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