On the behavior of strictly plurisubharmonic functions near real hypersurfaces
type: Item
instance of: scholarly article
title: On the behavior of strictly plurisubharmonic functions near real hypersurfaces
zbMATH Open document ID: 1148.32014
author: Björn Ivarsson
published in: Arkiv för Matematik
publication date: 2008-09-03T00:00:00Z
review text: The author describes the behavior of certain strictly plurisubharmonic functions near some real hypersurfaces in \({\mathbb{C}}^n\), \(n\geq 3\). His work is related, among others, to certain previous results of Walsh, of Caffarelli, Kohn, Nirenberg and Spruck, and of Błocki. The main result of the paper is, roughly, the following: If one can find complex curves which have sufficiently high contact order with the hypersurface, then the studied, certain type plurisubharmonic functions cannot be globally Lipschitz in an one sided neighborhood. To be more exact, the result of the author reads as follows: Theorem 2.2: Let \(M\) be a smooth real hypersurface in \({\mathbb{C}}^n\), with a defining function \(\rho\in {\mathcal{C}}^\infty (\Omega )\) (\(\Omega\) an open set in \({\mathbb{C}}^n\), containing \(M\)), thus \(M=\{z\in \Omega :\rho (z)=0\}\) with \(d\rho \neq 0\) on \(M\). Let \(M\) be pseudoconvex at \(p\in M\). Assume that there are complex curves \(\gamma _2, \dots ,\gamma _n\) whose order of contact with \(M\) at \(p\) is \(2l_2,\dots ,2l_n\), respectively, and that \(\gamma _2',\dots ,\gamma _n'\) are linearly independent. Suppose that \[ \sum_{j=2}^{n} \frac{1}{l_j}<n-2. \] Then \(p\) does not satisfy the local non-precipitousness condition (local NP-condition). That this local NP-condition of the author does not hold, means the following: If \(U\) is any neighborhood of \(p\) and \(u\) any smooth plurisubharmonic function defined on \(M^-\cap U\), where \(M^-:=\{ z\in \Omega :\rho (z)<0\}\), such that \(\lim_{z\rightarrow z_0}u(z)=0\) for all \(z_0\in M\cap U\), and \[ {\text{det}}\left( \frac{\partial ^2u}{\partial z_j\partial \overline{z}_k}\right)\geq 1, \] then \(u\) cannot be (globally) Lipschitz on \(M^-\cap U\). Theorem~2.2 gives information of course only in the case \(n\geq 3\). But, as the author points out in his interesting additional remarks, then the theorem gives plenty of examples of domains which do not satisfy the NP-condition. The probl…
reviewed by: Juhani Riihentaus
Mathematics Subject Classification ID: 32U15
Mathematics Subject Classification ID: 32U10
zbMATH DE Number: 5319809
zbMATH Keywords: plurisubharmonic functions
zbMATH Keywords: real hypersurfaces
zbMATH Keywords: Monge-Ampére mass
zbMATH Keywords: complex curves
zbMATH Keywords: Lipschitz continuity
MaRDI profile type: Publication
full work available at URL: s11512-006-0029-2
OpenAlex ID: W2081060070
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DOI: 10.1007/S11512-006-0029-2
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description: scientific article; zbMATH DE number 5319809