Commutator of fractional integral with Lipschitz functions associated with Schrodinger operator on local generalized Morrey spaces

Abstract

Let L = -Delta + V be a Schrodinger operator on R-n, where n >= 3 and the nonnegative potential V belongs to the reverse Holder class RHq1 for some q(1) > n/2. Let b belong to a new Campanato space Lambda(theta)(nu) (rho) and I-beta(L) be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators [b, I-beta(L)] with b is an element of Lambda(theta)(nu) (rho) on local generalized Morrey spaces LM rho,phi alpha,V,{x0} generalized Morrey spaces M-rho,phi(alpha,V) and vanishing generalized Morrey spaces VM rho,phi alpha,V associated with Schrodinger operator, respectively. When b belongs to Lambda(theta)(nu) (rho) with theta > 0, 0 < nu < 1 and (phi(1),phi(2)) satisfies some conditions, we show that the commutator operator [b,I-beta(L)] are bounded from LM rho,phi 1 alpha,V,{x0} to LM rho,phi 2 alpha,V,{x0} from LMq,phi 1 alpha,V to VMq,phi 2 alpha,V and from VM rho,phi 1 alpha,V to VM rho,phi 2 alpha,V, 1/p - 1/q = ( beta +nu)/n.