A Pedestrian’s Approach to Pseudodifferential Operators

Pseudodifferential operators are an indispensable tool for the study of partial differential equations and are therefore a branch of classical analysis. In this chapter we offer an approach using time-frequency methods. In this approach time-frequency representations that are standard in signal analysis are used to set up the formalism of pseudodifferential operators, and certain classes of function spaces and symbols, the so-called modulation spaces, arise naturally in the investigation. Although the approach is “pedestrian” and based more on engineering intuition than on “hard” analysis, strong results on boundedness and Schatten class properties are within its scope.