Singularities in linear wave propagation / Lars Garding

This book presents an application of the ideas of microlocal analysis to the examination of the singularities of solutions to linear hyperbolic equations. Chapter I is devoted to the construction of the fundamental solution of hyperbolic operators with constant coefficients. Here are discussed propagation cones, general conical refraction and the Herglotz-Petrovskiĭ formula. In Chapters 2 and 3 the basic facts from microlocal analysis including wave front sets, Fourier integral operators and the calculus for pseudodifferential operators are presented. Chapter 4 contains some material from symplectic geometry which is necessary for the exposition. In Chapter 5 the author proposes a new simple construction of a global parametrix for the fundamental solution of a first-order pseudodifferential operator. This construction leads to some oscillatory integrals paired in a suitable way. Chapters 6 and 7 contain a careful analysis of these paired oscillatory integrals. Here are defined sharp and diffuse fronts, Petrovskiĭ chains and cycles, and the singularities of such integrals are studied. The book is very well written, the exposition is clear and combined with historical remarks. These lecture notes are an excellent introduction to the analysis of singularities for hyperbolic operators. (MathSciNet)