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.