A note on continuity of strongly singular Calder\'on-Zygmund operators in Hardy-Morrey spaces

TL;DR

This paper investigates the continuity of strongly singular Calderón-Zygmund operators on Hardy-Morrey spaces under weaker kernel conditions, extending the understanding of their boundedness in harmonic analysis.

Contribution

It establishes the boundedness of these operators on Hardy-Morrey spaces with less restrictive kernel assumptions, including important pseudodifferential operators.

Findings

Operators are continuous on Hardy-Morrey spaces under weaker conditions.

Includes pseudodifferential operators in Hörmander classes as key examples.

Extends previous results on Calderón-Zygmund operators.

Abstract

In this note we address the continuity of strongly singular Calder\'on-Zygmund operators on Hardy-Morrey spaces $\mathcal{HM}_{q}^{\lambda}(\mathbb{R}^n)$, assuming weaker integral conditions on the associated kernel. Important examples that falls into this scope are pseudodifferential operators on the H\"ormander classes $OpS^{m}_{\sigma,\mu}(\mathbb{R}^n)$ with $0<\sigma \leq 1$, $0 \leq \mu <1$, $\mu \leq \sigma$ and $m\leq -n(1-\sigma)/2$.