On the Solvability of Linear Differential Operators on Vector Bundles over a Manifold

Abstract

A necessary and sufficient condition is established for the closedness of the range or surjectivity of a differential operator acting on smooth sections of vector bundles. For connected noncompact manifolds it is shown that these conditions are derived from the regularity conditions and the unique continuation property of solutions. An application of these results to elliptic operators (more precisely, to operators with a surjective principal symbol) with analytic coefficients, to second-order elliptic operators on line bundles with a real leading part, and to the Hodge–Laplace–de Rham operator is given. It is shown that the top de Rham (respectively, Dolbeault) cohomology group on a connected noncompact smooth (respectively, complex-analytic) manifold vanishes. For elliptic operators, we prove that solvability in smooth sections implies solvability in generalized sections.