TY - JOUR
T1 - A dual characterization of the C{script} 1 harmonic capacity and applications
AU - Mas, Albert
AU - Melnikov, Mark
AU - Tolsa, Xavier
PY - 2010/5/1
Y1 - 2010/5/1
N2 - 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.
AB - 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.
U2 - https://doi.org/10.1215/00127094-2010-019
DO - https://doi.org/10.1215/00127094-2010-019
M3 - Article
VL - 153
SP - 1
EP - 22
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
SN - 0012-7094
ER -