Solvability of invariant second-order differential operators on metabelian groups (Q1110662)

As the author points out, this paper is the third in a series applying group representations to obtain solvability properties of differential operators. Indeed, the reader will want to have ``Sobolev criteria for solvability of invariant differential operators'' [\textit{R. Lipsman}, Commun. Partial Differ. Equations 12, 327-349 (1987; Zbl 0615.58047)] available as a reference. Let the solvable group \(G=SN\), be a semidirect product where N is a normal simply connected abelian subgroup and dim S\(=1\). Let \({\mathcal N}\) and \({\mathcal S}\) be the corresponding Lie algebras and fix \(A\in {\mathcal S}\), \(A\neq 0\). Finally, let \(\Lambda\) be the matrix \(Ad_{{\mathcal N}} A.\) Theorem. Suppose that all the eigenvalues of \(\Lambda\) have positive real part. Then the heat operator, the Schrödinger operator, and the wave operator are globally solvable. - The proof of the theorem is based on a more technical lemma which guarantees global solvability for more general second order differential operators satisfying certain conditions.