Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan Volume 44, Number 2, 2018, Pages 304–317 FRACTIONAL MAXIMAL OPERATOR AND ITS COMMUTATORS IN GENERALIZED MORREY SPACES ON HEISENBERG GROUP AHMET EROGLU, JAVANSHIR V. AZIZOV, AND VAGIF S. GULIYEV Abstract. In this paper we study the boundedness of the fractional maximal operator M α on Heisenberg group H n in the generalized Mor- rey spaces M p,ϕ (H n ). We shall give a characterization for the strong and weak type Spanne and Adams type boundedness of M α on the general- ized Morrey spaces, respectively. Also we give a characterization for the Spanne and Adams type boundedness of fractional maximal commutator operator M b,α on the generalized Morrey spaces. 1. Introduction Heisenberg groups, in discrete and continuous versions, appear in many parts of mathematics, including Fourier analysis, several complex variables, geometry, and topology. We state some basic results about Heisenberg group. More detailed information can be found in [4, 7, 8] and the references therein. Let H n be the 2n + 1-dimensional Heisenberg group. That is, H n = C n × R, with multiplication (z,t) · (w, s)=(z + w, t + s +2Im(z · ¯ w)), where z · ¯ w = n ∑ j =1 z j ¯ w j . The inverse element of u =(z,t) is u −1 =(−z, −t) and we write the identity of H n as 0 = (0, 0). The Heisenberg group is a connected, simply connected nilpotent Lie group. We define one-parameter dilations on H n , for r> 0, by δ r (z,t)=(rz, r 2 t). These dilations are group automorphisms and the Jacobian determinant is r Q , where Q =2n + 2 is the homogeneous dimension of H n . A homogeneous norm on H n is given by |(z,t)| =(|z | 2 + |t|) 1/2 . With this norm, we define the Heisenberg ball centered at u =(z,t) with radius r by B(u, r)= {v ∈ H n : |u −1 v| <r}, and we denote by B r = B(0,r)= {v ∈ H n : |v| <r} the open ball centered at 0, the identity element of H n , with radius r. The volume of the ball B(u, r) is C Q r Q , where C Q is the volume of the unit ball B 1 . 2010 Mathematics Subject Classification. Primary 42B20, 42B25, 42B35. Key words and phrases. Heisenberg group, fractional maximal operator, generalized Morrey space. 304