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Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. (English) Zbl 1108.46012

K.Ball [Geom.Funct.Anal.2, No.2, 137–172 (1992; Zbl 0788.46050)] defined the notion of Markov type \(p\), \(1 \leq p < \infty\): a metric space \((X,d)\) has Markov type \(p\) if there is a constant \(K>0\) such that, for every stationary reversible Markov chain \((Z_t)_{t=0}^\infty\) on \(\{1, \dots, n\}\) and every map \(f \colon \{1,\dots, n\} \to X\), one has \({\mathbb E} d\big( f(Z_t), f(Z_0)\big)^p \leq K^p t\,{\mathbb E} d\big( f(Z_1), f(Z_0)\big)^p\), for every time \(t \geq 1\). For normed linear spaces, Markov type \(p\) implies the usual Rademacher type \(p\). In the above quoted paper, K.Ball proved the following extension theorem, which generalizes Maurey’s extension theorem to the nonlinear case: every Lipschitz map \(f \colon Z \to Y\) from a subspace \(Z\) of a metric space \(X\) of Markov type \(2\), with constant \(M_2 (X)\), into a Banach space \(Y\) having a modulus of convexity of power type \(2\), with constant \(K_2 (Y)\), can be extended to a Lipschitz function \(\tilde f \colon X \to Y\) such that \(\| \tilde f \| _{\text{ Lip}} \leq K\,\| f \| _{\text{ Lip}}\), with \(K \leq 6 M_2 (X) K_2 (Y)\).
Except for Hilbert spaces and metric spaces which bi-Lipschitz embed into Hilbert spaces, no other space with Markov type \(2\) was known.
In the present paper, the authors show that every normed space \(X\) with modulus of smoothness of power type \(q\), \(1 < q \leq 2\), with constant \(S_q (X)\), has Markov type \(q\), with constant \(M_q (X) \leq {8 \over (2^{q+1} - 4)^{1/q}}\,S_q (X)\). In particular, for \(2 \leq p < \infty\), \(L^p\) has Markov type \(2\), and \(M_2 (L^p) \leq 4 \sqrt{p -1}\). Combined with Ball’s extension theorem, this gives a theorem on the extension of Lipschitz maps from subsets of uniformly smooth Banach spaces with modulus of smoothness of power type \(2\) into uniformly convex Banach spaces with modulus of convexity of power type \(2\); in particular, from subsets of \(L^p\) into \(L^q\), when \(1 < q \leq 2 \leq p < \infty\); this answers a question of W.B.Johnson and J.Lindenstrauss [Contemp.Math.26, 189–206 (1984; Zbl 0539.46017)].
The present authors also show that other classes of metric spaces have Markov type \(2\):
First, they show that there is a universal constant \(C>0\) such that, for every \(\delta\)-hyperbolic metric space (Gromov hyperbolic space) \(X\), one has for every \(t \geq 1\): \[ {\mathbb E}d\big( f(Z_t), f(Z_0)\big)^2 \leq C^2 t\,{\mathbb E} d\big( f(Z_1), f(Z_0)\big)^2 + C^2\delta^2 (\log t)^2 \] for every stationary reversible Markov chain \((Z_t)_{t=0}^\infty\) on \(\{1, \dots, n\}\) and every map \(f : \{1, \dots, n\} \to X\). In particular, every \(\delta\)-hyperbolic group \(G\) with the word metric has Markov type \(2\), and \(M_2 (G) \leq C (1+ \delta)\), and every tree \(T\) (since trees are \(0\)-hyperbolic metric spaces) has Markov type \(2\), with \(M_2 (T)\) uniformly bounded (\(C=8\) suits; on the other hand, they show that there is a tree \(T\) such that \(M_2(T) \geq \sqrt{3}\)).
Then they show that every \(n\)-dimensional, complete, simply connected Riemannian manifold with pinched negative sectional curvature has Markov type \(2\), by showing that such a space bi-Lipschitz embeds into a product of a finite number of \({\mathbb R}\)-trees. They also show that Laakso graphs \(G_k\) have Markov type \(2\) and, moreover, \(\sup_k M_2 (G_k) < + \infty\).
The authors also define the notion of weak Markov type \(2\) and show that every doubling metric space has weak Markov type \(2\).
In the last section, they discuss and ask several questions, in particular, the connection between the usual Rademacher type, Enflo type, and Markov type, in particular for Banach lattices. It is also unknown whether, as for normed spaces, via Kahane’s inequalities, Markov type \(p\) implies Markov type \(q\) for every \(q<p\). Another open problem is the extension of Lipschitz maps from subsets of \(L^2\) into \(L^1\).
Reviewer: Daniel Li (Lens)

MSC:

46B09 Probabilistic methods in Banach space theory
54E35 Metric spaces, metrizability
05C05 Trees
05C12 Distance in graphs
20F67 Hyperbolic groups and nonpositively curved groups
46B20 Geometry and structure of normed linear spaces
53C20 Global Riemannian geometry, including pinching
60J99 Markov processes
46T20 Continuous and differentiable maps in nonlinear functional analysis
51F99 Metric geometry
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