Molecular characterization of anisotropic Musielak-Orlicz Hardy spaces and their applications

Let A be an expansive dilation on .n and . : .n . [0, .) . [0, .) an anisotropic Musielak.Orlicz function. Let H .A(.n) be the anisotropic Hardy space of Musielak.Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of H .A(.n). The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case (namely, A := 2I n.n ) coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space Hp(.n) with p . (0, 1] (in this case, A := 2I n.n , .(x, t) := t p for all x . .n and t . [0, .)), this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Calderón.Zygmund operators from H .A(.n) to L .(.n) or from H .A(.n) to itself.