The Köthe dual of mixed Morrey spaces and applications

Abstract

In this paper, we study the separable and weak convergence of mixed-norm Lebesgue spaces. Furthermore, we prove that the block space ${\mathcal{B}}_{\vec{p}{}^{\prime }}^{p_0^{\prime }}({\mathbb{R}}^n)$ is the Köthe dual of the mixed Morrey space ${\mathcal{M}}_{\vec{p}}^{p_0}({\mathbb{R}}^n)$ by the Fatou property of these block spaces. The boundedness of the Hardy--Littlewood maximal function is further obtained on the block space ${\mathcal{B}}_{\vec{p}{}^{\prime }}^{p_0^{\prime }}({\mathbb{R}}^n)$. As applications, the characterizations of $BMO({\mathbb{R}}^n)$ via the commutators of the fractional integral operator $I_{\alpha }$ on mixed Morrey spaces are proved as well as the block space ${\mathcal{B}}_{\vec{p}{}^{\prime }}^{p_0^{\prime }}({\mathbb{R}}^n)$.