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Properties of wave-front sets and non-tangential convergencePrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2011 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Växjö, Kalmar: Linnaeus University Press , 2011. , 161 p.
##### Series

Linnaeus University Dissertations, 58/2011
##### Keyword [en]

Fourier Banach spaces, Generalized free time-dependent Schrödinger equation, Micro-local, Modulation spaces, Non-tangential convergence, Pseudo-differential operators, Regularity, Wave-front sets.
##### National Category

Mathematical Analysis
##### Research subject

Mathematics, Mathematics
##### Identifiers

URN: urn:nbn:se:lnu:diva-12963ISBN: 978-91-86491-96-3 (print)OAI: oai:DiVA.org:lnu-12963DiVA: diva2:429609
##### Public defence

2011-10-20, Weber, Universitetsplatsen 1, Växjö, 13:15 (English)
##### Opponent

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##### Supervisors

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#####

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Available from: 2011-08-09 Created: 2011-06-23 Last updated: 2017-01-11Bibliographically approved
##### List of papers

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.

1. Local wave-front sets of Banach and Fréchet types, with applications to pseudo-differential operators$(function(){PrimeFaces.cw("OverlayPanel","overlay310410",{id:"formSmash:j_idt482:0:j_idt486",widgetVar:"overlay310410",target:"formSmash:j_idt482:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Global wave front set of modulation space type$(function(){PrimeFaces.cw("OverlayPanel","overlay426337",{id:"formSmash:j_idt482:1:j_idt486",widgetVar:"overlay426337",target:"formSmash:j_idt482:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Gabor pairs, and a discrete approach to wave-front sets$(function(){PrimeFaces.cw("OverlayPanel","overlay389205",{id:"formSmash:j_idt482:2:j_idt486",widgetVar:"overlay389205",target:"formSmash:j_idt482:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Association between temperate distributions and analytical functions in the context of wave-front sets$(function(){PrimeFaces.cw("OverlayPanel","overlay403763",{id:"formSmash:j_idt482:3:j_idt486",widgetVar:"overlay403763",target:"formSmash:j_idt482:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. A counter example on nontangential convergence for oscillatory integrals$(function(){PrimeFaces.cw("OverlayPanel","overlay310396",{id:"formSmash:j_idt482:4:j_idt486",widgetVar:"overlay310396",target:"formSmash:j_idt482:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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