A new proof of maximal theorem on Heisenberg groups

Abstract

Given $0\le \alpha <1$, we define \[\begin{array}{lr} \mathbf{M}_αf(u,v,t) = \sup_{ \mathbf{R} \ni (0,0,0)} {\rm vol} \{\mathbf{R}\}^{α-1} \iiint_\mathbf{R}\left|f [(u,v,t)\odot(ξ,η,τ)^{-1}]\right|dξdηdτ\end{array}\] where $\mathbf{R}\subset {\mathbb{R}}^{2n+1}$ is a rectangle parallel to the coordinates. Moreover, $\odot$ denotes the multiplication law on a real Heisenberg group. The ${\mathbf{L}}^p$-boundedness of ${\mathbf{M}}_0$ has been previously proved by M. Christ. We show ${\mathbf{M}}_{\alpha }:{\mathbf{L}}^p({\mathbb{R}}^{2n+1})\to {\mathbf{L}}^q({\mathbb{R}}^{2n+1})$ for $\alpha =\frac{1}{p}-\frac{1}{q},1<p\le q<\infty$ by applying a geometric covering lemma due to Córdoba and Fefferman.