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- W2949888424 abstract "Let $H= $ be the discrete Heisenberg group, equipped with the left-invariant word metric $d_W(cdot,cdot)$ associated to the generating set ${a,b,a^{-1},b^{-1}}$. Letting $B_n= {xin H: d_W(x,e_H)le n}$ denote the corresponding closed ball of radius $nin N$, and writing $c=[a,b]=aba^{-1}b^{-1}$, we prove that if $(X,|cdot|_X)$ is a Banach space whose modulus of uniform convexity has power type $qin [2,infty)$ then there exists $Kin (0,infty)$ such that every $f:Hto X$ satisfies {multline*} sum_{k=1}^{n^2}sum_{xin B_n}frac{|f(xc^k)-f(x)|_X^q}{k^{1+q/2}}le Ksum_{xin B_{21n}} Big(|f(xa)-f(x)|^q_X+|f(xb)-f(x)|^q_XBig). {multline*} It follows that for every $nin N$ the bi-Lipschitz distortion of every $f:B_nto X$ is at least a constant multiple of $(log n)^{1/q}$, an asymptotically optimal estimate as $ntoinfty$." @default.
- W2949888424 created "2019-06-27" @default.
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- W2949888424 date "2012-12-10" @default.
- W2949888424 modified "2023-09-27" @default.
- W2949888424 title "Vertical versus horizontal Poincar'e inequalities on the Heisenberg group" @default.
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