Generalized Morrey Spaces - Revisited

The generalized Morrey space M-p,M-phi(R-n) was defined by Mizuhara 1991 and Nakai in 1994. It is equipped with a parameter 0 < p < infinity and a function phi : R-n x (0, infinity) -> (0, infinity). Our experience shows that M-p,M-phi(R-n) is easy to handle when 1 < p < infinity. However, when 0 < p <= 1, the function space M-p,M-phi(R-n) is difficult to handle as many examples show. We propose a way to deal with M-p,M-phi(R-n) for 0 < p <= 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.