An Extension of Calder$\acute{\rm O}$n-Zygmund type singular integral

TL;DR

This paper extends classical Calderón-Zygmund singular integrals, providing $L^q$ estimates using a geometric approach, and recovers known estimates as special cases.

Contribution

It introduces a new extension of Calderón-Zygmund singular integrals and establishes $L^q$ bounds using a geometric method, broadening the scope of classical results.

Findings

Established $L^q$ estimates for the extended singular integral

Recovered classical Calderón-Zygmund estimates from the new framework

Applied geometric approach successfully in the proof

Abstract

In this paper, we consider a kind of singular integral which can be viewed as an extension of the classical Calder$\acute{\rm o}$n-Zygmund type singular integral. We establish an estimate of the singular integral in the $L^q$ space for $1<q<\infty$. In particular, the Calder$\acute{\rm o}$n-Zygmund estimate can be recovered from our obtained estimate. The proof of our main result is via the so called "geometric approach", which was applied in \cite{CP} on the $L^q$ estimate of the elliptic equations and in \cite{LW,Wang} on a new proof of the the Calder$\acute{\rm o}$n-Zygmund estimate.