Singularities, Supports and Lacunas

Abstract

These notes are a survey of three aspects of the modern theory of linear partial differential equations, and its generalization to the microlocal analysis of pseudodifferential operators. The first chapter is a study of the propagation of singularities of partial and pseudo differential equations, beginning with a sketch of the extensive background of pseudo differential and Fourier integral operators and wave front sets — the machinery of microlocal analysis in phase space. Selected results on equations with multiple characteristics are then discussed, in the involutive and non-involutive cases. The second chapter is a description of the work of C. Fefferman and others on the approximate simultaneous diagonalization of differential operators with variable coefficients, regarded as algebraic operators in phase space. The uncertainty principle, a title borrowed from Heisenberg’s quantum mechanics, limits the precision of this process, since a function and its Fourier transform cannot both have small supports. This area of investigation, which draws upon the full resources of microlocal analysis, appears to have interesting future prospects.