Large scale Sobolev inequalities on metric measure spaces and applications.
We introduce various notions of large-scale isoperimetric profile on a locally compact, compactly generated amenable group. These asymptotic quantities provide measurements of the degree of amenability of the group. We are particularly interested in a class of groups with exponential volume growth which are the most amenable possible in that sense. We show that these groups share various interesting properties such as the speed of on-diagonal decay of random walks, the vanishing of the reduced first...
Let G be a locally compact, non-compact group and f a function defined on G; we prove that, if f is uniformly continuous with respect to the left (right) structure on G and with a power integrable with respect to the left (right) Haar measure on G, then f must vanish at infinity. We prove that left and right cannot be mixed.