Fourier Integral Operators
Springer Science & Business Media, 29 nov. 1995 - 142 pages
More than twenty years ago I gave a course on Fourier Integral Op erators at the Catholic University of Nijmegen (1970-71) from which a set of lecture notes were written up; the Courant Institute of Mathematical Sciences in New York distributed these notes for many years, but they be came increasingly difficult to obtain. The current text is essentially a nicely TeXed version of those notes with some minor additions (e.g., figures) and corrections. Apparently an attractive aspect of our approach to Fourier Integral Operators was its introduction to symplectic differential geometry, the basic facts of which are needed for making the step from the local definitions to the global calculus. A first example of the latter is the definition of the wave front set of a distribution in terms of testing with oscillatory functions. This is obviously coordinate-invariant and automatically realizes the wave front set as a subset of the cotangent bundle, the symplectic manifold in which the global calculus takes place.
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amplitude asymptotic bicharacteristic curves bicharacteristic strips bijective called canonical relation canonical transformation Cauchy closed cone compact subset conic Lagrange manifold conic manifold conic neighborhood conic subset coordinate coordinatization defined Definition denoted densities of order diag diffeomorphism distribution equal ess supp fiber follows Fourier integral operator Fourier transform global graph Hamilton field homogeneous canonical homogeneous of degree Hormander implicit function theorem implies intersection invariant involutive isomorphism isotropic Lagrange space Lagrangian subspace Lemma line bundle linear mapping linear subspace linearly independent locally finite manifold in T*(X modulo n-dimensional noncharacteristic nondegenerate phase function Note obtain open subset operator of order oscillatory partition of unity principal symbol projection properly supported Proposition pseudodifferential operator respectively satisfies Section sgn Q smooth solution strictly hyperbolic submanifold symplectic manifold symplectic vector space tangent space Tm(M Tm(V transversal Tx(X unique vector field view of Theorem wave front set WF(u