Solvability of differential equations with linear coefficients of real type

Abstract:

Let $L$ be the infinitesimal generator associated with a flow on a manifold $M$. Regarding $L$ as an operator on a space of testfunctions we deal with the question if $L$ has closed range. (Questions of this kind are investigated in [4, 1, 2].) We provide conditions under which $L + \mu 1:\mathcal {S}(M) \to \mathcal {S}(M)$, $\mu \in {\mathbf {C}}$, has closed range, where $M = {{\mathbf {R}}^n} \times K$, $K$ being a compact manifold; here $\mathcal {S}(M)$ is the Schwartz space of rapidly decreasing smooth functions. As a consequence we show that the differential operator ${\Sigma _{i,j}}{a_{ij}}{x_j}(\partial /\partial {x_i}) + b$ defines a surjective mapping of the space $\mathcal {S}({{\mathbf {R}}^n})$ of tempered distributions onto itself provided that all eigenvalues of the matrix $({a_{ij}})$ are real. (In the case of imaginary eigenvalues this is not true in general [3].)