The lattice structure of negative Sobolev and extrapolation spaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866916641495842816 |
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| author | Arora, Sahiba Glück, Jochen Schwenninger, Felix L. |
| author_facet | Arora, Sahiba Glück, Jochen Schwenninger, Felix L. |
| contents | It is well-known that the Sobolev spaces $W^{k,p}(\mathbb R^d)$ are vector lattices with respect to the pointwise almost everywhere order if $k \in \{0,1\}$, but not if $k \ge 2$. In this note, we consider negative $k$ and show that the span of the positive cone in $W^{k,p}(\mathbb R^d)$ is a vector lattice in this case.
We also prove a related abstract result: if $(T(t))_{t \in [0,\infty)}$ is a positive $C_0$-semigroup on a Banach lattice $X$ with order continuous norm, then the span of the cone $X_{-1,+}$ in the extrapolation space $X_{-1}$ is a vector lattice. This complements results obtained by Bátkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The lattice structure of negative Sobolev and extrapolation spaces Arora, Sahiba Glück, Jochen Schwenninger, Felix L. Functional Analysis 46B40, 46B42, 46E35 It is well-known that the Sobolev spaces $W^{k,p}(\mathbb R^d)$ are vector lattices with respect to the pointwise almost everywhere order if $k \in \{0,1\}$, but not if $k \ge 2$. In this note, we consider negative $k$ and show that the span of the positive cone in $W^{k,p}(\mathbb R^d)$ is a vector lattice in this case. We also prove a related abstract result: if $(T(t))_{t \in [0,\infty)}$ is a positive $C_0$-semigroup on a Banach lattice $X$ with order continuous norm, then the span of the cone $X_{-1,+}$ in the extrapolation space $X_{-1}$ is a vector lattice. This complements results obtained by Bátkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems. |
| title | The lattice structure of negative Sobolev and extrapolation spaces |
| topic | Functional Analysis 46B40, 46B42, 46E35 |
| url | https://arxiv.org/abs/2404.02116 |