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We study the hypoellipticity properties of a class of time-periodic evolution operators, with coefficients globally defined on $\mathbb{R}^d$ and growing polynomially with respect to the space variable. To this aim, we introduce a class of time-periodic weighted Sobolev spaces, whose elements are characterised in terms of suitable Fourier expansions, associated with elliptic operators., Comment: 32 pages. This is the first version. Some changes might be performed in the next versions

We study structural properties of Wiener-Lebesgue spaces with respect to a slowly varying metrics and certain Lebesgue parameters. For $p\in (0,1]$, we deduce Schatten-$p$ properties for pseudo-differential operators whose symbols, together with their derivatives, obey suitable Wiener-Lebesgue-boundedness conditions. Especially, we perform such investigations for the Weyl-H\"ormander calculus. Finally, we apply our results to global-type SG and Shubin pseudo-differential operators., Comment: 21 pages. This is the first version. Some changes might be performed in the next versions

We treat some classes of stochastic partial differential equations of Schr\"odinger type within the framework of white noise analysis, combined with Wiener-It\^o chaos expansions and pseudodifferential operator methods. The initial data and potential term of the Schr\"odinger operator are assumed to be generalized stochastic processes that have both temporal and spatial dependence. We prove that the equations under consideration have unique solutions in the appropriate (intersections of weighted) Sobolev-Kato-Kondratiev spaces., Comment: 18 pages. This is the first version. It is expected that some revisions will be performed in the future

We deduce continuity and (global) wave-front properties of classes of Fourier multipliers, pseudo-differential, and Fourier integral operators when acting on Orlicz spaces, or more generally, on Orlicz-Sobolev type spaces. In particular, we extend H{\"o}rmander's improvement of Mihlin's Fourier multiplier theorem to the framework of Orlicz spaces. We also show how Young functions $\Phi$ of the Orlicz spaces are linked to properties of certain Lebesgue exponents $p_\Phi$ and $q_\Phi$ emerged from $\Phi$., Comment: 22 pages. Several parts of the first version (contained 14 pages) are performed. Especially some correction and completion of earlier insufficient arguments are performed

We study a class of weakly hyperbolic Cauchy problems on $\mathbb{R}^d$, involving linear operators with characteristics of variable multiplicities, whose coefficients are unbounded in the space variable. The behaviour in the time variable is governed by a suitable "shape function". We develop a parameter-dependent symbolic calculus, corresponding to an appropriate subdivision of the phase space. By means of such calculus, a parametrix can be constructed, in terms of (generalized) Fourier integral operators naturally associated with the employed symbol class. Further, employing the parametrix, we prove $\mathscr{S}(\mathbb{R}^{d})$-wellposedness and give results about the global decay and regularity of the solution, within a scale of weighted Sobolev space., Comment: 52 pages. Misprints and other minor corrections. Extension and improvement of some arguments and examples

We study the Cauchy problem for Schr\"odinger type stochastic partial differential equations with uniformly bounded coefficients on a curved space. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc., Comment: 10 pages. Minor modification and extension of the introduction. Misprints correction. arXiv admin note: text overlap with arXiv:2010.07087

We study function-valued solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable parabolicity hypotheses. We provide conditions on the initial data and on the stochastic terms, namely, on the associated spectral measure, so that these mild solutions exist uniquely in suitably chosen functional classes., Comment: 14 pages

We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order $(1,1)$, on an asymptotically Euclidean manifold. We first prove a two term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator $Q=(1+|x|^2)(1-\Delta)$ on $\mathbb{R}^d$., Comment: 26 pages

We consider bilinear pseudo-differential operators whose symbols posses Gevrey type regularity and may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand-Shilov type spaces. We consider both Beurling and Roumieu type symbol classes and Gelfand-Shilov spaces., Comment: 31 pages

We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R^n. We prove well-posedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front sets describing the evolution of both smoothness and decay singularities of temperate distributions. Moreover, we can prove the existence of random-field solutions for the associated stochastic Cauchy problems. To this aim, we first discuss algebraic properties for iterated integrals of suitable parameter-dependent families of Fourier integral operators, associated with the characteristic roots, which are involved in the construction of the fundamental solution. In particular, we show that, also for this operator class, the involutiveness of the characteristics implies commutative properties for such expressions., Comment: 43 pages

We develop the notion of Lagrangian distribution on scattering manifolds, meaning on the compactified cotangent bundle, which is a manifold with corners equipped with a scattering symplectic structure. In particular, we study the notion of principal symbol of the arising class of distributions., Comment: 57 pages, 4 figures. Second version: new introduction, small changes to the text and structure

We deduce one-parameter group properties for pseudo-differential operators $\operatorname{Op} (a)$, where $a$ belongs to the class $\Gamma ^{(\omega _0)}_*$ of certain Gevrey symbols. We use this to show that there are pseudo-differential operators $\operatorname{Op} (a)$ and $\operatorname{Op} (b)$ which are inverses to each others, where $a\in \Gamma ^{(\omega _0)}_*$ and $b\in \Gamma ^{(1/\omega _0)}_*$. We apply these results to deduce lifting property for modulation spaces and construct explicit isomorpisms between them. For each weight functions $\omega ,\omega _0$ moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) $\operatorname{Tp} (\omega _0)$ is an isomorphism from $M^{p,q}_{(\omega )}$ onto $M^{p,q}_{(\omega /\omega _0)}$ for every $p,q \in (0,\infty ]$., Comment: 43 pages. This is the first version. It is expected that it will be some major changes in the future. arXiv admin note: text overlap with arXiv:0905.4954

We study mild solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable hyperbolicity hypotheses on the linear part. We provide conditions on the initial data and on the stochastic terms, namely, on the associated spectral measure, so that mild solutions exist and are unique in suitably chosen functional classes. More precisely, function-valued solutions are obtained, as well as a regularity result., Comment: 27 pages; major revision of the contents with focus on the semilinear case

Given two compact manifolds $X,Y,$ with boundary and a boundary preserving symplectomorphism $\chi:T^*Y\setminus0\to T^*X\setminus0$, which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with $\chi$. We study their mapping properties between Sobolev spaces, develop a calculus and prove a Egorov type theorem. We also introduce a notion of ellipticity which implies the Fredholm property. Finally, we show how -- in the spirit of a classical construction by A. Weinstein -- a Fredholm operator of this type can be associated with $\chi$ and a section of the Maslov bundle. If $\dim Y>2$ or the Maslov bundle is trivial, the index is independent of the section and thus an invariant of the symplectomorphism.

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 v1

We study a class of Fourier integral operators on compact manifolds with boundary, associated with a natural class of symplectomorphisms, namely, those which preserve the boundary. A calculus of Boutet de Monvel's type can be defined for such Fourier integral operators, and appropriate continuity properties established. One of the key features of this calculus is that the local representations of these operators are given by operator-valued symbols acting on Schwartz functions or temperate distributions. Here we focus on properties of the corresponding local phase functions, which allow to prove this result in a rather straightforward way.

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 pages

We introduce global wave-front sets $\operatorname{WF}_{{\mathcal B}} (f)$, $f\in {\mathscr S}^\prime(\textbf{R}^d)$, with respect to suitable Banach or Fr\'echet spaces ${\mathcal B}$. An important special case is given by the modulation spaces ${\mathcal B}=M(\omega,\mathscr B)$, where $\omega$ is an appropriate weight function and $\mathscr B$ is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to $\operatorname{WF}_{{\mathcal B}} (f)$. In particular, we prove that micro locality and microellipticity hold for a class of globally defined pseudo-differential operators $\operatorname{Op}_t(a)$, acting continuously on the involved spaces., Comment: 51 pages, mistakes and typos correction, reorganized materials

We introduce the wave-front set for distributions with respect to Fourier images of weighted translation invariant Banach function spaces. We prove that usual mapping properties for pseudo-differential operators hold in the context of such wave-front sets., Comment: Pre-versio, 29 pages

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