Boundedness of the fractional maximal operator in generalized Morrey space on the Heisenberg group

Abstract

In this paper we study the fractional maximal operator M (alpha) , 0 a parts per thousand currency sign alpha < Q on the Heisenberg group a"i (n) in the generalized Morrey spaces M (p, I center dot)(a"i (n) ), where Q = 2n + 2 is the homogeneous dimension of a"i (n) . We find the conditions on the pair (I center dot (1), I center dot (2)) which ensures the boundedness of the operator M (alpha) from one generalized Morrey space M (p, I center dot 1)(a"i (n) ) to another M (q, I center dot 2)(a"i (n) ), 1 < p < q < a, 1/p-1/q = alpha/Q, and from the space M (1, I center dot 1)(a"i (n) ) to the weak space WM (q, I center dot 2)(a"i (n) ), 1 < q < a, 1 - 1/q = alpha/Q. We also find conditions on the phi which ensure the Adams type boundedness of M (alpha) from to for 1 < p < q < a and from M (1, I center dot)(a"i (n) ) to for 1 < q < a. As applications we establish the boundedness of some Schrodinger type operators on generalized Morrey spaces related to certain nonnegative potentials V belonging to the reverse Holder class B (a)(" (n) ).