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 operator Mα, 0 ≤α 〈 Q on any Carnot group G （i.e., nilpotent stratified Lie group） in the generalized Morrey spaces Mp,φ（G）, where Q is the homogeneous dimension of G. We find the conditions on the pair （φ1, φ2） which ensures the boundedness of the operator Ms from one generalized Morrey space Mp,φ1 （G） to another Mq,φ2 （G）, 1. 〈 p ≤q 〈 ∞. 1/p - 1/q = α/Q, and from the space M1,φ1 （G） to the weak space Wq,φ2 （G）, 1 〈 q 〈 ∞, 1 - 1/q = α/Q. Also find conditions on the φ which ensure the Adams type boundedness of the Ms from M α （G） from Mp,φ^1/p（G）to Mq,φ^1/q（G） for 1 〈p〈q〈∞ and fromM1,φ（G） toWMq,φ^1/q（G）for 1〈q〈∞. In the case b ∈ BMO（G） and 1 〈 p 〈 q 〈 ∞, find the sufficient conditions on the pair （φ1, φ2） which ensures the boundedness of the kth-order commutator operator Mb,α,k from Mp,φ1 （G） to Mq,φ2（G） with 1/p - 1/q = α/Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb,α,k from Mp,φ^1/p（G） tom Mp,φ^1/p （G） for 1 〈p〈q〈∞. In all the cases the conditions for the boundedness of Mα are given it terms of supremaltype inequalities on （φ1, φ2） and φ , which do not assume any assumption on monotonicity of （φ1, φ2） and φ in r. As applications we consider the SchrSdinger operator -△G ＋ V on G, where the nonnegative potential V belongs to the reverse Holder class B∞（G）. The MB,φ1 - Mq,φ2 estimates for the operators V^γ（-△G ＋ V）^-β and V^γ△↓G（-△G ＋ V）^-β are obtained.