Plurisubharmonic domination in Banach spaces

TL;DR

The paper proves that in Banach spaces with a Schauder basis, any locally bounded function on a pseudoconvex open set can be dominated by the norm of a holomorphic function into another Banach space.

Contribution

It establishes a plurisubharmonic domination result in Banach spaces with Schauder bases, extending complex analysis techniques to infinite-dimensional settings.

Findings

Existence of a Banach space and holomorphic function dominating a given locally bounded function.

Extension of plurisubharmonic domination to Banach spaces with Schauder bases.

Framework for holomorphic domination in infinite-dimensional complex analysis.

Abstract

We show that if $X$ is a Banach space with a Schauder basis, $\Omega\subset X$ is a pseudoconvex open subset, and $u:\,\Omega\to(-\infty,\infty)$ is a locally bounded function, then there are a Banach space $Z$ and a holomorphic function $h:\,\Omega\to Z$ with $u(x)<\|h(x)\|$ for all $x\in\Omega$.