# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 36, Issue 1, 2017, pp. 17–35**

**DOI: 10.4171/ZAA/1577**

Published online: 2017-01-06

Generalized Morrey Spaces – Revisited

Ali Akbulut^{[1]}, Vagif Sabir Guliyev

^{[2]}, Takahiro Noi

^{[3]}and Yoshihiro Sawano

^{[4]}(1) Ahi Evran University, Kirsehir, Turkey

(2) Ahi Evran University, Kirsehir, Turkey

(3) Tokyo Metropolitan University, Japan

(4) Tokyo Metropolitan University, Japan

The generalized Morrey space ${\mathcal M}_{p,\phi}({\mathbb R}^n)$ was defined by Mizuhara 1991 and Nakai in 1994. It 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. We propose a way to deal with ${\mathcal M}_{p,\phi}({\mathbb R}^n)$ 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.

*Keywords: *Generalized Morrey spaces, decomposition, maximal operators

Akbulut Ali, Guliyev Vagif Sabir, Noi Takahiro, Sawano Yoshihiro: Generalized Morrey Spaces – Revisited. *Z. Anal. Anwend.* 36 (2017), 17-35. doi: 10.4171/ZAA/1577