Abstract
In this paper, we are interested in the Laguerre hypergroup K=[0, infinity]xR which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup (K)over-bar which can be topologically identified with the so-called Heisenberg fan, the subset of R-2:
boolean OR(j epsilon N){(lambda,mu) epsilon R-2: mu=vertical bar lambda vertical bar(2j+alpha+1), lambda not equal 0} boolean OR{0, mu) epsilon R-2: mu >= 0},
by means of which the maximal function is investigated. For 1 < p <= infinity, the L-p((K)over-bar)-boundedness and weak L-1((K)over-bar)-boundedness result for the maximal function is obtained.