A new proof of maximal theorem on Heisenberg groups

Given $0\leq\alpha<1$, we define $$\begin{array}{lr} \mathbf{M}_\alpha f(u,v,t) = \sup_{ \mathbf{R} \ni (0,0,0)} {\rm vol} \{\mathbf{R}\}^{\alpha-1} \iiint_\mathbf{R}\left|f [(u,v,t)\odot(\xi,\eta,\tau)^{-1}]\right|d\xi d\eta d\tau \end{array}$$ 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 $\mathbf{M}_\alpha\colon\mathbf{L}^p(\mathbb{R}^{2n+1}) \to \mathbf{L}^q(\mathbb{R}^{2n+1})$ for $\alpha={1\over p}-{1\over q},~ 1<p\leq q<\infty$ by applying a geometric covering lemma due to C\'{o}rdoba and Fefferman.