Propagation of Singularities in the Interior of a Domain

Abstract

In this chapter I consider microlocal propagation of singularities (i.e., oscillation front sets) in the interior of a domain. In general in this book I consider two types of microlocal operator, namely, microelliptic and microhyperbolic. For microelliptic operators there is no propagation of singularities at all and I prove the different variants of this fact which I need by means of either a parametrix construction or the Gårding inequality; sometimes I combine both methods. In this chapter I treat microhyperbolic operators. In section 2.1 I prove the main general theorem and certain of its corollaries which will be necessary for spectral asymptotics; this theorem is formulated in terms of auxiliary real-valued functions Ø _j (x, ξ) with j = 1,..., J. Then in section 2.2 I reformulate this theorem in terms of bicharacteristics and generalized bicharacteristics. Finally, in section 2.3 I study the propagation of singularities for scalar and related operators for a large time interval by quite different methods.