Lp-Liouville property for non-local operators

✍ Scribed by Jun Masamune; Toshihiro Uemura

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 193 KB
• Volume: 284
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200910211

✦ Synopsis

Abstract

The L^p^‐Liouville property of a non‐local operator \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {A}$\end{document} is investigated via the associated Dirichlet form \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$({\mathcal E},{\mathcal F})$\end{document}. We will show that any non‐negative continuous \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal E}$\end{document}‐subharmonic function \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$f \in {\mathcal F}_{\rm loc} \cap L^p$\end{document} are constant under a quite mild assumption on the kernel of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal E}$\end{document} if p ≥ 2. On the contrary, if 1 < p < 2, we need an additional assumption: either, the kernel has compact support; or f is Hölder continuous.