On the behavior of strictly plurisubharmonic functions near real hypersurfaces
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 problem here is, as the author again points out, just to find the curves \(\gamma _2, \dots ,\gamma_n\). There are few minor misprints in the paper.
scientific article; zbMATH DE number 5319809
scientific article