Metrics on the Phase Space
Nicolás Lerner · Birkhäuser Basel eBooks · 2010
In this chapter, we describe a general version of the pseudo-differential calculus, due to L. Hörmander ([69], Chapter 18 in [73]). That version followed some earlier generalizations due to R. Beals and C. Fefferman ([8]) and to R. Beals ([6]). It was followed by some other generalizations due to A. Unterberger [145] and to a joint work of J.-M. Bony and N. Lerner ([20]), whose presentation we follow. We also give a precised version of the Fefferman-Phong inequality, following [100] where we provide an upper bound for the number of derivatives sufficient to obtain that inequality. Finally, we study the Sobolev spaces naturally attached to an admissible metric on the phase space, essentially following the paper by J.-M. Bony and J.-Y. Chemin [19]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.