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 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 Ma 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 Mp,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 Ma from Mp,φ1p(G) to Mp,φ1q(G) for 1 <p <q < ∞ and from M1,φ(G) to W Mq,φ1q(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, a, k from Mp,φ1(G) to Mp,φ2(G) with 1/p − 1/q = a/Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb, α, k from Mp,φ1p(G) to Mq,φ1q(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 (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)−β and Vγ∇G(−ΔG+V)−β are obtained.