O razreshimosti lineynykh differentsial'nykh operatorov na vektornykh rassloeniyakh nad mnogoobraziem

Abstract

Necessary and sufficient condition is established for the closedness of the rangeor surjectivity of a differential operator acting on smooth sections of vector bundles. For connectednoncompact manifolds it is shown that these conditions are derived from the regularityconditions and the unique continuation property of solutions. An application of these results toelliptic operators (more precisely, to operators with a surjective principal symbol) with analyticcoefficients, to second-order elliptic operators on line bundles with a real leading part, and to theHodge–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 impliessolvability in generalized sections.