Homogeneous mixed Herz-Morrey spaces and its Applications

TL;DR

This paper introduces homogeneous mixed Herz-Morrey spaces, explores their properties, and investigates the boundedness of various operators and their commutators within these spaces.

Contribution

It defines a new class of function spaces and proves boundedness results for important operators, extending existing theory to these spaces.

Findings

Boundedness of sublinear operators in the new spaces

Boundedness of fractional type and Calderón-Zygmund operators

Boundedness of commutators in these spaces

Abstract

In this paper, we introduce homogeneous mixed Herz-Morrey spaces $M\dot{K}_{p,\vec{q}}^{\alpha,\lambda}(\mathbb{R}^n)$ and show it's some properties. Firstly, the boundedness of sublinear operators, fractional type operators in homogeneous mixed Herz-Morrey spaces is investigated. In particular, the above results are still valid for Calder$\acute{o}$n-Zygmund operators and fractional maximal operators. Lastly, the boundedness of their commutators in homogeneous mixed Herz-Morrey spaces is obtained.