Bounded point evaluation in $\textbf {C}^ n$

Abstract:

A positive Borel measure $\mu$ on a domain $\Omega \in {{\mathbf {C}}^n}$ is said to be in $\mathcal {R}(\Omega )$, if point evaluations at every $p \in \Omega$ are locally uniformly bounded in ${L^2}(\mu )$-norm. It is proved that the multiplication of a measure in $\mathcal {R}(\Omega )$ by a function decreasing no faster than a power of a holomorphic function produces a measure in $\mathcal {R}(\Omega )$. Some applications to classical Hardy and Bergman spaces are given.