A Pointwise Bound for a Holomorphic Function which is Square-Integrable with Respect to an Exponential Density Function

Abstract

Let $\varphi$ be a real-valued smooth function on $\mathbf{C}$ satisfying $0\le \Delta \varphi \le M$ for some $M\ge 0$. We consider the space of all holomorphic functions which are square-integrable with respect to the measure $e^{-\varphi (z)}dz$. In this paper, a pointwise bound for any function in this space is established. We show that there exists a constant $K$ depending only on $M$ such that $|f(z){|}^2\le K{e}^{\varphi (z)}\Vert f{\Vert }^2$ for any $f$ in this space and for any complex number $z$.

8 pages, no figure

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