SMOOTHING ESTIMATES OF THE RADIAL SCHRO ¨DINGER PROPAGATOR IN DIMENSIONS n≥2  

作　　者：李栋 张晓轶 

机构地区：[1]Department of Mathematics, University of Iowa [2]Academy of Mathematics and System Sciences, Chinese Academy of Sciences

出　　处：《Acta Mathematica Scientia》2010年第6期2103-2109,共7页数学物理学报（B辑英文版）

基　　金：supported in part by NSF Grant-0908032;a start-up funding from University of Iowa;supported by an Alfred P. Sloan fellowship

摘　　要：The usual Kato smoothing estimate for the Schrodinger propagator in 1D takes the form ｜｜｜δx｜1/2eitθxxu0｜｜ Lx∞Lt2〈∽｜｜u0｜｜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 inhomogeneons Kato smoothing holds true, the corresponding time-local inhomogeneous smoothing estimate cannot hold in general.The usual Kato smoothing estimate for the Schrodinger propagator in 1D takes the form ｜｜｜δx｜1/2eitθxxu0｜｜ Lx∞Lt2〈∽｜｜u0｜｜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 inhomogeneons Kato smoothing holds true, the corresponding time-local inhomogeneous smoothing estimate cannot hold in general.

关 键 词：SchrSdinger equation smoothing estimate 

分 类 号：O175[理学—数学]