Metrics on the Phase Space
In this chapter, we describe a general version of the pseudo-differential calculus, due to L. Hörmander (, Chapter 18 in ). That version followed some earlier generalizations due to R. Beals and C. Fefferman () and to R. Beals (). It was followed by some other generalizations due to A. Unterberger  and to a joint work of J.-M. Bony and N. Lerner (), whose presentation we follow. We also give a precised version of the Fefferman-Phong inequality, following  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 .
KeywordsPhase Space Sobolev Space Wigner Function Selfadjoint Operator Composition Formula
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