We primarily consider here the $L^2$ mapping properties of a class of strongly singular Radon transforms on the Heisenberg group $\mathbb{H}^n$; these are convolution operators on $\mathbb{H}^n$ with kernels of the form $M(z,t)=K(z)\delta_0(t)$, where $K$ is a strongly singular kernel on $\mathbb{C}^n$. Our results are obtained by using the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdélyi.

We also discuss the behaviour of related twisted strongly singular operators on $L^2(\mathbb{C}^n)$ and obtain results in this context independently of group Fourier transform methods. Key to this argument is a generalization of the results for classical strongly singular integrals on $L^2(\mathbb{R}^d)$.