Cutoff Sobolev inequalities for local and non-local $p$-energies on metric measure spaces

TL;DR

The paper develops a nonlinear subordination framework for local and non-local $p$-energies on metric measure spaces by establishing a network of equivalences between cutoff-Sobolev (CS) and cutoff-energy (CE) conditions. It proves that, under suitable geometric assumptions, cutoff functions with Hölder regularity can propagate regularity from a local $p$-energy satisfying PI and CS to all subordinated non-local $p$-energies with scaling functions strictly below the original, and conversely. The core contributions include (i) a precise subordination theorem relating local and non-local $p$-energies through scaling functions with strict growth relations, (ii) the equivalence CE$\Leftrightarrow$CS in both local and non-local settings, and (iii) robust oscillation tools (interior and boundary) for nonlinear $p$-Laplacian-type problems that handle tail terms in the non-local framework. These results extend the classical subordination principle beyond Dirichlet forms and provide a nonlinear, scalable scheme to transfer regularity and functional inequalities across energy types on complex spaces. The framework has potential impacts on elliptic regularity, Harnack-type estimates, and potential theory for nonlinear non-local operators on fractal-like geometries.

Abstract

For $p>1$, we study subordination phenomena for local and non-local regular $p$-energies on metric measure spaces. Under suitable geometric assumptions, we show that if a local regular $p$-energy satisfies a Poincaré inequality together with a cutoff Sobolev inequality with scaling function $Ψ$, then all associated stable-like non-local $p$-energies with scaling functions strictly below $Ψ$ are regular and satisfy the corresponding non-local cutoff Sobolev inequalities. Moreover, if a stable-like non-local regular $p$-energy with scaling function $Ψ$ satisfies the corresponding non-local cutoff Sobolev inequality, then the same conclusion holds for all associated stable-like non-local $p$-energies with scaling functions below $Ψ$. These results provide a non-linear extension of the classical subordination principle beyond the Dirichlet form framework.

Figures (1)

• Figure 1: Strategy of the proof of "$\text{CE}^{(\bullet)}_{\square}\Leftrightarrow\text{CS}^{(\bullet)}_{\square}$"

Theorems & Definitions (78)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Proposition 2.6
• Corollary 2.7
• Theorem 2.8
• Remark 2.9
• Lemma 3.1: BB11; see also Yan25d