On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators

TL;DR

This paper investigates the invariance of certain vanishing subspaces of Morrey spaces under classical harmonic analysis operators, revealing their stability and structural properties relevant to functional analysis.

Contribution

It demonstrates that specific vanishing subspaces of Morrey spaces remain invariant under key harmonic analysis operators, extending understanding of their structural stability.

Findings

Vanishing subspaces are invariant under Hardy-Littlewood maximal operator.

These subspaces are preserved under singular integral operators.

Vanishing properties are maintained under Riesz potential and fractional maximal operators.

Abstract

We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of $C_0^\infty(\mathbb{R}^n)$ in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy-Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.