In this paper we study the L^p-convergence of the Riesz means for the sublaplacian on the sphere S^{2n-1} in the complex n-dimensional space C^n. We show that the Riesz means of order delta of a function f converge to f in L^p(S^{2n-1}) when delta>de
lta(p):=(2n-1)|12-1p|. The index delta(p) improves the one found by Alexopoulos and Lohoue, $2n|12-1p|$, and it coincides with the one found by Mauceri and, with different methods, by Mueller in the case of sublaplacian on the Heisenberg group.
By using the notion of contraction of Lie groups, we transfer $L^p-L^2$ estimates for joint spectral projectors from the unit complex sphere $sfera$ in ${{mathbb{C}}}^{n+1}$ to the reduced Heisenberg group $h^{n}$. In particular, we deduce some estim
ates recently obtained by H. Koch and F. Ricci on $h^n$. As a consequence, we prove, in the spirit of Sogges work, a discrete restriction theorem for the sub-Laplacian $L$ on $h^n$.
