Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications

Abstract

Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations {A^k : k ∈ Z}, where A is a real n × n matrix with all its eigenvalues λ satisfy |λ|>1. The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let ϕ: Rⁿ×[0,∞) →[0,∞) be an anisotropic p-growth function with p ∈ (0, 1]. The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type {{\displaystyle H}}_A^{\phi ,\infty }({ℝ}^n) and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to {{\displaystyle H}}_A^{\phi ,\infty }({ℝ}^n) and the boundedness of the anisotropic Calderón-Zygmund operator from {{\displaystyle H}}_A^{\phi ,\infty }({ℝ}^n) to L^{\phi ,\infty }({ℝ}^n). It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of {{\displaystyle H}}_A^{\phi ,\infty }({ℝ}^n) and the interpolation theorem.

Key words： Expansive dilation; Muckenhoupt weight; weak Hardy space; Musielak-Orlicz function; atomic decomposition; Calderón-Zygmund operator

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