Atomic Decomposition for Morrey Spaces

Abstract

The Hardy space $H^p({\mathbb{R}}^n)$ substitutes for the Lebesgue space $L^p({\mathbb{R}}^n)$. When $p>1$, then the Hardy space $H^p({\mathbb{R}}^n)$ coincides with the Lebesgue spaces $L^p({\mathbb{R}}^n)$. This is shown by using the reflexivity of the function spaces. The atomic decomposition is readily available for $H^p({\mathbb{R}}^n)$ with $0<p<\infty$. This idea can be applied to many function spaces. As example of such an attempt, we now propose here a non-smooth decomposition of Morrey spaces. As applications, we consider the Olsen inequality. In the end of this article, we compare our results with existing ones and propose some possibility of extensions, which are left as future works.