Optimal Forward and Reverse Estimates of Morawetz and Kato–Yajima Type with Angular Smoothing Index
Optimal Forward and Reverse Estimates of Morawetz and Kato–Yajima Type with Angular Smoothing Index
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DOI：
10.1007/s00041-014-9371-0
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发表时间：
2014-09
影响因子：
1.2
通讯作者：
N. Bez;M. Sugimoto
中科院分区：
文献类型：
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作者：
N. Bez;M. Sugimoto
对于自由Schrödinger方程的解，我们得到了正、反向平滑估计的最优常数和特征极值，它们在空间和时间上是全局的，在空间变量中包含齐次和径向权，并包含一定的角正则性。这将遵循一个更一般的结果，该结果允许类似的尖锐的正向和反向平滑估计以及自由Klein-Gordon方程和波动方程解的极值值的特征。极值的性质对尺寸和相对于尺寸的平滑指数的大小都很敏感。此外，在四个空间维度和某些特殊的平滑指标下，我们得到了这些演化方程的精确恒等式。
For the solution of the free Schrödinger equation, we obtain the optimal constants and characterise extremisers for forward and reverse smoothing estimates which are global in space and time, contain a homogeneous and radial weight in the space variable, and incorporate a certain angular regularity. This will follow from a more general result which permits analogous sharp forward and reverse smoothing estimates and a characterisation of extremisers for the solution of the free Klein–Gordon and wave equations. The nature of extremisers is shown to be sensitive to both the dimension and the size of the smoothing index relative to the dimension. Furthermore, in four spatial dimensions and certain special values of the smoothing index, we obtain an exact identity for each of these evolution equations.