Introduction What are Poincare and Sobolev inequalities? Poincare inequalities, pointwise estimates, and Sobolev classes Examples and necessary conditions Sobolev type inequalities by means of Riesz potentials Trudinger inequality A version of the Sobolev embedding theorem on spheres Rellich-Kondrachov Sobolev classes in John domains Poincare inequality: examples Carnot-Caratheodory spaces Graphs Applications to PDE and nonlinear potential theory Appendix References.

Sobolev met Poincaré

Part I. The Type Problem: 1. Basic facts 2. Recurrence and transience of infinite networks 3. Applications to random walks 4. Isoperimetric inequalities 5. Transient subtrees, and the classification of the recurrent quasi transitive graphs 6. More on recurrence Part II. The Spectral Radius: 7. Superharmonic functions and r-recurrence 8. The spectral radius 9. Computing the Green function 10. Spectral radius and strong isoperimetric inequality 11. A lower bound for simple random walk 12. Spectral radius and amenability Part III. The Asymptotic Behaviour of Transition Probabilities: 13. The local central limit theorem on the grid 14. Growth, isoperimetric inequalities, and the asymptotic type of random walk 15. The asymptotic type of random walk on amenable groups 16. Simple random walk on the Sierpinski graphs 17. Local limit theorems on free products 18. Intermezzo 19. Free groups and homogenous trees Part IV. An Introduction to Topological Boundary Theory: 20. Probabilistic approach to the Dirichlet problem, and a class of compactifications 21. Ends of graphs and the Dirichlet problem 22. Hyperbolic groups and graphs 23. The Dirichlet problem for circle packing graphs 24. The construction of the Martin boundary 25. Generalized lattices, Abelian and nilpotent groups, and graphs with polynomial growth 27. The Martin boundary of hyperbolic graphs 28. Cartesian products.

Random Walks on Infinite Graphs and Groups

A number of authors have defined contractive type mappings on a complete metric space X which are generalizations of the well-known Banach contraction, and which have the property that each such mapping has a unique fixed point. The fixed point can always be found by using Picard iteration, beginning with some initial choice xo E X. In this paper we compare this multitude of definitions. X denotes a complete metric space with distance function d, and f a function mapping X into itself. 1. Definitions of contractive type mappings. (1) (Banach) There exists a number a, 0 6 a < 1, such that, for each x,y E X, d(f (x),f (y)) < ad(xY)(2) (Rakotch [21]) There exists a monotone decreasing function a: (0, oo) [0, 1) such that, for each x, y E= X, x # y, d(f (x)j(y)) <ad(x,Y)(3) (Edelstein [10]) For each x, y E X, x # y, d(f (x),f (y)) < d(x,y). (4) (Kannan [18]) There exists a number a, 0 < a < !, such that, for each x,y E X, d(f (x),f (y)) < a[d(x,f (x)) + d(y,f (y))]. (5) (Bianchini [3]) There exists a number h, 0 < h < 1, such that, for each x,,y E X, d(f (x),f (y)) < h max{d(x,f (x)), d(y,f (y))} (6) For each x, y E X, x # y, Received by the editors June 4, 1974 and, in revised form, March 5, 1975 and November 18, 1975. AMS (MOS) subject classifications (1970). Primary 47H10. c American Maithematical Society 1977 257 This content downloaded from 157.55.39.17 on Fri, 02 Sep 2016 04:42:04 UTC All use subject to http://about.jstor.org/terms

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A comparison of various definitions of contractive mappings

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the max-cut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.

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Some applications of Laplace eigenvalues of graphs

Kirchhoff's laws.- Finite networks.- Currents and potentials withfinite energy.- Uniqueness and related topics.- Some examples and computations.- Royden's compactification.- Rough isometries.

Potential Theory on Infinite Networks

Amenability, unimodularity, and the spectral radius of random walks on infinite graphs

A fixed point theorem for nonexpansive mappings in dual Banach spaces is proved. Applications in certain Banach lattices are given. 1. Suppose K is a subset of a Banach space X and T: K -* K is a nonexpansive mapping, i.e. II T(x) T(y)jj S lix -yIj, x, y E K. A wellknown theorem due to Kirk [1] states that, if K is convex weakly compact (weak* compact when X is a dual space) and has normal structure, then T has a fixed point in K. In particular, if X = LP (1 0. Here and in the sequel V and A denote the least upper bound and the greatest lower bound respectively. X is said to be order complete if each set A C X with an upper bound has a least upper bound. A complex AM-space is defined as the complexification of an AM-space. Suppose X is an order complete AM-space with unit (i.e. an element e such that the unit ball at zero is the order interval [e, e]); then X is isometrically lattice isomorphic to the space CR (S) of all continuous real-valued functions defined on a compact Stonian space S. For these and other facts about Banach lattices we refer to Schaefer's book [4]. Received by the editors March 19, 1978. AMS (MOS) subject classifications (1970). Primary 47H10; Secondary 46B99, 46E05.

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Existence of fixed points of nonexpansive mappings in certain Banach lattices

We show that the LiouvilleD
 p
 -property is invariant under rough isometries between a Riemannian manifold of bounded geometry and a graph of bounded degree.

$p$-harmonic functions on graphs and manifolds

Harmonic functions on graphs and on other discrete structures are interesting not only for their own sake but also because they are closely related to harmonic functions on noncompact Riemannian manifolds. For instance, if the local geometry of M is controlled, then the parabolicity of M , i.e. the nonexistence of Green’s function on M , is characterized by the parabolicity of certain discrete structures, called ε-nets, of M . Furthermore, M has nonconstant harmonic functions of finite energy if and only if any ε-net of M does so. Analogous statements hold also in the case of p-harmonic functions. Recall that an ε-net of M is a graph whose vertex set X is a maximal ε-separated subset of M , with a fixed constant ε > 0, and whose edge set consists of all (unoriented) pairs x, y ∈ X such that 0 < d(x, y) ≤ 3ε , where d stands for the Riemannian distance. Above x and y are also said to be neighbors. Furthermore, a real-valued function on X is harmonic if and only if its value at each point x ∈ X is the average of its values at the neighbors of x . We refer to [V1], [K2], [H2], and [HS] for the proofs and precise formulations of the above statements. In [LS] Lyons and Sullivan considered a slightly different discretization of a manifold M . We mention here just one result among the many in [LS]. It states that, for certain discrete sets X ⊂ M and for all y ∈ M , there are positive probability measures νy on X such that every bounded harmonic function h on M satisfies h(y) = ∑

Paolo M. Soardi

Papers

Potential Theory on Infinite Networks

Amenability, unimodularity, and the spectral radius of random walks on infinite graphs

Existence of fixed points of nonexpansive mappings in certain Banach lattices

$p$-harmonic functions on graphs and manifolds

A strong Liouville theorem for $p$-harmonic functions on graphs