On the continuity of strongly singular Calder\'on-Zygmund-type operators on Hardy spaces

TL;DR

This paper proves the continuity of strongly singular Calderón-Zygmund operators on Hardy spaces under weaker conditions, including applications to pseudodifferential operators and weighted boundedness.

Contribution

It establishes new continuity results for a class of singular integral operators on Hardy spaces with weaker kernel conditions, extending their boundedness to weighted spaces.

Findings

Operators are bounded on Hardy spaces under weaker H"ormander conditions.

Includes classes of pseudodifferential operators and standard δ-kernels.

Strongly singular operators are bounded on weighted Hardy and Lebesgue spaces.

Abstract

In this work, we establish results on the continuity of strongly singular Calder\'on-Zygmund operators of type $\sigma$ on Hardy spaces $H^p(\mathbb{R}^n)$ for $0<p\leq 1$ assuming a weaker $L^{s}-$type H\"ormander condition on the kernel. Operators of this type include appropriated classes of pseudodifferential operators $OpS^{m}_{\sigma,b}(\mathbb{R}^n)$ and operators associated to standard $\delta$-kernels of type $\sigma$ introduced by \'Alvarez and Milman. As application, we show that strongly singular Calder\'on-Zygmund operators are bounded from $H^{p}_{w}(\mathbb{R}^n)$ to $L^{p}_{w}(\mathbb{R}^n)$, where $w$ belongs to a special class of Muckenhoupt weight.