Lp-Liouville Property for Non-Local Operators

TL;DR

This paper investigates the conditions under which non-negative continuous Lp E-subharmonic functions are constant for non-local operators, revealing different criteria depending on whether p is above or below 2.

Contribution

It establishes new conditions for the Lp-Liouville property of non-local operators, including mild kernel assumptions for p ≥ 2 and additional support or regularity conditions for 1 < p < 2.

Findings

Non-negative continuous Lp E-subharmonic functions are constant under mild kernel assumptions for p ≥ 2.

For 1 < p < 2, either kernel compact support or Holder continuity ensures the property.

The paper delineates the different criteria needed for the Lp-Liouville property depending on p.

Abstract

The Lp-Liouville property of a non-local operator A is investigated via the associated Dirichlet form. We will show that any non-negative continuous Lp E-subharmonic functions are constant under a quite mild assumption on the kernel of E if p is not less than 2. On the contrary, if 1 < p < 2, we need an additional assumption: either, the kernel has compact support; or f is Holder continuous.