A New Proof of Global SmoothingEstimates for Dispersive Equations
Part of the
Operator Theory: Advances and Applications
book series (OT, volume 155)
The aim of this article is to provide a new method to prove global smoothing estimates for dispersive equations such as Schrödinger equations. For the purpose, the Egorov-type theorem via canonical transformation in the form of a class of Fourier integral operators is established, and their weighted L2-boundedness is also proved. The boundedness result is not covered by previous one such as Asada and Fujiwara . By using them, a different proof for the result obtained by Ben-Artzi & Klainerman  is provided. This new idea gives a clear understanding of smoothing effects of dispersive equations, and further developments are also expected. In fact, some extended results based on the same idea are also announced.
KeywordsDispersive equation smoothing effect canonical transformation
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