A class of domains with noncompact \(\overline{\partial }\)-Neumann operator (Q2839305)

Let \(n_1,n_2\) be positive integers, and let \(n=n_1+n_2\). For a point \(z=(z_1, \dots, z_n)\in \mathbb{C}^n\), we write \('z=(z_1, \dots , z_{n_1})\) and \(z'=(z_{n_1+1}, \dots , z_n)\). Let \(\alpha >0\), and consider the bounded pseudoconvex domain \(H\) (Hartogs triangle) given by \(H= \big\{ ('z,z') \in \mathbb{C}^n : \|'z\|_{(1)} < (\|z'\|_{(2)})^\alpha <1\big \}\), where \(\|.\|_{(1)}\) and \(\|.\|_{(2)}\) are arbitrary norms for \(\mathbb{C}^{n_1} \) and \(\mathbb{C}^{n_2}\), respectively. The boundary \(bH\) of \(H\) contains the piece \(S =\big\{ ('z,z')\in \mathbb{C}^n : \|'z\|_{(1)} < 1, \, \|z'\|_{(2)}=1 \big\}= \mathbb{B}_1 \times b\mathbb{B}_2\), where \(\mathbb{B}_1\) and \(\mathbb{B}_2\) are the unit balls in \(\mathbb{C}^{n_1} \) and \(\mathbb{C}^{n_2}\) respectively. Let \(\Omega\) be a bounded pseudoconvex domain in \(\mathbb{C}^n\). Suppose that there exists a biholomorphic map \(F: H \longrightarrow F(H)\subset \Omega\), which extends to a \(\mathcal{C}^\infty\)-diffeomorphism of a neighborhood of \(\overline H\) onto a neighborhood of \(\overline{F(H)}\) in \(\mathbb{C}^n\), and suppose that \(F\) itself extends biholomorphically to \(S\) in such a way that \(F(S)\subset b\Omega\). It is shown that the \(\overline \partial\)-Neumann operator of \(\Omega\) acting on \(L^2_{(0,q)}(\Omega)\) is noncompact for \(1\leq q \leq n_1.\) The proof uses a generalization of a method for the polydisc introduced by \textit{S. G. Krantz} [Proc. Am. Math. Soc. 103, No. 4, 1136--1138 (1988; Zbl 0736.35071)].