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Plurisubharmonic domination. (English) Zbl 1042.32013

The main result of the paper is the following domination theorem.
Let \(X\) be a Banach space with countable unconditional basis and let \(\Omega\subset X\) be pseudoconvex. Let \(u:\Omega\to\mathbb R\) be locally bounded from above. Then there exist a separable Banach space \(V\) and a holomorphic mapping \(f:\Omega\to V\) such that \(u(x)\leq\| f(x)\| \), \(x\in\Omega\).
The author presents various applications of the above result. The idea is the following one. Given a function \(u\) which pointwise controls a parameter of an analytic object (in such a way that \(u\) is locally bounded from above), we construct a corresponding exhaustion \(\Omega_j:=\{x\in\Omega: \| f(x)\| <j\}\), \(j=1,2,\dots\), of \(\Omega\) by pseudoconvex open sets.

MSC:

32U05 Plurisubharmonic functions and generalizations
46G20 Infinite-dimensional holomorphy
32T35 Exhaustion functions
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