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

Kamthorn Chailuek·Wicharn Lewkeeratiyutkul
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Abstract

Let $ϕ$ be a real-valued smooth function on $\mathbf{C}$ satisfying $0 \le Δϕ\le M$ for some $M \ge 0$. We consider the space of all holomorphic functions which are square-integrable with respect to the measure $e^{-ϕ(z)} dz$. 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 $|f(z)|^2 \le Ke^{ϕ(z)}\|f\|^2$ for any $f$ in this space and for any complex number $z$.

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Kamthorn Chailuek, Wicharn Lewkeeratiyutkul. 2003-12-17. A Pointwise Bound for a Holomorphic Function which is Square-Integrable with Respect to an Exponential Density Function. https://arxiv.org/abs/math/0312341

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