Elliptic Harnack inequalities for mixed local and nonlocal $p$-energy form on metric measure spaces

TL;DR

This work develops a unified, analytic framework for mixed local and nonlocal $p$-energy forms on general metric measure spaces and proves weak and elliptic Harnack inequalities under a standard package of structural conditions, including $VD$, $RVD$, $CS$, $PI$, and mild jump-kernel hypotheses. The approach extends the De Giorgi–Nash–Moser scheme to nonlinear, nonlocal energies by carefully decomposing the energy into local and nonlocal parts and establishing sharp inequalities (Caccioppoli, growth, logarithmic lemmas) adapted to tails and jumps. A key contribution is the self-improvement of the cutoff Sobolev inequality and the FK–SI–STI equivalences, which undergird the Harnack theory in this nonlinear, nonlocal setting and even accommodate killing terms in a capacity-compatible way. The paper provides concrete Euclidean and ultrametric examples, showing the framework recovers classical local p-Laplacian results, nonlocal $p$-Laplacians, and mixed operators, thereby broadening the scope of elliptic regularity on metric measure spaces and enabling tail-aware estimates in diverse geometries.

Abstract

We derive the weak elliptic Harnack inequality and the elliptic Harnack inequality for mixed local and nonlocal $p$-energy form on metric measure spaces. The proof relies on the Poincaré inequality, the cutoff Sobolev inequality, and some mild conditions on the jump measure. Our approach is based on the De Giorgi-Nash-Moser method, extending the corresponding result for Dirichlet forms without a killing part (the case $p=2$).

Theorems & Definitions (97)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Proposition 2.5
• proof
• Remark 2.6
• Proposition 2.7: KS25
• Proposition 2.8