Metrics on the phase space and non-selfadjoint pseudo-differential operators

This book is devoted to the study of pseudo-differential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space. We expose the most recent developments of the theory with its applications to local solvability and semi-classical estimates for nonselfadjoint operators. The first chapter is introductory and gives a presentation of classical classes of pseudo-differential operators. The second chapter is dealing with the general notion of metrics on the phase space. We expose some elements of the so-called Wick calculus and introduce general Sobolev spaces attached to a pseudo-differential calculus. The third and last chapter, is devoted to the topic of non-selfadjoint pseudo-differential operators. After some introductory examples, we enter into the discussion of estimates with loss of one derivative, starting with the proof of local solvability with loss of one derivative under condition (P). We show that an estimate with loss of one derivative is not a consequence of condition (Psi). Finally, we give a proof of an estimate with loss of 3/2 derivatives under condition (Psi). This book is accessible to graduate students in Analysis, and provides an up-todate overview of the subject, hopefully useful to researchers in PDE and Semi-classical Analysis.