Albert Mas, Xavier Tolsa
Published 2011-01-10, updated 2011-09-02Version 2
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.
On the compactness of oscillation and variation of commutators
$L^p$-to-$L^q$ boundedness for commutators of the Cauchy transform
Rectifiable measures, square functions involving densities, and the Cauchy transform
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