scholarly journals Wave Front Sets with respect to the Iterates of an Operator with Constant Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
C. Boiti ◽  
D. Jornet ◽  
J. Juan-Huguet
Keyword(s):  
Differential Operator ◽  
Wave Front ◽  
Partial Differential Operator ◽  
Regularity Theorem ◽  
Wave Front Set ◽  
Open Set ◽  
Wave Front Sets ◽  
Constant Coefficients ◽  
New Type ◽  
Classical Distribution

We introduce the wave front setWF*P(u)with respect to the iterates of a hypoelliptic linear partial differential operator with constant coefficients of a classical distributionu∈𝒟′(Ω)in an open set Ω in the setting of ultradifferentiable classes of Braun, Meise, and Taylor. We state a version of the microlocal regularity theorem of Hörmander for this new type of wave front set and give some examples and applications of the former result.

The wave front set of the solution of a simple initial-boundary value problem with glancing rays

1976 ◽  
Vol 79 (1) ◽  
pp. 145-159 ◽  
Author(s):  
F. G. Friedlander
Keyword(s):  
Wave Front ◽  
Partial Differential Operator ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Zero Section ◽  
Hamiltonian Vector Field ◽  
Partial Differential ◽  
Wave Front Set ◽  
Open Set ◽  
Wave Front Sets

1. The singularities of solutions of linear partial differential equations are most appropriately described in terms of wave front sets of distributions. (For the definition and properties of wave front sets, see (4) and (5).) In particular, let Ω ⊂ RN be an open set and let P be a linear partial differential operator defined on Ω, with smooth coefficients, real principal symbol p(x, ξ), and simple characteristics. Then if u ∈ ′ (Ω) and Pu = 0, the wave front set of u, WF(u), is a set in the cotangent bundle T*Ω (minus the zero section) that is contained in p−1(0) and is invariant under the flow generated in p−1(0) by the Hamiltonian vector field (gradξp, − gradxp). In other words, WF(u) is made up of bicharacteristic strips. This is a special case of an important theorem due to Hörmander (5), and Duistermaat and Hörmander (1), which generalizes classical results on the relation between discontinuities of solutions of a partial differential equation and the characteristics of the equation.

Motivic Wave Front Sets

2019 ◽  
Author(s):  
Michel Raibaut
Keyword(s):  
Tensor Product ◽  
Wave Front ◽  
Lie Groups ◽  
Wave Front Set ◽  
Singular Support ◽  
Valued Field ◽  
Wave Front Sets ◽  
Multiplicative Subgroups ◽  
Products Of Distributions ◽  
Lambda Wave

Abstract 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.

scholarly journals Some Spaces are not the Domain of a Closed Linear Operator in a Banach Space

1980 ◽  
Vol 23 (4) ◽  
pp. 501-503
Author(s):  
Peter Dierolf ◽  
Susanne Dierolf
Keyword(s):  
Banach Space ◽  
Differential Operator ◽  
Linear Operator ◽  
Partial Differential Operator ◽  
Linear Partial Differential Operator ◽  
Closed Linear Operator ◽  
Partial Differential ◽  
Minimal Operator ◽  
Constant Coefficients

Let be a linear partial differential operator with C∞- coefficients. The study of P(∂) as an operator in L2(ℝn) usually starts with the investigation of the minimal operator P0 which is the closure of P(∂) acting on . In the case of constant coefficients it is known that the domain D(P0) of P0 at least contains the space (cf. Schechter [4, p. 58, Lemma 1.2]).

scholarly journals On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets

2006 ◽  
Vol 133 (31) ◽  
pp. 115-136 ◽  
Author(s):  
Claudia Garetto ◽  
G. Hormann
Keyword(s):  
Wave Front ◽  
Pseudodifferential Operators ◽  
Symbol Calculus ◽  
Wave Front Set ◽  
Colombeau Algebras ◽  
Generalized Complex Numbers ◽  
Topological Modules ◽  
Wave Front Sets ◽  
Basic Facts

Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functional and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis. AMS Mathematics Subject Classification (2000): 46F30, 46A20, 47G30.

scholarly journals Note on Hypoellipticity of a First Order Linear Partial Differential Operator

1968 ◽  
Vol 32 ◽  
pp. 323-330
Author(s):  
Yoshio Kato
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Differentiable Function ◽  
Partial Differential Operator ◽  
Linear Partial Differential Operator ◽  
Partial Differential ◽  
First Order ◽  
Open Set ◽  
Euclidian Space ◽  
Infinitely Differentiable Function

Let Ω be a domain in the (n + 1)-dimensional euclidian space Rn+1. A linear partial differential operator P with coefficients in C∞(Ω) (resp. in Cω(Ω)) will be termed hypoelliptic (resp. analytic-hypoelliptic) in Ω if a distribution u on Ω (i.e. u ∈ D′(Ω)) is an infinitely differentiable function (resp. an analytic function) in every open set of Ω where Pu is an infinitely differentiable function (resp. an analytic function).

scholarly journals WAVE FRONT HOLONOMICITY OF -CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDS

2020 ◽  
Vol 8 ◽  
Author(s):  
AVRAHAM AIZENBUD ◽  
RAF CLUCKERS
Keyword(s):  
Fourier Transform ◽  
Wave Front ◽  
Fourier Transforms ◽  
Stability Result ◽  
Local Fields ◽  
Wave Front Set ◽  
Class A ◽  
Subanalytic Functions ◽  
Wave Front Sets ◽  
The Fourier Transform

Many phenomena in geometry and analysis can be explained via the theory of $D$ -modules, but this theory explains close to nothing in the non-archimedean case, by the absence of integration by parts. Hence there is a need to look for alternatives. A central example of a notion based on the theory of $D$ -modules is the notion of holonomic distributions. We study two recent alternatives of this notion in the context of distributions on non-archimedean local fields, namely $\mathscr{C}^{\text{exp}}$ -class distributions from Cluckers et al. [‘Distributions and wave front sets in the uniform nonarchimedean setting’, Trans. Lond. Math. Soc.5(1) (2018), 97–131] and WF-holonomicity from Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)]. We answer a question from Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)] by showing that each distribution of the $\mathscr{C}^{\text{exp}}$ -class is WF-holonomic and thus provides a framework of WF-holonomic distributions, which is stable under taking Fourier transforms. This is interesting because the $\mathscr{C}^{\text{exp}}$ -class contains many natural distributions, in particular, the distributions studied by Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)]. We show also another stability result of this class, namely, one can regularize distributions without leaving the $\mathscr{C}^{\text{exp}}$ -class. We strengthen a link from Cluckers et al. [‘Distributions and wave front sets in the uniform nonarchimedean setting’, Trans. Lond. Math. Soc.5(1) (2018), 97–131] between zero loci and smooth loci for functions and distributions of the $\mathscr{C}^{\text{exp}}$ -class. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory.

scholarly journals Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potentials

2015 ◽  
Vol 27 (01) ◽  
pp. 1550001 ◽  
Author(s):  
Elena Cordero ◽  
Fabio Nicola ◽  
Luigi Rodino
Keyword(s):  
Wave Front ◽  
Modulation Spaces ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Wave Front Set ◽  
Local Propagation ◽  
The Cauchy Problem ◽  
Wave Front Sets ◽  
Propagation Of Singularities ◽  
Gabor Wave Front Set

We consider Schrödinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may not even be differentiable. The well-posedness of the Cauchy problem is proved in the frame of the modulation spaces, and results of micro-local propagation of singularities are given in terms of Gabor wave front sets.

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