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1. Feneuil, Joseph. Analyse harmonique sur les graphes et les groupes de Lie : fonctionnelles quadratiques, transformées de Riesz et espaces de Besov : Harmonic analysis on graphs and Lie groups : quadratic functionals, Riesz transforms and Besov spaces.

Degree: Docteur es, Mathématiques, 2015, Grenoble Alpes

Ce mémoire est consacré à des résultats d'analyse harmonique réelle dans des cadres géométriques discrets (graphes) ou continus (groupes de Lie).Soit Γ un graphe (ensemble de sommets et d'arêtes) muni d'un laplacien discret Δ=I-P, où P est un opérateur de Markov.Sous des hypothèses géométriques convenables sur Γ, nous montrons la continuité Lp de fonctionnelles de Littlewood-Paley fractionnaires. Nous introduisons des espaces de Hardy H1 de fonctions et de 1-formes différentielles sur Γ, dont nous donnons plusieurs caractérisations, en supposant seulement la propriété de doublement pour le volume des boules de Γ. Nous en déduisons la continuité de la transformée de Riesz sur H1. En supposant de plus des estimations supérieures ponctuelles (gaussiennes ou sous-gaussiennes) sur les itérées du noyau de l'opérateur P, nous obtenons aussi la continuité de la transformée de Riesz sur Lp pour 1<p<2.Nous considérons également l'espace de Besov Bp,q_α(G) sur un groupe de Lie unimodulaire G muni d'un sous-laplacien Δ. En utilisant des estimations du noyau de la chaleur associé à Δ, nous donnons plusieurs caractérisations des espaces de Besov, et montrons une propriété d'algèbre pour Bp,q_α(G) \cap L^∞(G), pour α>0, 1 ≤  p ≤ +∞ et 1 ≤  q ≤  +∞. Les résultats sont valables en croissance polynomiale ou exponentielle du volume des boules.

This thesis is devoted to results in real harmonic analysis in discrete (graphs) or continuous (Lie groups) geometric contexts.Let Γ be a graph (a set of vertices and edges) equipped with a discrete laplacian Δ=I-P, where P is a Markov operator.Under suitable geometric assumptions on Γ, we show the Lp boundedness of fractional Littlewood-Paley functionals. We introduce H1 Hardy spaces of functions and of 1-differential forms on Γ, giving several characterizations of these spaces, only assuming the doubling property for the volumes of balls in Γ. As a consequence, we derive the H1 boundedness of the Riesz transform. Assuming furthermore pointwise upper bounds for the kernel (Gaussian of subgaussian upper bounds) on the iterates of the kernel of P, we also establish the Lp boundedness of the Riesz transform for 1<p<2.We also consider the Besov space Bp,q_α(G) on a unimodular Lie group G equipped with a sublaplacian Δ.Using estimates of the heat kernel associated with Δ, we give several characterizations of Besov spaces, and show an algebra property for Bp,q_α(G) \cap L^∞(G) for α>0, 1 ≤  p ≤ +∞ and 1 ≤  q ≤  +∞.These results hold for polynomial as well as for exponential volume growth of balls.

Advisors/Committee Members: Russ, Emmanuel (thesis director).

Subjects/Keywords: Graphes; Groupes de lie; Fonctionnelles quadratiques; Transformée de Riesz; Espaces de Besov; Espaces de Hardy; Estimations de Gaffney; Noyau de la chaleur; Estimations sous-gaussiennes; Estimations gaussiennes; Paraproduits; Graphs; Lie groups; Quadratic functionals; Riesz transforms; Besov spaces; Hardy spaces; Heat kernel; Gaffney estimates; Gaussian estimates; Sub-Gaussian estimates; Paraproducts; 510

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Feneuil, J. (2015). Analyse harmonique sur les graphes et les groupes de Lie : fonctionnelles quadratiques, transformées de Riesz et espaces de Besov : Harmonic analysis on graphs and Lie groups : quadratic functionals, Riesz transforms and Besov spaces. (Doctoral Dissertation). Grenoble Alpes. Retrieved from http://www.theses.fr/2015GREAM040

Chicago Manual of Style (16th Edition):

Feneuil, Joseph. “Analyse harmonique sur les graphes et les groupes de Lie : fonctionnelles quadratiques, transformées de Riesz et espaces de Besov : Harmonic analysis on graphs and Lie groups : quadratic functionals, Riesz transforms and Besov spaces.” 2015. Doctoral Dissertation, Grenoble Alpes. Accessed January 28, 2020. http://www.theses.fr/2015GREAM040.

MLA Handbook (7th Edition):

Feneuil, Joseph. “Analyse harmonique sur les graphes et les groupes de Lie : fonctionnelles quadratiques, transformées de Riesz et espaces de Besov : Harmonic analysis on graphs and Lie groups : quadratic functionals, Riesz transforms and Besov spaces.” 2015. Web. 28 Jan 2020.

Vancouver:

Feneuil J. Analyse harmonique sur les graphes et les groupes de Lie : fonctionnelles quadratiques, transformées de Riesz et espaces de Besov : Harmonic analysis on graphs and Lie groups : quadratic functionals, Riesz transforms and Besov spaces. [Internet] [Doctoral dissertation]. Grenoble Alpes; 2015. [cited 2020 Jan 28]. Available from: http://www.theses.fr/2015GREAM040.

Council of Science Editors:

Feneuil J. Analyse harmonique sur les graphes et les groupes de Lie : fonctionnelles quadratiques, transformées de Riesz et espaces de Besov : Harmonic analysis on graphs and Lie groups : quadratic functionals, Riesz transforms and Besov spaces. [Doctoral Dissertation]. Grenoble Alpes; 2015. Available from: http://www.theses.fr/2015GREAM040


Australian National University

2. Morris, Andrew Jordan. Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds .

Degree: 2012, Australian National University

The connection between quadratic estimates and the existence of a bounded holomorphic functional calculus of an operator provides a framework for applying harmonic analysis to the theory of differential operators. This is a generalization of the connection between Littlewood – Paley – Stein estimates and the functional calculus provided by the Fourier transform. We use the former approach in this thesis to study first-order differential operators on Riemannian manifolds. The theory developed is local in the sense that it does not depend on the spectrum of the operator in a neighbourhood of the origin. When we apply harmonic analysis to obtain estimates, the local theory only requires that we do so up to a finite scale. This allows us to consider manifolds with exponential volume growth in situations where the global theory requires polynomial volume growth. A holomorphic functional calculus is constructed for operators on a reflexive Banach space that are bisectorial except possibly in a neighbourhood of the origin. We prove that this functional calculus is bounded if and only if certain local quadratic estimates hold. For operators with spectrum in a neighbourhood of the origin, the results are weaker than those for bisectorial operators. For operators with a spectral gap in a neighbourhood of the origin, the results are stronger. In each case, however, local quadratic estimates are a more appropriate tool than standard quadratic estimates for establishing that the functional calculus is bounded. This theory allows us to define local Hardy spaces of differential forms that are adapted to a class of first-order differential operators on a complete Riemannian manifold with at most exponential volume growth. The local geometric Riesz transform associated with the Hodge – Dirac operator is bounded on these spaces provided that a certain condition on the exponential growth of the manifold is satisfied. A characterisation of these spaces in terms of local molecules is also obtained. These results can be viewed as the localisation of those for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ. Finally, we introduce a class of first-order differential operators that act on the trivial bundle over a complete Riemannian manifold with at most exponential volume growth and on which a local Poincaré inequality holds. A local quadratic estimate is established for certain perturbations of these operators. As an application, we solve the Kato square root problem for divergence form operators on complete Riemannian manifolds with Ricci curvature bounded below that are embedded in Euclidean space with a uniformly bounded second fundamental form. This is based on the framework for Dirac type operators that was introduced by Axelsson, Keith and McIntosh.

Subjects/Keywords: holomorphic functional calculi; quadratic estimates; sectorial operators; local Hardy spaces; Riemannian manifolds; differential forms; Hodge – Dirac operators; local Riesz transforms; off-diagonal estimates; Davies – Gaffney estimates; Kato square-root problems; submanifolds; divergence form operators; first-order differential operators

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Morris, A. J. (2012). Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds . (Thesis). Australian National University. Retrieved from http://hdl.handle.net/1885/8864

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Morris, Andrew Jordan. “Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds .” 2012. Thesis, Australian National University. Accessed January 28, 2020. http://hdl.handle.net/1885/8864.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Morris, Andrew Jordan. “Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds .” 2012. Web. 28 Jan 2020.

Vancouver:

Morris AJ. Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds . [Internet] [Thesis]. Australian National University; 2012. [cited 2020 Jan 28]. Available from: http://hdl.handle.net/1885/8864.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Morris AJ. Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds . [Thesis]. Australian National University; 2012. Available from: http://hdl.handle.net/1885/8864

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.