L^p-Liouville property for non-local operators

The L^p-Liouville property of a non-local operator A is investigated via the associated Dirichlet form (ε, F). We will show that any non-negative continuous ε-subharmonic function f ∈ F_loc ∩ L^p are constant under a quite mild assumption on the kernel of ε 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.