Logarithmic Convexity for Supremum Norms of Harmonic Functions

Abstract

We prove the following convexity property for supremum norms of harmonic functions. Let Ω be a domain in Rn, Ω0 and E a subdomain and a compact sebset of Ω,respectively. Then there exists a constant α = α(E, Ω0, Ω) ε(0, 1) such that for all harmonic functions u on Ω, the inequality ‖ u ‖ E ⩽ ‖ u ‖ Ω 0 α ‖ u ‖ Ω 1 − α is valid. The case of concentric balls Ω ⊂ E ⊂ Ω plays a key role in the proof. For positive harmonic funcitons ono osuch balls, we determine the sharp constant α in the inequlity.