Bochner--Riesz summability for analytic functions on the \(m\)-complex unit sphere and for cylindrically symmetric functions on \(\mathbb{R}^{n-1} \times \mathbb{R}\) (Q707464)

Let \(S^{2m-1}\) be the boundary of the unit ball \(\mathcal{B}\)\(_{m}\) in \(\mathbb{C}^m\) and \(H^p(S^{2m-1})\) the subspace of \(L^p(S^{2m-1})\) consisting of analytic functions on \(S^{2m-1}\). This paper shows that the spectral projections of the Laplace-Beltrami operator \(E_{\Delta_{S^{2m-1}}}([0,\mathbb{R}))\), as operators from \(H^p(S^{2m-1})\) to \(L^p(S^{2m-1})\), are uniformly bounded for all \(p\in(1,\infty)\). It also shows that the Bochner-Riesz conjecture is true when restricted to cylindrically symmetric functions on \(\mathbb{R}^{n-1}\times \mathbb{R}\).