Esther’s title and abstract: 

Title: Coercive Inequalities and U-Bounds

Abstract: In the setting of step-two Carnot groups, we prove Poincaré and $\beta$-Logarithmic Sobolev inequalities for probability measures as a function of various homogeneous norms. To do that, the key idea is to obtain an intermediate inequality called the U-Bound inequality (based on joint work with B. Zegarlinski).  Using this U-Bound inequality, we show that certain infinite dimensional Gibbs measures- with unbounded interaction potentials as a function of homogeneous norms- on an infinite product of Carnot groups satisfy the Poincaré inequality (based on joint work with Y. Qiu, B. Zegarlinski, and M. Zhang).

We also enlarge the class of measures as a function of the Carnot-Carathéodory distance that gives us the q−Logarithmic Sobolev inequality in the setting of Carnot groups. As an application, we use the Hamilton-Jacobi equation in that setting to prove the p−Talagrand inequality and hypercontractivity.

Yifu’s title and abstract:

Title: Higher Order Coercive Inequalities

Abstract: We study the higher order $q$- Poincare and other coercive inequalities for a class probability measures satisfying Adam’s regularity condition. We will start from Adam’s regularity condition, and introduce the traditional Adam’s inequality. Then we will generalize the inequality and use the generalization to obtain norm equivalence in weighted Sobolev spaces. In the second part, we will introduce a few results related to higher order Poincare inequalities and relations between $L^p$ norms and Orlicz norms.