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

TL;DR

This paper extends the classical subordination principle to non-linear $p$-energies on metric measure spaces, showing how local and non-local regular energies relate through cutoff Sobolev inequalities under geometric conditions.

Contribution

It introduces a non-linear subordination principle connecting local and non-local $p$-energies via cutoff Sobolev inequalities, broadening the Dirichlet form framework.

Findings

Stable-like non-local $p$-energies inherit regularity from local energies.

Non-local cutoff Sobolev inequalities imply regularity for energies with smaller scaling functions.

Results apply under suitable geometric assumptions on metric measure spaces.

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 Poincar\'e inequality together with a cutoff Sobolev inequality with scaling function $\Psi$, then all associated stable-like non-local $p$-energies with scaling functions strictly below $\Psi$ are regular and satisfy the corresponding non-local cutoff Sobolev inequalities. Moreover, if a stable-like non-local regular $p$-energy with scaling function $\Psi$ 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 $\Psi$. These results provide a non-linear extension of the classical subordination principle beyond the Dirichlet form framework.