Bochner-Riesz Summability for Analytic Functions on the <i>m</i>-complex Unit Sphere and for Cylindrically Symmetric Functions on ℝ^{n - 1} x ℝ

Abstract

We prove that spectral projections of Laplace-Beltrami operator on the m-complex unit sphere E ∆ S 2m−1 ([0, R)) are uniformly bounded as operators fromWe also show that the Bochner-Riesz conjecture is true when restricted to cylindrically symmetric functions onSuppose that L is a positive definite, self-adjoint operator acting on L 2 (X, µ), where X is a measurable space with a measure µ. Such operator admits a spectral resolutionBy the spectral theorem, if F is a Borel bounded function on [0, ∞), then the operator F (L) given byis well-defined and bounded on L 2 (X, µ). One of the fundamental problems in the theory of spectral multipliers is to determine when F (L) is bounded on L p for some p = 2. An interesting example is the following family of functionsWe define the operator S δ R (L) using (1). S δ R (L) is called the Riesz mean or the Bochner-Riesz mean of order δ. The basic question in the theory

L p -summability of Riesz means for the sublaplacian on complex spheres

“…More recently, Sikora and Tao [27] studied the L p -convergence of the Riesz means associated with the Laplace-Beltrami operator in two particular situations: on the complex sphere S 2n−1 when restricted to the subspace H p (S 2n−1 ) ⊂ L p (S 2n−1 ) of analytic functions on S 2n−1 , and on R n = R n−1 × R, when restricted to the space of cylindrically symmetric functions on R n−1 × R. In both cases they obtained sharp results and were able to remove the condition |1/p − 1/2| 1/(n + 1) from the assumptions for the L p -convergence of the Riesz means.…”

“…It is interesting to recall that, in the case of minus the Laplace-Beltrami operator −Δ S 2n−1 , Sikora and Tao [27] showed that the Riesz means of index δ when restricted to the holomorphic functions, that is, to +∞ =0 H ,0 (or on its dual space), converge in the L p -norm if δ 0. That is, they were able to show that the Riesz means for the Laplace-Beltrami on the complex sphere behave much better when restricted to holomorphic functions than on the whole of L p (S 2n−1 ).…”