Fractional maximal function on the dual of Laguerre hypergroup
Miloud Assal, Taieb Ahmed · Integral Transforms and Special Functions · 2015
In this paper, we are interested in the Laguerre hypergroup K=[0,∞)×R 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 Kˆ which topologically can be identified with the so-called Heisenberg fan, the subset of R2: (∪m∈N{(λ,μ)∈R2:μ=|λ|(2m+α+1),λ≠0})∪{(0,μ)∈R2:μ≥0}, by means of which fractional maximal function is investigated also the necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator on the dual of Laguerre hypergroup from the spaces Lp(Kˆ) to the spaces Lq(Kˆ) and from the spaces L1(Kˆ) to the weak spaces WLq(Kˆ) is obtained.