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

TL;DR

The paper develops a unified $L^{p}$-theory for energy forms on measure spaces via the generalized $p$-contraction property, bridging $p$-Clarkson inequalities, strong subadditivity, and nonlinear Dirichlet-type potential theory for $p\neq 2$. It proves differentiability of $p$-energy forms under $p$-Clarkson inequalities, introduces a two-variable derivative $\mathcal{E}_{p}(f; g)$, and shows this derivative induces a homeomorphism between the quotient space $\mathcal{F}/\mathcal{E}^{-1}(0)$ and its dual. The framework extends to $p$-energy measures, their chain rules and locality properties, and to self-similar structures where self-similar $p$-energy forms and measures are constructed, extended, and fixed-pointed, yielding estimates for scaling factors and Hölder regularity of $p$-harmonic functions. By introducing $p$-resistance forms, the authors develop nonlinear potential theory on general and fractal spaces, including harmonic functions, traces, weak comparison principles, and sharp Hölder regularity, with applications to generalized Sierpiński carpets and higher-dimensional Sierpiński gaskets, culminating in results about $p$-walk dimensions and the monotonicity of conductivity parameters $\\sigma_{p}$ in self-similar settings.

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 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 $p$.

Figures (4)

• Figure 1.1: The Sierpiński gasket (left) and the Sierpiński carpet (right)
• Figure 1.2: From the left, a non-planar generalized Sierpiński carpet (Menger Sponge) and the $2$-dimensional level-$l$ Sierpiński gaskets ($l = 2,3,4$)
• Figure B.1: Examples of affine nested fractals that do NOT satisfy \ref{['fig.nonallorbit']}. From the left, $D$-dimensional level-$2$ Sierpiński gasket ($D = 2,3$), pentakun and hexagasket.
• Figure B.2: Examples of affine nested fractals that satisfy \ref{['fig.nonallorbit']}. From the left, $2$-dimensional level-$l$ Sierpiński gasket ($l = 3,4$), snowflake and a Sierpiński gasket-type fractal.

Theorems & Definitions (344)

• Definition 1.1: Generalized $p$-contraction property; Definition \ref{['defn.GC']}
• Proposition 1.2: Proposition \ref{['prop.GC-list']}
• Theorem 1.3: Proposition \ref{['prop.diffble']} and Theorem \ref{['thm.p-form']}
• Theorem 1.4: Propositions \ref{['prop.c-diff-em']}, \ref{['prop.em-holder']} and Theorem \ref{['thm.em-basic']}
• Definition 1.5: $p$-Resistance form; Definition \ref{['defn.RFp']}
• Theorem 1.6: Corollary \ref{['cor.tri']}
• Definition 1.7: $\mathcal{E}_{p}$-Harmonic function; see Definition \ref{['dfn:part-harmonic']}
• Proposition 1.8: Part of Theorem \ref{['thm.RF-exist']}
• Theorem 1.9: Trace of $p$-resistance form; part of Theorem \ref{['thm.RF-exist']}
• Theorem 1.10: Theorem \ref{['t:lip-harm']}