BOCHNER-RIESZ SUMMABILITY FOR ANALYTIC FUNCTIONS ON THE m-COMPLEX UNIT SPHERE AND FOR CYLINDRICALLY SYMMETRIC FUNCTIONS ON R × R ADAM SIKORA AND TERENCE TAO

We prove that spectral projections of Laplace-Beltrami operator on the m-complex unit sphere E∆ S2m−1 ([0, R)) are uniformly bounded as an operator from H(S) to L(S) for all p ∈ (1,∞). We also show that the Bochner-Riesz conjecture is true when restricted to cylindrically symmetric functions on R × R. Suppose that L is a positive definite, self-adjoint operator acting on L(X,μ), where X is a measurable space with a measure μ. Such operator admits a spectral resolution L = ∫ ∞ 0 λ dEL(λ). By the spectral theorem, if F is a Borel bounded function on [0,∞), then the operator F (L) given by