F.B.I. transformation: second microlocalization and semilinear caustics
During the last ten years, FBI transformation and second microlocalization have been used by several authors to solve different problems in the theory of linear or nonlinear partial differential equations. The aim of this book is to give an introduction to these topics, in the spirit of the work ofSj|strand, and to present their recent application to the propagation of conormal singularities for solutions of seminlinear hyperbolic equations, due to Lebeau. The text is quite self-contained and provides a useful entry to the subject and a bridging link to more specialized papers.
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assume Cauchy data Cauchy problem Chapter classical symbol compactly supported compactly supported distribution conic neighborhood conormal bundle contour coordinates cotangent bundle critical value deduce defined definition diffeomorphism dt(x equation exists FBI phase FBI transformation finite formula given holomorphic function homotopy hypersurface implies inclusion 3.4 inequality integral isomorphism lagrangian submanifold Let us choose Let us consider Let us denote linear Moreover neighborhood of xo open subset pluriharmonic Proceedings Proposition quadratic form rapidly decreasing real analytic function real analytic manifold real analytic map real analytic submanifold real number resp result right hand side satisfies second kind second microlocalization second microsupport second wave front semilinear sequence singularities smooth Sobolev spaces solution space SS(u Stokes formula stratification strictly plurisubharmonic subanalytic sets subanalytic subset symplectic Theorem 1.3 transverse unique critical point vector fields wave front set WF(u