Characterization and semiadditivity of the c<inf>1</inf>-harmonic capacity

Aleix Ruiz De Villa, Xavier Tolsa

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Abstract

The C1-harmonic capacity ?c plays a central role in problems of approximation by harmonic functions in the C1-norm in ℝn+1. In this paper we prove the comparability between the capacity ?c and its positive version ?c+ . As a corollary, we deduce the semiadditivity of ?c. This capacity can be considered as a generalization in ℝn+1 of the continuous analytic capacity α in ℂ. Moreover, we also show that the so-called inner boundary conjecture fails for dimensions n > 1, unlike in the case n = 1. © 2010 American Mathematical Society.
Original languageEnglish
Pages (from-to)3641-3675
JournalTransactions of the American Mathematical Society
Volume362
Issue number7
DOIs
Publication statusPublished - 1 Jul 2010

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    De Villa, A. R., & Tolsa, X. (2010). Characterization and semiadditivity of the c<inf>1</inf>-harmonic capacity. Transactions of the American Mathematical Society, 362(7), 3641-3675. https://doi.org/10.1090/S0002-9947-10-05105-6