Restricted weak type endpoint estimate for the spherical maximal operators on the Heisenberg group
Hyunwoo Jeon, Joonil Kim
TL;DR
This work proves a restricted weak type endpoint estimate for the spherical maximal operator $\mathcal{M}$ on the Heisenberg group $\mathbb{H}^n$ at $p=\frac{d}{d-1}$ for $d>2$. The authors introduce a full-frequency decomposition in $\mathbb{R}^{d+1}$ and a refined Calderón–Zygmund decomposition, combined with Bourgain interpolation, to overcome endpoint losses and establish both the weak-type bound and the required $L^2$ decay. Central innovations include a new doubling ball and Vitali/CZ framework, a Hörmander-type condition after removing the worst part, and detailed $L^2$ reductions via asymptotics of $d\sigma$ and dilation arguments. The results fill a gap in endpoint endpoint behavior for global spherical averages on two-step nilpotent/Métivier-type groups, with potential extensions to broader nilpotent settings and implications for endpoint endpoint harmonic analysis on noncommutative spaces.
Abstract
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 $p \in (\tfrac{d}{d-1}, \infty]$. In this paper, we establish a restricted weak type $(p,p)$ estimate at the endpoint $p = \tfrac{d}{d-1}$ for $\mathcal{M}$, provided $d \ge 3$.