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

Abstract

We prove that, for ρ>2, the ρ-variation and oscillation for the smooth truncations of the Cauchy transform on Lipschitz graphs are bounded in Lp for 1<p<∞. 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 L2 boundedness of the Cauchy transform on Lipschitz graphs by Coifman, McIntosh, and Meyer.