Morrey-type estimates for commutator of fractional integral associated with Schrodinger operators on the Heisenberg group

Abstract

Let L = - Delta(Hn) + V be a Schrodinger operator on the Heisenberg group H-n m where the nonnegative potential V belongs to the reverse Holder class RH q , for some q(1) >= Q/2, and Q is the homogeneous dimension of H-n Let b belong to a new Campanato space Lambda(theta)(nu)(rho), and let T-beta(i) 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 central generalized Morrey spaces LMp,phi 1(alpha,V)(H-n), generalized Morrey spaces Mp,phi(alpha,V)(H-n), and vanishing generalized Morrey spaces VMp,phi(alpha,V)(H-n) associated with Schrodinger operator, respectively. When b belongs to Lambda(theta)(nu)(rho) with theta > 0, 0 < v < 1 and (phi 1 , phi 2) satisfies some conditions, we show that the commutator operator [b,I-beta(L)] is bounded from LMp,phi 1(alpha,V)to LMp,phi 1 alpha,V(H-n), from Mp,phi 1(alpha,V)(H-n) to Mq,phi 2(alpha,V)(H-n) and from VMp,phi(alpha,V)(1)(H-n) to VMq,phi(alpha,V)(2)(H-n),1/p - 1/q = (beta+ nu)/Q.