Generalized Hardy–Morrey Spaces

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This paper is an off-spring of the contribution [Z. Anal. Anwend. 36 (2017)(1), 17–35]. We propose 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 what we have culminated for the Hardy–Littlewood maximal operator, we can easily refine the existing results. As an application, we consider bilinear estimates, which is the "so-called" Olsen inequality. In particular, our results complement the one in the 2014 paper by Iida, the fourth author and Tanaka [Z. Anal. Anwend. 33 (2014)(2), 149–170]; there were two mistakes. One lies in the decomposition result and another lies in the proof of the Olsen inequality.