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.
|Journal||Transactions of the American Mathematical Society|
|Publication status||Published - 1 Jul 2010|