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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2301.02617 |
| Etiquetas: |
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- We characterize the condition $(Ω)$ for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions $(PΩ)$ and $(P\overline{\overlineΩ})$ for distributional kernels are characterized in a similar way. By lifting theorems for Fréchet spaces and (PLS)-spaces, this provides characterizations of the problem of parameter dependence for smooth and distributional solutions of differential equations by shifted fundamental solutions. As an application, we give a new proof of the fact that the space $\{ f \in \mathscr{E}(X) \, | \, P(D)f = 0\}$ satisfies $(Ω)$ for any differential operator $P(D)$ and any open convex set $X \subseteq \mathbb{R}^d$.