The $L^2$-boundedness of the variational Calder\'on-Zygmund operators
Functional Analysis
2020-09-10 v1
Abstract
In this paper, we verify the -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 -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.
Keywords
Cite
@article{arxiv.2009.04066,
title = {The $L^2$-boundedness of the variational Calder\'on-Zygmund operators},
author = {Y. Chen and G. Hong},
journal= {arXiv preprint arXiv:2009.04066},
year = {2020}
}