\(q\)-plurisubharmonicity and \(q\)-pseudoconvexity in \(\mathbb C^n\)
Let \(\varOmega\subset\mathbb C^n\) be open. We say that an upper semicontinuous function \(u:\varOmega\longrightarrow[-\infty,+\infty)\) is \(q\)-plurisubharmonic (\(q\)-psh, \(u\in\mathcal{PSH}_q(\varOmega)\), \(0\leq q\leq n-1\)) if for every complex linear \((q+1)\)-dimensional subspace \(L\), for every closed ball \(B\subset L\cap\varOmega\), and for every smooth plurisuperharmonic function \(g\) defined on an open neighborhood \(U\subset L\cap\varOmega\) of \(B\), if \(u\leq g\) on \(\partial B\), then \(u\leq g\) on \(B\). We say that \(u\) is \textit{strictly \(q\)-psh} on \(\varOmega\) if for every point \(z_0\in\varOmega\) there exist a neighborhood \(U\) and \(\varepsilon>0\) such that the function \(U\ni z\longmapsto u(z)-\varepsilon\|z\|^2\) is \(q\)-psh. A domain \(\varOmega\subset\mathbb C^n\) is \(q\)-pseudoconvex if there exists a neighborhood \(U\) of \(\partial\varOmega\) such that the function \(U\cap\varOmega\ni z\longmapsto -\log\operatorname{dist}(z,\partial\varOmega)\) is \(q\)-psh. A bounded domain \(\varOmega\subset\mathbb C^n\) is \(q\)-hyperconvex if it has a negative continuous \(q\)-psh exhaustion function. A bounded domain \(\varOmega\subset\mathbb C^n\) is \(B_q\)-regular if for every \(\varphi\in\mathcal C(\partial\varOmega, \mathbb R)\) there exists a \(u\in\mathcal C(\overline\varOmega)\cap\mathcal{PSH}_q(\varOmega) \cap(-\mathcal{PSH}_{n-q-1}(\varOmega))\) with \(u=\varphi\) on \(\partial\varOmega\). A compact set \(K\subset\mathbb C^n\) is \(B_q\)-regular if \(\mathcal C(K)\) coincides with the closure in \(\mathcal C(K)\) of the space of all functions that are continuous on \(K\) and \(q\)-psh in a neighborhood of \(K\). The main results of the paper are the following two theorems. (1) Let \(\varOmega\subset\mathbb C^n\) be a bounded domain. (i) If \(\varOmega\) is \(q\)-convex and \(\partial\varOmega\) is \(B_r\)-regular (\(0\leq r\leq n-q\)), then \(\varOmega\) is \(B_{q+r}\)-regular. (ii) Assume that \(\varOmega\) is \(B_q\)-regular. Then \(\varOmega\) is \(q\)-hyperconvex. Moreover, \(\partial\varOmega\) is \(B_q\)-regular if at any point \(z_0\in\partial\varOmega\) one of the following conditions holds: (a) there exist a neighborhood \(U\) of \(z_0\) and \(u\in\mathcal C(U)\cap\mathcal{PSH}_q(U)\) with \(u(z_0)=1\), \(u(z)<1\), \(z\in U\cap\partial\varOmega\setminus\{z_0\}\); (b) there exist a neighborhood \(U\) of \(z_0\) and \(u\in\mathcal C(\overline{U\cap\varOmega})\cap\mathcal C^2(U\cap\varOmega)\) such that \(U\cap\partial\varOmega\) is \(\mathcal C^1\)-smooth, \(u(z_0)=1\), \(u(z)<1\), \(z\in \overline{U\cap\varOmega}\setminus\{z_0\}\), and at every point from \(U\cap\varOmega\) the Levi form of \(u\) is positive definite on a complex linear \(q\)-codimensional subspace. (2) Let \(\varOmega\subset\mathbb C^n\) be a \(q\)-pseudoconvex domain. Then for every \(u\in\mathcal{PSH}_r(\varOmega)\) there exists a sequence \((u_k)_{k=1}^\infty\) of piecewise smooth strictly \(\max\{q,r\}\)-psh functions on \(\varOmega\) such that \(u_k\searrow u\).
scientific article; zbMATH DE number 5140582