On global representation of lagrangian distributions and solutions of hyperbolic equations

Abstract

In this paper we develop a new approach to the theory of Fourier integral operators. It allows us to represent the Schwartz kernel of a Fourier integral operator by one oscillatory integral with a complex phase function. We consider Fourier integral operators associated with canonical transformations, having in mind applications to hyperbolic equations. As a by-product we obtain yet another formula for the Maslov index.

Global Fourier Integral Operators and Semiclassical Asymptotics

“…The simplicity of our construction is achieved by using, in our FIO's, complex phases with quadratic imaginary parts. Our approach is inspired by works [17] and [18], where the global parameterization of homogeneous Langrangian distributions was introduced for construction of fundamental solutions of wave equations. Oscillatory integrals with complex phases having quadratic imaginary parts were first considered in [1,2,3,8,9,16,20,25,27] in connection with the problem of propagation of Gaussian wave packets.…”

The symbol of a lagrangian distribution

“…As shown in [1] (see also Theorem 3.3 below), any oscillatory integral (1.4) can be rewritten (modulo C 00 } with an amplitude a(*;y,7/) independent of x. This amplitude a is called the (full) symbol of our Lagrangian distribution.…”

“…The fact that we allow our phase functions to be complex-valued is crucial, because otherwise we would not be able to use a global construction, see [1].…”

“…Properties of phase functions of the type described above were studied in [1]. Let us mention briefly some of these properties:…”

Preface to the Special Issue in Memory of Yuri Safarov

“…Yuri insisted on using his particular microlocal technique based on an invariantly defined complexvalued phase function. This technique originates from the paper [8] and allows one to construct oscillatory integrals globally in time, passing without problems through caustics and avoiding the need for tackling compositions of oscillatory integrals. Furthermore, this approach leads to a new, purely analytic and straightforward, definition of the Maslov index.…”