Boundedness of Fractional Maximal Operator and Their Higher Order Commutators in Generalized Morrey Spaces on Carnot Groups

In the article we consider the fractional maximal operatorM?, 0 ? ? < Q on any Carnot groupG (i.e., nilpotent stratified Lie group) in the generalized Morrey spacesMp,?(G), where Q is the homogeneous dimension ofG. We find the conditions on the pair ?1, ?2) which ensures the boundedness of the operator Ma from one generalized Morrey space Mp,?1(G) to another Mq,?2(G),1<p?q<?,1/p-1/q=?/Q, and from the space Mp,?1(G) to the weak space WMq, ?2(G),1?q<?,1-1/q=?/Q. Also find conditions on the ? which ensure the Adams type boundedness of the Ma from Mp,?1p(G) to Mp,?1q(G) for 1 < p < q < ? and from M1,?(G) to WMq,?1q(G) for 1 < q < ?. In the caseb?BMO(G) and 1 < p < q < ?, find the sufficient conditions on the pair (?1, ?2) which ensures the boundedness of the kth-order commutator operator Mb, a, k from Mp,?1(G) to Mp,?2(G) with 1/p - 1/q = a/Q. Also find the sufficient conditions on the ? which ensures the boundedness of the operator Mb, ?, k from Mp,?1p(G) to Mq,?1q(G) for 1 < p < q < ?. In all the cases the conditions for the boundedness of Ma are given it terms of supremal-type inequalities on (?1, ?2) and ?, which do not assume any assumption on monotonicity of (?1, ?2) and ? in r. As applications we consider the Schrödinger operator-?G+V onG, where the nonnegative potential V belongs to the reverse Hölder class B?(G). The Mp, ?1 - Mq, ?2 estimates for the operatorsV?(-?G+V)-ß and V??;G(-?G+V)-ß are obtained. © 2013 Wuhan Institute of Physics and Mathematics.Firat University Scientific Research Projects Management Unit: FEN 4001.12.0018 EIF-2010-1(1)-40/06-1 110T695, FEN 4001.12.0019 --*Received November 30, 2011; revised March 2, 2013. The research of V. Guliyev was partially supported by the grant of Ahi Evran University Scientific Research Projects (FEN 4001.12.0018) and by the grant of Science Development Foundation under the President of the Republic of Azerbaijan project EIF-2010-1(1)-40/06-1. V. Guliyev and A. Akbulut were partially supported by the Scientific and Technological Research Council of Turkey (TUBITAK Project No: 110T695). The research of A. Akbulut was partially supported by the grant of Ahi Evran University Scientific Research Projects (FEN 4001.12.0019). -