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Weighted pluripotential theory in CN
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 125, Number 1, February 2003
- pp. 57-103
- 10.1353/ajm.2003.0002
- Article
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Let K ⊂ CN 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 supζ→zVK,Q(ζ) where VK,Q(z) := sup{u(z): u plurisubharmonic in CN, 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 VK := VK,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 CN, then d(E) = d(F) if and only if V*E = V*F where d(E) denotes the transfinite diameter of E.