Let $L$ denote the sub-Laplacian on the Heisenberg group $\mathbb{H}_n$ and $T_r^\lambda := (1 - r L)^\lambda_+$ the corresponding Bochner-Riesz operator. Let $Q$ denote the homogeneous dimension and $D$ the Euclidean dimension of $\mathbb{H}_n$. We prove convergence a.e. of the Bochner-Riesz means $T_r^\lambda f$ as $r \rightarrow 0$ for $\lambda > 0$ and for all $f \in L^p(\mathbb{H}_n)$, provided that \frac{Q-1}{Q} \Big(\frac{1}{2} - \frac{\lambda}{D-1} \Big) < 1/p \le 1/2. Our proof is based on explicit formulas for the operators $\partial_{\omega^a}$ with $a \in \mathbb{C}$, defined on the dual of $\mathbb{H}_n$ by $\partial_{\omega^a} \widehat{f} := \widehat{\omega^a f}$, which may be of independent interest. Here $\omega$ is given by $\omega(z,u) := |z|^2 - 4iu$ for all $(z,u) \in \mathbb{H}_n$.

2000 Mathematical Subject Classification: 22E30, 43A80.

A spatial epidemic process where the individuals are located at positions in the Euclidean space R2 is considered. The infective individuals, with an infection period that is exponentially distributed with parameter µ, move in R2 according to a Brownian motion with a diffusion coefficient σ2. The susceptible individuals may also move. But we shall use the approximation that they remain unchanged in numbers and therefore assume that the averaged ‘density' of susceptibles per unit area is the same throughout space and time. The transition probability rate of infection of a susceptible in the infinitesimal element of area dy by an infective in dx is assumed to be a function h(x – y |) of the distance | x – y | between x and y. Then our process can be considered as a two-dimensional birth and death Brownian motion. Let be the number of infective individuals in the set D at time t and . The almost everywhere convergence of the random variables to a limit random variable W(D) is established.