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Category Archives: Global analysis in Dirichlet spaces
Intermittency for the parabolic Anderson model on spaces with sub-Gaussian heat kernel estimates
With Hongyi Chen, Li Chen and Søren Kjærsgaard, we have posted a new paper, Intermittency for the parabolic Anderson model on spaces with sub-Gaussian heat kernel estimates. This is a sequel to our earlier work on bounded domains of recurrent … Continue reading
Riesz transform on eventually Gaussian local trees
With Aobo Chen and Li Chen, we have posted a new paper, Riesz transform on eventually Gaussian local trees which is a companion paper to the paper discussed in the previous post to which we refer for background about Riesz transform … Continue reading
Riesz transforms on the Vicsek set
This post is about a long paper with Aobo Chen and Li Chen, Riesz transforms on the Vicsek set. I want to explain, without too much technicalities, what we prove and put the results into context. As a side note, … Continue reading
Caccioppoli sets, Part II
After the general introduction to the theory of Caccioppoli sets that was presented in the previous post. I will now sketch some elements of the theory that was developed in our works: Besov class via heat semigroup on Dirichlet spaces … Continue reading
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Caccioppoli sets, Part I
In the next two posts, longer than usual, I will explain some ideas of recent works written in collaboration with Patricia Alonso-Ruiz, Li Chen, Luke Rogers, Nageswari Shanmugalingam and Alexander Teplyaev about the study of bounded variation functions in the context of Dirichlet … Continue reading
Lecture notes: An Introduction to Dirichlet Spaces
Peter Gustav Lejeune Dirichlet An introduction to Dirichlet spaces: Lecture notes The outline of those (unpolished) lecture notes is the following: Chapter 1: Semigroups Chapter 2: Markovian semigroups and Dirichlet forms Chapter 3: Dirichlet spaces with Gaussian or sub-Gaussian heat … Continue reading
Lecture 8. Sobolev inequalities on Dirichlet spaces: The Varopoulos approach
In this lecture, we study Sobolev inequalities on Dirichlet spaces. The approach we develop is related to Hardy-Littlewood-Sobolev theory The link between the Hardy-Littlewood-Sobolev theory and heat kernel upper bounds is due to Varopoulos, but the proof I give below I … Continue reading
Lecture 7. Diffusion operators as Markovian operators
In this section, we consider a diffusion operator where and are continuous functions on and for every , the matrix is a symmetric and non negative matrix. Our goal is to prove that if is essentially self-adjoint, then the semigroup … Continue reading
Lecture 6. The Lp theory of Markovian semigroups
Our goal, in this lecture, is to define, for , on . This may be done in a natural way by using the Riesz-Thorin interpolation theorem that we recall below. Theorem: [Riesz-Thorin interpolation theorem] Let , and . Define by … Continue reading
Lecture 5. Dirichlet forms
As in the previous lecture, let be a good measurable space equipped with a -finite measure . Definition: A function on is called a normal contraction of the function if for almost every , Definition: Let be a densely defined … Continue reading
