Geometric microlocal analysis in Denjoy–Carleman
classes

Fürdös, Stefan

doi:10.2140/pjm.2020.307.303

Cited by 8 publications

(15 citation statements)

References 47 publications

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“…In [8], it is proved that a distribution on Ω belongs to   (Ω) if and only if for every 0 ∈ Ω there are ∈  ∞ (Ω), with ≡ 1 in an open neighborhood of 0 , ⊂ Ω an open neighborhood of 0 and a positive constant such that:…”

Section: Condition C) Meansmentioning

confidence: 99%

“…In [8], it is proved that a distribution u on Ω belongs to

{scriptC}^{scriptM} false(normalΩ false)

if and only if for every

x_{0} \in Ω

there are

χ \in {scriptC}_{c}^{\infty} false(normalΩ false)

, with

χ \equiv 1

in an open neighborhood of x 0 ,

U \subset Ω

an open neighborhood of x 0 and a positive constant A such that:

|scriptF ([], χ, u) (x, ξ)| \leq \frac{A^{k + 1} M_{k}}{{false| ξ false|}^{k}}, k \in {double-struckZ}_{+}, x \in U, ξ \in {double-struckR}^{N} ∖ false{0 false} .

This last inequality can be used to microlocalize the notion of

C^{M}

‐regularity. As usual, a subset

Γ \subset {double-struckR}^{N}

is said to be a cone if for every

x \in Γ

and every

t > 0

we have

t x \in Γ

.…”

Section: Denjoy–carleman Classesmentioning

confidence: 99%

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Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

Rodrigues

Silva

2021

Mathematische Nachrichten

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In this paper we study microlocal regularity of a  2 -solution of the equation) is ultradifferentiable in the variables ( , ) ∈ ℝ × ℝ and holomorphic in the variables () ∈ ℂ × ℂ . We proved that if   is a regular Denjoy-Carleman class (including the quasianalytic case) then:where WF  ( ) is the Denjoy-Carleman wave-front set of and Char( ) is the characteristic set of the linearized operator :

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Section: Condition C) Meansmentioning

confidence: 99%

“…In [8], it is proved that a distribution u on Ω belongs to

{scriptC}^{scriptM} false(normalΩ false)

if and only if for every

x_{0} \in Ω

there are

χ \in {scriptC}_{c}^{\infty} false(normalΩ false)

, with

χ \equiv 1

in an open neighborhood of x 0 ,

U \subset Ω

an open neighborhood of x 0 and a positive constant A such that:

|scriptF ([], χ, u) (x, ξ)| \leq \frac{A^{k + 1} M_{k}}{{false| ξ false|}^{k}}, k \in {double-struckZ}_{+}, x \in U, ξ \in {double-struckR}^{N} ∖ false{0 false} .

This last inequality can be used to microlocalize the notion of

C^{M}

‐regularity. As usual, a subset

Γ \subset {double-struckR}^{N}

is said to be a cone if for every

x \in Γ

and every

t > 0

we have

t x \in Γ

.…”

Section: Denjoy–carleman Classesmentioning

confidence: 99%

Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

Rodrigues

Silva

2021

Mathematische Nachrichten

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“…In this section we summarize the results for Denjoy-Carleman classes that we need in the following. For a more detailed presentation see [16]. Note that, unless stated otherwise, Ω ⊆ R n will be an open set.…”

Section: Regular Denjoy-carleman Classesmentioning

confidence: 99%

“…However using Dyn'kins characterization of ultradifferentiable functions by almost analytic extensions [12,11] we were able in [16] to develop a geometric theory for the ultradifferentiable wavefront set. In particular, if the weight sequence is regular, the ultradifferentiable wavefront set of a distribution on an ultradifferentiable manifold is shown to be well defined.…”

Section: Introductionmentioning

confidence: 99%

“…In section 3 we first recall the results from Dyn'kin [12,11] on the almost analytic extension of ultradifferentiable functions. Furthermore we give the definition and basic results on the ultradifferentiable wavefront set according to Hörmander [22] and close the section by presenting the geometric theory for the ultradifferentiable wavefront set given in [16].…”

Section: Introductionmentioning

confidence: 99%

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Ultradifferentiable CR manifolds

Fürdös¹

2018

Preprint

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Analytic States in Quantum Field Theory on Curved Spacetimes

Strohmaier,

Witten

2024

Ann. Henri Poincaré

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We discuss high energy properties of states for (possibly interacting) quantum fields in curved spacetimes. In particular, if the spacetime is real analytic, we show that an analogue of the timelike tube theorem and the Reeh–Schlieder property hold with respect to states satisfying a weak form of microlocal analyticity condition. The former means the von Neumann algebra of observables of a spacelike tube equals the von Neumann algebra of observables of a significantly bigger region that is obtained by deforming the boundary of the tube in a timelike manner. This generalizes theorems by Araki (Helv Phys Acta 36:132–139, 1963) and Borchers (Nuovo Cim (10) 19:787–793, 1961) to curved spacetimes.

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Geometric microlocal analysis in Denjoy–Carleman classes

Cited by 8 publications

References 47 publications

Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

Ultradifferentiable CR manifolds

Analytic States in Quantum Field Theory on Curved Spacetimes

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