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SUMMARY:Emanuel Milman (Techion)
DTSTART:20210312T150000Z
DTEND:20210312T160000Z
DTSTAMP:20260423T035745Z
UID:SRS/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/SRS/10/">Fun
 ctional Inequalities on sub-Riemannian manifolds via QCD</a>\nby Emanuel M
 ilman (Techion) as part of Sub-Riemannian Seminars\n\n\nAbstract\nWe are i
 nterested in obtaining Poincarè and log-Sobolev inequalities on domains i
 n sub-Riemannian manifolds (equipped with their natural sub-Riemannian met
 ric and volume measure).\n\nIt is well-known that strictly sub-Riemannian 
 manifolds do not satisfy any type of Curvature-Dimension condition CD(K\,N
 )\, introduced by Lott-Sturm-Villani some 15 years ago\, so we must follow
  a different path. Motivated by recent work of Barilari-Rizzi and Balogh-K
 ristàly-Sipos\, we show that in the ideal setting or for general corank 1
  Carnot groups\, these spaces nevertheless do satisfy a quasi-convex relax
 ation of the CD condition\, which we name QCD(Q\,K\,N). As a consequence\,
  these spaces satisfy numerous functional inequalities with exactly the sa
 me quantitative dependence (up to the slack parameter Q>1) as their CD cou
 nterparts. We achieve this by extending the localization paradigm to compl
 etely general interpolation inequalities\, and a one-dimensional compariso
 n of QCD densities with their "CD upper envelope".  We thus obtain the bes
 t known quantitative estimates for (say) the $L^p$-Poincare and log-Sobole
 v inequalities on domains in ideal sub-Riemannian manifolds and in general
  corank 1 Carnot groups\, which in particular are independent of the topol
 ogical dimension. For instance\, the classical Li-Yau / Zhong-Yang spectra
 l-gap estimate holds on all Heisenberg groups of arbitrary dimension up to
  a factor of 4.\n
LOCATION:https://researchseminars.org/talk/SRS/10/
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