Some properties of holomorphic generalized functions on \(C^ \infty\)-strictly pseudoconvex domains
For an open subset \(V\) of \(\mathbb{C}^n\) by \({\mathcal G}(V)\) will be denoted the Colombeau algebra of generalized functions on \(V\), and by \({\mathcal H}G(V)\) will be denoted the algebra of holomorphic generalized functions on \(V\), i.e. the elements of \({\mathcal G} \cap \text{Ker}(\overline{\partial})\). The main theorem in the present paper says that for every \(C^\infty\)-strictly pseudoconvex subdomain \(\Omega\) with \(\overline{\Omega} \subset V\) there exists a representative \(\widehat{f}\) of \(f|_\Omega\), \(f \in {\mathcal H} G(V)\) such that \(\widehat {f} (\varphi, \cdot)\) is holomorphic in \(\Omega\) for every \(\varphi \in A_0\). Applications to the problem of the \(\overline{\partial}\)-equation on open pseudoconvex domains in \(\mathbb{C}^n\) are given.
scientific article; zbMATH DE number 878123