Analytic capacity and projections

In this paper we study the connection between the analytic capacity of a set and the size of its orthogonal projections. More precisely, we prove that if E ⊂ C is compact and μ is a Borel measure supported on E, then the analytic capacity of E satisfies (equation presented) where c is some positive constant, I ⊂ (0; π) is an arbitrary interval, and Pθμ is the image measure of μ by Pθ , the orthogonal projection onto the line freiθ V r 2 Rg. This result is related to an old conjecture of Vitushkin about the relationship between the Favard length and analytic capacity. We also prove a generalization of the above inequality to higher dimensions which involves related capacities associated with signed Riesz kernels.