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Harmonic Analysis

Analytic capacity, the Cauchy transform, and curvature of measures

speaker: Xavier Tolsa (ICREA / Universitat Autònoma de Barcelona)

abstract: In the first lecture I will talk about the characterization of analytic capacity in terms of Menger curvature and its comparability with $\gamma_+$ (which in particular implies that analytic capacity is semiadditive.

In the second lecture I will talk about two facts: The first one is that analytic capacity is ``invariant'' under bilipschitz mappings. The second one is that if the Cauchy tranform is bounded on $L2(\mu)$, then any sufficiently smooth Calderon-Zygmund operator with odd kernel is also bounded in $L2(\mu)$. The proof of both results relies on a corona type decomposition for non doubling measures.

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