Generalized Hardy-Morrey spaces

TL;DR

This paper introduces a new approach to analyze generalized Hardy-Morrey spaces for 0<p≤1, refining existing estimates and decomposition methods, and correcting previous results, with applications to bilinear inequalities.

Contribution

It proposes a novel decomposition method for generalized Hardy-Morrey spaces and refines estimates for the Hardy-Littlewood maximal operator, especially for 0<p≤1.

Findings

Refined vector-valued estimates for Hardy-Morrey spaces.

Established a new decomposition framework for these spaces.

Corrected previous results related to bilinear estimates.

Abstract

The generalized Morrey space was defined independetly by T. Mizuhara 1991 and E. Nakai in 1994. Generalized Morrey space ${\mathcal M}_{p,\phi}({\mathbb R}^n)$ is equipped with a parameter $0<p<\infty$ and a function $\phi:{\mathbb R}^n \times (0,\infty) \to (0,\infty)$. Our experience shows that ${\mathcal M}_{p,\phi}({\mathbb R}^n)$ is easy to handle when $1<p<\infty$. However, when $0<p \le 1$, the function space ${\mathcal M}_{p,\phi}({\mathbb R}^n)$ is difficult to handle as many examples show. The aim of this paper is twofold. One of them is to propose a way to deal with ${\mathcal M}_{p,\phi}({\mathbb R}^n)$ for $0<p \le 1$. One of them is to propose here a way to consider the decomposition method of generalized Hardy-Morrey spaces. We shall obtain some estimates for these spaces about the Hardy-Littlewood maximal operator. Especially, the vector-valued estimates obtained in the earlier papers are refined. The key tool is the weighted Hardy operator. Much is known about the weighted Hardy operator. Another aim is to propose here a way to consider the decomposition method of generalized Hardy-Morrey spaces. Generalized Hardy-Morrey spaces emerged from generalized Morrey spaces. By means of the grand maximal operator and the norm of generalized Morrey spaces, we can define generalized Hardy-Morrey spaces. With this culmination, we can easily refine the existing results. In particular, our results complement the one the 2014 paper by Iida, the third author and Tanaka; there was a mistake there. As an application, we consider bilinear estimates, which is the \lq \lq so-called" Olsen inequality.