Duke Mathematical Journal

A dual characterization of the C1 harmonic capacity and applications

Albert Mas, Mark Melnikov, and Xavier Tolsa

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Abstract

The Lipschitz and C1 harmonic capacities κ and κc in Rn can be considered as high-dimensional versions of the so-called analytic and continuous analytic capacities γ and α (resp.). In this article we provide a dual characterization of κc in the spirit of the classical one for the capacity α by means of the Garabedian function. Using this new characterization, we show that κ(E)=κ(∂oE) for any compact set E⊂Rn, where ∂oE is the outer boundary of E, and we solve an open problem posed by A. Volberg, which consists in estimating from below the Lipschitz harmonic capacity of a graph of a continuous function.

Article information

Source
Duke Math. J., Volume 153, Number 1 (2010), 1-22.

Dates
First available in Project Euclid: 28 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1272480930

Digital Object Identifier
doi:10.1215/00127094-2010-019

Mathematical Reviews number (MathSciNet)
MR2641938

Zentralblatt MATH identifier
1196.31002

Subjects
Primary: 31B05: Harmonic, subharmonic, superharmonic functions
Secondary: 31B15: Potentials and capacities, extremal length

Citation

Mas, Albert; Melnikov, Mark; Tolsa, Xavier. A dual characterization of the ${\mathcal C}^{1}$ harmonic capacity and applications. Duke Math. J. 153 (2010), no. 1, 1--22. doi:10.1215/00127094-2010-019. https://projecteuclid.org/euclid.dmj/1272480930


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