Weighted Bernstein-Markov property in \(\mathbb{C}^n\) (Q2893042)

Let \(E\) be a Borel (not necessarily bounded) non-pluripolar subset of \({\mathbb C}^n\), \(w\) be a positive upper semicontinuous function defined on \(E\) and \(\mu\) be a positive Borel measure on \(E\). One say that the triple \((E,w,\mu)\) has the Bernstein-Markov property if there is a strongly comparability between \(L^2\) and \(L^1\) norms of weighted polynomials on \(E\). In this paper, the authors study conditions guaranteeing that the triple \((E,w,\mu)\) has the Bernstein-Markov property. In the first main theorem, certain sufficient condition in terms of convergence of sequences of weighted Greeen function is given. This is strongly related to the work of Bloom and Bloom-Levenberg on the same subject. The second main theorem deal essentially with the Bernstein-Markov property of \((E,w,\mu)\) when \(E\) is assumed to be locally regular in \({\mathbb C}^n\). Finally, as an application the authors follow the work of Bloom and Shiffman to prove the convergence of certain Bergman kernels to weighted Green functions.