Axiomatic Theory of Sobolev Spaces

We develop an axiomatic approach to the theory of Sobolev spaces on metric measure spaces and we show that this axiomatic construction covers the main known examples (Hajtasz Sobolev spaces, weighted Sobolev spaces, Upper-gradients, etc). We then introduce the notion of variational p-capacity and discuss its relation with the geometric properties of the metric space. The notions of p-parabolic and p-hyperbolic spaces are then discussed. I n t r o d u c t i o n Recent years have seen important developments in geometric analysis on metric measure spaces (MM-spaces). Motivating examples came from various subjects such as singular Riemannian manifolds, discrete groups and graphs, Carnot-Carath@odory geometries, hypoelliptic PDE's, ideal boundaries of Gromov-hyperbolic spaces, stochastic processes, fractal geometry etc. The recent books [22] and [19] are convenient references on the subject. Suppose we are given a metric measure space (X,d,#) ; how can we define in a natural way a first order Sobolev space WI,P(X) ? Here is a simple construction. Let ~" be the class of Lipschitz functions with compact support u : X --+ ~, and define for any u E 5 r and any point x C X the infinitesimal stretching constant lu (y) u ( z ) l L~,(x) := lim sup r--tO d(y,x) 0 for any non constant function u. We just observed that this condition is not always satisfied; it must therefore be assumed axiomatically in order to develop a general theory. An alternative notion of Sobolev spaces on metric spaces has been developed in [4], [24] and [39] starting from the notion of upper gradient (see section 2.6 for a description of this Sobolev space). This approach is well adapted to the case of length spaces (these are metric spaces such that the distance is defined in terms of the length of curves) or more generally to quasi-convex spaces. A Poincar~ inequality (see §sec.poinc) is often assumed or proved. It follows from this inequality that the Sobolev space is non degenerate. Another approach is the Sobolev space on metric measure space defined by Piotr Hajiasz in [18]. The Hajtasz Sobolev space is in some sense a globally defined Sobolev space (unlike the constructions above), it is always non degenerate. Other concepts of Sobolev spaces where motivated by the study of first order differential operators on homogeneous spaces (see for example the discussion in [11]) and by graph theory. Our goal in this paper, is to develop an axiomatic version of the theory of Sobolev spaces on metric measure spaces: This axiomatic description covers many examples such as the Hajtasz Sobolev spaces, the weighted Sobolev spaces, the Sobolev spaces based on HSrmander systems of vector fields and on more abstract upper gradients. The basic idea of this axiomatic description is the following: Given a metric space X with a measure #, we associate (by some unspecified mean) to each function u : X -+ ~ a set Din] of functions called the pseudo-gradient8 of u; intuitively a pseudo-gradient 9 C D[u] is a function which exerts some control on the variation of u (for instance in the classical case of ~n: D[u] = {g e L~oc(]~n): g >_ IVu[ a.e.}). A function u C LP(X) belongs then to WI,P(X) if it admits a pseudo-gradient g E D[u] N I / ( X ) . Depending on the type of control required, the construction yields different versions of Sobolev spaces in metric spaces. Instead of specifying how the pseudo-gradients are actually defined, we require them to satisfy some axioms. Our axioms can be divided in two independent Axiomatic Theory of Sobolev Spaces 291 groups: The first group (axioms A1-A4) is a formal description on the set D[u] of pseudo-gradients and the second group (axioms A5 and A6) gives a meaning to the p-integrabil i ty of the pseudo-gradients. A correspondence u --+ D[u] satisfying the axioms is called a D-structure on the metric measure space X. We look at such a structure as an ersatz for a theory of differentiation of the functions on the space (hence the name). A metric measure space (X, d, #) equipped with a D-structure is called a MMD-space. The p-Diriehlet energy Cp(U) of a function u is the greatest lower bound of the p ' th power of the Lp-norms of all the pseudo-gradients of u and the p-Dirichlet space £t 'p(X) is the space of locally integrable functions with finite p-energy. The Sobolev space is then the space WI,p(X) := £),P(X) A Lv(X). We can prove from the axioms that WI,P(X) is a Banach space; however, due to the fact that the definition of pseudo-gradient is not based on a linear operation, we can't generally prove that it is a reflexive Banach space for 1 < p < oo. Using this theory, we obtain a classification of metric spaces into p-parabolic/p-hyperbolic types similar to the case of Riemannian manifolds. We now briefly describe the content of the paper: In section 1, we give the axiomatic construction of Sobolev spaces on metric measure spaces and the basic properties of these spaces. The setting is the following: we fix a metric measure space (X, d, #) and we choose a Boolean ring ](: of bounded subsets of X which plays the role of relatively compact subsets in the classical situation (the precise conditions that ~ must satisfy are specified in the next section). The space L~oc(X ) is defined to be the space of all measurable functions u such that UlA E LP(A) for all sets A C ]C. We then define the notion of D-structure by a set of axioms and we list some basic properties of the axiomatic Sobolev spaces. In section 2, we show that familiar examples of Sobolev spaces on metric spaces such as the classical Sobolev spaces on Riemannian manifolds, weighted Sobolev spaces, Sobolev spaces on graphs, Hajtasz Sobolev spaces and Sobolev spaces based on upper gradients are examples of axiomatic Sobolev spaces. In section 3, we develop the basics of non linear potential theory on metric spaces. We denote by £~'P(X) the closure of the set of continuous functions u E £1'P(X) with support in a ]C-set and we define the variational p-capacity of a set F C ~ by Capp(F) := inf{$p(u)tu C Ap(F)} , where Ap(F) := {u E £~'P(X)I u >_ 0 and u > 1 on a neighbourhood of F } . A metric space X is said to be p-hyperbolic if it contains a set Q c E of positive p-capacity and p-parabolic otherwise. One of our results (Theorem 3.1) says that the space X is p-parabol ic if and only if 1 E £~'P(X). In the last section, we quote without proof a few recent results from the theory of Sobolev spaces on metric spaces. 292 V. Gol'dshtein and M. Troyanov Let us conclude this introduct ion by mentioning a few impor tan t , very active and related topics which are not discussed in this paper. First there is the theory of Sobolev mappings between two metric spaces which is a na tura l extension of the present work. Some papers on this subject are [32], [38] and [44]. Then there are papers dealing with a generalized notion of (co)tangent bundle on metric spaces such as [4], [37], [45] and [46]. Finally, there is the theory of Dirichlet forms and analysis in Wiener spaces such as exposed in [2] and [35]. Finally, it is our pleasure to thank Piot r Hajtasz and Khaled Gafa'iti for their friendly and useful comments . C o n t e n t : 1 Axiomatic theory of Sobolev Spaces 1.1 The basic setting 1.2 D-structure and Sobolev spaces 1.3 Poincar@ inequalities 1.4 Locality 1.5 Topology on the Dirichlet space 1.6 Minimal pseudo-gradient 1.7 Defining a D-structure by completion 1.8 Relaxed topology, m-topology and density 1.9 Linear D-structures 2 Examples of axiomatic Sobolev Spaces 2.1 Classical Sobolev space 2.2 Weighted Sobolev space 2.3 Hajtasz-Sobolev space 2.4 Graphs (combinatorial Sobolev spaces) 2.5 Infinitesimal Stretch 2.6 Upper Gradients 2.7 Comparing different D-structures 3 Capacities and Hyperbol ic i ty 3.1 Definition of the variational capacity 3.2 Growth of balls and parabolicity 4 A survey of some recent results Axiomatic Theory of Sobolev Spaces 293 1 Axiomatic theory of Sobolev Spaces 1.1 T h e b a s i c s e t t i n g An MM-spaee is a metric space (X, d) equipped with a Borel regular outer measure # such that 0 < #(B) < c~ for any ball B C X of positive radius (recall, that an outer measure # is Borel regular if every Borel set is p-measurable and for every set E C X, there is a Borel subset A C E such that #(A) = #(E), see [9, page 6]). Our first aim is to introduce a notion of local Lebesgue space L~oc(X). For this purpose we need the following concept : Definition 1.1 (a) A local Borel ring in the MM-space (X, d, #) is a Boolean ring 1 K of bounded Borel subsets of X satisfying the following three conditions: