Contraction properties and differentiability of $p$-energy forms with applications to nonlinear potential theory on self-similar sets
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arXiv
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| Natura: | Preprint |
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2024
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| author | Kajino, Naotaka Shimizu, Ryosuke |
| author_facet | Kajino, Naotaka Shimizu, Ryosuke |
| contents | We introduce a new contraction property, which we call the generalized $p$-contraction property, for $p$-energy forms as generalizations of many well-known inequalities, such as $p$-Clarkson's inequality, the strong subadditivity and the Markov property in the theory of nonlinear Dirichlet forms, and show that any $p$-energy form satisfying $p$-Clarkson's inequality is Fréchet differentiable. We also verify the generalized $p$-contraction property for $p$-energy forms on fractals constructed by Kigami [Mem. Eur. Math. Soc. 5 (2023)] and by Cao--Gu--Qiu [Adv. Math. 405 (2022), no. 108517]. As a general framework of $p$-energy forms taking the generalized $p$-contraction property into consideration, we introduce the notion of $p$-resistance form and investigate fundamental properties of $p$-harmonic functions with respect to $p$-resistance forms. In particular, some new estimates on scaling factors of self-similar $p$-energy forms on self-similar sets are obtained by establishing Hölder regularity estimates for $p$-harmonic functions, and the $p$-walk dimensions of any generalized Sierpiński carpet and the $D$-dimensional level-$l$ Sierpiński gasket are shown to be strictly greater than $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13668 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Contraction properties and differentiability of $p$-energy forms with applications to nonlinear potential theory on self-similar sets Kajino, Naotaka Shimizu, Ryosuke Functional Analysis Analysis of PDEs Metric Geometry Probability 28A80, 39B62, 31E05 (Primary) 31C45, 31C25, 46E36 (Secondary) We introduce a new contraction property, which we call the generalized $p$-contraction property, for $p$-energy forms as generalizations of many well-known inequalities, such as $p$-Clarkson's inequality, the strong subadditivity and the Markov property in the theory of nonlinear Dirichlet forms, and show that any $p$-energy form satisfying $p$-Clarkson's inequality is Fréchet differentiable. We also verify the generalized $p$-contraction property for $p$-energy forms on fractals constructed by Kigami [Mem. Eur. Math. Soc. 5 (2023)] and by Cao--Gu--Qiu [Adv. Math. 405 (2022), no. 108517]. As a general framework of $p$-energy forms taking the generalized $p$-contraction property into consideration, we introduce the notion of $p$-resistance form and investigate fundamental properties of $p$-harmonic functions with respect to $p$-resistance forms. In particular, some new estimates on scaling factors of self-similar $p$-energy forms on self-similar sets are obtained by establishing Hölder regularity estimates for $p$-harmonic functions, and the $p$-walk dimensions of any generalized Sierpiński carpet and the $D$-dimensional level-$l$ Sierpiński gasket are shown to be strictly greater than $p$. |
| title | Contraction properties and differentiability of $p$-energy forms with applications to nonlinear potential theory on self-similar sets |
| topic | Functional Analysis Analysis of PDEs Metric Geometry Probability 28A80, 39B62, 31E05 (Primary) 31C45, 31C25, 46E36 (Secondary) |
| url | https://arxiv.org/abs/2404.13668 |