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Revista Matemática Complutense

, Volume 32, Issue 1, pp 37–55 | Cite as

Surjectivity of differential operators and linear topological invariants for spaces of zero solutions

  • T. KalmesEmail author
Article

Abstract

We provide a sufficient condition for a linear differential operator with constant coefficients P(D) to be surjective on \(C^\infty (X)\) and \({\mathscr {D}}'(X)\), respectively, where \(X\subseteq \mathbb {R}^d\) is open. Moreover, for certain differential operators this sufficient condition is also necessary and thus a characterization of surjectivity for such differential operators on \(C^\infty (X)\), resp. on \({\mathscr {D}}'(X)\), is derived. Additionally, we obtain for certain surjective differential operators P(D) on \(C^\infty (X)\), resp. \({\mathscr {D}}'(X)\), that the spaces of zero solutions \(C_P^\infty (X)=\{u\in C^\infty (X);\, P(D)u=0\}\), resp. \({\mathscr {D}}_P'(X)=\{u\in {\mathscr {D}}'(X);\,P(D)u=0\}\) possess the linear topological invariant \((\Omega )\) introduced by Vogt and Wagner (Stud. Math. 68:225–240, 1980), resp. its generalization \((P\Omega )\) introduced by Bonet and Domański (J. Funct. Anal. 230:329–381, 2006).

Keywords

Surjectivity of differential operator Linear topological invariants for kernels of differential operators Differential operators on vector-valued spaces of functions and distributions Parameter dependence for solutions of linear partial differential equations 

Mathematics Subject Classification

Primary 35E10 46A63 Secondary 35E20 

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Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany

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