SG-Lagrangian submanifolds and their parametrization

We continue our study of tempered oscillatory integrals $I_\varphi(a)$, 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 $I_\varphi(a)$ 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 $\mathbb{B}^d\times\mathbb{B}^d$. 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.