Weak Harnack inequality and Cartan property for nonlocal $W^{s,1}$-minimizers

TL;DR

This paper proves a weak Harnack inequality and Cartan properties for nonlocal $W^{s,1}$-minimizers in metric spaces, extending classical results and showing new insights even in Euclidean spaces.

Contribution

It establishes weak Harnack inequalities and Cartan properties for nonlocal $W^{s,1}$-minimizers, generalizing local BV results to nonlocal and metric space contexts.

Findings

Proved weak Harnack inequality for nonlocal $W^{s,1}$-subminimizers.

Showed semicontinuity of $W^{s,1}$-subminimizers.

Established Cartan-type properties for $W^{s,1}$-superminimizers.

Abstract

We establish a weak Harnack inequality for nonlocal $W^{s,1}$-subminimizers in a complete, connected, doubling metric measure space where $0<s<1$. As a corollary, we prove that $W^{s,1}$-subminimizers are semicontinuous, up to a suitable choice of pointwise representative. We then prove \emph{Cartan-type properties} for $W^{s,1}$-superminimizers. The theory turns out to be mostly analogous with the local case of BV super- and subminimizers. Our results seem to be new even in the classical Euclidean setting.