Marcinkiewicz integrals associated with Schrodinger operator and their commutators on vanishing generalized Morrey spaces
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Let L = -Delta + V be a Schrodinger operator, where Delta is the Laplacian on R-n and the non-negative potential V belongs to the reverse Holder class RHq for q >= n/2. In this paper, we study the boundedness of the Marcinkiewicz integral operators mu(L)(j) and their commutators [b, mu(L)(j)] with b is an element of BMO theta (rho) on generalized Morrey spaces M-p,phi(alpha,V) (R-n) associated with Schrodinger operator and vanishing generalized Morrey spaces VMp,phi alpha,V (R-n) associated with Schrodinger operator. We find the sufficient conditions on the pair (phi(1), phi(2)) which ensure the boundedness of the operators mu(L)(j) from one vanishing generalized Morrey space VMp,phi 1 alpha,V to another VMp,phi 2 alpha,V, 1 < p < infinity and from the space VM1,phi 1 alpha,V to the weak space VWM1,phi 2 alpha,V. When b belongs to BMO theta (rho) and (phi(1), phi(2)) satisfies some conditions, we also show that [b, mu(L)(j)] is bounded from M-p,phi 1(alpha,V) to M-p,M-phi 2(alpha,V) and from VMp,phi 1 alpha,V to VMp,phi 2 alpha,V, 1 < p < infinity.