Analytic capacity and projections

2017/12/02 by Alan Chang, Xavier Tolsa, Chang, Alan +1 · 2 citations
Mathematics · #Mathematical Approximation and Integration #Advanced Harmonic Analysis Research #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.1712.00594

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 E⊂ \mathbb C is compact and μ is a Borel measure supported on E, then the analytic capacity of E satisfies γ(E) ≥ c (μ(E)²)/(∫_I ‖P_θμ‖₂² dθ), where c is some positive constant, I⊂ [0,π) is an arbitrary interval, and P_θμ is the image measure of μ by P_θ, the orthogonal projection onto the line \re^iθ:r∈\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.

Cited by

• Sharp Favard length of random Cantor sets
• Besicovitch-Federer projection theorem for measures

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