This is an announcement for the paper "On holomorphic domination, I" by Imre Patyi.

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 $X$.

Archive classification: math.CV math.FA

Mathematics Subject Classification: 32U05; 32L10; 46G20

The source file(s), holodom-I-3.tex: 35922 bytes, is(are) stored in gzipped form as 0910.0476.gz with size 12kb. The corresponding postcript file has gzipped size 82kb.

Submitted from: i355p113@speedpost.net

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