Acta mathematica scientia, Series B >
SMOOTHING ESTIMATES OF THE RADIAL SCHRÖDINGER PROPAGATOR IN DIMENSIONS n≥2
Received date: 2010-10-11
Online published: 2010-11-20
Supported by
D. Li is supported in part by NSF Grant-0908032 and a start-up funding from University of Iowa. X. Zhang is supported by an Alfred P. Sloan fellowship and also a start-up funding from University of Iowa.
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
LI Dong , ZHANG Xiao-Tie . SMOOTHING ESTIMATES OF THE RADIAL SCHRÖDINGER PROPAGATOR IN DIMENSIONS n≥2[J]. Acta mathematica scientia, Series B, 2010 , 30(6) : 2103 -2109 . DOI: 10.1016/S0252-9602(10)60194-4
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