Construction of $p$-energy and associated energy measures on Sierpiński carpets
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909082416316416 |
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| author | Shimizu, Ryosuke |
| author_facet | Shimizu, Ryosuke |
| contents | We establish the existence of a scaling limit $\mathcal{E}_p$ of discrete $p$-energies on the graphs approximating generalized Sierpiński carpets for $p > \dim_{\text{ARC}}(\textsf{SC})$, where $\dim_{\text{ARC}}(\textsf{SC})$ is the Ahlfors regular conformal dimension of the underlying generalized Sierpiński carpet. Furthermore, the function space $\mathcal{F}_{p}$ defined as the collection of functions with finite $p$-energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, $(\mathcal{E}_2, \mathcal{F}_2)$ recovers the canonical regular Dirichlet form constructed by Barlow and Bass or Kusuoka and Zhou. We also provide $\mathcal{E}_{p}$-energy measures associated with the constructed $p$-energy and investigate its basic properties like self-similarity and chain rule. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_13902 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Construction of $p$-energy and associated energy measures on Sierpiński carpets Shimizu, Ryosuke Metric Geometry Analysis of PDEs Probability Primary 28A80, 30L99, 31E99, Secondary 46E36, 31C45 We establish the existence of a scaling limit $\mathcal{E}_p$ of discrete $p$-energies on the graphs approximating generalized Sierpiński carpets for $p > \dim_{\text{ARC}}(\textsf{SC})$, where $\dim_{\text{ARC}}(\textsf{SC})$ is the Ahlfors regular conformal dimension of the underlying generalized Sierpiński carpet. Furthermore, the function space $\mathcal{F}_{p}$ defined as the collection of functions with finite $p$-energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, $(\mathcal{E}_2, \mathcal{F}_2)$ recovers the canonical regular Dirichlet form constructed by Barlow and Bass or Kusuoka and Zhou. We also provide $\mathcal{E}_{p}$-energy measures associated with the constructed $p$-energy and investigate its basic properties like self-similarity and chain rule. |
| title | Construction of $p$-energy and associated energy measures on Sierpiński carpets |
| topic | Metric Geometry Analysis of PDEs Probability Primary 28A80, 30L99, 31E99, Secondary 46E36, 31C45 |
| url | https://arxiv.org/abs/2110.13902 |