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 Mα from one generalized Morrey space Mp, φ1 (G) to another Mq, φ2 (G), 1 < p ≤ q < ∞, 1/p − 1/q = /Q, and from the space M1, φ1 (G) to the weak space W Mq, φ2/ (G), 1 ≤ q < ∞, 1 − 1/q =α /Q. Also find conditions on the φ which ensure the Adams type boundedness of the M from M
p, φ1/p(G) to Mq, φ1/q(G) for 1 < p < q < ∞and from M1, φ(G) to W M q, φ1/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 M p, φ1/p(G) to Mq, φ1/q(G) for 1 < p < q < ∞. In all the cases the conditions for the boundedness of M 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 SchrÖdinger operator −ΔG + V on G, where the nonnegative potential V belongs to the reverse HÖlder class B∞(G). The Mp, φ1 −Mq, φ2 estimates for the operators Vγ (−ΔG + V )− and V γ∇G(−ΔG + V )− are obtained.
