Integral Transforms and Special Functions Vol. 19, No. 9, September 2008, 633–641 (L p ,L q ) boundedness of the fractional maximal operator on the Laguerre hypergroup Vagif S. Guliyev a,b * and Mehriban N. Omarova b a Institute of Mathematics and Mechanics, Baku, Azerbaijan; b Baku State University, Department of Mathematical Analysis, Baku, Azerbaijan (Received 19 August 2007 ) Let K =[0, ∞) × R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator on the Laguerre hypergroup from the spaces L p (K) to the spaces L q (K) and from the spaces L 1 (K) to the weak spaces WL q (K). Keywords: Laguerre hypergroup; generalized translation operator; Fourier–Laguerre transform; frac- tional maximal operator; fractional integral operator 2000 Mathematics Subject Classifications: Primary 42B20, 42B25, 42B35 1. Introduction In this paper we define the fractional maximal function using harmonic analysis on Laguerre hypergroups which can be seen as a deformation of the hypergroup of radial functions on the Heisenberg group (see, for example [1,5,7–9]) and we study the fractional maximal function on the Laguerre hypergroup. We obtain the necessary and sufficient conditions for the boundedness of the fractional maximal operator on the Laguerre hypergroup from the spaces L p (K) to the spaces L q (K) and from the spaces L 1 (K) to the weak spaces WL q (K). The paper is organized as follows. In Section 2, we give the main result on the boundness of the fractional maximal function on the Laguerre hypergroup. In Section 3, we present some definitions and auxiliary results. In Section 4, we give polar coordinates in Laguerre hypergroup and some lemmas needed to facilitate the proofs of our theorems. The main result of the paper is the boundness of the fractional maximal operator on the Laguerre hypergroup, established in Section 5. We prove the boundedness of the fractional maximal operator from the spaces L p (K) to L q (K),1 <p<(2α + 4)/β ,1/p − 1/q = β/(2α + 4) and from the spaces L 1 (K) to the weak Lebesgue spaces WL q (K),1 − 1/q = β/(2α + 4) and from the spaces L (2α+4)/β (K) to L ∞ (K). We show that the conditions on the parameters ensuring that the boundedness cannot be weakened. *Corresponding author. Email: vagif@guliyev.com ISSN 1065-2469 print/ISSN 1476-8291 online © 2008 Taylor & Francis DOI: 10.1080/10652460801948882 http://www.informaworld.com