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

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 Poincare inequalities and Sobolev inequalities on Vicsek graphs and we show that they are optimal.