Mathematical Research Letters

Volume 13 (2006)

Number 6

Lipschitz harmonic capacity and Bilipschitz images of cantor sets

Pages: 865 – 884



John Garnett (University of California at Los Angeles)

Laura Prat (Universitat de Barcelona)

Xavier Tolsa (Universitat Autonoma de Barcelona)


For bilipschitz images of Cantor sets in $\Rd$ we estimate the Lipschitz harmonic capacity and prove that this capacity is invariant under bilipschitz homeomorphisms. A crucial step of the proof is an estimate of the $L^2$ norms of the Riesz tranforms on $L^2(G,p)$ where $p$ is the natural probability measure on the Cantor set $E$ and $G \subset E$ has $p(G) > 0.$

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