Anisotropic Hardy Spaces Of Musielak-Orlicz Type

The real-variable theory of Hardy spaces plays a key role in harmonic analysis, which has an important application in various fields of analysis and partial differential equations. Let φ:Rn×[0, oo)â†'[0, oo) be a Musielak-Orlicz function and A an expansive dilation. This article mainly studies the theory of anisotropic Hardy spaces of Musielak-Orlicz type, HAφ(Rn), and related operators. Precisely, this article is organized as follows.Chapter 1 is devoted to the background, current research situation and the main re-sults of HA(Rn) and related operators.In Chapter 2, we recall the definitions of the anisotropic growth function and HAφ(Rn), their atomic variants (denoted by HAφ,q,s(Rn)), and their variants via the Lusin-area func-tion (denoted by HS,Aφ(Rn)). Then we show HAφ,q,s(Rn)= HS,Aφ(Rn) with equivalent (quasi-norms by the anisotropic Calderon reproducing formula, which, together with the known conclusion HAφ,q,s(Rn)=HAφ(Rn), further implies that HAφ(Rn)=HS,Aφ(Rn) with equivalent quasi-)norms. The anisotropic Littlewood-Paley g-function and the gλ*-function char-acterizations of HAφ(Rn) are established via the discrete anisotropic Calderon reproducing formula, an anisotropic Peetre’s inequality of Musielak-Orlicz type and a Fefferman-Stein vector-valued inequality of Musielak-Orlicz type.In Chapter 3, the notion of the anisotropic Musielak-Orlicz tent space TAφ(Rn x Z) is introduced and its anisotropic atomic decompositions is obtained. The anisotropic φ-Carleson measure characterization of BMOAφ(Rn) is established via the anisotropic atomic decomposition of TAφ(Rn x Z).In Chapter 4, we obtain the molecular characterization of HAφ(Rn) via the radical maximal function characterization of HAφ(Rn). Then, as an application of the molecular characterization of HAφ(Rn), the boundedness of anisotropic Calderon-Zygmund operators from HAφ(Rn) to Lφ(Rn) or from HAφ(Rn) to itself is obtained.