On a class of \(\overline {\partial }\)-equations without solutions (Q2713588)

The author's approach to the problem of insolvability of the \(\overline {\partial }\)-equation is based on the Bochner-Martinelli integral. Usually for an open set \(D\in \mathbb C^n\) the quotient (\((0,q)\)-\(\overline {\partial }\)-cohomology) NEWLINE\[NEWLINE H^{(0,q)}_{\overline {\partial }}(D)= {\{(0,q)\text{-forms} v\;\text{in} D\;\text{with} \overline {\partial }v=0\}}/ {\{\overline {\partial }:\;u\;\text{is a} (0,q-1)\text{-form in} D\}} NEWLINE\]NEWLINE is used as a measure of ``insolvability'' of the \(\overline {\partial }\)-equation in \(D\) (for \((0,q)\)-forms). If \(K\subset \mathbb C^n\) is a convex compact set then \(H^{(0,n-1)}_{\overline {\partial }}(\mathbb C^n\setminus K)\) is infinite dimensional; the proof can be based on Martineau's theorem on representation of the \(\overline {\partial }\)-cohomology classes as holomorphic functions. NEWLINENEWLINENEWLINEThe proof given in the article works in more general settings (for example, the compact set \(K\) need not be convex). Construction of some classes of \(\overline {\partial }\)-equations without solution involves Bochner-Martinelli type kernels and differentiation with respect to certain parameters in appropriate directions. Other generalizations are included.