On Berndtsson's generalization of Prékopa's theorem
type: Item
instance of: scholarly article
title: On Berndtsson's generalization of Prékopa's theorem
zbMATH Open document ID: 1079.32020
author: D. Cordero-Erausquin
published in: Mathematische Zeitschrift
publication date: 2005-04-14T00:00:00Z
review text: Let \(V\subset\subset {\mathbb C}^n\) be a strictly pseudo-convex domain. Let \(\phi:{\mathbb C}_0\times \overline V\rightarrow {\mathbb R}\) be a smooth plurisubharmonic function such that \(\Phi(w):=\phi(0,w)\) is uniformly plurisubharmonic on \(V\). Assume that the function \[ w\rightarrow \frac{\partial \phi}{\partial z}(0,w) \] is orthogonal in \(L^2(e^{-\Phi})\) to \[ \left\{h\in\,L^2(e^{-\Phi}): \overline\partial h=0\text{ and } \int h(w)e^{-\Phi(w)}\,d\lambda(w)=0\right\}. \] In this paper, the author proves that the function \(\Psi\) which is given by \[ e^{-\Psi(z)}=\int_V e^{-\phi(z,w)}\,d\lambda(w) \] satisfies \(\Delta\Psi(0)\geq 0\). Here \(d\lambda\) is the Lebesgue measure on \({\mathbb C}^n\cong{\mathbb R}^{2n}\).
Mathematics Subject Classification ID: 32U05
zbMATH DE Number: 2156051
zbMATH Keywords: Prekopa's theorem
zbMATH Keywords: Berndtsson extension
zbMATH Keywords: Brunn-Minkowski theory
zbMATH Keywords: Kiselman's minimum principle
reviewed by: Der-Chen Chang
MaRDI profile type: Publication
full work available at URL: s00209-004-0704-6
OpenAlex ID: W2030909718
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DOI: 10.1007/S00209-004-0704-6
description: scientific article; zbMATH DE number 2156051