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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2301.02617 |
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| _version_ | 1866910646708207616 |
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| author | Debrouwere, Andreas Kalmes, Thomas |
| author_facet | Debrouwere, Andreas Kalmes, Thomas |
| contents | 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 $X \subseteq \mathbb{R}^d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_02617 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Linear topological invariants for kernels of differential operators by shifted fundamental solutions Debrouwere, Andreas Kalmes, Thomas Functional Analysis 46A63, 35E20, 46M18 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 $X \subseteq \mathbb{R}^d$. |
| title | Linear topological invariants for kernels of differential operators by shifted fundamental solutions |
| topic | Functional Analysis 46A63, 35E20, 46M18 |
| url | https://arxiv.org/abs/2301.02617 |