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Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is:

  • symmetric,
  • adapted (in the sense that there is no proper subgroup $H$ such that $\mu(H)=1$),
  • supported on a compact generating set of $G$,
  • absolutely continuous w.r.t. the Haar measure on $G$.

Is there a "Poincaré inequality" for this settings?

If not, maybe for connected Lie group $G$?

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  • $\begingroup$ Which kind of Poincaré inequality do you mean? A "local" Poincaré inequality (on balls)? A $L^2$ Poincaré inequality with the variance? Can you please write it? $\endgroup$
    – Nicolas Juillet
    Commented Jul 22, 2015 at 12:29
  • $\begingroup$ Yes, a local one. Of course, one would have to define the gradient of a function, which may only make sense in the case of manifolds, or of finitely generated groups. $\endgroup$
    – Snoop Catt
    Commented Jul 22, 2015 at 12:34
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    $\begingroup$ A classical reference is a paper of David Jerison of 1986 in Duke Math Journal. But your measure is not smooth. Do you have an ideas for dealing with continuous or absolutely continuous measures? Lebesgue density point theorem? Is it enough for you to have the Poincaré inequality for small balls (radii $r$ smaller than some $R>0$)? $\endgroup$
    – Nicolas Juillet
    Commented Jul 22, 2015 at 13:03
  • $\begingroup$ @iPe: if you have a notion of gradient for f.g. groups then it's likely to be extended to something in any compactly generated locally compact group. $\endgroup$
    – YCor
    Commented Jul 22, 2015 at 13:29
  • $\begingroup$ @Nicolas: Let it be smooth then, but I do need it for large $R>0$. $\endgroup$
    – Snoop Catt
    Commented Jul 22, 2015 at 14:31

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