Abstract: Let $L$ be a second order diffusion operator defined on a smooth manifold $M$ (or $\mathbb R^n$) satisfying the subelliptic estimate $ \| f \|_\epsilon ^2 \leq C(\left| \left< f , Lf \right> \right| + \| f\|_2^2 ) $ for all $f\in C^\infty_0(M)$, where $\|f \|_\epsilon=\left(\int |\hat{u}(\xi)|^2 (1+| \xi|^2 )^\epsilon d\xi \right)^{1/2}$ is the Sobolev norm of order $0<\epsilon<1$. In a local sense, Poincar\'e inequalities and volume doubling properties for balls with respect to $L$ are well-known. In this talk, we will discuss similar global statements when we add certain conditions on $L$. With the conditions, we will have a class of sub-Riemannian manifolds including CR Sasakian manifolds and carnot groups of step 2, but we will not rely on geometry here.

​Current & Upcoming Seminar Talks

      November 6 - Konstantinos Tsougkas (Uppsala)