# On holomorphic domination, I - Mathematics > Complex Variables

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Abstract: Let $X$ be a separable Banach space and $u{:} X\to\Bbb{R}$ locally upperbounded. We show that there are a Banach space $Z$ and a holomorphic function$h{:} X\to Z$ with $ux<\|hx\|$ for $x\in X$. As a consequence we find thatthe sheaf cohomology group $H^qX,\Cal{O}$ vanishes if $X$ has the boundedapproximation property i.e., $X$ is a direct summand of a Banach space with aSchauder 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 $X=L 10,1$ of summablefunctions, then there is a $C^1$-smooth function $u$ on $X$ with$\overline\partial u=f$ on $X$.

Author: ** Imre Patyi**

Source: https://arxiv.org/