Solvability of invariant differential operators on metabelian groups (Q1175086)

Let \(G=NS\) be a solvable Lie group which is a semidirect product of simply connected abelian groups \(N\) and \(S\), where \(N\) is normal in \(G\) and \(S\) is one dimensional. Let \(L\) be a left invariant differential operator on \(G\). By definition, \(L\) is semiglobally solvable if for every \(f\in C_ c^{\infty}(G)\) and for every relatively compact open set \(U\) there exists \(u\in C_ c^{\infty}(G)\) such that \(Lu=f\) on \(U\). This paper deals with the question of whether a given differential operator on such a Lie group is semiglobally solvable. Using the partial Fourier transform over \(N\), \(L\) is equivalent to a direct integral over \(\xi\) in the dual of the Lie algebra of \(N\) of operators \(\pi^{\xi}(L)\) acting on \(L^ 2(\mathbb{R})\). The idea is that the operators \(\pi^{\xi}(L)\) are more basic than \(L\) and the question is reduced to the solvability of these component operators. Theorem. Suppose there exists a polynomial \(p\) such that for every \(f\in C_ c^{\infty}(G)\), \(\pi^{\xi}(L)u_{\xi}=f_{\xi}\) is solvable with \(\xi\to p(\xi)u_{\xi}(t)\) a tempered distribution for a.e. \(t\in\mathbb{R}\). Then \(L\) is semiglobally solvable. The author discusses the history of the problem and the relationship between this result and other similar results in the literature. A significant part of the paper is devoted to applications of the theorem above. Specific elliptic, sub-elliptic and parabolic operators are shown to satisfy the hypotheses of the theorem.