Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds

Emmanuel Russ (1) , Yannick Sire (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

Let G be a real connected Lie group with polynomial volume growth endowed with its Haar measure dx. Given a C(2) positive bounded integrable function M on G, we give a sufficient condition for an L(2) Poincare inequality with respect to the measure M(x) dx to hold on G. We then establish a nonlocal Poincare inequality on G with respect to M(x) dx. We also give analogous Poincare inequalities on Riemannian manifolds and deal with the case of Hardy inequalities.

Mots clés: fractional powers of operators; Riemannian manifolds; Lie group; polynomial volume growth; Non local Poincaré inequality

Domaines: Analyse fonctionnelle [math.FA]

Emmanuel Russ, Yannick Sire. Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds. Studia Mathematica, 2011, 203 (2), pp.105-127. ⟨10.4064/sm203-2-1⟩. ⟨hal-00449531⟩