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 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.