Smoothing estimates of the radial schrödinger propagator in dimensions n ≥ 2

Abstract

The usual Kato smoothing estimate for the Schrödinger propagator in 1D takes the form {norm of matrix}{norm of matrix}{norm of matrix}∂x{norm of matrix} 1/2 eit∂xxu0{norm of matrix}{norm of matrix}Lx∞Lt2≲{norm of matrix}{norm of matrix}u0{norm of matrix}{norm of matrix}Lx2. In dimensions n ≥ 2 the smoothing estimate involves certain localization to cubes in space. In this paper we focus on radial functions and obtain Kato-type sharp smoothing estimates which can be viewed as natural generalizations of the 1D Kato smoothing. These estimates are global in the sense that they do not need localization in space. We also present an interesting counterexample which shows that even though the time-global inhomogeneous Kato smoothing holds true, the corresponding time-local inhomogeneous smoothing estimate cannot hold in general. © 2010 Wuhan Institute of Physics and Mathematics.