Lp-Liouville Property for Non-Local Operators

TL;DR

The paper develops Lp-Liouville theory for non-local operators via symmetric Dirichlet forms and jump kernels. A central contribution is the integral derivation property for the carré du champ ${\Gamma}$, enabling Liouville-type conclusions under precise moment conditions: for $p\ge 2$ the property holds if $M_q<\infty$, and for $1<p<2$ additional assumptions—either kernel truncation or Hölder regularity—are required. The results are proven in a general Euclidean setting and illustrated with examples such as symmetric stable-like processes and Lévy processes, showing explicit $p$-range thresholds tied to the jump structure. The findings connect analysis, potential theory, and stochastic processes, advancing understanding of when non-local operators enforce constancy of non-negative ${\mathcal E}$-subharmonic functions in $L^p$ spaces.

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.

Theorems & Definitions (27)

• Theorem 1: Theorem \ref{['MT2']}
• Theorem 2: Integral Derivation Property: Proposition \ref{['prop:deri']}
• Lemma 1: Example 1.2.4 FOT94 and U04
• Lemma 2
• proof
• Remark 3
• Definition 4: ${\mathcal{E}}$-Subharmonic functions
• Remark 5
• Proposition 1: Derivation property of integral type
• proof