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

TL;DR

This paper introduces a new contraction property for p-energy forms, proves their differentiability under certain inequalities, and applies these concepts to analyze p-harmonic functions and self-similar fractals.

Contribution

It defines the generalized p-contraction property, verifies it for specific fractals, and develops a framework for p-resistance forms and p-harmonic functions on self-similar sets.

Findings

p-energy forms satisfying p-Clarkson's inequality are Fréchet differentiable

Established the generalized p-contraction property for fractals by Kigami and Cao--Gu--Qiu

Derived new estimates on scaling factors and proved p-walk dimensions exceed p on certain fractals.

Abstract

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 Fr\'{e}chet 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 H\"{o}lder regularity estimates for $p$-harmonic functions, and the $p$-walk dimensions of any generalized Sierpi\'{n}ski carpet and the $D$-dimensional level-$l$ Sierpi\'{n}ski gasket are shown to be strictly greater than $p$.