Holomorphic Spaces in the Unit Ball of

We introduce the and vector spaces of holomorphic functions defined in the unit ball of , 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 be the open unit disk in the complex plane . For , let be the M？bius transformation defined by For , we denote Green's function of with logarithmic singularity at by The Bloch space is defined as the set of analytic functions , such that For , Aulaskari and Lappan [1] introduced in 1994 the spaces as the family of analytic functions satisfying For , this definition was extended in [2]. Theorem 1 (see [2]). Let and be an analytic function. Then, if and only if With the aim of generalizing and enclosing several weighted function spaces, Zhao in [3] introduced the spaces in the next way. Let , , , and be an analytic function. We will say that belongs to if satisfies the integral condition So far, Theorem 1 is generalized for functions of by the following. Theorem 2 (see [3]). Let , , , and be an analytic function. Then, if and only if Zhao has shown that for certain intervals of , , and , makes as a Banach space. In this paper, we present the spaces of holomorphic functions in the unit ball of , that generalize the 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 -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 and , that is, when we use in the integral representation, the invariant Green function, or when we use the biholomorphism . We remark that the generalization to several complex variables of requires, for and , 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 and in and , respectively. In Section 2.2, we present what is the most natural Bloch space associated to 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 and , generalizing Proposition 3.4 of [4] and