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

Abstract

We provide a sufficient condition for a linear differential operator with constant coefficients $P(D)$ to be surjective on $C^{\infty }(X)$ and ${\mathcal{D}}^{\prime }(X)$, 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 ${\mathcal{D}}^{\prime }(X)$, is derived. Additionally, we obtain for certain surjective differential operators $P(D)$ on $C^{\infty }(X)$, resp. ${\mathcal{D}}^{\prime }(X)$, that the spaces of zero solutions $C_P^{\infty }(X)=\{u\in C^{\infty }(X);P(D)u=0\}$, resp. ${\mathcal{D}}_P^{\prime }(X)=\{u\in {\mathcal{D}}^{\prime }(X);P(D)u=0\}$ possess the linear topological invariant $(\Omega )$ introduced by Vogt and Wagner in [27], resp. its generalization $(P\Omega )$ introduced by Bonet and Domański in [1].

16 pages. This updated version emphasizes the implications of our results for the spaces of zero solutions to possess certain linear topological invariants. Apart from a revised introduction this version contains an additional section on said invariants and surjectivity of differential operators on vector-valued functions/distributions. In our opinion, this update justifies a change of the title

References in corpus (2)

• The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective
• Every P-convex subset of ${\mathbb{R}}^2$ is already strongly P-convex

Cited by in corpus (6)

• Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions
• Linear topological invariants for kernels of convolution and differential operators
• The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes
• An approximation theorem of Runge type for kernels of certain non-elliptic partial differential operators
• Quantitative Runge type approximation theorems for zero solutions of certain partial differential operators
• Boundary values of zero solutions of hypoelliptic differential operators in ultradistribution spaces