Crystalline Measures and Wave Front Sets

The notion of a second microlocalization along an involutive submanifold WCIS introduced by M. Kashiwara [6] .Y. Laurent developed this notion and established a theory of the 2-microdifferential operators [8]. In [Ill, 3 . Sjostrand defined the second analytic wave front set (and even a kth wave front set) along the lagrangian submanifolds.So he was able to get a result of M. Kashiwara and to generalize it. This result provides very strict relations between the support and the wave front set, and, as a generalization, between the (k-l)th wave front set and the kth one. J.M.Bony gave a definition adapted to the frame of the C∞o-singularities. Finally G. Lebeau spread the notion to the isotropic manifolds and applied it to a degenerated diffraction problem.

The concept of wave front set was introduced in 1969–1970 by Sato in the hyperfunctions context [1, 34] and by Hörmander [23] in the $\mathcal C^{\infty }$ context. Howe in [25] used the theory of wave front sets in the study of Lie groups representations. Heifetz in [22] defined a notion of wave front set for distributions in the $p$-adic setting and used it to study some representations of $p$-adic Lie groups. In this article, we work in the $k\mathopen{(\!(} t \mathopen{)\!)}$-setting with $k$ a Characteristic 0 field. In that setting, balls are no longer compact but working in a definable context provides good substitutes for finiteness and compactness properties. We develop a notion of definable distributions in the framework of [13] and [14] for which we define notions of singular support and $\Lambda$-wave front sets (relative to some multiplicative subgroups $\Lambda$ of the valued field) and we investigate their behavior under natural operations like pullback, tensor product, and products of distributions.

We consider the spaces of ultradifferentiable functions $$\mathcal {S}_\omega $$ as introduced by Bjorck (and its dual $$\mathcal {S}'_\omega $$ ) and we use time-frequency analysis to define a suitable wave front set in this setting and obtain several applications: global regularity properties of pseudodifferential operators of infinite order and the micro-pseudolocal behaviour of partial differential operators with polynomial coefficients and of localization operators with symbols of exponential growth. Moreover, we prove that the new wave front set, defined in terms of the Gabor transform, can be described using only Gabor frames. Finally, some examples show the convenience of the use of weight functions to describe more precisely the global regularity of (ultra)distributions.

Wave front set for solutions to Schrödinger equations

In this paper we introduce and study global wave front sets in terms of the tau -Wigner transform in global ultradifferentiable classes of Beurling type modulated with weight functions in the sense of Braun, Meise, and Taylor, and we compare it with other wave front sets existing in the literature defined by different time-frequency analysis tools, such as the short-time Fourier transform or Gabor frames. Conditions for the equality of these wave front sets are provided and some examples are given.

A unipotent support for irreducible representations

On considere un systeme hyperbolique et on etudie la propagation des ensembles front d'onde dans des classes de Gevrey pour la solution U(t) sans supposer la multiplicite constante

Etude de la propagation des ensembles de fronts d'ondes dans les classes de Gevrey pour les solutions d'equations hyperboliques avec des racines caracteristiques de multiplicites variables

In this paper, a generalization of the sector property theorem first pioneered by Baouendi, Rothschild and Treves is given. The main contribution consists in showing that if a submanifold of $\C^n$ with higher codimension is locally presented in a weighted normal form, similar to that described by Bloom and Graham, then a characterization is given as to when a vector in the tangent space belongs to the analytic wave front set for the set of locally defined CR functions on the submanifold. This characterization is described in terms of the sign of the inner product of this vector with the local graphing functions for the submanifold on sectors of suitable size along complex lines in its tangent space. Examples are given to show that under certain circumstances, the results are sharp. Previous results by Baouendi et. al. contained a semi-rigidity assumption which is not assumed in the present paper. The hypoanalytic wave front set determines the cone of directions in which CR functions extend analytically to the ambient space and thus provides an explicit description of the local hull of holomorphy of a submanifold of $\C^n$.

We discuss in some detail the existence of global generating functions describing Lagrangian submanifolds connected with evolution problems for Hamilton–Jacobi (H–J) equations. First, we produce a physical application of a result by Viterbo: for generic (in a suitable sense) Hamiltonian functions and initial data, the envelopes, i.e., the wave front sets, related to Hamilton–Jacobi problems are globally finitely generated. Furthermore, we show how to compute global space–time generating functions with finite parameters for geometric solutions of a H–J equation of the evolution kind.

In this paper we show, for an example of hyperbolic system, that the wave front set in Gevrey class of a certain solution propagates precisely along “generalized broken null-bicharacteristic flow” defined by Kumano-go-Taniguchi [7] and Wakabayashi [10] [11].

The fundamental solution of the Cauchy problem is given in the form of Fourier integral operator, and when the problem is not C∞ well-posed, Cattabriga-Zanghirati2 have proved that the symbols of Fourier integral operators in the fundamental solution are not able to be estimated by finite order but able to be estimated by infinite order of the following type $$\begin{array}{*{20}{c}} {C\exp [c{{\xi }^{{1/\kappa }}}],} & {c > 0.} \\ \end{array}$$ (1) These symbols are estimeted from above by (1) for some constant tc satisfying k >1. Foureir integral operators and pseudo-differential operators with symbols estimated by (1) are investigated in Zanghirati26, Rodino-Zanghirati16, Cattabriga-Zanghirati2 and Aokil, and it is proved that they have the following properties: i) They map the ultradistributions into themselves; ii) pseudo-differential operators with symbols estimated from above by (1) make algebras; and iii) they propagate Gevrey wave front sets canonically.

For the two-point distribution of a quasi-free Klein-Gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space-time we prove the equivalence of (1) the global Hadamard condition, (2) the property that the Feynman propagator is a distinguished parametrix in the sense of Duistermaat and Hörmander, and (3) a new property referred to as the wave front set spectral condition (WFSSC), because it is reminiscent of the spectral condition in axiomatic quantum field theory on Minkowski space. Results in micro-local analysis such as the propagation of singularities theorem and the uniqueness up toC ∞ of distinguished parametrices are employed in the proof. We include a review of Kay and Wald's rigorous definition of the global Hadamard condition and the theory of distinguished parametrices, specializing to the case of the Klein-Gordon operator on a globally hyperbolic space-time. As an alternative to a recent computation of the wave front set of a globally Hadamard two-point distribution on a globally hyperbolic curved space-time, given elsewhere by Köhler (to correct an incomplete computation in [32]), we present a version of this computation that does not use a deformation argument such as that used in Fulling, Narcowich and Wald and is independent of the Cauchy evolution argument of Fulling, Sweeny and Wald (both of which are relied upon in Köhler's proof). This leads to a simple micro-local proof of the preservation of Hadamard form under Cauchy evolution (first shown by Fulling, Sweeny and Wald) relying only on the propagation of singularities theorem. In another paper [33], the equivalence theorem is used to prove a conjecture by Kay that a locally Hadamard quasi-free Klein-Gordon state on any globally hyperbolic curved space-time must be globally Hadamard.

In this paper, we study the propagation of singularities (in the sense of $\mathcal{C}^{\infty}$ wave front set) of the solution of a model case initial-boundary value problem with glancing rays for a concave domain on an asymptotically anti-de Sitter manifold. The main result addresses the diffractive problem and establishes that there is no propagation of singularities into the shadow for the solution, i.e. the diffractive result for codimension-1 smooth boundary holds in this setting. The approach adopted is motivated by the work done for a conformally related diffractive model problem by Friedlander, in which an explicit solution was constructed using the Airy function. This work was later generalized by Melrose and by Taylor, via the method of parametrix construction. Our setting is a simple case of asympotically anti-de Sitter spaces, which are Lorentzian manifolds modeled on anti-de Sitter space at infinity but whose boundary are not totally geodesic (unlike the exact anti-de Sitter space). Most technical difficulties of the problem reduce to studying and constructing a global resolvent for a semiclassical ODE on $\RR^+$, which at one end is a b-operator (in the sense of Melrose) while having a scattering behavior at infinity. We use different techniques near 0 and infinity to analyze the local problems: near infinity we use local resolvent bounds and near zero we build a local semiclassical parametrix. After this step, the `gluing' method by Datchev-Vasy serves to combine these local estimates to get the norm bound for the global resolvent.

There is showed how to define the second analytic wave front set in order to obtain propagation properties for differentiable, Gevrey, or analytic singularities of distributions without second analytic support.