关键词: Schr\"odinger equation, smoothing estimate

Abstract:

The usual Kato smoothing estimate for the Schr\"odinger propagator in 1D takes the form $\| |\partial_x|^{\frac 12} {\rm e}^{{\rm i}t\partial_{xx}} u_0 \|_{L_x^\infty L_t^2} \lesssim \| u_0 \|_{L_x^2}$. In dimensions $n\ge 2$ 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 eneralizations 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.

Key words: Schr\"odinger equation, smoothing estimate