The K\"othe dual of mixed Morrey spaces and applications

TL;DR

This paper explores the duality of mixed Morrey spaces, establishes boundedness of maximal functions, and characterizes BMO functions via commutators of fractional integrals, advancing the understanding of these function spaces.

Contribution

It proves the K"othe duality between mixed Morrey and block spaces and applies this to characterize BMO via fractional integral commutators.

Findings

The block space is the K"othe dual of the mixed Morrey space.

Boundedness of Hardy--Littlewood maximal function on the block space.

Characterizations of BMO via fractional integral commutators.

Abstract

In this paper, we study the separable and weak convergence of mixed-norm Lebesgue spaces. Furthermore, we prove that the block space $\mathcal{B}_{\vec{p}\,'}^{p'_0}(\mathbb{R}^n)$ is the K\"othe 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 $\mathcal{B}_{\vec{p}\,'}^{p'_0}(\mathbb{R}^n)$. 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 $\mathcal{B}_{\vec{p}\,'}^{p'_0}(\mathbb{R}^n)$.