Weighted vector-valued bounds for the singular integral operators with nonsmooth kernels

TL;DR

This paper establishes weak type endpoint estimates and refined weighted bounds for singular integral operators with nonsmooth kernels, extending the understanding of their behavior in vector-valued function spaces.

Contribution

It provides new weak type endpoint estimates and refined weighted bounds for nonsmooth kernel singular integral operators, advancing their theoretical analysis.

Findings

Proved weak type endpoint vector-valued estimates of L log L type.

Established refined weighted vector-valued bounds for the operator.

Extended the theory of singular integral operators with nonsmooth kernels.

Abstract

Let $T$ be a singular integral operator with non-smooth kernel which were introduced by Duong and McIntosh. In this paper, we prove that this operator and its corresponding grand maximal operator satisfies certain weak type endpoint vector-valued estimate of $L\log L$ type. As an application we established a refined weighted vector-valued bound for this operator.