We prove that spectral projections of Laplace-Beltrami operator on the
$m$-complex unit sphere $E_{\Delta_{S^{2m-1}}}([0,R))$ are
uniformly bounded as an
operator from $H^p(S^{2m-1})$ to $L^p(S^{2m-1})$ for all $p\in
(1,\infty)$.
We also show that the Bochner-Riesz conjecture is true when restricted
to cylindrically symmetric functions on $\R^{n-1} \times \R$.