JournalsjemsVol. 22, No. 12pp. 4121–4159

Analytic capacity and projections

  • Alan Chang

    Princeton University, USA
  • Xavier Tolsa

    ICREA and Universitat Autónoma de Barcelona, Bellaterra (Barcelona), Spain
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Abstract

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 ECE\subset \mathbb C is compact and μ\mu is a Borel measure supported on EE, then the analytic capacity of EE satisfies

γ(E)cμ(E)2IPθμ22dθ,\gamma(E) \geq c \frac{\mu(E)^2}{\int_I \|P_\theta\mu\|_2^2\,d\theta},

where cc is some positive constant, I[0,π)I\subset [0,\pi) is an arbitrary interval, and PθμP_\theta\mu is the image measure of μ\mu by PθP_\theta, the orthogonal projection onto the line {reiθ:rR}\{re^{i\theta}:r\in\mathbb R\}. 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. 4121–4159

DOI 10.4171/JEMS/1004