L-p MAPPING PROPERTIES FOR NONLOCAL SCHRODINGER OPERATORS WITH CERTAIN POTENTIALS

Abstract

In this paper, we consider nonlocal Schrodinger equations with certain potentials V is an element of RHq (q > n/2s > 1 and 0 < s < 1) of the form L(K)u Vu = f in R-n where L-K is an integro-differential operator. We denote the solution of the above equation by S-V f := u, which is called the inverse of the nonlocal Schrodinger operator L-K + V with potential V; that is, S-V = (L-K + V)(-1). Then we obtain an improved version of the weak Harnack inequality of non negative weak subsolutions of the nonlocal equation {L(K)u Vu = 0 in Omega, u = g in R-n \ Omega, where g is an element of H-S(R-n) and Omega is a bounded open domain in R-n with Lipschitz boundary, and also get an improved decay of a fundamental solution e(V) for L-K + V. Moreover, we obtain L-p and L-p - L-q mapping properties of the inverse S-V of the nonlocal Schrodinger operator L-K +V.