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Australian National University

1. Bandara, Lashi. Geometry and the Kato square root problem .

Degree: 2013, Australian National University

URL: http://hdl.handle.net/1885/10690

The primary focus of this thesis is to consider Kato square root problems for various divergence-form operators on manifolds. This is the study of perturbations of second-order differential operators by bounded, complex, measurable coefficients. In general, such operators are not self-adjoint but uniformly elliptic. The Kato square root problem is then to understand when the square root of such an operator, which exists due to uniform ellipticity, is comparable to its unperturbed counterpart. A remarkably adaptable operator-theoretic framework due to Axelsson, Keith and McIntosh sits in the background of this work. This framework allows us to take a powerful first-order perspective of the problems which we consider in a geometric setting. Through a well established procedure, we reduce these problems to the study of quadratic estimates. Under a set of natural conditions, we prove quadratic estimates for a class of operators on vector bundles over complete measure metric spaces. The first kind of estimates we prove are global, and we establish them on trivial vector bundles when the underlying measure grows at most polynomially. The second kind are local, and there, we allow the vector bundle to be non-trivial but bounded in an appropriate sense. Here, the measure is allowed to grow exponentially. An important consequence of obtaining quadratic estimates on measure metric spaces is that it allows us to consider subelliptic operators on Lie groups. The first-order perspective allows us to reduce the subelliptic problem to a fully elliptic one on a sub-bundle. As a consequence, we are able to solve a homogeneous Kato square root problem for perturbations of subelliptic operators on nilpotent Lie groups. For general Lie groups we solve a similar inhomogeneous problem.
In the situation of complete Riemannian manifolds, we consider uniformly elliptic divergence-form operators arising from connections on vector bundles. Under a set of assumptions, we show that the Kato square root problem can be solved for such operators. As a consequence, we solve this problem on functions under the condition that the Ricci curvature and injectivity radius are bounded. Assuming an additional lower bound for the curvature endomorphism on forms, we solve a similar problem for perturbations of inhomogeneous Hodge-Dirac operators. A theorem for tensors is obtained by additionally assuming boundedness of a second-order Riesz transform. Motivated by the study of these Kato problems, where for technical reasons it is useful to know the density of compactly supported functions in the domains of operators, we study connections and their divergence on a vector bundle. Through a first-order formulation, we show that this density property holds for the domains of these operators if the metric and connection are compatible and the underlying manifold is complete. We also show that compactly supported functions are dense in the second-order Sobolev space on complete manifolds under the sole assumption that the Ricci curvature is bounded below,…

Subjects/Keywords: Kato square root problem; quadratic estimates; elliptic operator; Lipschitz estimates; essentially self-adjoint; vector bundle; measure metric space; bounded measurable coefficients; Hodge-Dirac operator

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

APA (6^{th} Edition):

Bandara, L. (2013). Geometry and the Kato square root problem . (Thesis). Australian National University. Retrieved from http://hdl.handle.net/1885/10690

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Bandara, Lashi. “Geometry and the Kato square root problem .” 2013. Thesis, Australian National University. Accessed January 26, 2021. http://hdl.handle.net/1885/10690.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Bandara, Lashi. “Geometry and the Kato square root problem .” 2013. Web. 26 Jan 2021.

Vancouver:

Bandara L. Geometry and the Kato square root problem . [Internet] [Thesis]. Australian National University; 2013. [cited 2021 Jan 26]. Available from: http://hdl.handle.net/1885/10690.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Bandara L. Geometry and the Kato square root problem . [Thesis]. Australian National University; 2013. Available from: http://hdl.handle.net/1885/10690

Not specified: Masters Thesis or Doctoral Dissertation

2.
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, Université Grenoble Alpes (ComUE)

URL: http://www.theses.fr/2015GREAM040

Ce mémoire est consacré à des résultats d'analyse harmonique réelle dans des cadres géométriques discrets (graphes) ou continus (groupes de Lie).Soit Gamma un graphe (ensemble de sommets et d'arêtes) muni d'un laplacien discret Delta=I-P, où P est un opérateur de Markov. Sous des hypothèses géométriques convenables sur Gamma, 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 Gamma, dont nous donnons plusieurs caractérisations, en supposant seulement la propriété de doublement pour le volume des boules de Gamma. 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 B{p,q} alpha(G) sur un groupe de Lie unimodulaire G muni d'un sous-laplacien Delta. En utilisant des estimations du noyau de la chaleur associé à Delta, nous donnons plusieurs caractérisations des espaces de Besov, et montrons une propriété d'algèbre pour B{p,q} alpha(G) cap L nfty(G), pour alpha>0, 1leq p leq+infty et 1leq qleq +infty. 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 Gamma be a graph (a set of vertices and edges) equipped with a discrete laplacian Delta=I-P, where P is a Markov operator.Under suitable geometric assumptions on Gamma, we show the Lp boundedness of fractional Littlewood-Paley functionals. We introduce H1 Hardy spaces of functions and of 1-differential forms on Gamma, giving several characterizations of these spaces, only assuming the doubling property for the volumes of balls in Gamma. 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 B{p,q} alpha(G) on a unimodular Lie group G equipped with a sublaplacian Delta. Using estimates of the heat kernel associated with Delta, we give several characterizations of Besov spaces, and show an algebra property for B{p,q} alpha(G) cap L infty(G) for alpha>0, 1leq pleq+infty and 1leq qleq +infty.These results hold for polynomial as well as for exponential volume growth of balls.

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

Record Details Similar Records

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

APA (6^{th} 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). Université Grenoble Alpes (ComUE). Retrieved from http://www.theses.fr/2015GREAM040

Chicago Manual of Style (16^{th} 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, Université Grenoble Alpes (ComUE). Accessed January 26, 2021. http://www.theses.fr/2015GREAM040.

MLA Handbook (7^{th} 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. 26 Jan 2021.

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]. Université Grenoble Alpes (ComUE); 2015. [cited 2021 Jan 26]. 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]. Université Grenoble Alpes (ComUE); 2015. Available from: http://www.theses.fr/2015GREAM040

Australian National University

3.
Morris, Andrew Jordan.
Local Hardy spaces and *quadratic* *estimates* for Dirac type operators on Riemannian manifolds
.

Degree: 2010, Australian National University

URL: http://hdl.handle.net/1885/8864

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

Record Details Similar Records

❌

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

APA (6^{th} Edition):

Morris, A. J. (2010). 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

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Morris, Andrew Jordan. “Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds .” 2010. Thesis, Australian National University. Accessed January 26, 2021. http://hdl.handle.net/1885/8864.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Morris, Andrew Jordan. “Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds .” 2010. Web. 26 Jan 2021.

Vancouver:

Morris AJ. Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds . [Internet] [Thesis]. Australian National University; 2010. [cited 2021 Jan 26]. Available from: http://hdl.handle.net/1885/8864.

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; 2010. Available from: http://hdl.handle.net/1885/8864

Not specified: Masters Thesis or Doctoral Dissertation