Contraction properties and differentiability of \(p\)-energy forms with applications to nonlinear potential theory on self-similar sets

We introduce new contraction properties called the generalized \(p\)-contraction property for \(p\)-energy forms as generalizations of many well-known inequalities, such as Clarkson's inequalities, the strong subadditivity and the ``Markov property'' in the theory of nonlinear Dirichl...

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Bibliographic Details
Published in:arXiv.org 2024-04
Main Authors: Kajino, Naotaka, Shimizu, Ryosuke
Format: Article
Language:English
Subjects:
Dirichlet problem
Energy
Estimates
Harmonic functions
Inequalities
Potential theory
Scaling factors
Self-similarity
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Summary:We introduce new contraction properties called the generalized \(p\)-contraction property for \(p\)-energy forms as generalizations of many well-known inequalities, such as Clarkson's inequalities, the strong subadditivity and the ``Markov property'' in the theory of nonlinear Dirichlet forms, and show that any \(p\)-energy form satisfying Clarkson's inequalities is Fr\'{e}chet differentiable. We also verify the generalized \(p\)-contraction property for \(p\)-energy forms 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 into consideration the generalized \(p\)-contraction property, we introduce the notion of \(p\)-resistance form and investigate fundamental properties for \(p\)-harmonic functions with respect to \(p\)-resistance forms. In particular, some new estimates on scaling factors of \(p\)-energy forms are obtained by establishing H\"{o}lder regularity estimates for harmonic functions, and the \(p\)-walk dimensions of the generalized Sierpi\'{n}ski carpets and \(D\)-dimensional level-\(l\) Sierpi\'{n}ski gasket are shown to be strictly greater than \(p\).
ISSN:2331-8422