Characterization and semiadditivity of the 𝒞¹-harmonic capacity

This is the authors' abstract. We don't add key points for this paper.

The C 1 C^1 -harmonic capacity κ c κ ^c plays a central role in problems of approximation by harmonic functions in the C 1 C^1 -norm in R n + 1 Rⁿ⁺¹ . In this paper we prove the comparability between the capacity κ c κ ^c and its positive version κ + c κ ^c_+ . As a corollary, we deduce the semiadditivity of κ c κ ^c . This capacity can be considered as a generalization in R n + 1 Rⁿ⁺¹ of the continuous analytic capacity α α in C C . Moreover, we also show that the so-called inner boundary conjecture fails for dimensions n > 1 n>1 , unlike in the case n = 1 n=1 .