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

Let A be an expansive dilation on ℝ and φ : ℝ × [0, ∞) → [0, ∞) an anisotropic Musielak-Orlicz function. Let H (ℝ) 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 (ℝ). 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 := 2 I ) coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space H(ℝ) with p ∈ (0, 1] (in this case, A := 2 I , φ( x, t) := t for all x ∈ ℝ and t ∈ [0, ∞)), this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from H (ℝ) to L (ℝ) or from H (ℝ) to itself. [ABSTRACT FROM AUTHOR]