Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs

TL;DR

This paper establishes boundedness of r-variation and oscillation for certain singular integrals with odd kernels on Lipschitz graphs, extending classical L^2 results to L^p spaces for r>2.

Contribution

It proves new boundedness results for variation and oscillation of singular integrals with odd kernels on Lipschitz graphs, generalizing classical theorems.

Findings

Bounded r-variation and oscillation for Cauchy transform on Lipschitz graphs for r>2.

Extension of boundedness results to Riesz transforms and other odd kernel operators.

Strengthening of classical L^2 boundedness theorems to L^p for 1<p<∞.

Abstract

We prove that, for r>2, the r-variation and oscillation for the smooth truncations of the Cauchy transform on Lipschitz graphs are bounded in L^p for 1<p finite. The analogous result holds for the n-dimensional Riesz transform on n-dimensional Lipschitz graphs, as well as for other singular integral operators with odd kernel. In particular, our results strengthen the classical theorem on the L^2 boundedness of the Cauchy transform on Lipschitz graphs by Coifman, McIntosh, and Meyer.