Abstract
The Lipschitz and C{script} 1 harmonic capacities κ and κ c in R{double-struck} n can be considered as highdimensional versions of the so-called analytic and continuous analytic capacities γ and α (resp.). In this article we provide a dual characterization of κ c 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(∂ o E) for any compact set E ⊂ R{double-struck} n , where ∂ o E 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. © 2010.
Original language | English |
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Pages (from-to) | 1-22 |
Journal | Duke Mathematical Journal |
Volume | 153 |
DOIs | |
Publication status | Published - 1 May 2010 |