On holomorphic domination, I

TL;DR

This paper proves holomorphic domination for locally upper bounded functions on Banach spaces, leading to cohomology vanishing results and solving the $ar{ ext{d}}$-problem for smooth forms on $L_1[0,1]$.

Contribution

It introduces holomorphic domination in Banach spaces and derives new cohomology vanishing and $ar{ ext{d}}$-problem solutions.

Findings

Holomorphic domination for locally upper bounded functions on Banach spaces.

Vanishing of sheaf cohomology groups $H^q(X,\mathcal{O})$ under certain conditions.

Existence of smooth solutions to the $ar{\partial}$-problem on $L_1[0,1]$.

Abstract

Let $X$ be a separable Banach space and $u{:} X\to\Bbb{R}$ locally upper bounded. We show that there are a Banach space $Z$ and a holomorphic function $h{:} X\to Z$ with $u(x)<\|h(x)\|$ for $x\in X$. As a consequence we find that the sheaf cohomology group $H^q(X,\Cal{O})$ vanishes if $X$ has the bounded approximation property (i.e., $X$ is a direct summand of a Banach space with a Schauder basis), $\Cal{O}$ is the sheaf of germs of holomorphic functions on $X$, and $q\ge1$. As another consequence we prove that if $f$ is a $C^1$-smooth $\overline\partial$-closed $(0,1)$-form on the space $X=L_1[0,1]$ of summable functions, then there is a $C^1$-smooth function $u$ on $X$ with $\overline\partial u=f$ on $X$.