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Global wave front set of modulation space type
Department of mathematics, Turin's university, Italy.
Linnéuniversitetet, Fakultetsnämnden för naturvetenskap och teknik, Institutionen för datavetenskap, fysik och matematik, DFM.
Linnéuniversitetet, Fakultetsnämnden för naturvetenskap och teknik, Institutionen för datavetenskap, fysik och matematik, DFM.ORCID-id: 0000-0003-1921-8168
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
Abstract [en]

We introduce global wave-front sets WFB(f), f in S'(Rd), with respect to suitable Banach or Fréchet spaces B. An important special case is given by the modulation spaces B=M(ω,B), where ω is an appropriate weight function and B is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to WFB(f). In particular, we prove that microlocality and microellipticity hold for a class of globally defined pseudo-differential operators Opt(a), acting continuouslyon the involved spaces.

Emneord [en]
Wave-front, Fourier, Banach, modulation, micro-local
HSV kategori
Forskningsprogram
Naturvetenskap, Matematik
Identifikatorer
URN: urn:nbn:se:lnu:diva-12959OAI: oai:DiVA.org:lnu-12959DiVA, id: diva2:426337
Tilgjengelig fra: 2011-06-23 Laget: 2011-06-23 Sist oppdatert: 2017-01-11bibliografisk kontrollert
Inngår i avhandling
1. Properties of wave-front sets and non-tangential convergence
Åpne denne publikasjonen i ny fane eller vindu >>Properties of wave-front sets and non-tangential convergence
2011 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
Abstract [en]

In this thesis we consider regularity properties for solutions to partial differential equations and pseudo-differential equations. The thesis mainly concerns wave-front sets and micro-local properties. Regularity properties are also viewed in terms of nontangential convergence for the generalized free time-dependent Schrödinger equations, where the Laplace operator is replaced by more general functions.

Wave-front sets describe location of singularities and the directions of their propagation. We establish usual and convenient mapping properties for such wave-front sets under action of pseudodifferential operators with smooth symbols.

We define three components of wave-front sets with respect to appropriate Banach and Fréchet spaces, in order to describe local properties as well as behavior far away, including heavy oscillations. The union of these components is called the global wavefront set. For these wave-front sets, we establish micro-local and micro-ellipticity properties for pseudo-differential operators in appropriate symbol classes. We obtain the classical wave-front sets as special cases (cf. Hörmander [9]). For the type of wave-front sets which describe local properties we also prove equivalence between wave-front sets of Fourier Banach function and modulation space types.

To open up for numerical computations we introduce admissible lattices and Gabor pairs to define discrete versions of wave-front sets with respect to Fourier Lebesgue and modulation spaces. Furthermore, we prove that these wave-front sets agree with each other and with the corresponding wave-front sets of continuous type. We also consider the link between analytic functions and temperate distributions in terms of such wave-front sets.

The last part of this thesis concerns counter examples of nontangential convergence for the generalized time-dependent Schrödinger equation with initial data in Sobolev spaces.

sted, utgiver, år, opplag, sider
Växjö, Kalmar: Linnaeus University Press, 2011. s. 161
Serie
Linnaeus University Dissertations ; 58/2011
Emneord
Fourier Banach spaces, Generalized free time-dependent Schrödinger equation, Micro-local, Modulation spaces, Non-tangential convergence, Pseudo-differential operators, Regularity, Wave-front sets.
HSV kategori
Forskningsprogram
Matematik, Matematik
Identifikatorer
urn:nbn:se:lnu:diva-12963 (URN)978-91-86491-96-3 (ISBN)
Disputas
2011-10-20, Weber, Universitetsplatsen 1, Växjö, 13:15 (engelsk)
Opponent
Veileder
Tilgjengelig fra: 2011-08-09 Laget: 2011-06-23 Sist oppdatert: 2017-01-11bibliografisk kontrollert

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