[Paper Review] SG-Lagrangian submanifolds and their parametrization

Sandro Coriasco, René Schulz|arXiv (Cornell University)|Jun 7, 2014

Geometric Analysis and Curvature Flows21 references3 citations

TL;DR

This paper establishes a geometric framework for tempered oscillatory integrals on non-compact spaces by introducing SG-Lagrangian submanifolds—Lagrangian submanifolds extending to the boundary and corners of a compactified cotangent bundle $\mathbb{B}^d \times \mathbb{B}^d$. It proves that such submanifolds admit global parametrization via non-homogeneous, regular phase functions on $\mathbb{R}^d \times \mathbb{R}^s$, and characterizes equivalence of phase functions up to infinity, generalizing classical Fourier integral operator theory to include singularities at infinity.

ABSTRACT

We continue our study of tempered oscillatory integrals $I_φ(a)$, 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 $I_φ(a)$ 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 $T^*\mathbb{R}^d$ to $\mathbb{B}^d imes\mathbb{B}^d$. In particular, we show that such Lagrangians can always be parametrized by non-homogeneous, regular phase functions, globally defined on some $\mathbb{R}^d imes\mathbb{R}^s$. We also state how two such phase functions parametrizing the same Lagrangian may be considered equivalent up to infinity.

Motivation & Objective

• To extend classical Fourier integral operator theory to non-compact manifolds by incorporating singularities at infinity.
• To define and characterize Lagrangian submanifolds that extend to the boundary and corners of a compactification of $T^*\mathbb{R}^d$.
• To establish a global parametrization of such Lagrangians using non-homogeneous, regular phase functions on $\mathbb{R}^d \times \mathbb{R}^s$.
• To define equivalence relations between phase functions that parametrize the same SG-Lagrangian up to infinity.
• To provide a geometric foundation for tempered oscillatory integrals in the $\mathrm{SG}$-calculus framework, enabling global analysis of PDEs with growth and oscillatory behavior.

Proposed method

• Utilizes the $\mathrm{SG}$-calculus to define a symplectic structure at infinity on the compactified cotangent bundle $\mathbb{B}^d \times \mathbb{B}^d$, extending the classical symplectic form on $T^*\mathbb{R}^d \setminus \{0\}$.
• Introduces the concept of $\mathrm{SG}$-Lagrangian submanifolds as conic Lagrangian submanifolds in $\mathcal{W}_{\mathrm{SG}}$ that extend smoothly to the boundary and corners of the compactification.
• Applies the implicit function theorem on manifolds with corners to construct local parametrizations of $\mathrm{SG}$-Lagrangians via phase functions $\varphi(x,\theta)$ that are smooth and regular, but not necessarily homogeneous.
• Derives a global parametrization theorem showing that every $\mathrm{SG}$-Lagrangian arises as the stationary phase set $\Lambda_\varphi = \{(x, \nabla_x \varphi(x,\theta)) \mid \nabla_\theta \varphi(x,\theta) = 0\}$ for some non-homogeneous, regular phase function $\varphi$ defined on $\mathbb{R}^d \times \mathbb{R}^s$.
• Defines equivalence of phase functions up to infinity via a relation that preserves the associated Lagrangian and wave front set, generalizing classical phase equivalence in the $\mathrm{SG}$-setting.
• Employs the theory of manifolds with corners and neat submanifolds to ensure that the parametrizing phase functions and their associated Lagrangians respect the boundary structure of the compactified space.

Experimental results

Research questions

• RQ1 How can the classical theory of Lagrangian distributions and Fourier integral operators be extended to non-compact manifolds to include singularities at infinity?
• RQ2 What is the geometric structure of Lagrangian submanifolds that extend to the boundary and corners of the compactified cotangent bundle $\mathbb{B}^d \times \mathbb{B}^d$?
• RQ3 Can every such extended Lagrangian be globally parametrized by a non-homogeneous, regular phase function on $\mathbb{R}^d \times \mathbb{R}^s$?
• RQ4 What conditions define equivalence between two phase functions that generate the same $\mathrm{SG}$-Lagrangian submanifold?
• RQ5 How does the $\mathrm{SG}$-calculus provide a natural framework for wave front sets that encode both local and at-infinity singularities?

Key findings

• Every $\mathrm{SG}$-Lagrangian submanifold of $\mathbb{B}^d \times \mathbb{B}^d$ arises as the stationary phase set $\Lambda_\varphi$ of a non-homogeneous, regular phase function $\varphi$ globally defined on $\mathbb{R}^d \times \mathbb{R}^s$ for some $s \geq d$.
• The parametrization is globally valid and respects the boundary and corner structure of the compactified space, ensuring that the Lagrangian extends smoothly to the boundary.
• Two phase functions $\varphi_1$ and $\varphi_2$ parametrize the same $\mathrm{SG}$-Lagrangian if and only if they are equivalent up to infinity, meaning their difference satisfies specific asymptotic conditions at infinity.
• The wave front set of a tempered oscillatory integral $I_\varphi(a)$ coincides with the $\mathrm{SG}$-Lagrangian $\Lambda_\varphi$, generalizing Hörmander’s classical result to include singularities at infinity.
• The theory provides a geometric foundation for analyzing operators with kernels that are Legendrian or Lagrangian distributions in the $\mathrm{SG}$-setting, such as the two-point function in Klein-Gordon theory.
• The framework allows for a global extension of the composition calculus of Fourier integral operators to non-compact settings, with propagation of singularities now including behavior at infinity.