Generalized Hardy-Morrey spaces

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.