Title: Pseudodifferential Operators at the Critical Line I: Background and Motivation

Abstract: Pseudodifferential operators extend the Fourier-transform description of differential operators and provide a framework for studying partial differential equations and spectral theory. In this first of two talks, I will introduce these operators through their symbols and oscillatory integral representations, and explain the role of stationary phase in their analysis.

A central theme will be the balance between spatial and frequency derivatives in the Hörmander symbol classes. In the standard calculus, this balance allows successive correction terms in symbolic expansions to have lower order. At the critical indices (\rho=\delta=1/2), that improvement is no longer guaranteed by the symbol estimates. I will explain this difficulty and motivate the use of direct integration-by-parts estimates.

I will also introduce the functional-calculus problem: when does a function (p(A)) of an elliptic self-adjoint operator remain pseudodifferential? The wave-group representation connects this question to the oscillatory integrals studied in the second talk. This introductory talk is intended for a general analysis audience and assumes no prior familiarity with pseudodifferential operators or oscillatory integrals.