Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators - Baode Li; Dachun Yang; Wen Yuan (The Scientific World Journal)
Let φ:ℝn×[0,∞)→[0,∞) be a Musielak-Orlicz function and A an expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, HAφ(ℝn), via the grand maximal function. The authors then obtain some real-variable characterizations of HAφ(ℝn) in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy space HAp(ℝn) with p∈(0,1] and are new even for its weighted variant. Finally, the authors characterize these spaces by anisotropic atomic decompositions. The authors also obtain the finite atomic decomposition characterization of HAφ(ℝn), and, as an application, the authors prove that, for a given admissible triplet (φ,q,s), if T is a sublinear operator and maps all (φ,q,s)-atoms with q<∞ (or all continuous (φ,q,s)-atoms with q=∞) into uniformly bounded elements of some quasi-Banach spaces ℬ, then T uniquely extends to a bounded sublinear operator from HAφ(ℝn) to ℬ. These results are new even for anisotropic Orlicz-Hardy spaces on ℝn.