Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators. (16th March 2014) - Li, Baode; Yang, Dachun; Yuan, Wen
Abstract : Let φ : ℝ n × [ 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, H A φ ( ℝ n ), via the grand maximal function. The authors then obtain some real-variable characterizations of H A φ ( ℝ n ) in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy space H A p ( ℝ n ) 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 H A φ ( ℝ n ), 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 H A φ ( ℝ n ) to ℬ . These results are new even for anisotropic Orlicz-Hardy spaces on ℝ n .