Anisotropic Function Spaces And The Boundedness Of Related Operators

Anisotropy is a common attribute of the nature, which shows different characteri-zations 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 general discrete group of dilations{Ak:k ∈ Z}, where A is a real n×n matrix with all its eigenvalues A sat-isfy|λ|> 1. φ is an anisotropic Musielak-Orlicz function with growth assumptions. The aim of this article is to study the atomic decomposition characterization of the anisotropic Musielak-Orlicz weak Hardys,HAφ,∞(Rn) and their applications. Precisely, this article is organized as follows.In Chapter 1, we introduce the background, current research situation and the main results of this article of HAφ,∞(Rn) and related operators.In Chapter 2, we first recall the definitions of the Musielak-Orlicz functions, ex-pansive dilations and anisotropic Muckenhoupt weights. And then, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type,HAφ,∞(Rn), via anisotropic grand maximal function, and obtain its atomic decomposition.In Chapter 3, as applications, we obtain an interpolation theorem adapted to HAφ,∞(Rn) and the boundedness of anisotropic Calderon-Zygmund operator from HAφ,∞(Rn) to Lφ,∞(Rn).