Partial Dynamical Systems of $L^p$-Spaces and their Stability Spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908605020635136 |
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| author | Okeke, N. O. Egwe, M. E. |
| author_facet | Okeke, N. O. Egwe, M. E. |
| contents | Using the convolution product and weak derivatives, we consider the partial dynamical systems of the locally convex $L^p(Ω)$ spaces defined by the action of the smooth algebra $\mathscr{K}(Ω)$ through its nets. Slice analysis is then employed to show that the Sobolev spaces $W^{k,p}(Ω)$ are the stable states or space of these partial dynamical systems as limit spaces of the convolution actions of the smooth algebra $K(Ω)$ on the Banach spaces $L^p(Ω)$. Thus, the Sobolev spaces $W^{k,p}(Ω)$ are closed subspaces of the $Lp(Ω)$-spaces under convolution product and weak derivatives, with the weak derivative operators acting as equivariant maps of the slice spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18828 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partial Dynamical Systems of $L^p$-Spaces and their Stability Spaces Okeke, N. O. Egwe, M. E. Functional Analysis 46Fxx, 46L55, 46F10, 37Cxx Using the convolution product and weak derivatives, we consider the partial dynamical systems of the locally convex $L^p(Ω)$ spaces defined by the action of the smooth algebra $\mathscr{K}(Ω)$ through its nets. Slice analysis is then employed to show that the Sobolev spaces $W^{k,p}(Ω)$ are the stable states or space of these partial dynamical systems as limit spaces of the convolution actions of the smooth algebra $K(Ω)$ on the Banach spaces $L^p(Ω)$. Thus, the Sobolev spaces $W^{k,p}(Ω)$ are closed subspaces of the $Lp(Ω)$-spaces under convolution product and weak derivatives, with the weak derivative operators acting as equivariant maps of the slice spaces. |
| title | Partial Dynamical Systems of $L^p$-Spaces and their Stability Spaces |
| topic | Functional Analysis 46Fxx, 46L55, 46F10, 37Cxx |
| url | https://arxiv.org/abs/2403.18828 |