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.Comment: 45 pages, 1 figure, minor corrections and additions with respect to v

On the Spectral Asymptotics of Operators on Manifolds with Ends

We deal with the asymptotic behaviour for $\lambda\to+\infty$ of the counting function $N_P(\lambda)$ of certain positive selfadjoint operators $P$ with double order $(m,\mu)$, $m,\mu>0$, $m\not=\mu$, defined on a manifold with ends $M$. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier Integral Operators associated with weighted symbols globally defined on $\mathbb{R}^n$. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae for $N_P(\lambda)$ and show how their behaviour depends on the ratio $\frac{m}{\mu}$ and the dimension of $M$.Comment: Final version, 30 page

Global Lp continuity of Fourier integral operators

In this paper we establish global Lp regularity properties of Fourier integral operators. The orders of decay of the amplitude are determined for operators to be bounded on L^p(\Rn), $1<p<\infty$, as well as to be bounded from Hardy space H^1(\Rn) to L^1(\Rn). The obtained results extend local $L^p$ regularity properties of Fourier integral operators established by Seeger, Sogge and Stein (1991) as well as global L^2(\Rn) results of Asada and Fujiwara (1978) and Ruzhansky and Sugimoto (2006), to the global setting of L^p(\Rn). Global boundedness in weighted Sobolev spaces W^{\sigma,p}_s(\Rn) is also established. The techniques used in the proofs are the space dependent dyadic decomposition and the global calculi developed by Ruzhansky and Sugimoto (2006) and Coriasco (1999).Comment: 20 page