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

TL;DR

This paper proves a restricted weak type endpoint estimate for the spherical maximal operator on the Heisenberg group, extending understanding of its boundedness properties at critical exponents.

Contribution

It establishes a restricted weak type $(p,p)$ estimate at the endpoint for the spherical maximal operator on the Heisenberg group, a new result in harmonic analysis.

Findings

Proves restricted weak type estimate at the critical exponent

Extends boundedness results to the endpoint case

Provides new insights into maximal operators on non-commutative groups

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 $d \ge 3$.