PHASE SPACE TRANSFORMS AND MICROLOCAL ANALYSIS

The aim of this notes is to introduce a phase space approach to microlocal analysis. This is just a beginning, and there are many directions one can take from here. The main tool in our analysis is the Bargman transform, which is a phase space transform. In other words, it allows one to represent functions as smooth superpositions of elementary pieces, or coherent states. The coherent states are strongly localized both in position and in frequency, precisely on the scale of the uncertainty principle. This type of analysis has its origins in physics. On the mathematical side, a close relative, namely the FBI transform, was successfully used in the study of partial differential operators with analytic coefficients, see for instance [10]. For additional information about the FBI transform we refer the reader to Delort’s monograph [2], Folland’s book [5] and to the author’s article [15]. Also closely related related topics are discussed in [9], [8]. More recently, phase space transforms were used to construct parametrices for wave and Schrodinger operators with rough coefficients in [14], [12], [7]. We note that there is also an alternate approach to phase space analysis, namely to replace smooth decompositions with discrete decompositions. This was first outlined in Fefferman [4] and pursued later by a number of authors. Most notably, we should mention Smith [11]’s introduction of wave packets in the study of the wave equation. However, as the reader will see, there is a significant advantage in using smooth families of coherent states as opposed to any discrete methods. The road map for this article is as follows. First we introduce the Bargman transform and some simple properties. Then we use it to give a simple characterization of S 00 type pseudodifferential operators. Next, following [14] and [15], we introduce a higher order calculus and use it to prove some classical estimates, namely the sharp Garding and the Fefferman-Phong inequality.