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Quantitative property A, Poincaré inequalities, $L^p$-compression and $L^p$-distortion for metric measure spaces. (English)
Geom. Dedicata 136, 203-220 (2008).
A mapping $f:A\to B$ between two metric spaces is called a coarse embedding (sometimes uniform embedding) if there exist non-decreasing functions $ρ_1,ρ_2:[0,\infty)\to[0,\infty)$ such that $$\forall x,y\in A: ρ_1(d_A(x,y))\le d_B(f(x),f(y))\leρ_2(d_A(x,y)),\tag1$$ $$\lim_{r\to\infty}ρ_1(r)=\infty.\tag2$$ The supremum of all non-decreasing functions $ρ_1$ for which inequality (1) holds is called the compression of $f$. {\it G.‒L. Yu} [Invent. Math. 139, No.\,1, 201‒240 (2000; Zbl 0956.19004)] introduced Property A of discrete metric spaces as a sufficient condition for coarse embeddability into a Hilbert space. Since then, Property~A has found many other applications; e.g., it was discovered ({\it E.\,Guentner} and {\it J.\,Kaminker} [Topology 41, No.\,2, 411‒418 (2002; Zbl 0992.58002)] and {\it N.\,Ozawa} [C. R. Acad. Sci., Paris, Sér. I, Math. 330, No.\,8, 691‒695 (2000; Zbl 0953.43001)]) that, for discrete groups, Property~A is equivalent to the exactness of the reduced $C^*$-algebra. See {\it P.\,Nowak} and {\it G.‒L. Yu} [Notices Am. Math. Soc. 55, No.\,4, 474‒475 (2008; Zbl 1154.46010)] for a short summary of known facts. The author introduces the following quantitative version of Property~A. Let $X =(X, d,μ)$ be a metric measure space, $J :{\Bbb R}_+\to {\Bbb R}_+$ be some increasing function, and let $1\le p < \infty$. We say that $X$ has property $A(J, p)$ if for every $n\in{\Bbb N}$ and $x\in X$ there exists $ψ_{n,x}\in L_p(X)$ such that (a) $||ψ_{n,x}||_p\ge J(n)$ for every $x\in X$; (b) $\|ψ_{n,x}-ψ_{n,y}\|_p\le d(x,y)$; (c) $ψ_{n,x}$ is supported on $\{y\in X:~ d(x,y)\le n\}$. The author estimates the compression of coarse embeddings into $L_p$ in terms of the function $J$. We state one of his results: Theorem. Let $X$ be a metric measure space having property $A(J, p)$. Then, for every increasing function $f$ satisfying $$\int_1^\infty\left(\frac{f(t)}{J(t)}\right)^p\frac{dt}t<\infty,$$ there exists a large-scale Lipschitz coarse embedding $F$ of $X$ into an $L_p$-space with compression $ρ$ satisfying $f(t)\le Cρ(Ct)+C$ for some \$0 Reviewer: Mikhail Ostrovskii (Queens)