Quasi Riesz transforms, Hardy spaces and generalized sub-Gaussian heat kernel estimates - TEL - Thèses en ligne Access content directly
Theses Year : 2014

Quasi Riesz transforms, Hardy spaces and generalized sub-Gaussian heat kernel estimates

Quasi transformées de Riesz, espaces de Hardy et estimations sous-gaussiennes du noyau de la chaleur

Abstract

In this thesis, we mainly study Riesz transforms and Hardy spaces associated to operators. The two subjects are closely related to volume growth and heat kernel estimates. In Chapter 1, 2 and 4, we study Riesz transforms on Riemannian manifold and on graphs. In Chapter 1, we prove that on a complete Riemannian manifold, the quasi Riesz transform is always Lp bounded on for p strictly large than 1 and no less than 2. In Chapter 2, we prove that the quasi Riesz transform is also weak L1 bounded if the manifold satisfies the doubling volume property and the sub-Gaussian heat kernel estimate. Similarly, we show in Chapter 4 the same results on graphs. In Chapter 3, we develop a Hardy space theory on metric measure spaces satisfying the doubling volume property and different local and global heat kernel estimates. Firstly we define Hardy spaces via molecules and via square functions which are adapted to the heat kernel estimates. Then we show that the two H1 spaces via molecules and via square functions are the same. Also, we compare the Hp space defined via square functions with Lp. The corresponding Hp space for p large than 1 defined via square functions is equivalent to the Lebesgue space Lp. However, it is shown that in this situation, the Hp space corresponding to Gaussian estimates does not coincide with Lp any more. Finally, as an application of this Hardy space theory, we proved that quasi Riesz transforms are bounded from H1 to L1 on fractal manifolds. In Chapter 5, we consider Vicsek graphs. We prove generalised Poincaré inequalities and Sobolev inequalities on Vicsek graphs and we show that they are optimal.
Dans cette thèse nous étudions les transformées de Riesz et les espaces de Hardy associés à un opérateur sur un espace métrique mesuré. Ces deux sujets sont en lien avec des estimations du noyau de la chaleur associé à cet opérateur. Dans les Chapitres 1, 2 et 4, on étudie les transformées quasi de Riesz sur les variétés riemannienne et sur les graphes. Dans le Chapitre 1, on prouve que les quasi transformées de Riesz sont bornées dans Lp pour 1
Fichier principal
Vignette du fichier
VD2_CHEN_LI_24042014.pdf (1022.46 Ko) Télécharger le fichier
VD2_CHEN_LI_24042014_Synthese_en_francais_Annexes.pdf (979.68 Ko) Télécharger le fichier
Origin : Version validated by the jury (STAR)
Format : Other
Loading...

Dates and versions

tel-01001868 , version 1 (05-06-2014)

Identifiers

  • HAL Id : tel-01001868 , version 1

Cite

Li Chen. Quasi Riesz transforms, Hardy spaces and generalized sub-Gaussian heat kernel estimates. General Mathematics [math.GM]. Université Paris Sud - Paris XI; Australian national university, 2014. English. ⟨NNT : 2014PA112068⟩. ⟨tel-01001868⟩
408 View
432 Download

Share

Gmail Facebook X LinkedIn More