Esther Bou Dagher (Imperial College London)

Coercive Inequalities and U-Bounds

Abstract:

In the setting of step-two Carnot groups, we prove Poincaré and β-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.