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Coercive Inequalities and $U$-Bounds

Esther Bou Dagher

Imperial College London, United Kingdom   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak336a86cb59c7e9cca178b4aa26af836b').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy336a86cb59c7e9cca178b4aa26af836b = '&#101;sth&#101;r.b&#111;&#117;-d&#97;gh&#101;r17' + '&#64;'; addy336a86cb59c7e9cca178b4aa26af836b = addy336a86cb59c7e9cca178b4aa26af836b + '&#105;mp&#101;r&#105;&#97;l' + '&#46;' + '&#97;c' + '&#46;' + '&#117;k'; var addy_text336a86cb59c7e9cca178b4aa26af836b = '&#101;sth&#101;r.b&#111;&#117;-d&#97;gh&#101;r17' + '&#64;' + '&#105;mp&#101;r&#105;&#97;l' + '&#46;' + '&#97;c' + '&#46;' + '&#117;k';document.getElementById('cloak336a86cb59c7e9cca178b4aa26af836b').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy336a86cb59c7e9cca178b4aa26af836b + '\'>'+addy_text336a86cb59c7e9cca178b4aa26af836b+'<\/a>';

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.