Category Archives: Global analysis in Dirichlet spaces

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 →

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 →

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 →

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 →

In this lecture, we illustrate some of the concepts introduced earlier in the context of diffusion operators in . Throughout the lecture, we consider a second order differential operator that can be written where and are continuous functions on and … Continue reading →

Let be a Hilbert space and let be a densely defined operator on a domain . We have the following basic definitions.  The operator is said to be symmetric if for , The operator is said to be non negative … Continue reading →