(1,p)-Sobolev spaces based on strongly local Dirichlet forms
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929209513869312 |
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| author | Kuwae, Kazuhiro |
| author_facet | Kuwae, Kazuhiro |
| contents | In the framework of quasi-regular strongly local Dirichlet form $(\mathscr{E},D(\mathscr{E}))$ on $L^2(X;\mathfrak{m})$ admitting minimal $\mathscr{E}$-dominant measure $μ$, we construct a natural $p$-energy functional $(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$ on $L^p(X;\mathfrak{m})$ and $(1,p)$-Sobolev space $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ for $p\in]1,+\infty[$. In this paper, we establish the Clarkson type inequality for $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$. As a consequence, $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ is a uniformly convex Banach space, hence it is reflexive. Based on the reflexivity of $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$, we prove that (generalized) normal contraction operates on $(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$, which has been shown in the case of various concrete settings, but has not been proved for such general framework. Moreover, we prove that $(1,p)$-capacity ${\rm Cap}_{1,p}(A)<\infty$ for open set $A$ admits an equilibrium potential $e_A\in D(\mathscr{E}^{\,p})$ with $0\leq e_A\leq 1$ $\mathfrak{m}$-a.e. and $e_A=1$ $\mathfrak{m})$-a.e.~on $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11652 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | (1,p)-Sobolev spaces based on strongly local Dirichlet forms Kuwae, Kazuhiro Probability Functional Analysis 31C25, 30L15, 53C21, 58J35, 35K05, 42B05, 47D08 In the framework of quasi-regular strongly local Dirichlet form $(\mathscr{E},D(\mathscr{E}))$ on $L^2(X;\mathfrak{m})$ admitting minimal $\mathscr{E}$-dominant measure $μ$, we construct a natural $p$-energy functional $(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$ on $L^p(X;\mathfrak{m})$ and $(1,p)$-Sobolev space $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ for $p\in]1,+\infty[$. In this paper, we establish the Clarkson type inequality for $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$. As a consequence, $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ is a uniformly convex Banach space, hence it is reflexive. Based on the reflexivity of $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$, we prove that (generalized) normal contraction operates on $(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$, which has been shown in the case of various concrete settings, but has not been proved for such general framework. Moreover, we prove that $(1,p)$-capacity ${\rm Cap}_{1,p}(A)<\infty$ for open set $A$ admits an equilibrium potential $e_A\in D(\mathscr{E}^{\,p})$ with $0\leq e_A\leq 1$ $\mathfrak{m}$-a.e. and $e_A=1$ $\mathfrak{m})$-a.e.~on $A$. |
| title | (1,p)-Sobolev spaces based on strongly local Dirichlet forms |
| topic | Probability Functional Analysis 31C25, 30L15, 53C21, 58J35, 35K05, 42B05, 47D08 |
| url | https://arxiv.org/abs/2310.11652 |