z),
for each λ-integrable function f and ψ∈Aut(B).LetΔ~f(z)=Δ(f∘φz)(0) denote the invariant Laplacian of f [8], and let ∇~f(z)=∇(f∘φz)(0) denote the invariant gradient of f [7]. It is well known [7] that, for f∈𝒪,
(19)|∇~f(z)|2=(1-|z|2)(|∇f(z)|2-|Rf(z)|2),(20)|∇~(f∘φ)(z)|2=|(∇~f)∘φ(z)|2,
where φ∈Aut(B). In [7], the invariant Green's function is defined as G(z,a)=g(φa(z)), where
(21)g(z)=n+12n∫|z|1‍(1-t2)n-1t-2n+1dt.
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
(22)|f|ℬ=supz∈B|∇~f(z)|<+∞;
this is equivalent [4, Lemma 3.2] to, say,
(23)supz∈B(1-|z|2)|∇f(z)|<+∞.Proposition 5 (see [7]).
Letn≥2 be an integer; then there are positive constants C1 and C2, such that, for all z∈B∖{0},
(24)C1(1-|z|2)n|z|-2(n-1)≤g(z)≤C2(1-|z|2)n|z|-2(n-1).Proposition 6 (see [9]).
Forf∈𝒪, one has
(25)(1-|z|2)2|∇f(z)|2≤|∇~f(z)|2≤(1-|z|2)|∇f(z)|2≤|∇f(z)|2.Theorem 7 (see [7, Theorem 1.12]).
Suppose thatc is real and t>-1. Then, the integral
(26)Jc,t(z)=∫B‍(1-|w|2)tdv(w)|1-〈z,ζ〉|n+1+t+c,z∈B
has the following asymptotic properties.(i)
Ifc<0, Jc,t is bounded in B.(ii)
Ifc=0, then
(27)Jc,t(z)~ln11-|z|2,
as |z|→1-.(iii)
Ifc>0, then
(28)Jc,t(z)~1(1-|z|2)c,
as |z|→1-.Let0<p<∞,  -(p/2)<q. We say that p, q, and s satisfy condition (A) if
(29)1-p+2q2n<s<nn-1if0<p+2q2n<1,0<s<nn-1if1≤p+2q2n<∞;
and satisfy condition (B) if
(30)1-p+2q2n<s<∞if0<p+2q2n<1,0<s<∞if1≤p+2q2n<∞.Proposition 8.
Suppose thatp, q, and s satisfy condition (A). Then,
(31)sup|a|<1∫B‍(1-|z|2)(p/2)+q(|1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)<∞,lim|a|→1-∫B‍(1-|z|2)(p/2)+q(|1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)=0.Proof.
We have the following equalities:(32)∫B‍(1-|z|2)(p/2)+q(|1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)=∫B‍(1-|φa(w)|2)(p/2)+q(1-|w|2)ns|w|2(n-1)sdλ(w)=∫B‍(1-|a|2)(p/2)+q(1-|w|2)(p/2)+q+ns-n-1|1-〈w,a〉|p+2q|w|2(n-1)sdν(w)=(1-|a|2)(p/2)+q∫B‍(1-|w|2)(p/2)+q+ns-n-1|1-〈w,a〉|p+2q|w|2(n-1)sdν(w).
The singularity at 0 is integrable if s<n/(n-1). Now, if we estimate
(33)∫B‍(1-|w|2)(p/2)+q+ns-n-1|1-〈w,a〉|p+2qdν(w),
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 thatp, q, and s satisfy condition (B). Then,
(34)sup|a|<1∫B‍(1-|z|2)(p/2)+q(1-|φa(z)|2)nsdλ(z)<∞,lim|a|→1-∫B‍(1-|z|2)(p/2)+q(1-|φa(z)|2)nsdλ(z)=0.


## 2. Function SpacesF~G(p,q,s) and F~φ(p,q,s)

### 2.1. Main Definitions

In this section, we define the spacesF~G(p,q,s), F~φ(p,q,s) and their little associated spaces. We study some properties and relations between them.Suppose thatp,q, and s satisfy condition (A), and let f∈𝒪. Define
(35)I~p,q,sf(a)=∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z).
Suppose that p, q, and s satisfy condition (B), and let f∈𝒪. Define
(36)J~p,q,sf(a)=∫B‍|∇~f(z)|p(1-|z|2)q(1-|φa(z)|2)nsdλ(z).
It is clear that
(37)J~p,q,sf(a)≤I~p,q,sf(a),foreacha∈B.
Associated to these functions, we define the following vector spaces of holomorphic functions.  (i)
Suppose thatp, q, and s satisfy condition (A). f∈𝒪 belongs to F~G(p,q,s) if it satisfies
(38)supa∈BI~p,q,sf(a)<+∞
and belongs to the little space F~G,0(p,q,s) if it satisfies
(39)lim|a|→1-I~p,q,sf(a)=0.(ii)
Suppose thatp, q, and s satisfy condition (B). f∈𝒪 belongs to F~φ(p,q,s) if it satisfies
(40)supa∈BJ~p,q,sf(a)<+∞
and belongs to the little space F~φ,0(p,q,s) if it satisfies
(41)lim|a|→1-J~p,q,sf(a)=0.(iii)
Letn<(p+2q)/2. f∈𝒪 belongs to the Dirichlet space D~p,q if
(42)∫B‍|∇~f(z)|p(1-|z|2)qdλ(z)<∞.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 by(43)∥f∥G≔|f(0)|+supa∈B(I~p,q,sf(a))1/p,∥f∥φ:=|f(0)|+supa∈B(J~p,q,sf(a))1/p.
This is a direct consequence of the usual Lp(B,dμ) theory when we consider the measures dμ(z)=(1-|z|2)qGs(z,a)dλ(z),dμ(z)=(1-|z|2)q(1-|φa(z)|2)nsdλ(z), respectively. For 0<p<1 and nonnegative x, y, we have (x+y)p≤xp+yp; 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
(44)dG(f,h)=|f(0)-h(0)|+supa∈BI~p,q,s(f-h)(a),dφ(f,h)=|f(0)-h(0)|+supa∈BJ~p,q,s(f-h)(a),
respectively (see [10]). We will prove their completeness in Theorem 26.Example 10.
Let𝒪~ 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
(45)∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)≤maxz∈B¯|∇f(z)|p∫B‍(1-|z|2)(p/2)+qGs(z,a)dλ(z).
So, in the definitions, the parameters p, q, and s were chosen to guarantee that the vector spaces FG(p,q,s) and FG,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].
Letf:B→ℂ, with power series development given by
(46)f(z)=∑α‍aαzα,
such that ∑n=0∞‍n∑|α|=n‍|aα|<∞. Then, f belongs to each one of the previous spaces. By (37), we get the same result for Fφ(p,q,s) and Fφ,0(p,q,s).The spacesF~G(p,q,s) are Möbius invariant in the following sense.Theorem 11.
Suppose thatp, 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.
Leta,b∈B,w=φa(z), and c=φa(b). By the change of variables formula and the Möbius invariance of G(z,a), ∇,~ and dλ, we have
(47)∫B‍|∇~(f∘φa)(w)|p(1-|w|2)qGs(w,b)dλ(w)=∫B‍|(∇~f)∘φa(w)|p(1-|w|2)qGs(w,b)dλ(w)=∫B‍|(∇~f)∘φa(φa(z))|p(1-|φa(z)|2)qhhhhhhhh×Gs(φa(z),φa(c))dλ(z)=∫B‍|∇~f(z)|p(1-|φa(z)|2)qGs(z,c)dλ(z)≤(1+|a|1-|a|)|q|∫B‍|∇~f(z)|p(1-|z|2)qGs(z,c)dλ(z),
where we have used in the last line the estimations of Lemmas 3 and 4.Observe that ifq=0 and f∈F~G(p,0,s), then
(48)∫B‍|∇~(f∘φa)(w)|pGs(w,b)dλ(w)=∫B‍|∇~f(z)|pGs(z,c)dλ(z),
and these spaces are Möbius invariant in the classical way.Theorem 12.
Suppose thatp, q, and s satisfy condition (A). Suppose that
(49)I~p,q,sf(a)<∞,
for all a∈B. Then, the function I~p,q,sf is continuous on B.Proof.
Leta∈B be fixed and R, such that |a|<R<1. Consider a neighborhood Bδ(a) with δ=(R-|a|)/2. Let {am}m∈ℕ be a sequence contained in Bδ(a) and convergent to a. Consider now I~p,q,sf(am) and define dμ(z)=|∇~f(z)|p(1-|z|2)q+nsdλ(z). Then, Gs(z,am)(1-|z|2)-ns is bounded in (B∖BR)×Bδ(a). Thus, by bounded convergence theorem,
(50)∫B∖BR‍Gs(z,am)(1-|z|2)nsdμ(z)
converges to
(51)∫B∖BR‍Gs(z,a)(1-|z|2)nsdμ(z).
Now, integrating in BR and using change of variables,
(52)∫BR‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)=∫D(a,R)‍|∇~f(φa(w))|p(1-|φa(w)|2)qhhhhhhhhhh×gs(w)dλ(w)=∫D(a,R)‍|∇~f(φa(w))|p(1-|a|2)q|1-〈w,a〉|2qhhhhhhhhhhh×(1-|w|2)qgs(w)dλ(w).
Consider(53)|∇~f(φam(w))|p(1-|am|2)q|1-〈w,am〉|2q,
and define the measure dμ(w) by
(54)dμ(w)=(1-|w|2)qgs(w)dλ(w).
Now, if we take |w|<1, |a|<R, then 1-R≤|1-〈w,am〉|. Besides, on D(a,R), |∇~f(φam(w))| results are bounded by a constant M>0, because φam(w)∈BR. Now,
(55)∫D(am,R)‍|∇~f(φam(w))|p(1-|am|2)q|1-〈w,am〉|2qdμ(w)=∫B‍|∇~f(φam(w))|p(1-|am|2)q|1-〈w,am〉|2qhhhhhhhh×χφam(BR)(w)dμ(w),
where χφam(BR) denotes the characteristic function of the set D(am,R). As
(56)|∇~f(φam(w))|p(1-|am|2)q|1-〈w,am〉|2qχφam(BR)(w)≤M|1-R|2q,
then by Lebesgue’s dominated convergence theorem, we have that (55) converges to
(57)∫B‍|∇~f(φa(w))|p(1-|a|2)q|1-〈w,a〉|2qχφa(BR)(w)dμ(w).Theorem 13.
Suppose thatp, q, and s satisfy condition (B). Suppose that
(58)J~p,q,sf(a)<∞,
for all a∈B. Then, the function J~p,q,sf is continuous on B.Proof.
Leta∈B and δ>0, such that Bδ(a)¯⊂B. If K:=B¯×Bδ(a)¯, the function l:K→ℝ defined by
(59)(η,ξ)⟼l(η,ξ)=(1-|ξ|2)s|1-〈η,ξ〉|2s
results to be uniformly continuous on K. As f∈F~φ(p,q,s), then
(60)∫B‍|∇~f(z)|p(1-|z|2)q+nsdλ(z)=J~p,q,sf(0)<∞.
If f is nonconstant, given that ε>0, there exists ρ>0, such that |η′-η|<ρ and |ξ′-ξ|<ρ; then
(61)|l(η′,ξ′)-l(η,ξ)|<ε∫B‍|∇~f(z)|p(1-|z|2)q+nsdλ(z).
Thus, if |a-b|<ρ, then
(62)|J~p,q,sf(a)-J~p,q,sf(b)|≤∫B‍|∇~f(z)|p(1-|z|2)q+ns|l(η,a)-l(η,b)|dλ(z)<ε.Remark 14.
Last two theorems are the versions inℂn of Theorem 3.1 of [6].Some relations between the little spacesF~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 thatp, q, and s satisfy condition (A). If f∈F~G,0(p,q,s), then the function
(63)Gh(a)={I~p,q,sf(a),if|a|<1,0,if|a|=1,
is continuous on B¯.Proof.
By the Theorem12, 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-I~p,q,sf(a)=0. So, given ε>0, there exists δ>0, such that if ||a|-1|<δ, then |I~p,q,sf(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 thatp, q, and s satisfy condition (A). Then, F~G,0(p,q,s) is contained in F~G(p,q,s).Proof.
In fact iff∈F~G0(p,q,s), as the functionGh is uniformly continuous on B¯, then is bounded and so
(64)maxa∈B∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)<∞.With the same arguments, we getF~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, ℬα, are the families of holomorphic functions f:B→ℂ satisfying
(65)∥f∥ℬα=|f(0)|+supz∈B(1-|z|2)α|∇~f(z)|<+∞.
It is easy to verify that ∥·∥ℬα is a norm on the vector space ℬα and contains at least the holomorphic functions on domains Ω, with Ω⊃B¯. The respective little α-Bloch spaces ℬα,0 satisfy
(66)lim|z|→1-(1-|z|2)α|∇~f(z)|=0.
It is clear that ℬα,0⊂ℬα.Proposition 17.
(i) Suppose thatp, q, and s satisfy condition (A). Then,
(67)F~G(p,q,s)⊂ℬq/p,
and there exists C>0, such that
(68)∥f∥ℬq/p≤C∥f∥G.
(ii) Ifn<(p/2)+q. Then,
(69)D~p,q⊂Bq/p,0.Proof.
(i) Letf∈F~G(p,q,s); then by Propositions 5 and 6 and (20)
(70)∞>∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)≥∫D(a,R)‍|∇~f(z)|p(1-|z|2)qC1s(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)≥∫D(a,R)‍|∇~f(z)|p(1-|z|2)qC1s(1-R2)nsR2(n-1)sdλ(z)≥C1s(1-R2)nsR2(n-1)s∫D(a,R)‍|∇~f(z)|p(1-|z|2)qdλ(z)≥C1s(1-R2)nsR2(n-1)s∫BR‍|∇~(f∘φa)(w)|pHHHHHHHHHHHH×(1-|φa(w)|2)qdλ(w)≥C1s(1-R2)nsR2(n-1)s∫BR‍|∇~(f∘φa)(w)|p(1-|w|2)phhhhhhhhhhhhhhhh×(1-|a|2)q(1-|w|2)q-n-1-p|1-〈w,a〉|2qdν(w)≥C1s(1-R2)nsR2(n-1)sC2(p,q,n)×∫BR‍|∇~(f∘φa)(w)|p(1-|w|2)p(1-|a|2)qdν(w)=C(p,q,s,n,R)(1-|a|2)q∫BR‍|∇(f∘φa)(w)|pdν(w)≥C(p,q,s,n,R)(1-|a|2)q|∇(f∘φa)(0)|p=C(p,q,s,n,R)(1-|a|2)q|∇~f(a)|p=C(p,q,s,n,R)((1-|a|2)q/p|∇~f(a)|)p,
where we have used that |∇(f∘φa)|p is a subharmonic function; (i) follows from this estimation.
(ii) Letf∈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
(71)ε>∫A(r)‍|∇~f(z)|p(1-|z|2)qdλ(z)≥∫D(a,R)‍|∇~f(z)|p(1-|z|2)qdλ(z).
With the same kind of arguments used in (i), we obtain(72)ε≥C(p,q,n,R)((1-|a|2)q/p|∇~f(a)|)p,
for all a∈B, sufficiently close to 1, and we conclude the proof.The following result says that Dirichlet spaces are the limit case ofFG,0(p,q,s) and Fφ,0(p,q,s), when s→0+.Theorem 18.
(i) Suppose thatp, q, and s satisfy condition (A). Then,
(73)D~p,q⊂⋂0<s<3‍F~g,0(p,q,s).
(ii) Suppose thatp, q, and s satisfy condition (B). Then,
(74)D~p,q⊂⋂0<s<∞‍F~φ,0(p,q,s).Proof.
We prove the first inclusion. The proof of the second one is similar. Letf∈D~p,q and ε>0. By Proposition 17, there exists 0<ρ<1, such that
(75)∫A(ρ)‍|∇~f(z)|p(1-|z|2)qdλ(z)<ε,(76)(1-|z|2)q|∇~f(z)|p<ε,∀0<ρ<|z|<1.
Let 0<ρ<R<R′<1, such that if R′<|a|<1; then D(a,R)⊂A(R). Thus,
(77)Ip,q,sf(a)≤C2s∫BR‍|∇~f(z)|p(1-|z|2)q(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)+C2s∫A(R)∖D(a,R)‍|∇~f(z)|p(1-|z|2)q×(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)+C2s∫D(a,R)‍|∇~f(z)|p(1-|z|2)q×(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z).
By Proposition 8, the first integral goes to zero as |a|→1-. For the second one, consider (75) as following:
(78)∫A(R)∖D(a,R)‍|∇~f(z)|p(1-|z|2)q(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)≤1R2(n-1)s∫A(ρ)‍|∇~f(z)|p(1-|z|2)qdλ(z)<εR2(n-1)s.
Moreover, for the third, using the change of variable φa(w)=z,
(79)∫D(a,R)‍|∇~f(z)|p(1-|z|2)q(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)<∫D(a,R)‍ε(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)=ε∫BR‍(1-|w|2)ns|w|2(n-1)sdλ(w).
and this concludes the proof.Proposition 19.
Suppose thatp, q, and s satisfy condition (A), with 1<s<n/(n-1). Then,
(80)ℬq/p⊂F~G(p,q,s).Proof.
Letf∈ℬq/p. Then,
(81)supz∈B(1-|z|2)q/p|∇~f(z)|≤∥f∥ℬq/p<∞.
Using change of variables and by Proposition 5 we have
(82)∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)≤∫B‍∥f∥ℬq/pp(1-|z|2)q(1-|z|2)qGs(z,a)dλ(z)=∥f∥ℬq/pp∫B‍Gs(z,a)dλ(z)≤∥f∥ℬq/pp∫B‍C2s(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)=∥f∥ℬq/ppC2s∫B‍(1-|w|2)ns-n-1|w|2(n-1)sdν(w),
and this integral results convergent because
(83)∫01‍(1-r)ns-n-1r2(n-ns+s)-1dr<∞,
when 1<s<n/(n-1). Finally, ℬq/p⊆F~G(p,q,s) if 1<s<n/(n-1).Similar results to Propositions17 and 19 can be obtained for F~φ(p,q,s).Proposition 20.
Suppose thatp, q, and s satisfy condition (B), with 1<s<∞. Then,
(84)F~φ(p,q,s)⊂ℬq/p,
and there exists C>0, such that
(85)∥f∥ℬq/p≤C∥f∥φ.Proof.
Letf∈F~φ(p,q,s); then
(86)∞>∫B‍|∇~f(z)|p(1-|z|2)q(|1-|φa(z)|2)nsdλ(z)=∫B‍|∇~(f∘φa)(w)|p(|1-|φa(w)|2)q×(1-|w|2)ns-n-1dν(w)≥∫B1/2‍|∇~(f∘φa)(w)|p(1-|w|2)p(1-|a|2)q(1-|w|2)q|1-〈w,a〉|2q×(1-|w|2)ns-n-1+pdν(w)≥C(p,q,s,n)∫B1/2‍|∇~(f∘φa)(w)|p(1-|w|2)p(1-|a|2)qdν(w)≥C(p,q,s,n)(1-|a|2)q∫B1/2‍|∇(f∘φa)(w)|pdν(w),
and we finish the proof as in the Proposition 17.Proposition 21.
Suppose thatp, q, and s satisfy condition (A), with 1<s<∞. Then,
(87)ℬq/p⊂F~φ(p,q,s).Proof.
Letf∈ℬq/p. By hypothesis,
(88)supz∈B(1-|z|2)q/p|∇~f(z)|≤∥f∥ℬα<∞.
Thus,
(89)∫B‍|∇~f(z)|p(1-|z|2)q(1-|φa(z)|2)nsdλ(z)≤∫B‍∥f∥ℬαp(1-|z|2)q(1-|z|2)q(1-|φa(z)|2)nsdλ(z)=∥f∥ℬαp∫B‍(1-|a|2)ns(1-|z|2)ns-n-1|1-〈z,a〉|2nsdν(z)=∥f∥ℬαp(1-|a|2)ns∫B‍(1-|z|2)ns-n-1|1-〈z,a〉|2nsdν(z),
and by Theorem 7, we obtain that if s>1,
(90)∫B‍(1-|z|2)ns-n-1|1-〈z,a〉|2nsdν(z)≈1(1-|a|2)ns,
and the result follows.We can synthesize the last propositions in the following.Theorem 22.
Suppose thatp, q, and s satisfy condition (B). The following statements are equivalent.(i)
f∈ℬq/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 spacesF~G,0(p,q,s), F~φ,0(p,q,s), and ℬq/p,0.For these cases, the inclusionsF~G,0(p,q,s),F~φ,0(p,q,s)⊆ℬq/p,0 follow immediately from the previous estimations. For the other inclusions we have the following.Proposition 23.
Suppose thatp, q, and s satisfy condition (A) and 1<s<n/(n-1). Then,
(91)ℬq/p,0⊂F~G,0(p,q,s).Proof.
Since1<s<n/(n-1), then C=∫B‍Gs(z,a)dλ(z)<∞. Suppose that f∈ℬq/p,0; then
(92)lim|z|→1-(1-|z|2)q/p|∇~f(z)|=0.
Given ε>0, there exists 0<R<1, such that
(93)|∇~f(z)|p<εC(1-|z|2)q
if R<|z|<1. We split the integral
(94)∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z),
as
(95)∫BR‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)+∫B∖BR‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z).
By hypothesis, the second integral satisfies
(96)∫B∖BR‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)≤εC∫B∖BR‍Gs(z,a)dλ(z)<ε.
Now,
(97)∫BR‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)≤C2s∫BR‍∥f∥ℬα(1-|z|2)q(1-|z|2)q(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)≤C2s∥f∥ℬα(1-|a|2)ns×∫BR‍(1-|z|2)ns-n-1|1-〈z,a〉|2(n-1)s1|φa(z)|2(n-1)sdν(z),
and the function
(98)(a,z)⟼(1-|z|2)ns-n-1|1-〈z,a〉|2(n-1)s|φa(z)|2(n-1)s
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(99)ℬq/p,0⊂F~φ,0(p,q,s)
is similar.Lemma 24.
Let-(1/2)≤α<∞ and f∈ℬα. 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.
Letγ:[0,1]→B be a piecewise smooth curve. Thus, the lenght of γ in the Bergman metric ρ of B is
(100)l(γ)=∫01‍〈Q(γ(t))γ′(t),γ′(t)〉dt,
where Q(z) is the Bergman matrix of B (see [7]). Let f:B→ℂ be a holomorphic function. Then,
(101)|∇~f(z)|=sup{|〈∇f(z),w¯〉|〈Q(z)w,w〉:w∈ℂn-{0}}.
Now, if γ:[0,1]→B is a geodesic in the Bergman metric, with γ(0)=z0, γ1(1)=z1,…, and f∈ℬα, we have
(102)|f(z1)-f(z0)|≤∫01‍|(f∘γ)′(t)|dt=∫01‍|〈(∇f)(γ(t)),γ′(t)¯〉|dt≤∫01‍|∇~f(γ(t))|〈Q(γ(t)γ′(t),γ′(t)〉dt≤∫01‍|∇~f(γ(t))|(1-|γ(t)|2)q/p(1-|γ(t)|2)q/phhhh×〈Q(γ(t)γ′(t),γ′(t)〉dt≤∥f∥ℬα(1-supt∈[0,1]|γ(t)|2)q/p×∫01‍〈Q(γ(t)γ′(t),γ′(t)〉dt=∥f∥ℬα(1-supt∈[0,1]|γ(t)|2)q/pρ(z0,z1),
and the proof is complete.Theorem 25.
Let-(1/2)≤α<∞. The spaces ℬα and ℬα,0 are complete metric spaces.Proof.
Let{fk}⊂ℬα be a Cauchy sequence. Given ε>0, there exists N∈ℕ, such that ∥fl-fk∥ℬα<ε,   for all   k>l>N. In particular, {fk(0)} is convergent. Let 0<r<1. We apply the estimation of Lemma 24 to fl-fk,z0=0, andz1=z∈Br. Thus,
(103)|fl(z)-fk(z)-fl(0)+fk(0)|≤∥fl-fk∥ℬα(1-r2)q/plog1+r1-r.
Then,{fk} converges uniformly on compact subsets of B to some holomorphic function, say f. Therefore, ∂fk/∂zj converges uniformly on compact subsets of B to ∂f/∂zj, for j=1,…,n. So, we can get |∇~(fl-fk)|p converges to |∇~(f-fk)|p uniformly on compact sets of B when l→∞. For k,l>N, we have
(104)|fk(0)-fl(0)|+(1-|z|2)α|∇~(fk-fl)(z)|≤∥fk-fl∥ℬα<ε,
and taking limit when k→∞, we get
(105)|f(0)-fl(0)|+(1-|z|2)α|∇~(f-fl)(z)|≤ε,
that is, f-fl∈ℬα and hence f=(f-fl)+fl∈ℬα.
Let{fk}⊂ℬα,0 be a Cauchy sequence. Then, fk converges to f∈ℬα. Given ε>0, there exists N∈ℕ, such that ∥f-fk∥ℬα<ε/2, for all k>N. Moreover, there exists 0<r<1, such that
(106)(1-|z|2)α|∇~(fN+1)(z)|<ε2,∀r<|z|<1.
Thus,
(107)(1-|z|2)α|∇~(f)(z)|≤(1-|z|2)α|∇~(f-fN+1)(z)|+(1-|z|2)α|∇~fN+1(z)|<ε,
for all r<|z|<1; therefore, ℬα,0 is a complete space.Theorem 26.
Suppose thatp, 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 case1≤p<∞. Let {fk}⊂F~G(p,q,s) be a Cauchy sequence. Since {fk} is too a Cauchy sequence in ℬq/p, then converges to some f∈ℬq/p.
We will prove thatf∈F~G(p,q,s). Given ε>0, there exists N∈ℕ, such that
(108)I~p,q,s(fl-fk)(a)≤∥fl-fk∥Gp<εp,∀l>k>N.
By Fatou's Lemma, as l>N,
(109)I~p,q,s(fl-f)(a)=∫B‍|∇~(fl-f)(z)|p(1-|z|2)qGs(z,a)dλ(z)=∫B‍limk→∞|∇~(fl-fk)(z)|p(1-|z|2)qGs(z,a)dλ(z)≤liminfk→∞∫B‍|∇~(fl-fk)(z)|p(1-|z|2)qGs(z,a)dλ(z)≤limk→∞∥fl-fk∥Gp≤εp.
Since {fk(0)} converges to {f(0)}, if we take supremum on B and p roots in both sides of the last inequality, we obtain
(110)limk→∞∥fk-f∥G=0.
Since f=(f-fk)+fk, we get that f∈F~G(p,q,s).
If{fk} is a Cauchy sequence in F~G,0(p,q,s), then {fk} converges to f∈F~G(p,q,s). Given ε>0, there exists N∈ℕ, such that if k>N,
(111)I~p,q,sf(a)≤2pI~p,q,s(f-fk)(a)+2pI~p,q,sfk(a)≤2pεp+2pI~p,q,sfk(a).
Thus,
(112)lim|a|→1-I~p,q,sf(a)≤lim|a|→1-2pεp+lim|a|→1-2pI~p,q,sfk(a)=2pεp;
therefore, f∈F~G,0(p,q,s).
In the case0<p<1, the proof is similar.As a consequence of the previous results and Proposition17, we have the following.Corollary 27.
Suppose that1≤p<∞, q, and s satisfy condition (A). The spaces F~G(p,q,s), F~G,0(p,q,s), ℬq/p, and ℬq/p,0 are Banach spaces, and the inclusion operators I:F~G(p,q,s)→ℬq/p and I:F~G,0(p,q,s)→ℬq/p,0 are bounded.Similar results are valid for the spacesF~φ(p,q,s) and F~φ,0(p,q,s).


## 2.1. Main Definitions

In this section, we define the spacesF~G(p,q,s), F~φ(p,q,s) and their little associated spaces. We study some properties and relations between them.Suppose thatp,q, and s satisfy condition (A), and let f∈𝒪. Define
(35)I~p,q,sf(a)=∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z).
Suppose that p, q, and s satisfy condition (B), and let f∈𝒪. Define
(36)J~p,q,sf(a)=∫B‍|∇~f(z)|p(1-|z|2)q(1-|φa(z)|2)nsdλ(z).
It is clear that
(37)J~p,q,sf(a)≤I~p,q,sf(a),foreacha∈B.
Associated to these functions, we define the following vector spaces of holomorphic functions.  (i)
Suppose thatp, q, and s satisfy condition (A). f∈𝒪 belongs to F~G(p,q,s) if it satisfies
(38)supa∈BI~p,q,sf(a)<+∞
and belongs to the little space F~G,0(p,q,s) if it satisfies
(39)lim|a|→1-I~p,q,sf(a)=0.(ii)
Suppose thatp, q, and s satisfy condition (B). f∈𝒪 belongs to F~φ(p,q,s) if it satisfies
(40)supa∈BJ~p,q,sf(a)<+∞
and belongs to the little space F~φ,0(p,q,s) if it satisfies
(41)lim|a|→1-J~p,q,sf(a)=0.(iii)
Letn<(p+2q)/2. f∈𝒪 belongs to the Dirichlet space D~p,q if
(42)∫B‍|∇~f(z)|p(1-|z|2)qdλ(z)<∞.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 by(43)∥f∥G≔|f(0)|+supa∈B(I~p,q,sf(a))1/p,∥f∥φ:=|f(0)|+supa∈B(J~p,q,sf(a))1/p.
This is a direct consequence of the usual Lp(B,dμ) theory when we consider the measures dμ(z)=(1-|z|2)qGs(z,a)dλ(z),dμ(z)=(1-|z|2)q(1-|φa(z)|2)nsdλ(z), respectively. For 0<p<1 and nonnegative x, y, we have (x+y)p≤xp+yp; 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
(44)dG(f,h)=|f(0)-h(0)|+supa∈BI~p,q,s(f-h)(a),dφ(f,h)=|f(0)-h(0)|+supa∈BJ~p,q,s(f-h)(a),
respectively (see [10]). We will prove their completeness in Theorem 26.Example 10.
Let𝒪~ 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
(45)∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)≤maxz∈B¯|∇f(z)|p∫B‍(1-|z|2)(p/2)+qGs(z,a)dλ(z).
So, in the definitions, the parameters p, q, and s were chosen to guarantee that the vector spaces FG(p,q,s) and FG,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].
Letf:B→ℂ, with power series development given by
(46)f(z)=∑α‍aαzα,
such that ∑n=0∞‍n∑|α|=n‍|aα|<∞. Then, f belongs to each one of the previous spaces. By (37), we get the same result for Fφ(p,q,s) and Fφ,0(p,q,s).The spacesF~G(p,q,s) are Möbius invariant in the following sense.Theorem 11.
Suppose thatp, 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.
Leta,b∈B,w=φa(z), and c=φa(b). By the change of variables formula and the Möbius invariance of G(z,a), ∇,~ and dλ, we have
(47)∫B‍|∇~(f∘φa)(w)|p(1-|w|2)qGs(w,b)dλ(w)=∫B‍|(∇~f)∘φa(w)|p(1-|w|2)qGs(w,b)dλ(w)=∫B‍|(∇~f)∘φa(φa(z))|p(1-|φa(z)|2)qhhhhhhhh×Gs(φa(z),φa(c))dλ(z)=∫B‍|∇~f(z)|p(1-|φa(z)|2)qGs(z,c)dλ(z)≤(1+|a|1-|a|)|q|∫B‍|∇~f(z)|p(1-|z|2)qGs(z,c)dλ(z),
where we have used in the last line the estimations of Lemmas 3 and 4.Observe that ifq=0 and f∈F~G(p,0,s), then
(48)∫B‍|∇~(f∘φa)(w)|pGs(w,b)dλ(w)=∫B‍|∇~f(z)|pGs(z,c)dλ(z),
and these spaces are Möbius invariant in the classical way.Theorem 12.
Suppose thatp, q, and s satisfy condition (A). Suppose that
(49)I~p,q,sf(a)<∞,
for all a∈B. Then, the function I~p,q,sf is continuous on B.Proof.
Leta∈B be fixed and R, such that |a|<R<1. Consider a neighborhood Bδ(a) with δ=(R-|a|)/2. Let {am}m∈ℕ be a sequence contained in Bδ(a) and convergent to a. Consider now I~p,q,sf(am) and define dμ(z)=|∇~f(z)|p(1-|z|2)q+nsdλ(z). Then, Gs(z,am)(1-|z|2)-ns is bounded in (B∖BR)×Bδ(a). Thus, by bounded convergence theorem,
(50)∫B∖BR‍Gs(z,am)(1-|z|2)nsdμ(z)
converges to
(51)∫B∖BR‍Gs(z,a)(1-|z|2)nsdμ(z).
Now, integrating in BR and using change of variables,
(52)∫BR‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)=∫D(a,R)‍|∇~f(φa(w))|p(1-|φa(w)|2)qhhhhhhhhhh×gs(w)dλ(w)=∫D(a,R)‍|∇~f(φa(w))|p(1-|a|2)q|1-〈w,a〉|2qhhhhhhhhhhh×(1-|w|2)qgs(w)dλ(w).
Consider(53)|∇~f(φam(w))|p(1-|am|2)q|1-〈w,am〉|2q,
and define the measure dμ(w) by
(54)dμ(w)=(1-|w|2)qgs(w)dλ(w).
Now, if we take |w|<1, |a|<R, then 1-R≤|1-〈w,am〉|. Besides, on D(a,R), |∇~f(φam(w))| results are bounded by a constant M>0, because φam(w)∈BR. Now,
(55)∫D(am,R)‍|∇~f(φam(w))|p(1-|am|2)q|1-〈w,am〉|2qdμ(w)=∫B‍|∇~f(φam(w))|p(1-|am|2)q|1-〈w,am〉|2qhhhhhhhh×χφam(BR)(w)dμ(w),
where χφam(BR) denotes the characteristic function of the set D(am,R). As
(56)|∇~f(φam(w))|p(1-|am|2)q|1-〈w,am〉|2qχφam(BR)(w)≤M|1-R|2q,
then by Lebesgue’s dominated convergence theorem, we have that (55) converges to
(57)∫B‍|∇~f(φa(w))|p(1-|a|2)q|1-〈w,a〉|2qχφa(BR)(w)dμ(w).Theorem 13.
Suppose thatp, q, and s satisfy condition (B). Suppose that
(58)J~p,q,sf(a)<∞,
for all a∈B. Then, the function J~p,q,sf is continuous on B.Proof.
Leta∈B and δ>0, such that Bδ(a)¯⊂B. If K:=B¯×Bδ(a)¯, the function l:K→ℝ defined by
(59)(η,ξ)⟼l(η,ξ)=(1-|ξ|2)s|1-〈η,ξ〉|2s
results to be uniformly continuous on K. As f∈F~φ(p,q,s), then
(60)∫B‍|∇~f(z)|p(1-|z|2)q+nsdλ(z)=J~p,q,sf(0)<∞.
If f is nonconstant, given that ε>0, there exists ρ>0, such that |η′-η|<ρ and |ξ′-ξ|<ρ; then
(61)|l(η′,ξ′)-l(η,ξ)|<ε∫B‍|∇~f(z)|p(1-|z|2)q+nsdλ(z).
Thus, if |a-b|<ρ, then
(62)|J~p,q,sf(a)-J~p,q,sf(b)|≤∫B‍|∇~f(z)|p(1-|z|2)q+ns|l(η,a)-l(η,b)|dλ(z)<ε.Remark 14.
Last two theorems are the versions inℂn of Theorem 3.1 of [6].Some relations between the little spacesF~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 thatp, q, and s satisfy condition (A). If f∈F~G,0(p,q,s), then the function
(63)Gh(a)={I~p,q,sf(a),if|a|<1,0,if|a|=1,
is continuous on B¯.Proof.
By the Theorem12, 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-I~p,q,sf(a)=0. So, given ε>0, there exists δ>0, such that if ||a|-1|<δ, then |I~p,q,sf(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 thatp, q, and s satisfy condition (A). Then, F~G,0(p,q,s) is contained in F~G(p,q,s).Proof.
In fact iff∈F~G0(p,q,s), as the functionGh is uniformly continuous on B¯, then is bounded and so
(64)maxa∈B∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)<∞.With the same arguments, we getF~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, ℬα, are the families of holomorphic functions f:B→ℂ satisfying
(65)∥f∥ℬα=|f(0)|+supz∈B(1-|z|2)α|∇~f(z)|<+∞.
It is easy to verify that ∥·∥ℬα is a norm on the vector space ℬα and contains at least the holomorphic functions on domains Ω, with Ω⊃B¯. The respective little α-Bloch spaces ℬα,0 satisfy
(66)lim|z|→1-(1-|z|2)α|∇~f(z)|=0.
It is clear that ℬα,0⊂ℬα.Proposition 17.
(i) Suppose thatp, q, and s satisfy condition (A). Then,
(67)F~G(p,q,s)⊂ℬq/p,
and there exists C>0, such that
(68)∥f∥ℬq/p≤C∥f∥G.
(ii) Ifn<(p/2)+q. Then,
(69)D~p,q⊂Bq/p,0.Proof.
(i) Letf∈F~G(p,q,s); then by Propositions 5 and 6 and (20)
(70)∞>∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)≥∫D(a,R)‍|∇~f(z)|p(1-|z|2)qC1s(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)≥∫D(a,R)‍|∇~f(z)|p(1-|z|2)qC1s(1-R2)nsR2(n-1)sdλ(z)≥C1s(1-R2)nsR2(n-1)s∫D(a,R)‍|∇~f(z)|p(1-|z|2)qdλ(z)≥C1s(1-R2)nsR2(n-1)s∫BR‍|∇~(f∘φa)(w)|pHHHHHHHHHHHH×(1-|φa(w)|2)qdλ(w)≥C1s(1-R2)nsR2(n-1)s∫BR‍|∇~(f∘φa)(w)|p(1-|w|2)phhhhhhhhhhhhhhhh×(1-|a|2)q(1-|w|2)q-n-1-p|1-〈w,a〉|2qdν(w)≥C1s(1-R2)nsR2(n-1)sC2(p,q,n)×∫BR‍|∇~(f∘φa)(w)|p(1-|w|2)p(1-|a|2)qdν(w)=C(p,q,s,n,R)(1-|a|2)q∫BR‍|∇(f∘φa)(w)|pdν(w)≥C(p,q,s,n,R)(1-|a|2)q|∇(f∘φa)(0)|p=C(p,q,s,n,R)(1-|a|2)q|∇~f(a)|p=C(p,q,s,n,R)((1-|a|2)q/p|∇~f(a)|)p,
where we have used that |∇(f∘φa)|p is a subharmonic function; (i) follows from this estimation.
(ii) Letf∈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
(71)ε>∫A(r)‍|∇~f(z)|p(1-|z|2)qdλ(z)≥∫D(a,R)‍|∇~f(z)|p(1-|z|2)qdλ(z).
With the same kind of arguments used in (i), we obtain(72)ε≥C(p,q,n,R)((1-|a|2)q/p|∇~f(a)|)p,
for all a∈B, sufficiently close to 1, and we conclude the proof.The following result says that Dirichlet spaces are the limit case ofFG,0(p,q,s) and Fφ,0(p,q,s), when s→0+.Theorem 18.
(i) Suppose thatp, q, and s satisfy condition (A). Then,
(73)D~p,q⊂⋂0<s<3‍F~g,0(p,q,s).
(ii) Suppose thatp, q, and s satisfy condition (B). Then,
(74)D~p,q⊂⋂0<s<∞‍F~φ,0(p,q,s).Proof.
We prove the first inclusion. The proof of the second one is similar. Letf∈D~p,q and ε>0. By Proposition 17, there exists 0<ρ<1, such that
(75)∫A(ρ)‍|∇~f(z)|p(1-|z|2)qdλ(z)<ε,(76)(1-|z|2)q|∇~f(z)|p<ε,∀0<ρ<|z|<1.
Let 0<ρ<R<R′<1, such that if R′<|a|<1; then D(a,R)⊂A(R). Thus,
(77)Ip,q,sf(a)≤C2s∫BR‍|∇~f(z)|p(1-|z|2)q(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)+C2s∫A(R)∖D(a,R)‍|∇~f(z)|p(1-|z|2)q×(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)+C2s∫D(a,R)‍|∇~f(z)|p(1-|z|2)q×(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z).
By Proposition 8, the first integral goes to zero as |a|→1-. For the second one, consider (75) as following:
(78)∫A(R)∖D(a,R)‍|∇~f(z)|p(1-|z|2)q(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)≤1R2(n-1)s∫A(ρ)‍|∇~f(z)|p(1-|z|2)qdλ(z)<εR2(n-1)s.
Moreover, for the third, using the change of variable φa(w)=z,
(79)∫D(a,R)‍|∇~f(z)|p(1-|z|2)q(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)<∫D(a,R)‍ε(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)=ε∫BR‍(1-|w|2)ns|w|2(n-1)sdλ(w).
and this concludes the proof.Proposition 19.
Suppose thatp, q, and s satisfy condition (A), with 1<s<n/(n-1). Then,
(80)ℬq/p⊂F~G(p,q,s).Proof.
Letf∈ℬq/p. Then,
(81)supz∈B(1-|z|2)q/p|∇~f(z)|≤∥f∥ℬq/p<∞.
Using change of variables and by Proposition 5 we have
(82)∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)≤∫B‍∥f∥ℬq/pp(1-|z|2)q(1-|z|2)qGs(z,a)dλ(z)=∥f∥ℬq/pp∫B‍Gs(z,a)dλ(z)≤∥f∥ℬq/pp∫B‍C2s(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)=∥f∥ℬq/ppC2s∫B‍(1-|w|2)ns-n-1|w|2(n-1)sdν(w),
and this integral results convergent because
(83)∫01‍(1-r)ns-n-1r2(n-ns+s)-1dr<∞,
when 1<s<n/(n-1). Finally, ℬq/p⊆F~G(p,q,s) if 1<s<n/(n-1).Similar results to Propositions17 and 19 can be obtained for F~φ(p,q,s).Proposition 20.
Suppose thatp, q, and s satisfy condition (B), with 1<s<∞. Then,
(84)F~φ(p,q,s)⊂ℬq/p,
and there exists C>0, such that
(85)∥f∥ℬq/p≤C∥f∥φ.Proof.
Letf∈F~φ(p,q,s); then
(86)∞>∫B‍|∇~f(z)|p(1-|z|2)q(|1-|φa(z)|2)nsdλ(z)=∫B‍|∇~(f∘φa)(w)|p(|1-|φa(w)|2)q×(1-|w|2)ns-n-1dν(w)≥∫B1/2‍|∇~(f∘φa)(w)|p(1-|w|2)p(1-|a|2)q(1-|w|2)q|1-〈w,a〉|2q×(1-|w|2)ns-n-1+pdν(w)≥C(p,q,s,n)∫B1/2‍|∇~(f∘φa)(w)|p(1-|w|2)p(1-|a|2)qdν(w)≥C(p,q,s,n)(1-|a|2)q∫B1/2‍|∇(f∘φa)(w)|pdν(w),
and we finish the proof as in the Proposition 17.Proposition 21.
Suppose thatp, q, and s satisfy condition (A), with 1<s<∞. Then,
(87)ℬq/p⊂F~φ(p,q,s).Proof.
Letf∈ℬq/p. By hypothesis,
(88)supz∈B(1-|z|2)q/p|∇~f(z)|≤∥f∥ℬα<∞.
Thus,
(89)∫B‍|∇~f(z)|p(1-|z|2)q(1-|φa(z)|2)nsdλ(z)≤∫B‍∥f∥ℬαp(1-|z|2)q(1-|z|2)q(1-|φa(z)|2)nsdλ(z)=∥f∥ℬαp∫B‍(1-|a|2)ns(1-|z|2)ns-n-1|1-〈z,a〉|2nsdν(z)=∥f∥ℬαp(1-|a|2)ns∫B‍(1-|z|2)ns-n-1|1-〈z,a〉|2nsdν(z),
and by Theorem 7, we obtain that if s>1,
(90)∫B‍(1-|z|2)ns-n-1|1-〈z,a〉|2nsdν(z)≈1(1-|a|2)ns,
and the result follows.We can synthesize the last propositions in the following.Theorem 22.
Suppose thatp, q, and s satisfy condition (B). The following statements are equivalent.(i)
f∈ℬq/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 spacesF~G,0(p,q,s), F~φ,0(p,q,s), and ℬq/p,0.For these cases, the inclusionsF~G,0(p,q,s),F~φ,0(p,q,s)⊆ℬq/p,0 follow immediately from the previous estimations. For the other inclusions we have the following.Proposition 23.
Suppose thatp, q, and s satisfy condition (A) and 1<s<n/(n-1). Then,
(91)ℬq/p,0⊂F~G,0(p,q,s).Proof.
Since1<s<n/(n-1), then C=∫B‍Gs(z,a)dλ(z)<∞. Suppose that f∈ℬq/p,0; then
(92)lim|z|→1-(1-|z|2)q/p|∇~f(z)|=0.
Given ε>0, there exists 0<R<1, such that
(93)|∇~f(z)|p<εC(1-|z|2)q
if R<|z|<1. We split the integral
(94)∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z),
as
(95)∫BR‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)+∫B∖BR‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z).
By hypothesis, the second integral satisfies
(96)∫B∖BR‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)≤εC∫B∖BR‍Gs(z,a)dλ(z)<ε.
Now,
(97)∫BR‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)≤C2s∫BR‍∥f∥ℬα(1-|z|2)q(1-|z|2)q(1-|φa(z)|2)ns|φa(z)|2(n-1)sdλ(z)≤C2s∥f∥ℬα(1-|a|2)ns×∫BR‍(1-|z|2)ns-n-1|1-〈z,a〉|2(n-1)s1|φa(z)|2(n-1)sdν(z),
and the function
(98)(a,z)⟼(1-|z|2)ns-n-1|1-〈z,a〉|2(n-1)s|φa(z)|2(n-1)s
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(99)ℬq/p,0⊂F~φ,0(p,q,s)
is similar.Lemma 24.
Let-(1/2)≤α<∞ and f∈ℬα. 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.
Letγ:[0,1]→B be a piecewise smooth curve. Thus, the lenght of γ in the Bergman metric ρ of B is
(100)l(γ)=∫01‍〈Q(γ(t))γ′(t),γ′(t)〉dt,
where Q(z) is the Bergman matrix of B (see [7]). Let f:B→ℂ be a holomorphic function. Then,
(101)|∇~f(z)|=sup{|〈∇f(z),w¯〉|〈Q(z)w,w〉:w∈ℂn-{0}}.
Now, if γ:[0,1]→B is a geodesic in the Bergman metric, with γ(0)=z0, γ1(1)=z1,…, and f∈ℬα, we have
(102)|f(z1)-f(z0)|≤∫01‍|(f∘γ)′(t)|dt=∫01‍|〈(∇f)(γ(t)),γ′(t)¯〉|dt≤∫01‍|∇~f(γ(t))|〈Q(γ(t)γ′(t),γ′(t)〉dt≤∫01‍|∇~f(γ(t))|(1-|γ(t)|2)q/p(1-|γ(t)|2)q/phhhh×〈Q(γ(t)γ′(t),γ′(t)〉dt≤∥f∥ℬα(1-supt∈[0,1]|γ(t)|2)q/p×∫01‍〈Q(γ(t)γ′(t),γ′(t)〉dt=∥f∥ℬα(1-supt∈[0,1]|γ(t)|2)q/pρ(z0,z1),
and the proof is complete.Theorem 25.
Let-(1/2)≤α<∞. The spaces ℬα and ℬα,0 are complete metric spaces.Proof.
Let{fk}⊂ℬα be a Cauchy sequence. Given ε>0, there exists N∈ℕ, such that ∥fl-fk∥ℬα<ε,   for all   k>l>N. In particular, {fk(0)} is convergent. Let 0<r<1. We apply the estimation of Lemma 24 to fl-fk,z0=0, andz1=z∈Br. Thus,
(103)|fl(z)-fk(z)-fl(0)+fk(0)|≤∥fl-fk∥ℬα(1-r2)q/plog1+r1-r.
Then,{fk} converges uniformly on compact subsets of B to some holomorphic function, say f. Therefore, ∂fk/∂zj converges uniformly on compact subsets of B to ∂f/∂zj, for j=1,…,n. So, we can get |∇~(fl-fk)|p converges to |∇~(f-fk)|p uniformly on compact sets of B when l→∞. For k,l>N, we have
(104)|fk(0)-fl(0)|+(1-|z|2)α|∇~(fk-fl)(z)|≤∥fk-fl∥ℬα<ε,
and taking limit when k→∞, we get
(105)|f(0)-fl(0)|+(1-|z|2)α|∇~(f-fl)(z)|≤ε,
that is, f-fl∈ℬα and hence f=(f-fl)+fl∈ℬα.
Let{fk}⊂ℬα,0 be a Cauchy sequence. Then, fk converges to f∈ℬα. Given ε>0, there exists N∈ℕ, such that ∥f-fk∥ℬα<ε/2, for all k>N. Moreover, there exists 0<r<1, such that
(106)(1-|z|2)α|∇~(fN+1)(z)|<ε2,∀r<|z|<1.
Thus,
(107)(1-|z|2)α|∇~(f)(z)|≤(1-|z|2)α|∇~(f-fN+1)(z)|+(1-|z|2)α|∇~fN+1(z)|<ε,
for all r<|z|<1; therefore, ℬα,0 is a complete space.Theorem 26.
Suppose thatp, 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 case1≤p<∞. Let {fk}⊂F~G(p,q,s) be a Cauchy sequence. Since {fk} is too a Cauchy sequence in ℬq/p, then converges to some f∈ℬq/p.
We will prove thatf∈F~G(p,q,s). Given ε>0, there exists N∈ℕ, such that
(108)I~p,q,s(fl-fk)(a)≤∥fl-fk∥Gp<εp,∀l>k>N.
By Fatou's Lemma, as l>N,
(109)I~p,q,s(fl-f)(a)=∫B‍|∇~(fl-f)(z)|p(1-|z|2)qGs(z,a)dλ(z)=∫B‍limk→∞|∇~(fl-fk)(z)|p(1-|z|2)qGs(z,a)dλ(z)≤liminfk→∞∫B‍|∇~(fl-fk)(z)|p(1-|z|2)qGs(z,a)dλ(z)≤limk→∞∥fl-fk∥Gp≤εp.
Since {fk(0)} converges to {f(0)}, if we take supremum on B and p roots in both sides of the last inequality, we obtain
(110)limk→∞∥fk-f∥G=0.
Since f=(f-fk)+fk, we get that f∈F~G(p,q,s).
If{fk} is a Cauchy sequence in F~G,0(p,q,s), then {fk} converges to f∈F~G(p,q,s). Given ε>0, there exists N∈ℕ, such that if k>N,
(111)I~p,q,sf(a)≤2pI~p,q,s(f-fk)(a)+2pI~p,q,sfk(a)≤2pεp+2pI~p,q,sfk(a).
Thus,
(112)lim|a|→1-I~p,q,sf(a)≤lim|a|→1-2pεp+lim|a|→1-2pI~p,q,sfk(a)=2pεp;
therefore, f∈F~G,0(p,q,s).
In the case0<p<1, the proof is similar.As a consequence of the previous results and Proposition17, we have the following.Corollary 27.
Suppose that1≤p<∞, q, and s satisfy condition (A). The spaces F~G(p,q,s), F~G,0(p,q,s), ℬq/p, and ℬq/p,0 are Banach spaces, and the inclusion operators I:F~G(p,q,s)→ℬq/p and I:F~G,0(p,q,s)→ℬq/p,0 are bounded.Similar results are valid for the spacesF~φ(p,q,s) and F~φ,0(p,q,s).


## 3. The EqualityF~G(p,q,s)=F~φ(p,q,s)

In this section, we prove the equalityF~G(p,q,s)=F~φ(p,q,s).Theorem 28.
Let0<p<∞, q∈ℝ, 0<s<n/(n-1), 1/2<R<1, and f∈𝒪. Then,
(113)∫B1/2‍|∇f(z)|p(1-|z|2)qgs(z)dλ(z)≤t∫BR‍|∇f(z)|p(1-|z|2)q+nsdλ(z),
where t=t(q,s,n,R).Proof.
Let1/2<R<1. Define
(114)0<1τ(q,s,n,R)=∫1/2R‍(1-r2)q+ns-n-1r2n-1dr.
For u∈S2n-1, as |∇f|p is subharmonic on B, we have
(115)1τ(q,s,n,R)∫S2n-1‍|∇f(u2)|pdσ=∫1/2R‍(1-r2)q+ns-n-1r2n-1dr·∫S2n-1‍|∇f(u2)|pdσ≤∫1/2R‍(1-r2)q+ns-n-1r2n-1dr·∫S2n-1|∇f(ru)|pdσ=∫BR∖B1/2‍|∇f(z)|p(1-|z|2)q+nsdλ(z)≤∫BR‍|∇f(z)|p(1-|z|2)q+nsdλ(z).
Thus,
(116)∫S2n-1|∇f(u2)|pdσ≤τ(q,s,n,R)∫BR‍|∇f(z)|p(1-|z|2)q+nsdλ(z).
By Proposition 5,
(117)∫01/2‍(1-r2)q-n-1r2n-1gs(ru)dr≤C2s∫01/2‍(1-r2)q-n-1r2n-1(1-r2)nsr2(n-1)sdr=C2s∫01/2‍(1-r2)q+ns-n-1r2(n-ns+s)-1dr.
This integral results convergent because 0<s<n/(n-1). Denote the last integral by τ′(q,s,n). Then,
(118)∫B1/2‍|∇f(z)|p(1-|z|2)qgs(z)dλ(z)≤C2s∫01/2‍∫S2n-1‍|∇f(u2)|p×(1-r2)q+ns-n-1r2(n-ns+s)-1dσdr≤C2sτ′(q,s,n)τ(q,s,n,R)×∫BR‍|∇f(z)|p(1-|z|2)q+nsdλ(z),
and the result follows.Theorem 29.
Suppose thatp, q, and s satisfy condition (A). If f∈F~G(p,q,s), then
(119)∫B‍|∇~f(z)|p(1-|z|2)qgs(z)dλ(z)≤t∫B‍|∇~f(z)|p(1-|z|2)q+nsdλ(z),
where t=t(p,q,s,n,R) for some 1/2<R<1.Proof.
From Proposition6 and (113),
(120)∫B1/2‍|∇~f(z)|p(1-|z|2)qgs(z)dλ(z)≤∫B1/2‍|∇f(z)|p(1-|z|2)qgs(z)dλ(z)≤C2sτ(q,s,n)τ(q,s,n,R)×∫BR‍|∇f(z)|p(1-|z|2)p(1-|z|2)p(1-|z|2)q+nsdλ(z)≤C2sτ(q,s,n)τ(q,s,n,R)(1-R2)p×∫BR‍|∇~f(z)|p(1-|z|2)q+nsdλ(z).
Now, if1/2≤|z|<1,
(121)∫B∖B1/2|∇~f(z)|p(1-|z|2)qgs(z)dλ(z)≤C2s∫B∖B1/2‍|∇~f(z)|p(1-|z|2)q+ns|z|2(n-1)sdλ(z)≤C2s22(n-1)s∫B‍|∇~f(z)|p(1-|z|2)q+nsdλ(z).
Finally,(122)∫B‍|∇~f(z)|p(1-|z|2)qgs(z)dλ(z)≤t∫B‍|∇~f(z)|p(1-|z|2)q+nsdλ(z),
with t:=t(p,q,s,n,R):=C2s(τ′(q,s,n) τ(q,s,n,R)+22(n-1)s)/(1-R2)p.By the Möbius invariance of the spaceF~G(p,q,s), if we apply the last result to f∘φa, we obtain the following.Corollary 30.
Suppose thatp, q, and s satisfy condition (A) and f∈F~G(p,q,s). Then, for 1/2<R<1,
(123)∫B‍|∇~(f∘φa)(z)|p(1-|z|2)qgs(z)dλ(z)≤t(p,q,s,n,R)∫B‍|∇~(f∘φa)(z)|p(1-|z|2)q+nsdλ(z).Theorem 31.
Suppose thatp, q, and s satisfy condition (A) and f∈F~G(p,q,s). Then,
(124)supa∈B∫B‍|∇~f(w)|p(1-|w|2)qGs(w,a)dλ(w)<∞,
if and only if
(125)supa∈B∫B‍|∇~f(w)|p(1-|w|2)q(1-|φa(w)|2)nsdλ(w)<∞.Proof.
We will abbreviatet=t(p,q,s,n,R), then from the last corollary and (20)
(126)0≤∫B‍|∇~(f∘φa)(z)|p(1-|z|2)q×[t(1-|z|2)ns-gs(z)]dλ(z)=∫B‍|(∇~f)(φa(z))|p(1-|z|2)q×[t(1-|z|2)ns-gs(z)]dλ(z);
using the change of variables z=φa(w), we obtain
(127)0≤∫B‍|∇~f(w)|p(1-|φa(w)|2)q×[t(1-|φa(w)|2)ns-Gs(w,a)]dλ(w).
Then,
(128)0≤∫B‍|∇~f(w)|p(1-|a|2)q(1-|w|2)q|1-〈w,a〉|2q×[t(1-|φa(w)|2)ns-Gs(w,a)]dλ(w)≤(1+|a|)q(1-|a|)q∫B‍|∇~f(w)|p(1-|w|2)q×[t(1-|φa(w)|2)ns-Gs(w,a)]dλ(w),
and from here we obtain the result.Corollary 32.
Suppose thatp, q, and s satisfy condition (A). Then,
(129)F~G(p,q,s)=F~φ(p,q,s),F~G,0(p,q,s)=F~φ,0(p,q,s).


## 4. Harmonic Majorants inF~G(p,q,s) and F~φ(p,q,s)

Forz∈B and u∈S2n-1, the function P defined by
(130)P(z,u)=1-|z|2|z-u|2n
is the real Poisson kernel for the ball B⊂ℝ2n. If h(u) is a real function, integrable on S2n-1, we define the Poisson integral formula of h as
(131)H(z)=1σ(S2n-1)∫S2n-1h(u)P(z,u)dσ(u)=1σ(S2n-1)∫S2n-1h(u)1-|z|2|z-u|2ndσ(u),
where σ(S2n-1) is the area measure of S2n-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 S2n-1 (see [12]).Theorem 33 ((Harnack) see [13]).
ConsiderB. Let {Hm} be a monotone sequence of real harmonic functions on B. Suppose that {Hm(x0)} is bounded for some x0∈B. Then, the sequence {Hm} converges uniformly on compact sets to a harmonic function on B.Proposition 34.
Leth:B→ℝ+∪{0} be a continuous function on B. For 0<R<1, let the function hR:S2n-1→ℝ be defined by
(132)hR(u)=max0≤r≤Rh(ru).
Then, hR results in a uniformly continuous function on S2n-1. Besides, if 0<R1<R2<1, then
(133)hR1(u)≤hR2(u),
for every u∈S2n-1.Proof.
The functionh∣B¯ris uniformly continuous; that is, given that ε>0, there exists δ>0, such that |h(z)-h(z′)|<ε, for all z,z′∈BR¯, with |z-z′|<δ. Let u,v∈S2n-1. There exist ru,rv∈[0,R], such that hR(u)=h(ru) and hR(v)=h(rv). Thus, if |u-v|<δ, then |h(ruu)-h(ruv)|<ε and |h(rvv)-h(rvu)|<ε. In particular, hR(u)<ε+h(ruv)≤ε+hR(v) and hR(v)<ε+h(rvu)≤ε+hR(u). Therefore, if |u-v|<δ, then |hR(u)-hR(v)|<ε.
It follows directly that if0<R1<R2, then
(134)max0≤r≤R1h(ru)≤max0≤r≤R2h(ru),
so the result follows.Observe that Poisson's integral formula defines a continuous harmonic functionHR on B, with boundary values given by hR.Corollary 35.
Ifh is as in the previous proposition and for every u∈S2n-1, sup0<R<1hR(u)<∞, define h1:S2n-1→ℝ as
(135)h1(u)=sup0<R<1hR(u).
Then, one has besides that
(136)h1(u)=limR→1-hR(u).Theorem 36.
Ifh:B→[0,∞) is a continuous function and h1:S2n-1→ℝ is the function defined in the previous corollary, let H1:B→ℝ be defined by
(137)H1(z)=sup0≤R<1HR(z),
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,
(138)limR→1-HR(u)=h1(u).
If h1(u)≥0, then the nontangential limits
(139)limz→uH1(z)=h1(u)
exist almost everywhere on S2n-1.Proof.
Let{Rm} be an increasing sequence of positive numbers in (0,1), such that limm→∞Rm=1. For every m∈ℕ, we define the harmonic continuous function
(140)HRm(z)=1σ(S2n-1)∫S2n-1hRm(u)1-|z|2|z-u|2ndσ(u).
Then, if R1≤R2, it follows that
(141)HR1(z)≤HR2(z).
Then, the sequence{HRm} is increasing, so
(142)limm→∞HRm(z)=H1(z),
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
(143)HR(u)=hR(u)≤h1(u),
it follows that HR(u)→h1(u) if R→1- and limz→uH1(u) there exists almost everywhere.
For everyz∈B fixed, let
(144)1σ(S2n-1)hRm(u)1-|z|2|z-u|2n,
this is a sequence of positive Lebesgue’s integrable functions, such that for Rm1<Rm2,
(145)0≤1σ(S2n-1)hRm1(u)1-|z|2|z-u|2n≤1σ(S2n-1)hRm2(u)1-|z|2|z-u|2n.
Then, by Lebesgue's monotone convergence theorem,(146)0≤1σ(S2n-1)∫S2n-1hRm1(u)1-|z|2|z-u|2ndσ(u)⟶1σ(S2n-1)∫S2n-1h1(u)1-|z|2|z-u|2ndσ(u),
as m1→∞.LetΩ⊂ℂn 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 kGhR:S2n-1→ℝ defined by
(147)kGhR(u):=max0≤r≤RlGh(ru)h=maxa=ru,0≤r≤R∫B‍|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z).
By Theorem 12 and Proposition 34, this function is well defined for every 0<R<1 and results in a continuous function on S2n-1. Moreover, as f∈F~G(p,q,s), for 0<R1<R2<1,
(148)kGhR1(u)≤kGhR2(u)≤lGh1(u)≤supu∈S2n-1jGh1(u)={f}p,q,s.LetkGHR be the harmonic function associated with kGhR, that is, the solution of Dirichlet's problem, with boundary values given by kGhR. It follows from Proposition 34 that
(149)supz∈BlGHR(z)≤supz∈BjGH1(z)={f}p,q,s.In the same way as we did forkGhR, we introduce the family of functions hφhR:S2n-1→ℝ defined by
(150)hφhR(u):=max0≤r≤Rhφh(ru)h=maxa=ru,0≤r≤R∫B‍|∇~f(z)|p(1-|z|2)qhhhhhhhhhh×(|1-|φa(z)|2)nsdλ(z).
In this case we obtain Theorem 13 and Proposition 34.Consider(151)supz∈BiφHR(z)≤supz∈BiφH1(z)=supu∈S2n-1iφh1(u)=[f]p,q,s.So, we can summarize the previous results in the next theorems.Theorem 37.
Suppose thatp, q, and s satisfy condition (A). Then, f belongs to the class F~G(p,q,s) if and only if the function iGh1:S2n-1→ℝ defined by
(152)hGh1(u)=supa=ru,0≤r<1∫B|∇~f(z)|p(1-|z|2)qGs(z,a)dλ(z)
is bounded. The function hGH1 is then the Dirichlet solution, with boundary values given by hGh1. Further, hGH1 is the least harmonic majorant of the family of harmonic functions {iGHR}, 0<R<1.Analogous result is obtained for{jφHR} and iφH1.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 thatp, 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 hGH1 is the classical Dirichlet solution with boundary values given by iGh1;(iii)
φh1 is bounded and jφH1 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 hGH1 and kφH1 and their corresponding boundary values lGh1 and lφh1, for the invariant measure.



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