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.