On compactness of the \(\bar{\partial}\)-Neumann operator on \(r\)-convex domains
Let \(\Omega \subset \mathbb C^n\) be a domain. An upper semicontinuous function \(u : \Omega \to [-\infty, \infty)\), \(u \not\equiv -\infty\), is called \(r\)-subharmonic if for every \(r\)-dimensional complex plane \(L \subset \mathbb C^n\), \(u|L\) is a subharmonic function on \(L \cap \Omega\). The set of all \(r\)-subharmonic functions on \(\Omega\) is denoted by \(SH_r (\Omega)\). A domain \(\Omega \subset \mathbb C^n\) is said to be \(r\)-convex if it has an \(r\)-subharmonic exhaustion function. Let \(SH_q (\partial\Omega)\) denote the boundary values of \(q\)-subharmonic functions in the sense that \(f \in SH_q (\partial\Omega)\) if \(f\) is an upper semicontinuous function on \(\partial\Omega\) and there exists a compact \(K \subset \Omega\) and \(u \in SH_q (\Omega \;K)\) such that \[ \lim_ {\Omega\backslash K \ni z\to w} u(z) = f (w) \] for all \(w \in \partial\Omega\). A closed subset \(S\) of \(\partial\Omega\) is said to be a \(q\)-peak set if there exists \(u \in SH_q (\partial\Omega)\) such that \(u = 0\) on \(S\) and \(u < 0\) on \(\partial\Omega \backslash S\). The main result of the paper is the following theorem. Theorem. Let \(r \leq q \leq n\) and \(\Omega\) be a bounded \(r\)-convex domain in \(\mathbb C^n\). Assume that for every \(z \in \partial\Omega\) there exists a complex affine subspace \(L_z\) of dimension less than or equal to \(q - 1\) through \(z\) such that \(L_z \cap \partial\Omega\) is a \(q\)-peak set. Then the \(\overline{\partial}\)-Neumann operator \(N_q\) is compact on the square integrable \((0, q)\)-forms.
compactness of \(\bar{\partial}\)-Neumann operator
scientific article; zbMATH DE number 6388863
scientific article