FRACTIONAL INTEGRAL ASSOCIATED WITH SCHRODINGER OPERATOR ON VANISHING GENERALIZED MORREY SPACES

Abstract

Let L= -Delta + V be a Schrodinger operator, where the non-negative potential V belongs to the reverse Holder class RHn/2, let b belong to a new BMO theta(rho) space, and let I-beta(L) be the fractional integral operator associated with L. In this paper, we study the boundedness of the operator I-beta(L) and its commutators [b, I-beta(L)] with b is an element of BMO theta(rho) on generalized Morrey spaces associated with Schrodinger operator M-p,phi(alpha,V) and vanishing generalized Morrey spaces associated with Schrodinger operator VMp,phi alpha,V. We find the sufficient conditions on the pair (phi(1), phi(2)) which ensures the boundedness of the operator I-beta(L) from M-p,phi 1(alpha,V) to M-q,phi 2(alpha,V) and from VMp,phi 1 alpha,V to VMq,phi 2 alpha,V, 1/p - 1/q = beta/n. When b belongs to BMO theta(rho) and (phi(1), phi(2)) satisfies some conditions, we also show that the commutator operator [b, I-beta(L)] is bounded from M-p,phi 1(alpha,V) to M-q,phi 2(alpha,V) and from VMp,phi 1 alpha,V to VMq,phi 2 alpha,V, 1/p - 1/q = beta/n.