BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

Abstract

In the article we consider the fractional maximal operator M-alpha, 0 <= alpha < Q on any Carnot group G (i.e., nilpotent stratified Lie group) in the generalized Morrey spaces M-p,M-phi(G), where Q is the homogeneous dimension of G. We find the conditions on the pair (phi(1), phi(2)) which ensures the boundedness of the operator M-alpha from one generalized Morrey space M-p,M-phi 1(G) to another M-q,M-phi 2(G), 1 < p <= q < infinity, 1/p - 1/q = alpha/Q, and from the space M-1,M-phi 1(G) to the weak space WMq,phi 2(G), 1 <= q < infinity, 1 - 1/q = alpha/Q. Also find conditions on the phi which ensure the Adams type boundedness of the M-alpha from M-p,M-phi 1/p(G) to M-q,M-phi 1/q(G) for 1 < p < q < infinity and from M-1,M-phi(G) to WMq,phi 1/q(G) for 1 < q < infinity. In the case b is an element of BMO(G) and 1 < p < q < infinity, find the sufficient conditions on the pair (phi(1), phi(2)) which ensures the boundedness of the kth-order commutator operator M-b,M-alpha,M-k from M-p,M-phi 1(G) to M-q,M-phi 2(G) with 1/p - 1/q = alpha/Q. Also find the sufficient conditions on the phi which ensures the boundedness of the operator M-b,M-alpha,M-k from M-p,M-phi 1/p(G) to M-q,M-phi 1/q(G) for 1 < p < q < infinity. In all the cases the conditions for the boundedness of M-alpha are given it terms of supremal-type inequalities on (phi(1), phi(2)) and phi, which do not assume any assumption on monotonicity of (phi(1), phi(2)) and phi in r. As applications we consider the Schrodinger, operator -Delta(G) + V on G, where the nonnegative potential V belongs to the reverse Holder class B-infinity(G). The M-p,M-phi 1 - M-q,M-phi 2 estimates for the operators V-gamma(-Delta(G) + V)(-beta) and V-gamma del(G)(-Delta(G) + V)(-beta) are obtained.