Hypoellipticité pour l'opérateur \({\bar\partial}\). (Hypoellipticity of the \({\bar\partial}\)-operator)
Let \(\Omega \subset \subset {\mathbb{C}}^ n\) be a pseudoconvex domain with \({\mathcal C}^{\infty}\) boundary. The operator \({\bar \partial}\) is hypoelliptic in \(\Omega\) for (0,1)-forms if when \(\alpha\) is a (0,1)- form, \({\bar \partial}\) closed, with coefficients in \(L^ 2_{loc}(\Omega)\) there is \(u\in L^ 2_{loc}(\Omega)\) such that \(\partial u=\alpha\) and \(\sin g \sup p u=\sin g \sup p \alpha.\) We prove that \({\bar \partial}\) is hypoelliptic for (0,1)-forms if and only if for every compact \(K\subset {\bar \Omega}\) we have \((1)\quad K\cap \partial \Omega =\hat K\cap \partial \Omega.\) Here \(\hat K\) denotes the hull of K with respect to functions continuous in \({\bar \Omega}\) and plurisubharmonic in \(\Omega\). The necessity of the condition is due to Catlin. A result of propagation of singularities is given when condition (1) is not satisfied. Under a condition stronger than (1) it is possible to construct a solution u in \(L^ 2(\Omega)\) when \(\alpha\) is with coefficients in \(L^ 2(\Omega)\) and such that \(\sin g \sup p u=\sin g \sup p \alpha.\)
hypoelliptic \({\bar \partial }\)-operator
scientific article; zbMATH DE number 3946606