The $L^2$-boundedness of the variational Calder\'on-Zygmund operators

TL;DR

This paper establishes the $L^2$-boundedness of jump functions and variations of Calderón-Zygmund operators with certain kernel conditions, providing a foundational result for variational inequalities beyond convolution kernels.

Contribution

It introduces the first general criteria for variational inequalities for non-convolution Calderón-Zygmund kernels, expanding the scope of $L^2$-boundedness results.

Findings

Proves $L^2$-boundedness for jump functions and variations of Calderón-Zygmund operators.

Identifies kernel conditions ensuring boundedness beyond convolution type.

Lays groundwork for weighted norm inequalities in harmonic analysis.

Abstract

In this paper, we verify the $L^2$-boundedness for the jump functions and variations of Calder\'on-Zygmund singular integral operators with the underlying kernels satisfying \begin{align*}\int_{\varepsilon\leq |x-y|\leq N} K(x,y)dy=\int_{\varepsilon\leq |x-y|\leq N}K(x,y)dx=0\; \forall 0<\varepsilon\leq N<\infty,\end{align*} in addition to some proper size and smooth conditions. This result should be the first general criteria for the variational inequalities for kernels not necessarily of convolution type. The $L^2$-boundedness assumption that we verified here is also the starting point of the related results on the (sharp) weighted norm inequalities appeared in many recent papers.