On geometric properties of harmonic Lip₁-capacity

Pertti Mattila and P. V. Paramonov

Vol. 171 (1995), No. 2, 469–491

DOI: 10.2140/pjm.1995.171.469

Abstract

We shall study geometric properties of the harmonic Lip₁-capacity κ′_n(E), E ⊂ Rⁿ. It is related to functions which are harmonic outside E and locally Lipschitzian everywhere. We shall show that κ′_n+1(E × I) is comparable to κ′_n(E) for E ⊂ Rⁿ and for intervals I ⊂ R. We shall also show that if E lies on a Lipschitz graph, then κ′_n(E) is comparable to the (n − 1)-dimensional Hausdorff measure ℋⁿ⁻¹(E). Finally we give some general criteria to guarantee that κ′_n(E) = 0 although ℋⁿ⁻¹(E) > 0.

Mathematical Subject Classification 2000

Primary: 31B05

Secondary: 28A75, 42B20

Milestones

Received: 9 March 1993

Published: 1 December 1995