Calder\'{o}n-Zygmund Operators with Non-diagonal Singularity

TL;DR

This paper extends Calderón-Zygmund operators to cases where the singularity set is a hyper curve rather than the diagonal, maintaining key boundedness properties.

Contribution

It introduces a new class of singular integral operators with non-diagonal singularities and proves their boundedness properties similar to classical Calderón-Zygmund operators.

Findings

Operators are of weak-type (1,1)

Operators are of strong type (p,p) for 1<p<∞

Generalizes classical Calderón-Zygmund theory

Abstract

In this paper, we introduce a class of singular integral operators which generalize Calder\'on-Zygmund operators to the more general case, where the set of singular points of the kernel need not to be the diagonal, but instead, it can be a general hyper curve. We show that such operators have similar properties as ordinary Calder\'on-Zygmund operators. In particular, we prove that they are of weak-type $(1, 1)$ and strong type $(p,p)$ for $1<p<\infty$.