A sharp regularity estimate for the Schrödinger propagator on the sphere

Abstract

Let ΔSn denote the Laplace-Beltrami operator on the n-dimensional unit sphere Sn, n≥2. In this paper we show that‖eitΔSnf‖L4([0,2π)×Sn)≤C‖f‖Wα,4(Sn) holds if α>(n−2)/4. The range of α is sharp in the sense that the estimate fails for α<(n−2)/4. As a consequence, we obtain space-time Lp-estimates for eitΔSn for 2≤p≤∞. We also prove that the maximal operator f→sup0≤t<2π⁡|eitΔSnf| is bounded from Wα,2(Sn) to L6n/(3n−2)(Sn) for α>1/3 whenever f are zonal functions on Sn.