Fractional maximal function on the dual of Laguerre hypergroup

Abstract

In this paper, we are interested in the Laguerre hypergroup $\mathbb{K}=[0,\mathrm{\infty })\times \mathbb{R}$ 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 $\hat{\mathbb{K}}$ which topologically can be identified with the so-called Heisenberg fan, the subset of ${\mathbb{R}}^2$: $\left({\cup }_{m\in \mathbb{N}}\right\{(\lambda ,\mu )\in {\mathbb{R}}^2:\mu =|\lambda |(2m+\alpha +1),\lambda \ne 0\left\}\right)\cup \left\{\right(0,\mu )\in {\mathbb{R}}^2:\mu \ge 0\},$ 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 $L_p\left(\hat{\mathbb{K}}\right)$ to the spaces $L_q\left(\hat{\mathbb{K}}\right)$ and from the spaces $L_1\left(\hat{\mathbb{K}}\right)$ to the weak spaces $W{L}_q\left(\hat{\mathbb{K}}\right)$ is obtained.

Keywords:

• dual of Laguerre hypergroup
• generalized translation operator
• Fourier–Laguerre transform
• Fractional maximal function

Mathematics Subject Classification:

• 42B20
• 42B25
• 42B35

Disclosure statement

No potential conflict of interest was reported by the authors.