Commutator of fractional integral with Lipschitz functions related to Schrodinger operator on local generalized mixed Morrey spaces

Abstract

Let L = - triangle + V be the Schrodinger operator on R-n, where V not equal 0 is a non-negative function satisfying the reverse Holder class RHq1 for some q(1) > n/2. triangle is the Laplacian on R-n. Assume that b is a member of the Campanato space Lambda(theta)(nu)(rho) and that the fractional integral operator associated with L is I-beta(L). We study the boundedness of the commutators [b , I-beta(L)] with b is an element of Lambda(theta)(nu)(rho) on local generalized mixed Morrey spaces. Generalized mixed Morrey spaces M-(p) over right arrow, phi(alpha , V) , vanishing generalized mixed Morrey spaces VM(p) over right arrow, phi alpha , V , and LM(p) over right arrow, phi alpha , V, {x0}, are related to the Schrodinger operator, in that order. We demonstrate that the commutator operator [b, I-beta(L)] is satisfied when b b belongs to Lambda(theta)(nu)(rho) with theta > 0, 0 < nu < 1, and (phi(1) , phi(2)) satisfying certain requirements are bounded from LM(p) over right arrow, phi 1 alpha , V, {x0}, to LM(p) over right arrow, phi 2(alpha , V, {x0}); from M (alpha , V)((p) over right arrow, phi 1) to M-(p) over right arrow, phi 2(alpha , V), and from V-(p) over right arrow, phi 1(alpha , V) to V-(p) over right arrow, phi 2(alpha , V), Sigma(n)(i = 1) 1/p(i) - Sigma(n)(i = 1)1/q(i) = beta + nu .