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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Solvability of differential equations with linear coefficients of real type


Author: Rainer Felix
Journal: Trans. Amer. Math. Soc. 293 (1986), 583-591
MSC: Primary 58G05; Secondary 22E30, 35A05
DOI: https://doi.org/10.1090/S0002-9947-1986-0816311-9
MathSciNet review: 816311
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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].)


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DOI: https://doi.org/10.1090/S0002-9947-1986-0816311-9
Keywords: Divergences, invariant distributions, differential operators with critical points
Article copyright: © Copyright 1986 American Mathematical Society