Weighted pluripotential theory in C N

Abstract

Let K ⊂ C N be compact and let w be a nonnegative, uppersemicontinuous function on K with { z ∈ K : w ( z ) > 0} nonpluripolar. Let Q := - log w and define the weighted pluricomplex Green function V * K,Q ( z ) = lim sup ζ→ z V K,Q (ζ) where V K,Q ( z ) := sup{ u ( z ): u plurisubharmonic in C N , u ( z ) ≤ log + | z | + C, u ≤ Q on K } ( C depends on u ). If w ≡ 1; i.e., Q ≡ 0, we are in the unweighted case and we write V K := V K ,0 . We prove weighted generalizations of several results in pluripotential theory, and we prove a version of a unique continuation property of maximal psh functions. These results are used to show that if E ⊂ F are compact subsets of C N , then d ( E ) = d ( F ) if and only if V * E = V * F where d ( E ) denotes the transfinite diameter of E .