Global L ²-Boundedness Theorems for a Class of Fourier Integral Operators

Abstract

The local L 2 -mapping property of Fourier integral operators has been established in Hörmander [14] and in Eskin [12]. In this paper, we treat the global L 2 -boundedness for a class of operators that appears naturally in many problems. As a consequence, we will improve known global results for several classes of pseudodifferential and Fourier integral operators, as well as extend previous results of Asada and Fujiwara [1] or . As an application, we show a global smoothing estimate to generalized Schrödinger equations which extends the results of Ben-Artzi and Devinatz [2], Walther [27], and [28].

On the boundedness of certain bilinear oscillatory integral operators

“…In the context of our current investigation, the following result regarding the global boundedness of linear oscillatory integral operators will be useful to us. It is a reformulation of Theorem 2.1 in [18]. Theorem 2.3.…”

“…However, we will make use of the stronger assumptions stated here to prove commutator estimates, so for simplicity state the result as we will use it, rather than in the full generality of [18].…”

“…Since multiplication by e iϕ 1 (x,0)+iϕ 2 (x,0) is an isometry, and ϕ j will inherit directly from ϕ j exactly the same properties that ϕ j satisfies (namely the strong non-degeneracy condition and (9)), the question of boundedness is reduced to the case of an operator with phases which satisfy (18). Let us introduce two smooth cut-off functions χ, ν : R 2n → R, such that χ(ξ, η) = 1 for |(ξ, η)| 1 and χ(ξ, η) = 0 for |(ξ, η)| 2, and ν(ξ, η) = 0 for 2|ξ| |η| and ν(ξ, η) = 1 for 2|η| |ξ|.…”

“…In [18], M. Ruzhansky and M. Sugimoto considered a class of phases which, in particular, included those satisfying (9).…”

On global inversion of homogeneous maps

“…Generalising such results to the case of homogeneous phase as in Theorem 1.1 is also important but it is not straightforward because of the singularity at the origin. The global L 2 boundedness of oscillatory integrals with phases as in Theorem 1.1 has been analysed by the authors in [7] and applied to questions of global smoothing for partial differential equations in [8]. The application of Theorem 1.1 to the global analysis of hyperbolic partial differential equations and the corresponding Hamiltonian flows will appear elsewhere.…”