Lipschitz harmonic capacity and bilipschitz images of Cantor sets
The Lipschitz harmonic capacity \(k(E)\) of a compact set \(E\) in \(\mathbb R^n,\;n\geq 2\), plays a role in the theory of Lipschitz harmonic functions, analogous to the role of analytic capacity in the theory of bounded holomorphic functions. In particular \(k(E)=0\) if and only if \(E\) is removable for Lipschitz harmonic functions. The authors conjecture that if \(T:\mathbb R^n\to \mathbb R^n\) is a biLipschitz homeomorphism with constant \(A\), then \(k(T(E))\leq C(A)\,k(E);\) (an analogous result for analytic capacity has recently been proved by Tolsa). They prove this conjecture when \(E\) is a generalized Cantor set. A crucial step of the proof is an estimate of the \(L^2\) norms of the Riesz transforms on \(L^2(G,p)\), where \(p\) is a natural probability measure on \(E\) and \(G\subset E\) has \(p(G)>0\).
scientific article; zbMATH DE number 5121803