Restricted weak type endpoint estimate for the spherical maximal operators on the Heisenberg group

Let ${\mathbb{H}}^n$ denote the Heisenberg group, identified with ${\mathbb{R}}^d\times \mathbb{R}$, where $d=2n$ and $n\in \mathbb{N}$. We consider the spherical maximal operator $\mathcal{M}$ associated with the sphere $S^{d-1}$ embedded in the horizontal subspace ${\mathbb{R}}^d\times \{0\}$ of ${\mathbb{H}}^n$. It is known that $\mathcal{M}$ is bounded on $L^p({\mathbb{H}}^n)$ if and only if $p\in (\frac{d}{d-1},\mathrm{\infty }]$. In this paper, we establish a restricted weak type $(p,p)$ estimate at the endpoint $p=\frac{d}{d-1}$ for $\mathcal{M}$, provided $d\ge 3$.