arXiv subjects

Sandro Coriasco

Publications and source records attributed to Sandro Coriasco.

29 records · Page 2Linked to original sources

arXiv2018

Deterministic and Stochastic Cauchy problems for a class of weakly hyperbolic operators on R^n

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.

math.AP↗

arXiv2018

Solution theory to Semilinear Hyperbolic Stochastic Partial Differential Equations with polynomially bounded coefficients

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.

math.AP↗

arXiv2018

Lagrangian distributions on asymptotically Euclidean manifolds

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.

math.AP↗

arXiv2017

Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey type pseudo-differential operators

We deduce one-parameter group properties for pseudo-differential operators $\operatorname{Op} (a)$, where $a$ belongs to the class $Γ^{(ω_0)}_*$ 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 $a\in Γ^{(ω_0)}_*$ and $b\in Γ^{(1/ω_0)}_*$. We apply these results to deduce lifting property for modulation spaces and construct explicit isomorpisms between them. For each weight functions $ω,ω_0$ moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) $\operatorname{Tp} (ω_0)$ is an isomorphism from $M^{p,q}_{(ω)}$ onto $M^{p,q}_{(ω/ω_0)}$ for every $p,q \in (0,\infty ]$.

math.FA↗

arXiv2015

SG-Lagrangian submanifolds and their parametrization

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 $\mathbb{B}^d\times\mathbb{B}^d$. 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.

math.FA↗

arXiv2014

Fourier Integral Operators of Boutet de Monvel Type

Given two compact manifolds $X,Y,$ with boundary and a boundary preserving symplectomorphism $χ:T^*Y\setminus0\to T^*X\setminus0$, which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with $χ$. 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 $χ$ 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.

math.FA↗

arXiv2014

Global wave-front sets of Banach, Fr{é}chet and Modulation space types, and pseudo-differential operators

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 Fréchet spaces ${\mathcal B}$. An important special case is given by the modulation spaces ${\mathcal B}=M(ω,\mathscr B)$, where $ω$ 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.

math.FA↗

arXiv2014

On the Spectral Asymptotics of Operators on Manifolds with Ends

We deal with the asymptotic behaviour for $λ\to+\infty$ of the counting function $N_P(λ)$ of certain positive selfadjoint operators $P$ with double order $(m,μ)$, $m,μ>0$, $m\not=μ$, 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 $N_P(λ)$ and show how their behaviour depends on the ratio $\frac{m}μ$ and the dimension of $M$.

math.FA↗

arXiv2014

On a Class of Fourier Integral Operators on Manifolds with Boundary

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.

math.OA↗

arXiv2009

Wave-front sets of Banach function types

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.

math.FA↗

arXiv2009

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 $W^{σ,p}_s(\Rn)$ 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).

math.FA↗