A sharp regularity estimate for the Schrodinger propagator on the sphere

Abstract

Let delta(Sn) denote the Laplace-Beltrami operator on the n-dimensional unit sphere S-n, n >= 2. In this paper we show that ?e(it delta Sn)f?L-([0,2 pi)xSn)(4)<= C?f?(W alpha,4(Sn)) holds if alpha >(n-2)/4. The range of alpha is sharp in the sense that the estimate fails for alpha <(n-2)/4. As a consequence, we obtain space-time L-p-estimates for e(it delta Sn) for 2 <= p <=infinity. We also prove that the maximal operator f -> sup(0 <= t < 2 pi?)|e(it delta Sn)f| is bounded from W-alpha,W-2(S-n) to L6n/(3n-2)(S-n) for alpha > 1/3 whenever f are zonal functions on S-n. (C) 2022 Elsevier Masson SAS. All rights reserved.