subject:(off diagonal estimates)

Australian National University

1. Amenta, Alex. Extensions of the theory of tent spaces and applications to boundary value problems .

Degree: 2016, Australian National University

► We extend the theory of tent spaces from Euclidean spaces to various types of metric measure spaces. For doubling spaces we show that the usual… (more)

Subjects/Keywords: tent spaces; metric measure spaces; interpolation; hardy-sobolev spaces; besov spaces; elliptic boundary value problems; functional calculus; off-diagonal estimates; cauchy-riemann systems; semigroups of operators

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

APA (6^th Edition):

Amenta, A. (2016). Extensions of the theory of tent spaces and applications to boundary value problems . (Thesis). Australian National University. Retrieved from http://hdl.handle.net/1885/102564

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):

Amenta, Alex. “Extensions of the theory of tent spaces and applications to boundary value problems .” 2016. Thesis, Australian National University. Accessed January 19, 2021. http://hdl.handle.net/1885/102564.

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

MLA Handbook (7^th Edition):

Amenta, Alex. “Extensions of the theory of tent spaces and applications to boundary value problems .” 2016. Web. 19 Jan 2021.

Vancouver:

Amenta A. Extensions of the theory of tent spaces and applications to boundary value problems . [Internet] [Thesis]. Australian National University; 2016. [cited 2021 Jan 19]. Available from: http://hdl.handle.net/1885/102564.

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

Council of Science Editors:

Amenta A. Extensions of the theory of tent spaces and applications to boundary value problems . [Thesis]. Australian National University; 2016. Available from: http://hdl.handle.net/1885/102564

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

2. Magniez, Jocelyn. Etude de la bornitude des transformées de Riesz sur Lp via le Laplacien de Hodge-de Rham : Boundedness of the Riesz transforms on Lp via the Hodge-de Rham Laplacian.

Degree: Docteur es, Mathematiques pures, 2015, Bordeaux

URL: http://www.theses.fr/2015BORD0161

►

Cette thèse comporte deux sujets d’étude mêlés. Le premier concerne l’étude de la bornitude sur Lp de la transformée de Riesz d∆-½ , où ∆… (more)

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Cette thèse comporte deux sujets d’étude mêlés. Le premier concerne l’étude de la bornitude sur Lp de la transformée de Riesz d∆-½ , où ∆ désigne l’opérateur de Laplace-Beltrami (positif). Le second traite de la régularité de Sobolev W1,p de la solution de l’équation de la chaleur non perturbée. Nous établissons également quelques résultats concernant les transformées de Riesz d’opérateurs de Schrödinger avec un potentiel comportant éventuellement une partie négative.Dans le cadre de ces travaux, nous nous plaçons sur une variété riemanienne (M, g) complète et non compacte. Nous supposons que M satisfait la propriété de doublement de volume (de constante de doublement égale à D) ainsi qu’une estimation gaussienne supérieure pour son noyau de la chaleur (celui associé à l’opérateur ∆). Nous travaillons avec le laplacien de Hodge-de Rham, noté ∆, agissant sur les 1-formes différentielles de M. En s’appuyant sur la formule de Bochner, liant ∆ à la courbure de Ricci de M, nous assimilons ∆ à un opérateur de Schrödinger à valeurs vectorielles. C’est un argument de dualité, basé sur une formule de commutation algébrique, qui lie l’étude de ∆ à celle de ∆. [...]

This thesis has two main parts. The first one deals with the study of the boundedness on Lp of the Riesz transform d∆-½ , where ∆ denotes the nonnegative Laplace-Beltrami operator. The second one deals with the Sobolev regularity W1,p of the solution of the heat equation. We also establish some results on the Riesz transforms of Schrödinger operators with a potential possibly having a negative part. In this work, we consider a complete non-compact Riemannian manifold (M, g). We assume that M satisfies the volume doubling property (with doubling constant equal to D) as well as a Gaussian upper estimate for its heat kernel associated to the operator ∆. We work with the Hodge-de Rham Laplacian ∆, acting on 1-differential forms of M. With the Bochner formula, linking ∆to the Ricci curvature of M, we see ∆ has a vector-valued Schrödinger operator. It is a duality argument, based on a commutation formula, which links the study of ∆to the one of ∆. [...]

Advisors/Committee Members: Ouhabaz, El Maati (thesis director).

Subjects/Keywords: Variété riemanienne,; Opérateurs de Schrödinger,; Laplacien de Hodge-de Rham; Transformées de Riesz; Régularité de Sobolev; Noyaux de la chaleur; Estimées hors-diagonales.; Riemannian manifolds; Schrödinger operators; Hodge-de Rham Laplacian; Riesz transforms; Sobolev regularity; Heat kernels; Off-diagonal estimates

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

APA (6^th Edition):

Magniez, J. (2015). Etude de la bornitude des transformées de Riesz sur Lp via le Laplacien de Hodge-de Rham : Boundedness of the Riesz transforms on Lp via the Hodge-de Rham Laplacian. (Doctoral Dissertation). Bordeaux. Retrieved from http://www.theses.fr/2015BORD0161

Chicago Manual of Style (16^th Edition):

Magniez, Jocelyn. “Etude de la bornitude des transformées de Riesz sur Lp via le Laplacien de Hodge-de Rham : Boundedness of the Riesz transforms on Lp via the Hodge-de Rham Laplacian.” 2015. Doctoral Dissertation, Bordeaux. Accessed January 19, 2021. http://www.theses.fr/2015BORD0161.

MLA Handbook (7^th Edition):

Magniez, Jocelyn. “Etude de la bornitude des transformées de Riesz sur Lp via le Laplacien de Hodge-de Rham : Boundedness of the Riesz transforms on Lp via the Hodge-de Rham Laplacian.” 2015. Web. 19 Jan 2021.

Vancouver:

Magniez J. Etude de la bornitude des transformées de Riesz sur Lp via le Laplacien de Hodge-de Rham : Boundedness of the Riesz transforms on Lp via the Hodge-de Rham Laplacian. [Internet] [Doctoral dissertation]. Bordeaux; 2015. [cited 2021 Jan 19]. Available from: http://www.theses.fr/2015BORD0161.

Council of Science Editors:

Magniez J. Etude de la bornitude des transformées de Riesz sur Lp via le Laplacien de Hodge-de Rham : Boundedness of the Riesz transforms on Lp via the Hodge-de Rham Laplacian. [Doctoral Dissertation]. Bordeaux; 2015. Available from: http://www.theses.fr/2015BORD0161

Delft University of Technology

3. Teuwen, J.J.B. Shedding new light on Gaussian harmonic analysis.

Degree: 2016, Delft University of Technology

URL: http://resolver.tudelft.nl/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; urn:NBN:nl:ui:24-uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; 10.4233/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; urn:NBN:nl:ui:24-uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; http://resolver.tudelft.nl/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f

► This dissertation consists out of two rather disjoint parts. One part concerns some results on Gaussian harmonic analysis and the other on an optimization problem… (more)

▼ This dissertation consists out of two rather disjoint parts. One part concerns some results on Gaussian harmonic analysis and the other on an optimization problem in optics. In the first part we study the Ornstein–Uhlenbeck process with respect to the Gaussian measure. We focus on two areas. One is on “Gaussian” analogues of classical results in harmonic analysis, and in the second area we study the higher time-derivatives of the integral kernels associated to the Ornstein–Uhlenbeck operator. After introducing the necessary preliminaries on Hermite polynomials, we look at the non-tangential maximal function for the Ornstein–Uhlenbeck semigroup and we prove analogues to classical results. An important distinction here with the classical case is that the maximal function result for the Laplacian allows for t > 0, whereas our result only holds for certain 0 < t < 1 where the exact range depends on the position in space. Next, we compute an explicit formula for the higher time-derivatives of the integral kernel related to the Ornstein–Uhlenbeck operator. As an application we show several kernel bounds using our formula. Finally, as far as the mathematical part is concerned, we study off-diagonal estimates related to the Ornstein–Uhlenbeck operator. It is well-known that these hold for the Laplacian with respect to the Lebesgue measure, but do these hold for the Ornstein–Uhlenbeck operator with respect to the Gaussian measure? It is known that we always would have L2-L2 bounds, even for all t > 0, but in applications one often wants L2-L1 bounds. Even though these do hold for the Laplacian, we show that these cannot hold for the Ornstein–Uhlenbeck operator even for small 0 < t < 1. Moreover, our proof shows that letting the maximal t depend on the position in space will not work either. In the final part of this dissertation we study a problem in theoretical optics. Here we study the optimization of the electric field induced by light as a plane wave in a disk with given radius. We study the electric fields in the lens pupil and the focal region for several radii in the orde of magnitude of the wave length of the light used. Advisors/Committee Members: van Neerven, J.M.A.M., Urbach, H.P..

Subjects/Keywords: Ornstein-Uhlenbeck semigroup; Mehler kernel; Gaussian maximal function; admissible cones; Mehler kernel bounds; off-diagonal estimates; infinite-dimensional optimization

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

APA (6^th Edition):

Teuwen, J. J. B. (2016). Shedding new light on Gaussian harmonic analysis. (Doctoral Dissertation). Delft University of Technology. Retrieved from http://resolver.tudelft.nl/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; urn:NBN:nl:ui:24-uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; 10.4233/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; urn:NBN:nl:ui:24-uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; http://resolver.tudelft.nl/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f

Chicago Manual of Style (16^th Edition):

Teuwen, J J B. “Shedding new light on Gaussian harmonic analysis.” 2016. Doctoral Dissertation, Delft University of Technology. Accessed January 19, 2021. http://resolver.tudelft.nl/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; urn:NBN:nl:ui:24-uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; 10.4233/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; urn:NBN:nl:ui:24-uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; http://resolver.tudelft.nl/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f.

MLA Handbook (7^th Edition):

Teuwen, J J B. “Shedding new light on Gaussian harmonic analysis.” 2016. Web. 19 Jan 2021.

Vancouver:

Teuwen JJB. Shedding new light on Gaussian harmonic analysis. [Internet] [Doctoral dissertation]. Delft University of Technology; 2016. [cited 2021 Jan 19]. Available from: http://resolver.tudelft.nl/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; urn:NBN:nl:ui:24-uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; 10.4233/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; urn:NBN:nl:ui:24-uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; http://resolver.tudelft.nl/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f.

Council of Science Editors:

Teuwen JJB. Shedding new light on Gaussian harmonic analysis. [Doctoral Dissertation]. Delft University of Technology; 2016. Available from: http://resolver.tudelft.nl/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; urn:NBN:nl:ui:24-uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; 10.4233/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; urn:NBN:nl:ui:24-uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f ; http://resolver.tudelft.nl/uuid:bcfb2d09-5828-4ad8-bfe6-e6cfe3b83c3f

Australian National University

4. 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… (more)

▼ 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 (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

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):

Morris, Andrew Jordan. “Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds .” 2010. Thesis, Australian National University. Accessed January 19, 2021. 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 (7^th Edition):

Morris, Andrew Jordan. “Local Hardy spaces and quadratic estimates for Dirac type operators on Riemannian manifolds .” 2010. Web. 19 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 19]. 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; 2010. 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

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Open Access Theses and Dissertations