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 is compact and is a Borel measure supported on , then the analytic capacity of satisfies
where is some positive constant, is an arbitrary interval, and is the image measure of by , the orthogonal projection onto the line . 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.
Cite this article
Alan Chang, Xavier Tolsa, Analytic capacity and projections. J. Eur. Math. Soc. 22 (2020), no. 12, pp. 4121–4159DOI 10.4171/JEMS/1004