Monge-Ampère measures associated to extremal plurisubharmonic functions in 𝐶ⁿ

Abstract

We consider the extremal plurisubharmonic functions L E ∗ L_E^\ast and U E ∗ U_E^\ast associated to a nonpluripolar compact subset E E of the unit ball B ⊂ C n B \subset {{\mathbf {C}}^n} and show that the corresponding Monge-Ampère measures ( d d c L E ∗ ) n {(d{d^c}L_E^\ast )^n} and ( d d c U E ∗ ) n {(d{d^c}U_E^\ast )^n} are mutually absolutely continuous. We then discuss the polynomial growth condition ( L ∗ ) ({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 E and the ( L ∗ ) ({L^\ast }) condition.