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

Emmanuel Russ, Yannick Sire (2011)

Studia Mathematica

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.

Nonstandard hulls of locally uniform groups

Isaac Goldbring (2013)

Fundamenta Mathematicae

We present a nonstandard hull construction for locally uniform groups in a spirit similar to Luxemburg's construction of the nonstandard hull of a uniform space. Our nonstandard hull is a local group rather than a global group. We investigate how this construction varies as one changes the family of pseudometrics used to construct the hull. We use the nonstandard hull construction to give a nonstandard characterization of Enflo's notion of groups that are uniformly free from small subgroups. We...

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