SG-Lagrangian submanifolds and their parametrization

Abstract

We continue our study of tempered oscillatory integrals $I_{\phi }(a)$, here investigating the link with a suitable symplectic structure at infinity, which we describe in detail. We prove adapted versions of the classical theorems, which show that tempered distributions of the type $I_{\phi }(a)$ are indeed linked to suitable Lagrangians extending to infinity, that is, extending up to the boundary and in particular the corners of a compactification of $T^{\ast }{\mathbb{R}}^d$ to ${\mathbb{B}}^d\times {\mathbb{B}}^d$. In particular, we show that such Lagrangians can always be parametrized by non-homogeneous, regular phase functions, globally defined on some ${\mathbb{R}}^d\times {\mathbb{R}}^s$. We also state how two such phase functions parametrizing the same Lagrangian may be considered equivalent up to infinity.

45 pages, 1 figure, minor corrections and additions with respect to v1

References in corpus (1)

• Calculus, continuity and global wave-front properties for Fourier integral operators on ${\mathbf{R}}^d$

Cited by in corpus (1)

• Fourier integral operators algebra and fundamental solutions to hyperbolic systems with polynomially bounded coefficients on R^n