Generalized Morrey Spaces – Revisited

Abstract

The generalized Morrey space ${\mathcal{M}}_{p,\varphi }({\mathbb{R}}^n)$ was defined by Mizuhara 1991 and Nakai in 1994. It is equipped with a parameter $0<p<\infty$ and a function $\varphi :{\mathbb{R}}^n\times (0,\infty )\to (0,\infty )$. Our experience shows that ${\mathcal{M}}_{p,\varphi }({\mathbb{R}}^n)$ is easy to handle when $1<p<\infty$. However, when $0<p\le 1$, the function space ${\mathcal{M}}_{p,\varphi }({\mathbb{R}}^n)$ is difficult to handle as many examples show. We propose a way to deal with ${\mathcal{M}}_{p,\varphi }({\mathbb{R}}^n)$ for $0<p\le 1$, in particular, to obtain some estimates of the Hardy–Littlewood maximal operator on these spaces. Especially, the vector-valued estimates obtained in the earlier papers are refined. The key tool is the weighted dual Hardy operator. Much is known on the weighted dual Hardy operator.