Boundedness of singular integral operators with variable kernels on weighted weak Hardy spaces

TL;DR

This paper establishes the boundedness of variable kernel singular integral operators on weighted weak Hardy spaces using atomic decomposition, under certain Dini-type conditions on the kernel.

Contribution

It introduces new boundedness results for $T_ abla$ on weighted weak Hardy spaces with variable kernels, expanding understanding of these operators' behavior.

Findings

Boundedness of $T_ abla$ on weighted weak Hardy spaces is proven.

Atomic decomposition techniques are effectively used for analysis.

Results depend on Dini-type smoothness conditions of the kernel.

Abstract

Let $T_\Omega$ be the singular integral operator with variable kernel $\Omega(x,z)$. In this paper, by using the atomic decomposition theory of weighted weak Hardy spaces, we will obtain the boundedness properties of $T_\Omega$ on these spaces, under some Dini type conditions imposed on the variable kernel $\Omega(x,z)$.