Monge-Ampère measures associated to extremal plurisubharmonic functions in $\textbf {C}^ n$

Abstract:

We consider the extremal plurisubharmonic functions $L_E^\ast$ and $U_E^\ast$ associated to a nonpluripolar compact subset $E$ of the unit ball $B \subset {{\mathbf {C}}^n}$ and show that the corresponding Monge-Ampère measures ${(d{d^c}L_E^\ast )^n}$ and ${(d{d^c}U_E^\ast )^n}$ are mutually absolutely continuous. We then discuss the polynomial growth condition $({L^\ast })$, a generalization of Leja’s polynomial condition in the plane, and study the relationship between the asymptotic behavior of the orthogonal polynomials associated to a measure on $E$ and the $({L^\ast })$ condition.