A new proof of global smoothing estimates for dispersive equations

The aim of this article is to provide a new method to prove global smoothing estimates for dispersive equations such as Schrodinger 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 L-2-boundedness is also proved. The boundedness result is not covered by previous one such as Asada and Fujiwara [1]. By using them, a different proof for the result obtained by Ben-Artzi & Klainerman [2] 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.