Applications of Fourier Integral Operators
نویسندگان
چکیده
Fourier integral operators, for the calculus of which I refer to Hörmander [17], have been applied in essentially two ways: as similarity transformations and in the description of the solutions of genuinely nonelliptic (pseudo-) differential equations. The first application is based on the observation of Egorov [12] that if P9 resp. g, is a pseudo-differential operator with principal symbol equal to p9 resp. q9 and Po A = AoQ for an elliptic Fourier integral operator A defined by the homogeneous canonical transformation C, then p = #°C. This idea, or rather its local version in conic open subsets of the cotangent bundle of the manifolds on which the operators are defined, is a much more powerful tool for bringing operators locally into standard form than merely by coordinate changes in the base space. It has been used not only to reduce the study of wide classes of operators to simple ones like 3/9*1, 3/3*1 + ?'3/3*2> 3/3*i + ?*i3/3*2> but also in more subtle problems it has been a helpful trick. A rather complete impression of this sort of application can be obtained by looking at the papers of Egorov [13], [14], Nirenberg and Treves [26], Hörmander [18], Duistermaat and Hörmander [7], Sato, Kawai and Kashiwara [27], Duistermaat and Sjöstrand [8], Sjöstrand [28], Boutet de Monvel [3], and Weinstein [29]. In this respect the following conjecture of Singer seems interesting. Let L{X) denote the space of pseudo-differential operators of order m9 Lr°°{X) the space of smoothing operators. I restrict here to operators for which the total symbol has an asymptotic expansion in homogeneous terms of integer order. If A is an elliptic Fourier integral operator defined by a canonical transformation C: T*Y\0^> T*X\09 then P H* A~ PA is an isomorphism of filtered algebras:
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