Hypoellipticity and local solvability of pseudolocal continuous linear operators in Gevrey classes
The author considers a linear map on the Gevrey functions of order \(s> 1\) and the corresponding ultradistributions, \(P: G^s_0(\Omega)\to G^s(\Omega)\), \(E_s'(\Omega)\to D_s'(\Omega)\), \(\Omega\subset\mathbb{R}^n\). Assume that \(P\) is \(s\)-pseudo-local in the sense that the following inclusions hold for the Gevrey \(s\)-singular supports: \[ s\text{-sing supp}(Pu)\subset s\text{-sing supp}\,u,\quad u\in E_s'(\Omega). \] The first result of the paper states that if \(P\) is \(s\)-hypoelliptic in \(\Omega\), then the transposed operator \({^tP}\) is \(s\)-locally solvable in \(\Omega\). This generalizes a result of \textit{A. A. Albanese}, \textit{A. Corli} and \textit{L. Rodino} [Math. Nachr. 242, 5--16 (2002; Zbl 1056.35049)] concerning the case when \(P\) is a linear partial differential operator. New interesting applications are given here to \(s\)-hypoelliptic pseudo-differential operators. Further results concern the existence of a fundamental kernel \(K(x, y)\in D_s'(\Omega\times\Omega)\) for \({^tP}\), under the \(s\)-hypoellipticity assumption for \(P\).
scientific article; zbMATH DE number 2150094