Capacity convergence results and applications to a Berstein-Markov inequality

Abstract:

Given a sequence $\{E_{j}\}$ of Borel subsets of a given non-pluripolar Borel set $E$ in the unit ball $B$ in $\mathbf {C}^{N}$ with $E \subset \subset B$, we show that the relative capacities $C(E_{j})$ converge to $C(E)$ if and only if the relative (global) extremal functions $u_{E_{j}}^{*}$ ($V_{E_{j}}^{*}$) converge pointwise to $u_{E}^{*}$ ($V_{E}^{*}$). This is used to prove a sufficient mass-density condition on a finite positive Borel measure with compact support $K$ in $\mathbf {C}^{N}$ guaranteeing that the pair $(K,\mu )$ satisfy a Bernstein-Markov inequality. This implies that the $L^{2}-$orthonormal polynomials associated to $\mu$ may be used to recover the global extremal function $V_{K}^{*}$.