Duality of two balayage-space structures and Dirichlet integrals on balayage spaces

Abstract

We discuss the duality (mutual adjointness) of two balayage-space structures ℌ and $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathfrak{H}} $$ on a locally compact spaceX via measure representations σ for ℌ and ĝs for $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathfrak{H}} $$ . Then we show that some properties of the Dirichlet integral δ f (X) of functionsf onX, which have been shown for harmonic spaces having an adjoint harmonic structure, still hold for a balayage space having an adjoint balayage-space structure. In particular, we obtain the relations 0≤δ f (X)+1/2∫f 2 dσ(1)≤∫fdσ(f)<∞ forf=p 1−p 2 with bounded continuous potentialsp j such that ∫pjdσ(p j)<∞,j=1,2.