# CAPACITY

The ${\cal C}^{1}\text{-harmonic}$ capacity κ c plays a central role in problems of approximation by harmonic functions in the ${\cal C}^{1}\text{-norm}$ in ${\Bbb R}^{n+1}$ . In this paper we prove the comparability between the capacity κ c and its positive version $\kappa _{+}^{c}$ . As a... Full description

1st Person: DE VILLA, ALEIX RUIZ TOLSA, XAVIER in Transactions of the American Mathematical Society Vol. 362, No. 7 (2010), p. 3641-3675 More Articles Article English 2010 Volltext
Summary: The ${\cal C}^{1}\text{-harmonic}$ capacity κ c plays a central role in problems of approximation by harmonic functions in the ${\cal C}^{1}\text{-norm}$ in ${\Bbb R}^{n+1}$ . In this paper we prove the comparability between the capacity κ c and its positive version $\kappa _{+}^{c}$ . As a corollary, we deduce the semiadditivity of κ c . This capacity can be considered as a generalization in ${\Bbb R}^{n+1}$ of the continuous analytic capacity α in ${\Bbb C}$ . Moreover, we also show that the so-called inner boundary conjecture fails for dimensions n > 1, unlike in the case n = 1. Copyright: © 2010 American Mathematical Society Online-Ressource 0002-9947

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