Bochner-Riesz means on the Heisenberg group

We prove new $L^p$ boundedness results for Bochner-Riesz means associated with the spectral decomposition of the sub-Laplacian on the Heisenberg group ${\mathbb{H}}_n$. Our results hold for a range $1\le p\le p_n$ where $p_n\to 2$ as $n\to \mathrm{\infty }$. As shown by the first named author in 1990 a Stein-Tomas type Fourier restriction theorem fails to hold on ${\mathbb{H}}_n$ and thus previous results based on the approach by Fefferman and Stein from the Euclidean setting only allowed to cover the cases $p=1$ and $p=\mathrm{\infty }$. Our results on Bochner-Riesz means follow from a more general $p$-sensitive spectral multiplier theorem which is the main result of this article. This is obtained as a consequence of $L^p$ estimates for square functions associated with the Heisenberg wave operator.