Let G be a real connected Lie group with polynomial volume growth endowed with its Haar measuredx. Given a C² positive bounded integrable function M on G, we give a sufficient condition for an L² Poincaré inequality with respect to the measure M(x)dx to hold on G. We then establish a nonlocal Poincaré inequality on G with respect to M(x)dx. We also give analogous Poincaré inequalities on Riemannian manifolds and deal with the case of Hardy inequalities.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm203-2-1,
author = {Emmanuel Russ and Yannick Sire},
title = {Nonlocal Poincar\'e inequalities on Lie groups with polynomial volume growth and Riemannian manifolds},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {105-127},
zbl = {1223.22006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm203-2-1}
}
Emmanuel Russ; Yannick Sire. Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds. Studia Mathematica, Tome 204 (2011) pp. 105-127. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm203-2-1/