Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential
aa r X i v : . [ m a t h . M G ] F e b CURVEWISE CHARACTERIZATIONS OF MINIMAL UPPER GRADIENTSAND THE CONSTRUCTION OF A SOBOLEV DIFFERENTIAL
SYLVESTER ERIKSSON-BIQUE AND ELEFTERIOS SOULTANIS
Abstract.
We represent minimal upper gradients of Newtonian functions, in the range 1 ≤ p < ∞ ,by maximal directional derivatives along “generic” curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of p -weakcharts and prove that every Newtonian function admits a differential with respect to such charts,yielding a linear approximation along p -almost every curve. The differential can be computedcurvewise, is linear, and satisfies the usual Leibniz and chain rules.The arising p -weak differentiable structure exists for spaces with finite Hausdorff dimension andagrees with Cheeger’s structure in the presence of a Poincar´e inequality. It is moreover compatiblewith, and gives a geometric interpretation of, Gigli’s abstract differentiable structure, whenever itexists. The p -weak charts give rise to a finite dimensional p -weak cotangent bundle and pointwisenorm, which recovers the minimal upper gradient of Newtonian functions and can be computed bya maximization process over generic curves. As a result we obtain new proofs of reflexivity anddensity of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimalHilbertianity in terms of the pointwise norm. Contents
1. Introduction 12. Preliminaries 83. Curvewise (almost) optimality of minimal upper gradients 114. Charts and differentials 195. The p -weak differentiable structure 296. Relationship Cheeger’s and Gigli’s differentiable structures 33Appendix A. General measure theory 37References 401. Introduction
Overview.
Minimal weak upper gradients of Sobolev type functions on metric measure spaceswere first introduced by Cheeger [12], building on the notion of upper gradients from [30]. Shan-mugalingam [42] developed
Newtonian spaces N ,p ( X ) using the modulus perspective of [30], andproved that they coincide with the Sobolev space defined by Cheeger up to modification of itselements on a set of measure zero. Further notions of Sobolev spaces, based on test plans, weredeveloped by Ambrosio–Gigli–Savare [6], with a corresponding notion of minimal gradient. Ear-lier, Haj lasz [28] had introduced a Sobolev space whose associated minimal gradient however lacks suitable locality properties. While the various Sobolev spaces (with the exception of Haj lasz’sdefinition) are equivalent for generic metric measure spaces, Newtonian spaces consist of represen-tatives which are absolutely continuous along generic curves, a property central to the results inthis paper.The minimal p -weak upper gradient g f ∈ L p ( X ) of a Newtonian function f ∈ N ,p ( X ) on ametric measure space X is a Borel function characterized (up to a null-set) as the minimal functionsatisfying(1.1) | ( f ◦ γ ) ′ t | ≤ g f ( γ t ) | γ ′ t | for a.e. t ∈ I for all absolutely continuous γ : I → X outside a curve family of zero p -modulus. Here | γ ′ t | denotesthe metric derivative of γ for a.e. t , see Section 2. When X = R n and f ∈ C ∞ c ( R n ), g f is given by g f = k∇ f k ; in this case, for each x ∈ X , there exists a (smooth) gradient curve γ : ( − ε, ε ) → X with γ = x , satisfying ( f ◦ γ ) ′ = g f ( x ) | γ ′ | . (1.2)In general however, despite the minimality of g f , the equality in (1.2) is not always attained. Forexample the fat Sierpi´nski carpet (with the Hausdorff 2-measure and Euclidean metric) constructedin [36] with a sequence in ℓ \ ℓ , as pointed out in the introduction of [36], gives zero p -modulus( p >
1) to the family of curves parallel to the x -axis, and thus to the family of gradient curves ofthe function f ( x, y ) = x . We remark that the example above is measure doubling and supports aPoincar´e inequality; in this context an approximate form of (1.2) for Lipschitz functions was provenin [13, Theorem 4.2].Towards a positive answer for generic spaces, an “integral formulation” of (1.2) given by a resultof Gigli [25, Theorem 3.14] states that, when p > f ∈ N ,p ( X ), there exist probabilitymeasures η on C ( I ; X ) (known as test plans representing the gradient of f ) such thatlim t → Z f ( γ t ) − f ( γ ) t d η = lim t → Z t Z t g f ( γ s ) | γ ′ s | d s d η . In this paper we obtain a “pointwise” variant of (1.2) for general metric measure spaces using acombination of plan-modulus duality, developed in [3, 21, 23], and disintegration techniques. Theo-rem 1.1 below expresses the minimal weak upper gradient of a Newtonian function as the supremumof directional derivatives along generic curves passing through a given point. Here, it is crucial touse Newtonian functions, which are absolutely continuous along almost every curve.This curvewise characterization of minimal upper gradients yields the existence of an abundanceof curves in a given region of the space, provided the region supports non-trivial Newtonian func-tions. The idea of constructing an abundance of curves goes back to Semmes [41] in the presenceof a Poincar´e inequality. Under this assumption Cheeger showed that g f = Lip f , where Lip f de-notes the pointwise Lipschitz constant of a Lipschitz function f . Note that inequality Lip f ≤ g for continuous upper gradients g of a Lipschitz function f on a geodesic space is a direct, but central,observation made in [12, p. 432–433].The work of Cheeger lead to many developments, including the paper [13] pioneering the ideaof using directional derivatives along curves (and the early version of Theorem 1.1 in [13, Theorem4.2]) as well as the development and detailed analysis of Lipschitz differentiability spaces , cf. [8, 9,14, 34, 39, 40]. In the latter, curves are replaced by curve fragments whose abundance is expressed
URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 3 using
Alberti representations . Alberti representations are similar to plans used in this paper. Theconnection between such representations and the ideas in [13] was first observed by Preiss, see [8,p.2], and can be used to prove the self-improvement of the Lip − lip inequality to the Lip − lipequality, cf. [8, 14, 39].Similarly the abundance of curves, obtained here using duality, yields geometric informationon Sobolev functions on general metric measure spaces. (Indeed, duality, in the disguise of aminimax principle, was previously used to find Alberti representations in Lipschitz differentiabilityspaces, see [8, Theorem 5.1] which uses [38, Lemma 9.4.3].) As an important first application,we use curvewise directional derivatives to define p -weak charts and a differential for Newtonianfunctions with respect to such charts. The arising p -weak differentiable structure , i.e. a covering by p -weak charts, exists for any Lipschitz differentiability space but also in more general situations,see Proposition 5.4. With the aid of Theorem 1.1 we adapt Cheeger’s construction to producea measurable L ∞ -bundle, called the p -weak cotangent bundle , over spaces admitting a p -weakdifferentiable structure; differentials of Newtonian functions are sections over this bundle. Whilethe Cheeger differential d C f yields a linearization of a Lipschitz function f , our p -weak differentiald f is given by a linearization along p -almost every curve, and the pointwise norm of d f recoversthe minimal weak upper gradient, cf. Theorem 1.7.This definition of a weak differentiable structure seems to be the natural extension of Cheeger’sseminal work in [12] to settings without a Poincar´e inequality and yields a ”partial differentiablestructure”, which has been the aim of many authors previously, cf. [1,14,15,39]. Namely, the p -weakcotangent bundle measures and makes precise the set of accessible directions (for positive modulus)in the space. By constructing the differential using directional derivatives along curves, we give it aconcrete geometric meaning. A sequence of recent work has sought such concrete descriptions, seee.g. [15, 32]. Our approach yields a new unification of the concrete and abstract cotangent modulesof Cheeger [12] and Gigli [26], respectively; the p -weak cotangent bundle is compatible with Gigli’scotangent module when the latter is locally finitely generated, and with Cheeger’s cotangent bundlewhen the space satisfies a Poincar´e inequality, see Theorems 1.11 and 1.8.The geometric approach in this paper has many natural applications. We mention here the tensorization of Cheeger energy , pursued in [6, 7, 15], and the identification of abstract tangentbundles with geometric tangent cones, cf. [1,15]. Our methods give tools to generalize and refine theresults mentioned above, and moreover enable a blow-up analysis to study analogues of generalizedlinearity considered for example in [12, 14]. Indeed, blow-ups of plans that define the pointwisenorm on a p -weak chart (see Lemmas 4.1, 4.3 and 4.2) along a sequence of re-scaled spaces yieldcurves in the limiting space along which limiting maps of rescaled Newtonian maps behave linearly.In this context we highlight [39], which gives a similar geometric and blow-up analysis in the contextof abstract Weaver derivations. We leave the detailed exploration and development of these ideasfor future work.1.2. Curvewise characterization of minimal upper gradients.
Throughout the paper, we fixa metric measure space X = ( X, d, µ ), that is, a complete separable metric space (
X, d ) togetherwith a Radon measure µ which is finite on bounded sets. A plan is a finite measure η on C ( I ; X )that is concentrated on the set AC ( I ; X ) of absolutely continuous curves. The natural evaluationmap e : C ( I ; X ) × I → X, ( γ, t ) γ t gives rise to the barycenter d η := e ∗ ( | γ ′ t | d t d η ) of η . If SYLVESTER ERIKSSON-BIQUE AND ELEFTERIOS SOULTANIS d η = ρ d µ for some ρ ∈ L q ( µ ), we say that η is a q -plan. (Not to be confused with q -test plans,see e.g. Section 2 or [3].) Every finite measure π on C ( I ; X ) × I admits a disintegration withrespect to e : for e ∗ π -almost every x ∈ X , there exists a (unique) measure π x ∈ P ( C ( I ; X ) × I ),concentrated on e − ( x ), such that the collection { π x } satisfies π ( B ) = Z π x ( B )d( e ∗ π )( x )for all Borel sets B ⊂ C ( I ; X ) × I . We refer to [4, 11] for more details.We use these notions to define a “generic curve”: if η is a q -plan on X and { π x } the disintegrationof d π := | γ ′ t | d t d η with respect to e , then π x -a.e.-curve passes through x , for e ∗ π -a.e. x ∈ X . Inthe forthcoming discussion, we omit the reference to e in the disintegration. We now formulate ourfirst result, in which the equality in (1.2) is obtained as an essential supremum with respect to thedisintegration for almost every point. In the statement below we denoteDiff( f ) = { ( γ, t ) ∈ AC ( I ; X ) × I : f ◦ γ ∈ AC ( I ; R ) , ( f ◦ γ ) ′ t and | γ ′ t | > } for a µ -measurable function f : X → [ −∞ , ∞ ]. Theorem 1.1.
Let ≤ p < ∞ , and let < q ≤ ∞ satisfy /p + 1 /q = 1 . Suppose f ∈ N ,p ( X ) , g f is a Borel representative of the minimal p -weak upper gradient of f , and D := { g f > } . Thereexists a q -plan η with µ | D ≪ η so that the disintegration { π x } of d π := | γ ′ t | d t d η is concentratedon e − ( x ) ∩ Diff( f ) and g f ( x ) = (cid:13)(cid:13)(cid:13)(cid:13) ( f ◦ γ ) ′ t | γ ′ t | (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( π x ) (1.3) for µ -almost every x ∈ D . Remark 1.2.
The statement also holds when f ∈ N ,ploc ( X ), that is, f | B ( x,r ) ∈ N ,p ( B ( x, r )) foreach ball B ( x, r ) ⊂ X . Indeed, a localization argument, replacing f by f η n with η n a sequence ofLipschitz functions with bounded support and η n | B ( x ,n − = 1 for some x , reduces the statementfor f ∈ N ,ploc ( X ) to Theorem 1.1. Similarly, other notions in this paper, such as charts, could use alocal Sobolev space, but to avoid technicalities we do not discuss this point further. A reader cansee Lemma 4.5 and its proof for a prototypical form of such a localization argument.In particular, we have the following corollary. Corollary 1.3.
Let p, q and f, g f , D be as in Theorem 1.1. There exists a q -plan η and, for every ε > , a Borel set B = B ε ⊂ Diff( f ) with the following property. If { π x } denotes the disintegrationof d π := | γ ′ t | d t d η , then π x ( B ) > and (1 − ε ) g f ( x ) | γ ′ t | ≤ ( f ◦ γ ) ′ t ≤ g f ( x ) | γ ′ t | for every ( γ, t ) ∈ e − ( x ) ∩ B for µ -a.e. x ∈ D . Theorem 1.1 notably covers the case p = 1. In Section 3.3 we also prove a variant (Theorem 3.6)when p >
1, using test plans representing a gradient instead of plan-modulus duality.
URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 5
Application: p -weak differentiable structure. Cheeger [12] showed that PI-spaces (met-ric measure spaces with a doubling measure supporting some Poincar´e inequality) admit a countablecover by
Cheeger charts , also called a
Lipschitz differentiable structure (see [35]). Let LIP( X ) de-note the collection of Lipschitz functions on X , and let LIP b ( X ) consist those Lipschitz functionswith bounded support. A Cheeger chart ( U, ϕ ) of dimension n consists of a Borel set U with µ ( U ) >
0, and a Lipschitz function ϕ : X → R n such that, for every f ∈ LIP( X ) and µ -a.e. x ∈ U there exists a unique linear map d C,x f : R n → R , called the Cheeger differential of f , such that f ( y ) − f ( x ) = d C,x f ( ϕ ( y ) − ϕ ( x )) + o ( d ( x, y )) as y → x. (1.4)Not every space admits Lipschitz differentiable structure, as shown by the so called Rickman’s rug X := [0 , equipped with the metric d (( x , y ) , ( x , y )) = | x − x | + | y − y | α , where α ∈ (0 , µ = L | X . Indeed, a Weierstrass-type function in the y -variable combined with [39, Theorem 1.14]would yield non-horizontal rectifiable curves if the space were a differentiability space, contradictingthe fact that all rectifiable curves in X are horizontal.Here, we introduce p -weak differentiable structures , which exist in much more generality (in-cluding Rickman’s rug, see the discussion after Definition 1.4), adapting Cheeger’s construction bysubstituting (1.4) for a weaker curve-wise control. To accomplish this, we replace the pointwiseLipschitz constant by the minimal p -weak upper gradient in the definition of “infinitesimal linearindependence” (1.5) and use Theorem 1.1 to circumvent the difficulties arising from the fact thatthe latter is defined only up to a null-set.In the remainder of the introduction, we use the notation | Df | p for the minimal p -weak uppergradient of f ∈ N ,ploc ( X ) and refer to Section 2 for more discussion on this notation. Given p ≥ N ∈ N , we say that a Sobolev map ϕ ∈ N ,ploc ( X ; R N ) is p -independent in U ⊂ X , ifess inf v ∈ S N − | D ( v · ϕ ) | p > µ -a.e. in U, (1.5)and p -maximal in U , if no Lipschitz map into a higher dimensional target is p -independent in apositive measure subset of U . Here, we use the essential infimum of an uncountable collection,which agrees µ -a.e. with the pointwise infimum over any countable dense collection of S N − , seeAppendix A.2. Note that p -maximality does not depend on the particular map ϕ but rather thedimension of its target space. Definition 1.4. An N -dimensional p -weak chart ( U, ϕ ) of X consists of a Borel set U ⊂ X withpositive measure and a Lipschitz function ϕ : X → R N which is p -independent and p -maximal in U . We say that X admits a p -weak differentiable structure if it can be covered up to a null set bycountably many p -weak charts.By convention, zero dimensional p -weak charts satisfy ϕ ≡ | Df | p = 0 µ -a.e on U for every f ∈ LIP b ( X ) (see also Proposition4.4). In Section 4.6 we briefly discuss a lower regularity requirement in Definition 1.4 and the factthe resulting notion yields essentially the same p -weak differentiable structure. We also show thatand N -dimensional p -weak chart ( U, ϕ ) satisfies N ≤ dim H U , where dim H U denotes the Hausdorffdimension of U , see Proposition 4.13. In particular, we have the following theorem. Theorem 1.5.
A metric measure space of finite Hausdorff dimension admits a p -weak differentiablestructure for any p ≥ . SYLVESTER ERIKSSON-BIQUE AND ELEFTERIOS SOULTANIS
We refer to Proposition 5.4 for a more technical statement, which immediately implies the the-orem. Next, we give an analogue of the Cheeger differential (1.4) using p -weak charts. Definition 1.6.
Given an N -dimensional p -weak chart ( U, ϕ ) of X , a Newtonian function f ∈ N ,p ( X ) has a p -weak differential d f : U → ( R N ) ∗ with respect to ( U, ϕ ), whose values at x ∈ U are denoted d x f , such that f ( γ s ) − f ( γ t ) = d γ t f ( ϕ ( γ s ) − ϕ ( γ t )) + o ( | t − s | ) a.e. t ∈ γ − ( U ) , as s → t (1.6)for p -a.e. absolutely continuous curve γ in X . The map d f is called a p -weak differential (for f ).If the curve γ does not enter U , or only spends zero length in the set, then condition (1.6) becomesvacuously satisfied with both sides vanishing. The p -weak differential is uniquely determined almosteverywhere equivalence by (1.6), cf. Lemma 4.3. Further, it is also local, i.e. if g ∈ N ,p ( X ) and f | A = g | A on a positive measure subset A ⊂ U , then d f | A = d g | A . The differential satisfies naturalcomputation rules, see Propositions 4.10 and 5.7. Theorem 1.7.
Suppose p ≥ , and ϕ : X → R N is a p -weak chart on U . Then any f ∈ N ,p ( X ) has a p -weak differential d f : U → ( R N ) ∗ with respect to ϕ , and the map f d f is linear.Moreover, for µ -a.e. x ∈ U , there is a norm | · | x on ( R N ) ∗ , such that x
7→ | ξ | x is Borel for every ξ ∈ ( R N ) ∗ , and | d f | x = | Df | p ( x ) µ -a.e. x ∈ X, for every f ∈ N ,p ( X ) . Whereas Lipschitz functions are differentiable with respect to Cheeger charts, (1.5) yields onlythe curvewise control (1.6). Indeed, if there are very few or no rectifiable curves, or if the curves onlypoint into certain directions, then the p -weak differential vanishes, or measures only these directions,respectively. For example, given a fat Cantor set K ⊂ R n with L n ( K ) > X := ( K, d
Eucl , L n | K ) isa Lipschitz differentiability space but the minimal weak upper gradient of every Lipshitz function iszero. On the other hand, Rickman’s rug admits non-trivial p -weak charts ϕ ( x, y ) = x . The p -weakdifferential in this case can be identified with the x -derivative, d f ≡ ∂ x f , and the only curveswith positive p -modulus are those which are horizontal. These examples demonstrate that p -weakdifferentiable structures might exist for spaces not admitting a Cheeger structure, but the two neednot coincide even if both exist. However, if a Poincar´e inequality is present, the two structurescoincide. Theorem 1.8.
Suppose X is a p -PI space for p ≥ . Then any p -weak chart ( U, ϕ ) of X is aCheeger chart. It follows from the discussion after Definition 1.4 that a p -PI space admits p -weak charts. InSubsection 1.4, we obtain a precise statement on the relationship between the p -weak and Lips-chitz differentiable structure, as well as a characterization of the existence of p -weak differentiablestructures in terms of Gigli’s cotangent module, cf. [26]. Here we mention a noteworthy corollaryof the existence of a p -weak differentiable structure. Theorem 1.9.
Let p ≥ . If X admits a p -weak differential structure, then LIP b ( X ) is norm-densein N ,p ( X ) . Theorem 1.9 has been obtained by other methods for p > p = 1.In particular, we highlight that the density holds if X has finite Hausdorff dimension. URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 7
Connections to Cheeger’s and Gigli’s differentiable structures.
Together with thepointwise norm from Theorem 1.7, a p -weak differentiable structure gives rise to a p -weak cotangentbundle T ∗ p X over X , analogous to the measurable L ∞ -cotangent bundle T ∗ C X arising from theLipschitz differentiable structure [12, 35], which is equipped with the pointwise norm | ξ | C,x := Lip( ξ ◦ ϕ )( x ) , ξ ∈ ( R N ) ∗ for µ -a.e. x ∈ U , where ( U, ϕ ) is an N -dimensional Cheeger chart. For any f ∈ LIP b ( X ), thedifferentials d f and d C f are sections of the cotangent bundles T ∗ p X and T ∗ C X , respectively. Werefer to Section 5 for the precise definition of measurable L ∞ -bundles and their sections.In the next theorem we show that there is a submetric bundle map T ∗ C X → T ∗ p X , and give acondition under which the bundle map is an isometric isomorphism. See Section 5 for the definitionof bundle maps. In the statement, a modulus of continuity is an increasing continuous function ω : [0 , ∞ ) → [0 , ∞ ) with ω (0) = 0, and a linear submetry between normed spaces V and W is asurjective linear map L : V → W with L ( B V ( r )) = B W ( r ). Theorem 1.10.
Suppose X admits a Cheeger structure and let p ≥ . There is a bundle map π = π C,p : T ∗ C X → T ∗ p X which is a linear submetry µ -a.e. and satisfies π x (d C,x f ) = d x f, µ − a.e. x ∈ X (1.7) for every f ∈ LIP b ( X ) . If there exists a collection { ω x } x ∈ X of moduli of continuity satisfying Lip f ( x ) ≤ ω x ( | Df | p ( x )) µ − a.e. on X for every f ∈ LIP b ( X ) , then π C,p is an isometric bijection µ -a.e. Theorem 1.10 follows from [32, Theorem 1.1] and the following theorem, which identifies thespace Γ p ( T ∗ p X ) of p -integrable sections of the p -weak cotangent bundle T ∗ p X with Gigli’s cotangentmodule L p ( T ∗ X ). We refer to Section 6 for the relevant definitions, and remark here that Gigli’sconstruction is the most general in the sense that L p ( T ∗ X ) can be defined for any metric measurespace. It is a priori defined only as an abstract L p -normed L ∞ -module in the sense of [25, 26].We say that L p ( T ∗ X ) is locally finitely generated if X has a countable Borel partition B so thateach B ∈ B admits a finite generating set in B . Here, a collection V ⊂ L p ( T ∗ X ) is a generatingset in B (or generates L p ( T ∗ X ) in B ) if χ B L p ( T ∗ X ) is the smallest closed submodule of L p ( T ∗ X )containing χ B v for every v ∈ V . Gigli’s cotangent modules admit a dimensional decomposition ,i.e. a Borel partition { A N } N ∈ N ∪{∞} of X so that L p ( T ∗ X ) admits a generating set of cardinality N (and no smaller) in A N , for each N . For N = ∞ , no finite set generates L p ( T ∗ X ) in A N . Thedimensional decomposition is uniquely determined up to µ -negligible sets.In the statement below, we denote by d G f and | · | G the abstract differential and pointwisenorm in the sense of Gigli, see Theorem 6.1. A morphism between L p -normed L ∞ -modules (i.e.a continuous L ∞ -linear map) is said to be an isometric isomorphism if it preserves the pointwisenorm, and has an inverse that is a morphism. Theorem 1.11.
Let X be a metric measure space and p ≥ . Then X admits a p -weak differentiablestructure if and only if L p ( T ∗ X ) is locally finitely generated. In this case, (a) there exists an isometric isomorphism ι : Γ p ( T ∗ p X ) → L p ( T ∗ X ) of normed modules, satis-fying ι (d f ) = d G f for every f ∈ N ,p ( X ) and uniquely determined by this property, SYLVESTER ERIKSSON-BIQUE AND ELEFTERIOS SOULTANIS (b) each set A N in the dimensional decomposition of X can be covered up to a null-set by N -dimensional p -weak charts, (c) N ≤ dim H ( A N ) for each N ∈ N . Theorem 1.11 gives a concrete interpretation of Gigli’s cotangent module, and bounds the Haus-dorff dimension of the sets in the dimensional decomposition. As corollaries we obtain the reflexivityof N ,p ( X ) when p >
1, and a characterization of infinitesimal Hilbertianity in terms of the point-wise norm of Theorem 1.7 when p = 2, for spaces admitting a p -weak differentiable structure, seeCorollary 6.7. Reflexivity could also be obtained directly from Theorem 1.7 following the argumentin [12, Section 4].1.5. Acknowledgements.
The first author was partially supported by National Science Founda-tion under Grant No. DMS-1704215 and by the Finnish Academy under Research postdoctoralGrant No. 330048. The second author was supported by the Swiss National Science FoundationGrant 182423. Throughout the project the authors have had insightful discussions with NageswariShanmugalingam, which have been tremendously useful. A further thanks goes to Jeff Cheegerfor helpful comments and inspiration for the project. The authors thank IMPAN for hosting thesemester “Geometry and analysis in function and mapping theory on Euclidean and metric mea-sure space” where this research was started. Through this workshop the authors were partiallysupported by grant
Preliminaries
Throughout this paper X = ( X, d, µ ) will be a complete separable metric measure space equippedwith a Radon measure µ finite on balls. We denote by C ( I ; X ) the space of continuous curves γ : I → X equipped with the metric of uniform convergence, and by AC ( I ; X ) the subset ofabsolutely continuous curves in X . Mostly, we will be concerned with statements independent ofparametrization, thus the choice of I is immaterial. However, when we need to refer to the endpoints of the curve, then we will take I = [0 , γ is a curve, its value at t ∈ I is denoted by γ t := γ ( t ). If f : X → R N is a function, we alsouse this notation as ( f ◦ γ ) t = f ( γ t ). The derivative of f in the direction of γ at γ t , when it exists,is denoted ( f ◦ γ ) ′ t = ( f ◦ γ ) ′ ( t ). The metric derivative of the curve, in the sense of say [4, SectionI.1], is defined as | γ ′ t | = lim h → d ( γ t + h ,γ t ) h , when it exists. The metric derivative is defined almosteverywhere on I for γ ∈ AC ( I ; X ).2.1. Plans and modulus.
A finite measure η on C ( I ; X ) is called a plan if it is concentratedon AC ( I ; X ), and a q -plan if the barycenter d η := e ∗ ( | γ ′ t | d t d η ) satisfies d η = ρ d µ for some ρ ∈ L q ( µ ). We denote by AC q ( I ; X ) the space of curves γ ∈ AC ( I ; X ) satisfying Z | γ ′ t | q d t < ∞ ,and say that a q -plan η ∈ P ( C ( I ; X )) is a q -test plan , if it is concentrated on AC q ( I ; X ) and e t ∗ η ≤ Cµ for every t ∈ I , and Z Z | γ ′ t | q d t d η < ∞ URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 9 for some constant
C >
0. Here e t : C ( I ; X ) → X is the map e t ( γ ) = γ t . Remark 2.1.
Every q -test plan is a also a q -plan. However, the converse can fail for two reasons. A q -test plan fixes a given parametrization for curves (with an integrability condition on the speed),and insists on a compression bound e t ∗ ( η ) ≤ Cµ . However, for each q -plan supported on Γ ⊂ AC ( I ; X ), one can construct associated q -test plans supported on reparametrized curves, which aresubcurves of curves in Γ.The argument for this is a combination of two observations in [3]. First, for each q -plan onecan re-parametrize curves to get a plan with a good ”parametric barycenter” [3, Definition 8.1 andTheorem 8.3]. The parametric barycenter depends on the parametrization, while the barycenter η does not. The second point concerns the compression bound, where given the previous plan,one can take sub-segments of curves and average these over shifts to get a compression bound,which is explained as part of the proof of [3, Theorem 9.4].This remark would allow, for example, to phrase Theorem 1.1 with test plans instead of plans,if one were so inclined.If Γ ⊂ C ( I ; X ) is a family of curves, then a Borel function ρ : X → [0 , ∞ ] is called admissible,if R γ ρ d s ≥ γ ∈ Γ. In particular, if there are no rectifiable curves, then thiscondition is vacuous. We define, for p ∈ [1 , ∞ )Mod p (Γ) = inf ρ Z X ρ p d µ, where the infimum is over all admissible ρ . We remark, that due to Vitali-Caratheodory, suchan infimum can always be taken with respect to lower semi-continuous functions. Notice thatthe modulus is supported on rectifiable curves and is independent of the parameterization of suchcurves. We say that a property holds for p -almost every curve if there is a family of curves Γ B so that Mod p (Γ B ) = 0, and the property holds for all γ ∈ C ( I ; X ) \ Γ B . Modulus is invariantof the parametrization of curves, but some of our statements depend on a parametrization. Inthose cases, we will say that the property holds for p -almost every absolutely continuous curve in X (or p a.e. γ ∈ AC ( I ; X )) to emphasize that the property holds for each γ ∈ AC ( I ; X ) \ Γ B with Mod p (Γ B ) = 0. The reader may consult [31, Sections 4–7] for a more in-depth treatment ofmodulus, upper gradients and Vitali-Caratheodory. Remark 2.2.
A crucial fact we will use is that if Γ satisfies Mod p (Γ) = 0, then for any q -plan η we have η (Γ) = 0 (which holds for p ∈ [1 , ∞ ) and q its dual exponent). The converse is alsotrue for p ∈ (1 , ∞ ). See the arguments and discussion in [3, Sections 4,9]. One point here is thatif we used q -test plans, this relationship would be more complex, and we would need to consider”stable” families of curves, see [3, Theorem 9.4]. The case of p = 1 is also somewhat subtle, andwe will deal with a special case of this issue in Section 3. The argument of Proposition 2.3 wouldgive the converse for compact families of curves and p = 1. See also, [23] for a much more detailedexploration of this borderline case.The previous remark concerns an inequality relating modulus and q -plans. However, there is acloser connection, and in a sense these are dual to each other. Previously, this has been exploredin [3, Theorem 5.1] for p >
1, and in [23, Theorem 6.3] for p = 1. Due to its importance for us, wesummarize one main consequence of these results. We further briefly describe the main steps of a direct proof from [16, Proposition 4.5]. A similar argument appeared previously in a more specificcontext in [21, Theorem 3.7]. Proposition 2.3.
Let p ∈ [1 , ∞ ) and q its dual exponent with p − + q − = 1. If K ⊂ C ( I ; X ) isa compact family of curves, and Mod p ( K ) ∈ (0 , ∞ ), then there exists a q -plan η with spt( η ) ⊂ K . Proof.
A power of the modulus Mod p ( K ) /p arises from a convex optimization problem on ρ witha constraint for every curve γ ∈ K . A dual formulation of this corresponds to a variable foreach constraint, i.e. a measure ν supported on K . Thus, it is reasonable to consider a modifiedLagrangian defined by Φ( ρ, ν ) = || ρ || L p − Mod p ( K ) /p Z K Z γ ρ d s d ν γ , where ρ : X → [0 , ∞ ] is a function and ν is a probability measure supported on K . Let P ( K ) bethe collection of these probability measures supported on K equipped with the topology of weak*convergence. In order to obtain required continuity, we will restrict to ρ ∈ G with G := { ρ : X → [0 , , ρ compactly supported and continuous in X } . The set G is equipped with the topology ofuniform convergence. Then Φ : G × P ( K ) → R is a functional with two properties: a) Φ( · , ν ) isconvex and continuous for each ν ∈ P ( K ), b) Φ( ρ, · ) is concave and upper semi-continuous for each ρ : X → [0 , P ( K ) is compact and convex in the weak* topology and G is a convexsubset.By Sion’s minimax theorem, see e.g. statement in [16, Theorem 4.7], we havesup ν ∈ P ( K ) inf ρ ∈ G G ( ρ, ν ) = inf ρ ∈ G sup ν ∈ P ( K ) G ( ρ, ν ) . We can compute inf ρ ∈ G sup ν ∈ P ( K ) G ( ρ, ν ) ≥
0. Indeed, given any ρ ∈ G , we can use the definitionof modulus to find a γ ∈ K with R γ ρ d s ≤ || g || p Mod p ( K ) − /p . If we choose ν = δ γ , a Dirac measure on γ ,the bound immediately follows.Therefore, we have also sup ν ∈ P ( K ) inf ρ ∈ G G ( ρ, ν ) ≥
0. But, up to showing that this supremum isattained, there must be some η ∈ P ( K ) for which we get inf ρ ∈ G G ( ρ, η ) ≥
0. After unwinding thedefinition of a q -plan, and an application of Radon-Nikodym on X , the measure η is our desired q -plan. (cid:3) Sobolev spaces and functions.
A function f : ( X, d X ) → ( Y, d Y ) between two metric spacesis called Lipschitz if LIP( f ) := sup x,y ∈ X,x = y d Y ( f ( x ) ,f ( y )) d X ( x,y ) < ∞ . A bijection f : X → Y is calledBi-Lipschitz if f and f − are Lipschtiz. Further, if x ∈ X , we define the local Lipschitz constant asLip f ( x ) := lim sup y → x,y = x d Y ( f ( x ) , f ( y )) d X ( x, y ) . Let LIP b ( X ) be the collection of Lipschitz maps f : X → R with bounded support. Definition 2.4.
Let f : X → R ∪ {±∞} be measurable, g : X → [0 , ∞ ] a Borel function, and γ : I → X a rectifiable path. We say that g is an upper gradient of f along a rectifiable curve URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 11 γ : [0 , → X , if R γ g d s < ∞ and | f ( γ t ) − f ( γ s ) | ≤ Z γ | [ s,t ] g, for each s < t with s, t ∈ I with the convention ∞ − ∞ = ∞ . We say that g is an upper gradientof f , if it is an upper gradient along every rectifiable curve, and a p -weak upper gradient if g is anupper gradient of f along p -a.e. rectifiable curve.The space N ,p ( X ) is defined as all µ -measurable functions f ∈ L p ( X ) which have an uppergradient g in L p ( X ). The (semi-)norm on this space is defined as k f k N ,p = (cid:0) k f k pL p + inf k g k pL p (cid:1) /p , where the infimum is taken over all L p -integrable upper gradients g of f . The theory of these spaceswas largely developed in [42], see also [31] for most of the classical theory. By the results therecombined with an observation of Haj lasz [29] in the case of p = 1, one can show that there alwaysexists a unique minimal g f , which is an upper gradient along p -almost every path, and for which k f k N ,p = (cid:0) k f k pL p + k g f k pL p (cid:1) /p . We call g f the minimal p -upper gradient . Similarly, we can define f ∈ N ,ploc ( X ) if f η ∈ N ,p whenever η ∈ LIP b ( X ). In these cases we also can define a minimal p -upper gradient g f , so that ηg f ∈ L p ( X ) for every η ∈ LIP b ( X ). In other words, g f ∈ L ploc ( X ).We denote by N ,p ( X ; R N ) ≃ N ,p ( X ) N the space of functions ϕ : X → R N so that each com-ponent is in N ,p . Similarly, we define LIP b ( X ; R N ) ≃ LIP b ( X ) N .Another notion of Sobolev space can be defined using q -test plans and we denote it W ,p ( X ),with | Df | p denoting the minimal gradient of f ∈ W ,p ( X ). Namely, a function f ∈ L p ( µ ) belongsto the Sobolev space W ,p ( X ) if there exists g ∈ L p ( µ ) such that Z | f ( γ ) − f ( γ ) | d η ≤ Z Z g ( γ t ) | γ ′ t | d t d η for every q -test plan η on X . The space has a norm k f k W ,p = (cid:0) k f k pL p + inf g k g k pL p (cid:1) /p , where theinfimum is over all such functions g . We refer to [19] for details.Note that any representative of an element of W ,p ( X ) still belongs to W ,p ( X ), whilst a represen-tative of an element in N ,p ( X ) belongs to N ,p ( X ) if and only if they agree outside a p -exceptionalset. The next theorem says that up to this ambiguity of representatives, the two approaches pro-duce the same object. The measurability conclusion is also a corollary of [22]. We refer to [2] for aproof. Theorem 2.5.
Let p ∈ (1 , ∞ ) . If f ∈ N ,p ( X ) , then f ∈ W ,p ( X ) and g f = | Df | p µ -a.e.Conversely, if f ∈ W ,p ( X ) , then f has a Borel representative ¯ f ∈ N ,p ( X ) with g ¯ f = | Df | p µ -a.e. Curvewise (almost) optimality of minimal upper gradients
Upper gradients with respect to plans.
Given a plan η , we can speak of a gradient alongits curves. Definition 3.1. If η is a q -plan, and f ∈ N ,p then a Borel function g is an η -upper gradient iffor η -almost every γ , g is an upper gradient of f along γ . The following lemma gives a notion of a minimal η -upper gradient and shows how to computeit by using derivatives along curves. Lemma 3.2.
Suppose g f is a minimal upper gradient, and η is any q -plan and d π = d η | γ ′ t | d t withdisintegration π x , then (1) g η = || ( f ◦ γ ) ′ t /γ ′ t || L ∞ ( π x ) is a η -upper gradient. (2) g η ≤ g for any other η -upper gradient for almost every x ∈ X . (3) g η ≤ g f for almost every x ∈ X . (4) Suppose η ′ is another q -plan, and η ≪ η ′ , then g η ≤ g η ′ .Proof. Let g f be the minimal p -upper gradient for f . By Lemma A.2 there is a Borel familyΓ ⊂ C ( I ; X ), so that f is absolutely continuous on each curve γ Γ c with upper gradient g f and so that η (Γ ) = 0. By Corollary A.3 and Lemma A.1 there is a set N ⊂ C ( I ; X ) × I so thatfor π ( N ) = 0, and for each ( γ, t ) N both ( f ◦ γ ) ′ ( t ) and | γ ′ t | are defined and measurable. Let M = Γ × I ∪ N , we get π ( M ) = 0. For each curve γ Γ the function f is absolutely continuouswith upper gradient ( f ◦ γ ) ′ t | γ ′ t | . Since g η ( γ t ) ≥ ( f ◦ γ ) ′ t | γ ′ t | for π -almost every ( γ, t ) ∈ M , we have that g η is an η upper gradient.If g is any other Borel η -upper gradient, then the set of ( γ, t ) ∈ Diff( f ) \ M with ( f ◦ γ ) ′ t / | γ ′ t | >g ( γ t ) must have null measure, and thus the claim follows by Fubini and the definition in (1).The function g f is an upper gradient for f on curves in Γ c , and thus the claim follows again fromcurvewise absolute continuity and by showing that the set of ( γ, t ) with ( f ◦ γ ) ′ t / | γ ′ t | > g f ( γ t ) musthave null π -measure. The final claim follows since g η ′ must be a η -upper gradient for f . (cid:3) Proof of Theorem 1.1.
In this subsection we prove Theorem 1.1. The idea is that for each q -plan η we can associate a gradient ”along” the curves of such a plan. Each such gradient must beless than the minimal upper gradient, and thus the task is to show that by varying over differentplans η we can obtain the minimal upper gradient through maximization. In order to show equalityof the result of this maximization, we argue by contradiction, that if it were not a minimal uppergradient, then we could witness this by a given plan. This is the core of the following result.It should be compared to [3, Sections 9–11], where a similar analysis is done, but with differentterminology and only for p >
1. In the following statement we will need to refer to end points ofcurves, and thus choose the domain of curves as I = [0 , Lemma 3.3.
Let p ∈ [1 , ∞ ) , and q be its dual exponent. Let f ∈ N ,p ( X ) . Suppose g is anynon-negative Borel function so that A = { g < g f } has positive measure, then there exists a q -plan η , so that for η -almost every curve γ : [0 , → X we have (3.1) | f ( γ ) − f ( γ ) | > Z γ g d s. Proof.
By by Vitali-Caratheodory we may find a lower semicontinuous ˜ g ≥ g which is integrable andso that ˜ A = { ˜ g < g f } has positive measure. We will suppress the tildes below to simplify notationand thus only consider the case of g lower semicontinuous. Since g < g f on a positive measuresubset, g cannot be a minimal upper gradient, and thus there must exist a family Γ ⊂ C ( I ; X ) of URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 13 curves with Mod p (Γ) >
0, so that Estimate (3.1) holds for each γ ∈ Γ. Modulus is invariant underreparametrization of curves and so we may only the subset of those γ ∈ Γ which are Lipschitz. Wewant to find a plan supported on Γ. However, the issue with this is that since p = 1 is allowed thefamily Γ may not be compact, the duality of modulus and q -plans may fail. So, we seek to “cover”Γ, up to a null modulus family by compact families. This covering is done in an iterative way.Fix an R so that the modulus of Γ R of those curves in Γ, which are contained in a ball B ( x , R )for some fixed x ∈ X , is positive. Since f is measurable and X is complete and separable,Egorov’s theorem implies the existence of an increasing sequence of compact sets K n satisfying µ ( B ( x, R ) \ S K n ) = 0 for which f | K n is continuous for each n . Define µ ( B ( x, R ) \ K n ) = ǫ n . Bypassing to a sub-sequence of n , we may assume that P n √ ǫ n < γ ∈ Γ R so that f is absolutely continuous on γ , and that H ( γ \ ( S ∞ n =1 K n )) = 0. This holds for Mod p -almost every curve, since f ∈ N ,p and since p -almostevery curve spends measure zero in the null set X \ S ∞ n =1 K n . Thus, Mod p (Γ) > m be those curves γ : I → X , which are m -Lipschitz, so that Len( γ ) ≤ m | b − a | , diam( γ ) ≥ m , γ , γ ∈ K m and Estimate (3.1) holds. We will show that every γ ∈ Γ containsa subcurve, up to reparametrization, in S ∞ m =1 Γ m . From this, and Lemma [10, Lemma 1.34], itfollows that Mod p ( S ∞ m =1 Γ m ) >
0, and thus there is some
M > p (Γ M ) >
0. It is easyto show that Γ m is a closed family of curves in C ( I ; X ) with respect to uniform convergence, since g is taken to be lower semicontinuous (see e.g. [33, Proposition 4]).To obtain the previous fact, consider a non-constant curve γ ∈ Γ. We have | f ( γ ) − f ( γ ) | > Z γ g d s. We may also parametrize γ by constant speed as the claim is invariant under reparametrizations.By parametrization with unit speed, we have that | I \ S ∞ n =1 γ − ( K n ) | = 0 and that f ◦ γ iscontinuous. Since R γ g d s < ∞ , and f ◦ γ is continuous, we can find (for all n ≥ N for some N ∈ N )sequences a n , b n ∈ [0 ,
1] so that lim n →∞ a n = a , γ a n ∈ K n , γ b n ∈ K n and lim n →∞ b n = b . Then, forsufficiently large n | f ( γ b n ) − f ( γ a n ) | > Z γ | [ an,bn ] g d s. For n large enough we also have Len( γ [ a n ,b n ] ) ≤ n | b − a | , diam( γ [ a n ,b n ] )) ≥ n . Since the curves areparametrized by unit speed, they are then n -Lipschitz. So γ ′ = γ [ a n ,b n ] is, up to a reparametrization,in Γ n for n large enough, and the claim follows.Fix M > p (Γ M ) >
0. Next, choose δ < min(Mod p (Γ M ) , δ n = ǫ / pn .Choose N so that P ∞ n = N √ ǫ n < δ p / Mt be the family of curves γ ∈ Γ M so that R γ X \ K n d s ≤ δδ n for each n ≥ N . Since (cid:16)P n ≥ N (cid:16) X \ Kn δδ n (cid:17) p (cid:17) /p is a function admissible for Γ M \ Γ Mt , we haveMod p (Γ M \ Γ Mt ) ≤ X n ≥ N ǫ n δ p δ pn < δ/ . Thus, by sub-additivity of modulus, see e.g. [24, Theorem 1], Mod p (Γ Mt ) ≥ Mod p (Γ M ) − Mod p (Γ M \ Γ Mt ) > δ/
2. By Lemma 3.4, since Γ M is closed, the family Γ Mt ⊂ Γ M is a compactfamily of curves in a complete space. Then, by Proposition 2.3 there exists a q -plan η supportedon Γ Mt . Each curve γ ∈ Γ Mt satisfies Inequality (3.1), and thus the claim follows. (cid:3) For the following proof, recall that if
A, B ⊂ X , then d ( A, B ) := inf a ∈ A inf b ∈ B d ( a, b ), and N ǫ ( A ) := ∪ a ∈ A B ( a, ǫ ) for ǫ > Lemma 3.4.
Suppose that K n are a compact sets, η n > constants with lim n →∞ η n = 0 , L > and let Γ ⊂ C ( I ; X ) be a closed family of curves in a complete space X . Let Γ t be the family ofcurves γ ∈ Γ for which Len( γ ) ≤ L , diam( γ ) ≥ L and which are L -Lipschitz with R γ X \ K n d s ≤ η n for each n ∈ N . Then Γ t is compact.Proof. Let I = [ a, b ]. Since Γ and Γ t are closed, it suffices to show pre-compactness.Let γ ∈ Γ t . We may suppose that η n < L by restricting to large enough n . Then, we have foreach n Z γ K n d s = Z γ s − Z γ X \ K n d s ≥ diam( γ ) − η n > L − η n . Thus γ ∩ K n = ∅ . Moreover, if t ∈ I , and d ( γ t , K n ) = s , then there will be a sub-segment oflength at least min( s, diam( γ k ) /
2) in X \ K n . This gives min( s, diam( γ ) / ≤ η n < L . This isonly possible if s ≤ η n , since diam( γ ) / ≥ L . Indeed, we have d ( γ, K n ) ≤ η n .To run the usual proof of Arzel`a-Ascoli, since we have equicontinuity with the Lipschitz bound,we only need to show that for each t ∈ I the set A t = { γ t : γ ∈ Γ t } is pre-compact. However,since X is complete, it suffices to show that A t is totally bounded. Fix ǫ >
0. We concludedthat d ( γ, K n ) ≤ η n for all n ∈ N . Thus, we have for some large n that η n ≤ ǫ/ A t ⊂ N η n ( K n ) ⊂ N ǫ/ ( K n ). Since K n is compact, it is totally bounded, and the claim follows bycovering K n by finitely many ǫ/ ǫ > (cid:3) Proof of Theorem 1.1.
Let Π q be the set of all q - plans, and for each η ∈ Π q with its disintegrationbeing given by π x define g η ( x ) = (cid:13)(cid:13)(cid:13)(cid:13) ( f ◦ γ ) ′ t | γ ′ t | (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( π x ) . Finally, define | D π f | = ess sup η ∈ Π ∞ g π ( x ) . Claim 1:
There is a q -plan ˜ η so that | D π f | = g ˜ η . By Lemma A.5, we can find a sequence η n so that g η n → | D π f | almost everywhere. Consider the measures d π n := | γ ′ t | d η n d t on AC ( I ; X ) × I . Set a n = 1 + η n ( C ( I ; X )) + || d η dµ || L q + π n ( AC ( I ; X ) × I ), where η is the barycenter of η , which is absolutely URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 15 continuous with respect to µ . Let ˜ η = P ∞ n =1 a − n − n η n . This will be a plan with g ˜ η ≥ g η n foreach n by Lemma 3.3. For µ -almost every x , we have g ˜ η ≥ | D π f | . Then, by Lemma 3.3 we have k ( f ◦ γ ) ′ t | γ ′ t | k L ∞ ( π x ) = | D π f | , as stated. Claim 2:
We have | D π f | = g f almost everywhere.Since g f is an p -weak upper gradient, Lemma 3.3 gives | D π f | ≤ g f . Suppose for the sake ofcontradiction then that | D π f | < g f on a positive measure subset. Then, by Lemma 3.3, thereexists a plan η ′ so that | f ( γ ) − f ( γ ) | > Z γ | D π f | d s for η ′ -almost every γ .However, by the definition of a plan upper gradient, we have for η ′ almost every curve that | f ( γ ) − f ( γ ) | ≤ Z γ g η ′ d s. Now, as g η ′ ≤ | D π f | almost everywhere and as η ′ is a q -plan, we have for η ′ -almost every curve γ that Z γ g η ′ d s ≤ Z γ | D π f | d s, which contradicts the above inequalities.Finally, since | D π f | = g ˜ η = g f , then we must have µ | D ≪ ˜ η . Indeed, otherwise there wouldbe a non-null Borel set E ⊂ D for which µ ( E ) > η ( E ) = 0. However, then g ˜ η | E = 0,contradicting the equality µ -almost everywhere. (cid:3) We now prove Corollary 1.3.
Proof.
Let f ∈ N ,p and consider the plan η ′ obtained from Theorem 1.1. Let η ′′ = r ∗ ( η ′ ), where r : C ( I ; X ) → C ( I ; X ) is the reversal-map which reverses the orientation of every path. Define η = η ′′ + η ′ . Fix ǫ >
0, and define B = { ( γ, t ) ∈ Diff( f ) : g f ( x ) ≥ ( f ◦ γ ) ′ t | γ ′ t | ≥ (1 − ǫ ) g f } . Since (cid:13)(cid:13)(cid:13) ( f ◦ γ ) ′ t | γ ′ t | (cid:13)(cid:13)(cid:13) L ∞ ( π ′ x ) = g f ( x ) for µ -almost every x ∈ D , where π ′ x is the disintegration for η ′ , we have π x ( B ) > µ -almost every x ∈ D where π x is the disintegration corresponding to η . Note that,we can remove the absolute values from the supremum norm since for each path γ in the supportof η ′ we include also its reversal, and r preserves η . (cid:3) Alternative curvewise characterizations of upper gradients when p > . In thissubsection we assume that p, q ∈ (1 , ∞ ) satisfy 1 /p + 1 /q = 1 and prove a variant Theorem 1.1using test plans representing gradients, introduced by Gigli.Given f ∈ N ,p ( X ), a q -test plan η represents g f , if f ◦ e t − f ◦ e t e E /qt → g f ◦ e and e E /pt → g f ◦ e in L p ( η ) , where e E t ( γ ) = 1 t Z t | γ ′ s | q d s, γ ∈ AC ( I ; X ) , e E t ( γ ) = + ∞ otherwise.A test plan η representing the gradient of a Sobolev map f ∈ N ,p ( X ) is concentrated on “gradientcurves” of f in an asymptotic and integrated sense. We refer to [25,37] for discussion of the definitionwe are using here. The following result of Gigli states that Sobolev functions always possess testplans representing their gradient. In the statement, P q ( X ) denotes probability measures ν on X with R d ( x , x ) q d ν ( x ) < ∞ for some and thus any x ∈ X . Theorem 3.5 ( [25], Theorem 3.14) . If f ∈ N ,p ( X ) and ν ∈ P q ( X ) satisfies ν ≤ Cµ for some C > , there exists a q -test plan η representing g f , with e ∗ η = ν . We now state the main result of this subsection.
Theorem 3.6.
Let f ∈ N ,p ( X ) , and g f be a Borel representative of the minimal p -weak uppergradient of f , with D := { g f > } of positive µ -measure. Let η be a q -test plan representing g f with µ | D ≪ e ∗ η ≪ µ | D .For every ε > there exists a Borel set B ⊂ Diff( f ) such that d π := χ B | γ ′ t | d t d η is a positive(finite) measure with µ | D ≪ e ∗ π ≪ µ | D , whose disintegration { π x } with respect to e satisfies (1 − ε ) g f ( x ) ≤ ( f ◦ γ ) ′ t | γ ′ t | ≤ g f ( x ) and (1 − ε ) g f ( x ) p/q ≤ | γ ′ t | ≤ (1 + ε ) g f ( x ) p/q π x − a.e. ( γ, t ) for µ -almost every x ∈ D . For the proof, we present the following three elementary lemmas. Denote D t ( γ ) = f ( γ t ) − f ( γ ) t , and G t ( γ ) = 1 t Z t g f ( γ s ) p d s, γ ∈ AC ( I ; X ) , and + ∞ otherwise. The following observation is essentially made in [37, Lemma 1.19] (we areusing different notation for our purposes). See Lemma A.1(3) for the Borel measurability of thefunctionals in the claim. Lemma 3.7.
Suppose f ∈ N ,p ( X ) and suppose η is a q -test plan representing g f . Then D t , G t , e E t → g pf ◦ e in L ( η ) . Proof.
Since e E /pt → g pf ◦ e in L p ( η ), it follows that e E t → g pf ◦ e in L ( η ). The convergence D t → g pf ◦ e is proven in [37, Lemma 1.19], while G t → g pf ◦ e in L ( η ) follows from [25, Proposition2.11]. (cid:3) Lemma 3.8.
For every ε > there exists δ > with the following property: if a, b > and a p p + b q q ≤ ab − δ , then (cid:12)(cid:12)(cid:12) a p/q b − (cid:12)(cid:12)(cid:12) < ε .Proof. The function h : (0 , ∞ ) → (0 , ∞ ), given by h ( t ) = p t + q t − q/p , has a global minimum at t = 1, with h (1) = 1. Thus h | (0 , and h | [1 , ∞ ) have continuous inverses and it follows that for every ε > δ > | − h ( t ) | < δ then | − t | < ε (expressing the fact that both URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 17 inverses are continuous at 1). The claim follows from this by noting that if a p p + b q q ≤ ab − δ then0 ≤ h ( t ) − < δ where t := a p/q /b . (cid:3) Lemma 3.9.
Let h ≤ g be two integrable functions on I with lim inf n →∞ T n Z T n g d s =: A > and lim n →∞ T n Z T n [ g − h ]d s = 0 . Then for every ε > and n , the set { (1 − ε ) g < h } ∩ [0 , T n ] has positive L -measure.Proof. For large enough n we have that 0 < A/ < T n Z T n g d s and 0 ≤ T n Z T n [ g − h ]d s < εA/ n for which 1 T n Z T n [ g − h ]d s < εT n Z T n g d s , for each n > n . It followsthat Z T n [(1 − ε ) g − h ]d s < n > n , and the claim follows from this. (cid:3) We will also need the following technical result, compare Lemma 3.7.
Lemma 3.10.
Let E ⊂ X be a Borel set, t > , and let D E,t ( γ ) := 1 t Z t χ E ( γ s )( f ◦ γ ) ′ s d s, γ ∈ Γ( f ) . Then D E,t → ( χ E g pf ) ◦ e in L ( η ) .Proof. Denote F t ( γ ) := 1 t Z t g f ( γ s ) | γ ′ s | d s. Since D t ≤ F t ≤ p G t + q e E t η -almost everywhere, Lemma 3.7 implies that F t → g pf ◦ e and thus( χ E ◦ e ) F t → ( χ E g pf ) ◦ e in L ( η ). We show that ( χ E ◦ e ) F t − D E,t → L ( η ).For η -almost every γ we have that | χ E ( γ ) F t ( γ ) − D E,t ( γ ) | = (cid:12)(cid:12)(cid:12)(cid:12) t Z t [ χ E ( γ ) g f ( γ s ) | γ ′ s | − χ E ( γ s )( f ◦ γ ) ′ s ]d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ t Z t (cid:18) | χ E g f ( γ s ) − χ E g f ( γ ) || γ ′ s | + χ E ( γ ) | g f ( γ s ) − g f ( γ ) || γ ′ s | + χ E ( γ s )[ g f ( γ s ) | γ ′ s | − ( f ◦ γ ) ′ s ] (cid:19) d s ≤ "(cid:18) t Z t | χ E g f ( γ s ) − χ E g f ( γ ) | p d s (cid:19) /p + (cid:18) t Z t | g f ( γ s ) − g f ( γ ) | p d s (cid:19) /p t Z t | γ ′ s | q d s (cid:19) /q + F t ( γ ) − D t ( γ ) . This estimate, together with the H¨older inequality and Lemma 3.7, yieldslim sup t → Z | ( χ E ◦ e ) F t − D E,t | d η ≤ lim sup t → "(cid:18)Z t Z t | g f ( γ s ) − g f ( γ ) | p d s d η (cid:19) /p + (cid:18)Z t Z t | χ E g f ( γ s ) − χ E g f ( γ ) | p d s d η (cid:19) /p × (cid:18)Z g pf ◦ e d η (cid:19) /q = lim sup t → "(cid:18) t Z t k g f ◦ e s − g f ◦ e k pL p ( η ) d s (cid:19) /p + (cid:18) t Z t k χ E g f ◦ e s − χ E g f ◦ e k pL p ( η ) d s (cid:19) /p × (cid:18)Z g pf ◦ e d η (cid:19) /q . Since s h ◦ e s is continuous in L p ( η ) whenever h ∈ L p ( µ ) (cf. [27, Proposition 2.1.4]) all termsabove tend to zero, proving the claimed convergence. (cid:3) Proof of Theorem 3.6.
Let C := N c , where N is as in Corollary A.3. The function A ( γ, t ) = 1 p g f ( γ t ) p + 1 q | γ ′ t | q , ( γ, t ) ∈ C, A ( γ, t ) = + ∞ , ( γ, t ) / ∈ C is Borel. Let η represent g f and satisfy µ | D ≪ e ∗ η ≪ µ | D . Fix ε >
0, let δ > δ = min { ε, δ } . We define the Borel function H ( γ, t ) = (1 − δ ) A ( γ, t ) − ( f ◦ γ ) ′ t , ( γ, t ) ∈ C, H = + ∞ otherwise , cf. Corollary A.3. The set B := { H ≤ } is Borel and, for ( γ, t ) ∈ C , we have( f ◦ γ ) ′ t ≤ g f ( γ t ) | γ ′ t | ≤ A ( γ, t ) . (3.2)Note that H ( γ, t ) ≤ − ε ) g f ( γ t ) | γ ′ t | ≤ ( f ◦ γ ) ′ t and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − g f ( γ t ) p/q | γ ′ t | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε, (3.3)cf. (3.2) and Lemma 3.8. Once we show that d π := χ B | γ ′ t | d t d η satisfies µ | D ≪ e ∗ π ≪ µ | D , it follows from (3.2) and (3.3) that π ′ := π / π ( C ( I ; X ) × I ) ∈ P ( C ( I ; X ) × I ) satisfies(1 − ε ) g f ( γ t ) | γ ′ t | ≤ ( f ◦ γ ) ′ t ≤ g f ( γ t ) and g ( γ t ) p/q ε ≤ | γ ′ t | ≤ g ( γ t ) p/q − ε for π ′ -almost every ( γ, t ), which readily implies the inequalities in the theorem.To prove e ∗ π ≪ µ | D observe that (3.3) implies χ B | γ ′ t | d t d η ≤ (1 + ε ) g ( γ t ) p/q d t d η and thus Z Z χ B ( γ, t ) χ E ( γ t ) | γ ′ t | d t d η ≤ (1 + ε ) Z Z X χ E g p/qf e t ∗ (d η )d t ≤ C Z E g p/qf d µ URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 19 for any Borel set E ⊂ X .It remains to prove that µ | D ≪ e ∗ π . Let E ⊂ D be a Borel set with µ ( E ) >
0. Then e ∗ η ( E ) = η ( { γ : γ ∈ E } ) >
0. Since0 ≤ p Z t χ E ( γ s ) A ( γ, s )d s − D E,t ( γ ) ≤ p G t ( γ ) + 1 q e E t ( γ ) − D t ( γ ) t → −→ ,D E,t t → −→ χ E g pf ◦ e in L ( η ), cf. Lemmas 3.7 and 3.10 respectively, there exist a sequence T n → η -almost every γ ∈ e − ( E ) the functions h γ ( s ) := χ E ( γ s )( f ◦ γ ) ′ s , g γ ( s ) := χ E ( γ s ) A ( γ, s )satisfy the hypotheses of Lemma 3.9. It follows that for η -almost every γ ∈ e − ( E ) the sets I nγ := { s ∈ [0 , T n ] : (1 − δ ) g γ ( s ) < h γ ( s ) } = { s ∈ [0 , t n ] : γ s ∈ E, H ( γ, s ) ≤ } have positive measure for all n . Notice that for η -almost every γ , if s ∈ I nγ then γ s ∈ E and | γ ′ s | > , g f ( γ s ) > < ( f ◦ γ ) ′ s ≤ g f ( γ t ) | γ ′ s | ). Consequently Z χ B ( γ, s ) χ E ( γ s ) | γ ′ s | d s ≥ Z I nγ | γ ′ s | d s > η -almost every γ ∈ e − ( E ) which in turn implies that e ∗ π ( E ) >
0. Since E ⊂ D is an arbitraryBorel set with positive µ -measure, this completes the proof. (cid:3) Charts and differentials
Notational remarks.
In what follows, define for any set U ⊂ X the set of curves whichspend positive length in U : Γ + U = { γ ∈ AC ( I ; X ) : Z γ χ U d s > } . Having positive length in U is more restrictive than assuming that γ − ( U ) has positive measure.We will also discuss p -weak differentials and co-vector fields of the form d f : U → ( R N ) ∗ or ξ : U → ( R N ) ∗ for measurable subsets U ⊂ X . The value of such a map at x ∈ U is denotedd x f, ξ x , respectively.4.2. Canonical minimal gradients.
Let p ≥ N ≥ ϕ ∈ N ,ploc ( X ; R N ) ≃ N ,ploc ( X ) N , with the convention N ,ploc ( X ; R N ) = N ,ploc ( X ) N = { } when N = 0. Our aim is to construct a “canonical” representative of the minimal weak upper gradients | D ( ξ ◦ ϕ ) | p of the functions ξ ◦ ϕ . We will use a plan to represent it. Lemma 4.1.
There exists a q -plan η and a Borel set D with µ | D ≪ η such that (4.1) Φ ξ ( x ) := χ D ( x ) (cid:13)(cid:13)(cid:13)(cid:13) ξ (( ϕ ◦ γ ) ′ t ) | γ ′ t | (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( π x )0 SYLVESTER ERIKSSON-BIQUE AND ELEFTERIOS SOULTANIS is a representative of | D ( ξ ◦ ϕ ) | p for every ξ ∈ ( R N ) ∗ . Here { π x } is the disintegration of d π := | γ ′ t | d η d t with respect to the evaluation map e .Proof. Let { ξ , ξ , . . . } ⊂ ( R N ) ∗ be a countable dense set and, for each n ∈ N , choose Borel repre-sentatives ρ n of | D ( ξ n ◦ ϕ ) | p and denote D n := { ρ n > } . By Theorem 1.1 and the Borel regularityof µ , for each n ∈ N there exists a q -plan η n and a Borel set B n ⊂ D n with µ ( D n \ B n ) = 0 suchthat the disintegration { π nx } of d π n := | γ ′ t | d η n d t satisfies (cid:13)(cid:13)(cid:13)(cid:13) ξ n (( ϕ ◦ γ ) ′ t ) | γ ′ t | (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( π nx ) = ρ n ( x )for every x ∈ B ξ .Define D := S n ∈ N B n and η = P n − n a − n η n , where a n = 1 + η n ( C ( I ; X )) + || d η n dµ || L q + π n ( AC ( I ; X ) × I ). Then µ | D ≪ η . Define Φ ξ ( x ) as in Equation (4.1). By Lemma 3.2 wehave that ρ n = Φ ξ n µ -a.e. on X and thus the claim holds for every ξ n ∈ A .We prove the claim in the statement for arbitrary ξ ∈ ( R N ) ∗ . Let ( ξ n l ) l ⊂ A be a sequence with | ξ n l − ξ | < − l and denote by ϕ , . . . , ϕ N ∈ N ,p ( X ) the component functions of ϕ . Since || D ( ξ n l ◦ ϕ ) | p − | D ( ξ ◦ ϕ ) | p | ≤ | D (( ξ n l − ξ ) ◦ ϕ ) | p ≤ | ξ n l − ξ | N X k | Dϕ k | p µ -a.e., we have that | D ( ξ ◦ ϕ ) | p = lim l →∞ Φ ξ nl µ -a.e. on X . In particular, | D ( ξ ◦ ϕ ) | p = 0 µ -a.e. on X \ D . On the other hand, for p -a.e. curve γ , we have that | ξ n l (( ϕ ◦ γ ) ′ t ) − ξ (( ϕ ◦ γ ) ′ t ) | ≤ | ξ n l − ξ | N X k | Dϕ k | p ( γ t ) | γ ′ t | a.e. t. Since η is a q -plan with µ | D ≪ η this implies thatlim sup l →∞ (cid:12)(cid:12)(cid:12)(cid:12) ξ n l (( ϕ ◦ γ ) ′ t ) | γ ′ t | − ξ (( ϕ ◦ γ ) ′ t ) | γ ′ t | (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim sup l →∞ | ξ n l − ξ | N X k | Dϕ k | p ( x ) = 0 π x − a.e. ( γ, t )for µ -a.e. x ∈ D . Thus Φ ξ ( x ) = lim l →∞ Φ ξ nl ( x ) for µ -a.e. x ∈ D . Since Φ ξ = 0 = | D ( ξ ◦ ϕ ) | p µ -a.e.on X \ D , the proof is completed. (cid:3) In the next two lemmas we collect the properties of the Borel function constructed above.
Lemma 4.2.
The map
Φ : ( R N ) ∗ × X → R given by (4.1) is Borel and satisfies the following. (1) For every ξ ∈ ( R N ) ∗ , Φ ξ := Φ( ξ, · ) is a representative of | D ( ξ ◦ ϕ ) | p , and (2) for every x ∈ X , Φ x := Φ( · , x ) is a seminorm in ( R N ) ∗ .Moreover, there exists a path family Γ B with Mod p (Γ B ) = 0 and for each γ ∈ AC ( I ; X ) \ Γ B anull-set E γ ⊂ I so that, for every ξ ∈ ( R N ) ∗ , we have (3) Φ ξ is an upper gradient of ξ ◦ ϕ along γ , and (4) | ( ξ ◦ ϕ ◦ γ ) ′ t | ≤ Φ ξ ( γ t ) | γ ′ t | for t / ∈ E γ . URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 21
Proof of Lemma 4.2.
Borel measurability follows from Lemmas A.1 and Corollary A.3, and prop-erty (1) follows from Lemma 4.1, while (2) follows from Equation (4.1).Fix a countable dense set A ⊂ ( R N ) ∗ and one ξ ∈ A . We have that Φ( ξ, x ) is a weak uppergradient for ξ ◦ ϕ , so there is family of curves Γ ξ so that ξ ◦ ϕ is absolutely continuous with uppergradient | D ( ξ ◦ ϕ ) | p on each γ ∈ Γ i , and so that Mod p (Γ \ Γ i ) = 0. Let Γ ′ = ∩ ξ ∈ A Γ ξ , whosecomplement Γ B = AC ( I ; X ) \ Γ ′ has null p -modulus.Since ξ ◦ ϕ has as upper gradient Φ ξ ( x ) on γ for each ξ ∈ A , then by considering a sequence ξ l → ξ for ξ A we obtain the same conclusion.Finally, fixing an absolutely continuous curve γ Γ B there is a full measure set F γ , where thecomponents of ϕ ◦ γ t are differentiable at t ∈ F γ . Both sides of (4) are continuous and defined in ξ on the set F γ . Since Φ ξ ( x ) is an upper gradient for ξ ◦ ϕ along γ , there is a full measure subset F γ ⊂ F γ , where the inequality holds for ξ ∈ A . Continuity then extends it for all ξ ∈ ( R N ) ∗ and t ∈ F γ and the claim follows by setting E γ = I \ F γ . (cid:3) Next, we collect some basic properties of the canonical minimal gradient. Let Φ be the mapgiven by (4.1).
Lemma 4.3.
Set I ( ϕ )( x ) := inf k ξ k ∗ =1 Φ x ( ξ ) for µ -a.e. x ∈ X . Then (1) I ( ϕ ) = ess inf k ξ k ∗ =1 | D ( ξ ◦ ϕ ) | p µ -a.e. in X ; (2) If U ⊂ X and ξ : U → ( R N ) ∗ are Borel, then Φ x ( ξ x ) = 0 µ -a.e. x ∈ U if and only if ξ γ t (( ϕ ◦ γ ) ′ t ) = 0 a.e. t ∈ γ − ( U ) , for p -a.e. absolutely continuous γ in X . (3) If ϕ is p -independent on U and f ∈ N ,p ( X ) , then the p -weak differential d f with respectto ( U, ϕ ) , if it exists, must be unique.Proof of Lemma 4.3. First, we show (1). For any ξ in the unit sphere of ( R N ) ∗ , we have Φ ξ ( x ) = | D ( ξ ◦ ϕ ) | p almost everywhere by Lemma 4.1. Taking an infimum on the left then givesinf k ζ k ∗ =1 Φ ζ ( x ) ≤ | D ( ξ ◦ ϕ ) | p , i.e. inf k ζ k ∗ =1 Φ ζ ( x ) ≤ ess inf k ξ k ∗ =1 | D ( ξ ◦ ϕ ) | p almost everywhere by the definition of an essentialinfimum, see Definition A.4.On the other hand, if ξ n , for n ∈ N , is a countably dense collection in the unit sphere of ( R N ) ∗ ,then we have Φ ξ n ( x ) = | D ( ξ n ◦ ϕ ) | p ≥ ess inf k ξ k ∗ =1 | D ( ξ ◦ ϕ ) | p almost everywhere. By intersectingthe sets where this holds for different ξ n and since the collection is countable, we have that thesehold simultaneously on a full-measure set. Specifically, inf n ∈ N Φ ξ n ( x ) ≥ ess inf k ξ k ∗ =1 | D ( ξ ◦ ϕ ) | p .By Lemma 4.2, we have that ξ → Φ ξ ( x ) is Lipschitz. Thus, almost everywhere,inf k ξ k ∗ =1 Φ( ξ, x ) = inf n ∈ N Φ( ξ n , x ) ≥ ess inf k ξ k ∗ =1 | D ( ξ ◦ ϕ ) | p , which gives the claim.Next fix ξ : U → ( R N ) ∗ as in (2). Assume first that Φ x ( ξ x ) = 0 for µ -a.e. x ∈ U . Set C = { x : Φ x ( ξ γ x ) = 0 } with µ ( C ) = 0. Since µ ( C ) = 0, we have Mod p (Γ + C ) = 0. Let Γ B be thefamily of curves from Lemma 4.2. We will show the claim for γ ∈ AC ( I ; X ) \ (Γ B ∪ Γ + C ). By Lemma4.2(4), we obtain a null set E γ so that for any ξ ∈ ( R N ) ∗ we have | ( ξ ◦ ϕ ◦ γ ) ′ t | ≤ Φ ξ ( γ t ) | γ ′ t | and t / ∈ E γ . Let F γ be the set of t E γ so that | γ ′ t | > γ t ( ξ γ t ) = 0. Since 0 = R γ C d s ≥ R F γ | γ ′ t | d t ,we have that the measure of F γ is null. Now, if t E γ ∪ F γ , then either | γ ′ t | = 0 (and the conditionis vacuously satisfied), or the claim follows from Φ γ t ( ξ γ t ) = 0.On the other hand, suppose that ξ γ t (( ϕ ◦ γ ) ′ t ) = 0 for a.e. t ∈ γ − ( U ) and p -a.e. absolutelycontinuous curve γ . Let η be the q -plan from Lemma 4.1 and { π x } the disintegration given there.The equality ξ γ t (( ϕ ◦ γ ) ′ t ) = 0 holds then for η -a.e. curve and a.e. t ∈ γ − ( U ), since η is a q -plan(recall Remark 2.2). Then for µ -a.e. x we have Φ ξ ( x ) = 0 or we have Φ ξ x ( x ) = (cid:13)(cid:13)(cid:13) ξ x (( ϕ ◦ γ ) ′ t ) | γ ′ t | (cid:13)(cid:13)(cid:13) L ∞ ( π x ) .In the latter case, since η is a q -plan, we have for µ -a.e. such x and π x -a.e. ( γ, t ) ∈ Diff( f ) ∩ e − ( x )that ξ x (( ϕ ◦ γ ) ′ t ) = 0. Thus, the claim follows together with the properties of disintegrations andCorollary A.3, since the essential supremum then vanishes.The final claim about uniqueness follows since, if d i f were two p -weak differentials for i = 1 , ξ x = d f − d f k d f − d k x, ∗ when d f = d f and otherwise ξ x = 0. We then getimmediately form the definition and the second part that Φ x ( ξ ) = 0, for µ -a.e. x ∈ U . This wouldcontradict independence. (cid:3) Charts.
The presentation here should be compared to [12, Section 4], and specifically to theproof of [12, Theorem 4.38], where similar arguments are employed. We first consider 0-dimensional p -weak charts. These correspond to regions of the space where no curve spends positive time. Proposition 4.4.
Suppose (
U, ϕ ) is a 0-dimensional p -weak chart. ThenMod p (Γ + U ) = 0 . (4.2)Conversely, if U ⊂ X is Borel and satisfies (4.2), then ( U,
0) is a 0-dimensional p -weak chart of X . Proof.
Since (
U, ϕ ) is a 0-dimensional p -weak chart, we have that | Df | p = 0 µ − a.e. in U (4.3)for every f ∈ LIP b ( X ). Let { x n } ⊂ X be a countable dense subset, and f n := max { − d ( x n , · ) , } .By [4, Thm 1.1.2] (see also its proof) and (4.3) we have that | γ ′ t | = sup n | ( f n ◦ γ ) ′ t | ≤ sup n | Df n | p ( γ t ) | γ ′ t | = 0 a.e t ∈ γ − ( U )for p -a.e. γ ∈ AC ( I ; X ). It follows that R γ χ U d s = 0 for p -a.e. γ ∈ AC ( I ; X ), proving (4.2).In the converse direction, (4.2) implies, for any f ∈ LIP b ( X ), that Z χ U ( γ t ) | ( f ◦ γ ) ′ t | d t ≤ LIP( f ) Z χ U ( γ t ) | γ ′ t | d t = 0for p -a.e γ ∈ AC ( I ; X ). Thus | ( f ◦ γ ) ′ t | = 0 for p -a.e. γ ∈ AC ( I ; X ) and a.e. t ∈ γ − ( U ). Then, byTheorem 1.1, together with measurability considerations from Corollary A.3, this gives | Df | p = 0 µ -a.e. on U for every f ∈ LIP b ( X ), showing that ( U,
0) is a 0-dimensional p -weak chart. (cid:3) For the remainder of this subsection we assume that N ≥ U, ϕ ) is an N -dimensionalchart of X . Denote by Φ the canonical minimal gradient of ϕ (see Lemma 4.1). URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 23
Lemma 4.5.
The function ξ Φ x ( ξ ) is a norm on ( R N ) ∗ for µ -a.e. x ∈ U . Moreover, for every f ∈ LIP( X ) there exists a p -weak differential d f . That is, a Borel measurable map d f : U → ( R N ) ∗ satisfying ( f ◦ γ ) ′ t = d γ t f (( ϕ ◦ γ ) ′ t ) a.e. t ∈ γ − ( U ) , for p -a.e. absolutely continuous curve γ in X . The map d f is uniquely determined a.e. in U andsatisfies | Df | p ( x ) = Φ x (d f ) µ -a.e. in U . Remark 4.6.
The equation in the statement is an equivalent formulation of the definition of the p -weak differential in Definition 1.6. Indeed, the latter follows by integration of the first, andconversely, the first follows by Lebesgue differentiation. Further, it would be enough to consideronly p-a.e. curve γ ∈ Γ + U . Indeed, if a curve γ does not spend positive length in the set U , then | γ ′ t | = 0 for a.e. t ∈ γ − ( U ) and both sides of the equation vanish. Proof.
First, consider f ∈ LIP b ( X ). Since Φ x is a norm if and only if I ( ϕ )( x ) >
0, Lemma 4.3(1)and (1.5) imply that Φ x is a norm for µ -a.e. x ∈ U .Next, let f ∈ LIP b ( X ) and consider the map ψ = ( ϕ, f ) : X → R N +1 . Let Ψ be the canonicalminimal gradient of ψ . Given ξ ∈ ( R N ) ∗ and a ∈ R , we use the notation( ξ, a ) ∈ ( R N +1 ) ∗ , v = ( v ′ , v N +1 ) ξ ( v ′ ) + av N +1 . For µ -a.e. x ∈ U , we have that Ψ x ( ξ,
0) = Φ x ( ξ ) and Ψ x (0 , a ) = | a || Df | p ( x ) for every ξ ∈ ( R N ) ∗ , a ∈ R (cf. Lemma 4.2(3) and (4)). Since ϕ is a chart, we have that I ( ψ ) = 0 almost everywhere.Thus, given that I ( ϕ ) >
0, ker Ψ x is a 1-dimensional subspace of ( R N +1 ) ∗ .Thus for µ -a.e. x ∈ U there exists a unique ξ := d x f ∈ ( R N ) ∗ such that Ψ x (d x f, −
1) = 0, andthe map x d x f is Borel, see e.g. [11, Lemma 6.7.1]. By Lemma 4.3(2), d f : U → ( R N ) ∗ satisfies0 = (d γ t f, − ψ ◦ γ ) ′ t ) = d γ t f (( ϕ ◦ γ ) ′ t ) − ( f ◦ γ ) ′ t a.e. t ∈ γ − ( U ) , for p -a.e. γ . Moreover, we have || Df | p ( x ) − Φ x (d x f ) | ≤ | Ψ x (0 , − − Ψ x (d x f, | ≤ Ψ x (d x f, −
1) = 0for µ -a.e. x ∈ U , completing the proof in the case f ∈ LIP b ( X ).The case of f ∈ LIP( X ) follows through localization. Indeed, let x ∈ X be arbitrary, and con-sider the functions η n ( x ) := min { max { n − d ( x , d ) , } , } for n ∈ N . Then, define f n = η n f so that f n | B ( x ,n − = f | B ( x ,n − . For each f n we can define a differential d f n , and d f n | B ( x , min( m,n ) − =d f m | B ( x , min( m,n ) − (a.e.) for each n, m ∈ N . Thus, we can define d f ( x ) = d f n ( x ) for x ∈ B ( x , n −
1) with only an ambiguity on a null set. It is easy to check that d f is a differential. (cid:3) Differential and pointwise norm.
Denote | · | x := Φ x and defineΓ p ( T ∗ U ) = { ξ : U → ( R N ) ∗ Borel : k ξ k Γ p ( T ∗ U ) < ∞} , k ξ k Γ p ( T ∗ U ) := (cid:18)Z U | ξ | px d µ (cid:19) /p , (with the usual identification of elements that agree µ -a.e.). Then (Γ p ( T ∗ U ) , k · k Γ p ( T ∗ U ) ) is anormed space. Observe that, if V j := U ∩ { I ( ϕ ) ≥ /j } , the sets U j := V j \ S i Suppose ( f n ) ⊂ LIP b ( X ) is a sequence such that f n → f in L p ( X ) and d f n → ξ in Γ p ( T ∗ U ) for some f ∈ N ,p ( X ) and ξ ∈ Γ p ( T ∗ U ) . Then ξ is the (uniquely defined) differential of f in U , and lim n →∞ Z U | D ( f n − f ) | pp d µ = 0 . In particular, Φ( ξ , · ) = | Df | p µ -a.e. in U .Proof. By Lemma 4.5 and Fuglede’s Theorem [24, Theorem 3(f)] (applied to the sequence of func-tions h n = χ U ( γ t ) | d γ t f n − ξ γ t | γ t and f n ) we can pass to a subsequence so thatlim n →∞ Z χ U ( γ t ) | ( f n ◦ γ ) ′ t − ξ γ t (( ϕ ◦ γ ) ′ t ) | d t ≤ lim n →∞ Z χ U ( γ t ) | d γ t f n − ξ γ t | γ t | γ ′ t | d t = 0 , lim n →∞ Z | f n ( γ t ) − f ( γ t ) || γ ′ t | d t = 0(4.5)for p -a.e. γ ∈ AC ( I ; X ). Fix a curve γ where (4.5) holds and f n ◦ γ , f ◦ γ are absolutely continuous.We may assume that γ is constant speed parametrized. By (4.5) f n ◦ γ → f ◦ γ in L ([0 , f n ◦ γ ) ′ → g in L ( γ − ( U )), where g ( t ) := χ U ( γ t ) ξ γ t (( ϕ ◦ γ ) ′ t ). It follows that( f ◦ γ ) ′ t = ξ γ t (( ϕ ◦ γ ) ′ t ) a.e. t ∈ γ − ( U ) . This shows that ξ is the differential of f , and uniqueness follows from Lemma 4.3(3). The identity(( f − f n ) ◦ γ ) ′ t = ( ξ γ t − d f n )(( ϕ ◦ γ ) ′ t ) a.e. t ∈ γ − ( U ) for p -a.e. γ ∈ AC ( I ; X ), together withLemma 3.2(3), implies that Φ x ( ξ − d f n ) ≤ | D ( f − f n ) | p for µ -a.e. x ∈ U . By the convergenced( f m − f n ) → ξ − d f n (as m → ∞ ) we have that | D ( f m − f n ) | p → m →∞ Φ x ( ξ − d f n ) in L p ( U ), andthus | D ( f − f n ) | p ≤ Φ x ( ξ − d f n ) µ -a.e. in U . Thus | D ( f − f n ) | p = Φ x ( ξ − d f n ) converges to zeroin L p ( U ). The equality Φ ξ = | Df | p follows, completing the proof. (cid:3) We say that a sequence ( ξ n ) n ⊂ Γ p ( T ∗ U ) is equi-integrable if the sequence {| ξ n | x } n ⊂ L p ( U ) isequi-integrable. Recall, that a collection of integrable functions F is called equi-integrable, if thereis a M so that R X | f | p dµ ≤ M for every f ∈ F and if for every ǫ > 0, there is an δ > ǫ , so that for any measurable set E with µ ( E ) ≤ δ , we have R Ω cǫ ∪ E | f | p d µ ≤ ǫ foreach f ∈ F . By the Dunford-Pettis Theorem a set of L functions is equi-integrable if and only ifit is sequentially compact, see for example [20, Theorem IV.8.9]. Remark 4.8. It follows from (4.4) that, if ( ξ n ) n ⊂ Γ p ( T ∗ U ) is equi-integrable, then there exists ξ ∈ Γ p ( T ∗ U ) such that ξ n ⇀ ξ weakly in Γ p ( T ∗ U ) up to a subsequence and, by Mazur’s lemma,that a convex combination of ξ n ’s converges to ξ in Γ p ( T ∗ U ). Indeed, the p > p = 1 case uses the Dunford-Pettis argument above. URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 25 Next, we show that any Sobolev function f ∈ N ,p has a uniquely defined differential with respectto a chart. Note, however, that here we still postulate the existence of charts. Proof of Theorem 1.7. The measurable norm |·| x is given by Lemma 4.5. Let f ∈ N ,p ( X ). Lemma4.3(3) implies that d f , if it exists, is a.e. uniquely determined on U . Let ( f n ) ⊂ LIP b ( X ) be suchthat f n → f and | Df n | p → | Df | p in L p ( µ ) as n → ∞ , which exists by [22, Theorem 1.1]. ByLemma 4.5, (d f n ) n ⊂ Γ p ( T ∗ U ) is equi-integrable. It follows that there exists ξ ∈ Γ p ( T ∗ U ) suchthat d f n ⇀ ξ weakly in L p ( T ∗ U ), cf. Remark 4.8. By Mazur’s lemma, a sequence ( g n ) ⊂ LIP b ( X )of convex combinations of the f n ’s converges to f in L p ( µ ) and d g n → ξ in Γ p ( T ∗ U ). By Lemma 4.7, ξ =: d f is the differential of f . The linearity of f d f follows from the uniqueness of differentials,Lemma 4.3(3). (cid:3) The proof above also yields the following corollary. Note that, while the claim initially holdsonly after passing to a subsequence, since the limit is unique, the convergence holds along the fullsequence. Corollary 4.9. Let ( U, ϕ ) be a p -weak chart of X . Suppose that f ∈ N ,p ( X ) and ( f n ) ⊂ LIP b ( X ) converges to f in energy, that is, f n → L p f and | Df n | p → L p | Df | p . Then we have that d f n ⇀ d f weakly in Γ p ( T ∗ U ) . Using Lemma 4.3 we prove that the differential satisfies natural rules of calculation. Proposition 4.10. Let ( U, ϕ ) be an N -dimensional p -weak chart of X , f, g ∈ N ,p ( X ), and F : X → Y a Lipschitz map into a metric measure space ( Y, d, ν ) with F ∗ µ ≤ Cν for some C > V, ψ ) is a p -weak chart with ϕ | U ∩ V = ψ | U ∩ V then the p -weak differentials of f withrespect to both charts agree µ -a.e. on U ∩ V .(2) If f, g ∈ L ∞ ( X ), then d( f g ) = f d g + g d f µ -a.e. on U .(3) Let ( V, ψ ) be an M -dimensional p -weak chart of Y with ν ( U ∩ F − ( V )) > 0. For µ -a.e. U ∩ F − ( V ) there exists a unique linear map D x F : R N → R M satisfying the following:if h ∈ N ,p ( Y ) and E is the set of y ∈ V where the differential d y h does not exist, then µ ( U ∩ F − ( E )) = 0 andd x ( h ◦ F ) = d F ( x ) h ◦ D x F µ − a.e. x ∈ U ∩ F − ( V \ E ) . Proof. Claim (1) follows from Lemma 4.3(2) and the fact that ( ϕ ◦ γ ) ′ t = ( ψ ◦ γ ) ′ t a.e. t ∈ γ − ( U ∩ V ),for p -a.e. γ ∈ AC ( I ; X ).To prove (2) note that, since we have ( f g ◦ γ ) ′ t = g ( γ t )( f ◦ γ ) ′ t + f ( γ t )( g ◦ γ ) ′ t a.e t for p -a.e. curve γ ∈ AC ( I ; X ), it follows from (1.6) thatd γ t ( f g )(( ϕ ◦ γ ) ′ t ) = g ( γ t )d f γ t (( ϕ ◦ γ ) ′ t ) + f ( γ t )d g γ t (( ϕ ◦ γ ) ′ t ) a.e. t ∈ γ − ( U )for p -a.e. γ ∈ AC ( I ; X ). By Lemma 4.3(2) and (3) the claimed equality holds.Finally, for (3), let G = ( G , . . . , G M ) = ψ ◦ F ∈ LIP( X ; R M ) and define the expression D x F :=(d x G , . . . , d x G M ) : R N → R M for µ -a.e. x ∈ U ∩ F − ( V ). We have that( ψ ◦ F ◦ γ ) ′ t = D γ t F (( ϕ ◦ γ ) ′ t ) a.e. t ∈ γ − ( U ) for p -a.e. γ ∈ AC ( I ; X ). Note that if h and E are as in the claim, then µ ( U ∩ F − ( E )) ≤ Cν ( E ) = 0.To show the claimed identity, let Γ ⊂ C ( I ; Y ) be a path family with Mod p Γ = 0 such that( h ◦ α ) ′ t = d α t h (( ψ ◦ α ) ′ t ) a.e. t ∈ α − ( V )for every absolutely continuous α / ∈ Γ , and set Γ = F − Γ := { γ ∈ C ( I ; X ) : F ◦ γ ∈ Γ } . SinceMod p Γ ≤ C LIP( F ) p Mod p (Γ ) = 0 it follows from the two identities above that( h ◦ F ◦ γ ) ′ t = d F ( γ t ) h (( ψ ◦ F ◦ γ ) ′ t ) = d F ( γ t ) h ( D γ t (( ϕ ◦ γ ) ′ t )) a.e. t ∈ γ − ( U ∩ F − ( V ))for p -a.e. γ ∈ AC ( I ; X ). Lemma 4.3(2) and (3) imply the claim. (cid:3) Dimension bound. In this section we give a geometric condition which guarantees that finitedimensional weak p -charts exist. This involves a bound on the size of p -independent Lipschitz maps.As a technical tool we need the notion of a decomposability bundle V ( ν ) of a Radon measure ν on R m , see [1]. We will not fully define this here, as we only need some of its properties. Firstly, letGr( m ) be the set of linear subspaces of R m equipped with a metric d ( V, V ′ ) defined as the Hausdorffdistance of V ∩ B (0 , 1) to V ′ ∩ B (0 , V is denoted dim( V ).The decomposability bundle is then a certain Borel measurable map R m → Gr( m ), which associatesto every x ∈ R m a subspace V ( ν ) x ∈ Gr( m ). In a sense, this bundle measures the directions inwhich a Lipschitz function must be differentiable in (at almost every point). We collect the mainproperties we need for this bundle and briefly cite where the proofs of these claims can be found. Theorem 4.11. Suppose that ν is a Radon measure on R m , then there exists a decomposabilitybundle V ( ν ) with the following property. (1) If dim( V ( ν ) x ) = m for ν -a.e. x ∈ R m , then ν ≪ λ . (2) There is a Lipschitz function f : R m → R so that for ν -a.e. x ∈ R m we have that thedirectional derivative of f does not exist in the direction v for any v V ( ν ) x . (3) If ν ′ ≪ ν , then V ( ν ′ ) x = V ( ν ) x for ν ′ -a.e. x ∈ R m .Proof. The first follows from [17, Theorem 1.14] when combined with [1, Theorem 1.1(i)]. Thesecond claim follows from [1, Theorem 1.1(ii)]. Note that the second claim is vacuous for those points x ∈ R m where the decomposability bundle has dimension m . The third claim is [1, Proposition2.9(i)]. (cid:3) The following lemma gives a modulus perspective to the decomposability bundle. Lemma 4.12. Assume N ≥ , ϕ : X → R N is Lipschitz, U ⊂ X is a Borel set of bounded measureand ν = ϕ ∗ ( µ | U ) . Then, for p -a.e. curve γ and almost every t ∈ γ − ( U ) we have that ( ϕ ◦ γ ) ′ t exists and ( ϕ ◦ γ ) ′ t ∈ V ( ν ) ϕ ( γ t ) .Proof. By part (ii) of Theorem 4.11, there is a Lipschitz function f : R N → R , so that for ν -almost every x ∈ R N and any v V ( ν ) x we have that the directional derivative D v ( f ) =lim h → f ( x + hv ) − f ( x ) h does not exist. Let A ⊂ R N be a full ν -measure Borel set so that this claimholds.Let B = ϕ − ( R N \ A ) ∩ U , which is µ -null. The family Γ + B has null p -modulus. We will showthat the claim holds for p -a.e. γ ∈ AC ( I ; X ) \ Γ + B . The derivatives ( ϕ ◦ γ ) ′ t and ( f ◦ ϕ ◦ γ ) ′ t existfor almost every t ∈ γ − ( U ). Also, for a.e. t ∈ I we can either take | γ ′ t | = 0 or γ t B and so URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 27 ( ϕ ◦ γ ) t A , since γ Γ + B . If | γ ′ t | = 0, then ( ϕ ◦ γ ) ′ t = 0 ∈ V ( ν ) ϕ ( γ t ) . In the other case, when γ t B , the function f does not have a directional derivative for v V ( ν ) ( ϕ ◦ γ ) t . The only way forboth ( ϕ ◦ γ ) ′ t and ( f ◦ ϕ ◦ γ ) ′ t to exist then is if ( ϕ ◦ γ ) ′ t ∈ V ( ν ) ϕ ( γ t ) which gives the claim. (cid:3) The following should be compared to [12, Lemma 4.37]. Proposition 4.13. Suppose ϕ ∈ LIP( X ; R N ) is p -independent on U . Then N ≤ dim H U . Proof. By restriction to a subset of the form U ∩ B ( x , R ), for x ∈ X, R > 0, of positive measure,it suffices to assume that U has finite measure. The claim is automatic, if dim H U = ∞ . Thus,assume that the Hausdorff dimension is finite.Set ν = ϕ ∗ ( µ | U ) and let V ( ν ) be the decomposability bundle of ν . If V ( ν ) x has dimension N foralmost every x with respect to ν , then ν ≪ λ by Theorem 4.11(1) and thus H N ( ϕ ( U )) > 0, since ν is concentrated on ϕ ( U ). Then N ≤ dim H ( ϕ ( U )) ≤ dim H ( U ).Suppose then to the contrary, that there exists a subset A ⊂ U with positive ν -measure where V ( ν ) x has dimension less than dim H ( U ) for each x ∈ A . We can take A to be Borel. Consider µ ′ = µ | ϕ − ( A ) , which has push-forward ν ′ = ν | A = ϕ ∗ ( µ ′ ). By the third part in Theorem 4.11 wehave that V ( ν ′ ) x = V ( ν ) x for ν ′ -a.e. x ∈ A . Further ϕ − ( A ) ⊂ U , so ϕ is still p -independent on ϕ − ( A ) = U ′ . Now, by considering U ′ instead of U and ν ′ instead of ν , we have that V ( ν ′ ) ϕ ( x ) has dimension less than N for ν ′ -almost every x ∈ U . In the following, we simplify notation bydropping the primes, and restricting to the positive measure subset U ′ so constructed.For ν -almost every x ∈ U , we have that V ( ν ) ϕ ( x ) is a strict subspace of R N , and thus thereare vectors perpendicular to these. Since x → V ( ν ) ϕ ( x ) is Borel, we can choose a Borel map x → ξ x ∈ ( R N ) ∗ so that ξ x is a unit vector that vanishes on V ( ν ) ϕ ( x ) for µ -a.e. x ∈ U (seee.g. [11, Theorem 6.9.1] which is an instance of a Borel selection theorem). Let ˜ U ⊂ U be thefull measure subset where these properties hold for every x ∈ ˜ U . Now, by Lemma 4.12 we havefor p -a.e. curve γ that ( ϕ ◦ γ ) ′ t ∈ V ( ν ) ϕ ( γ t ) for almost every t ∈ γ − ( U ). The set U \ ˜ U has nullmeasure, and thus Γ + U \ ˜ U has null modulus.Thus, for p -a.e. curve γ ∈ AC ( I ; X ) and a.e. t ∈ γ − ( U ) we can further assume γ t ∈ U or | γ ′ t | = 0. Therefore, ξ γ t (( ϕ ◦ γ ) ′ t ) = 0 for almost every t ∈ γ − ( U ) and such curves γ . By part (2)of Lemma 4.3, we have that I ( ϕ ) ≤ Φ x ( ξ x ) = 0 for µ -a.e. x ∈ U . This contradicts p -independenceand proves the claim. (cid:3) Sobolev charts. By definition, a p -weak chart is a Lipschitz map which has target of maximaldimension with respect to Lipschitz maps. The notions of p -independence and maximality howeverare well-defined for any Sobolev map, and in fact p -weak charts could be required to have Sobolev(instead of Lipschitz) regularity. Despite the apparent difference of the alternative definition, theexistence of maximal p -independent Sobolev maps also guarantees the existence of p -weak chart ofthe same dimension. This follows from the energy density of Lipschitz functions, see [22], togetherwith results of the previous subsection. Proposition 4.14. Suppose p ≥ 1, and ϕ ∈ N ,p ( X ; R N ) is p -independent and p -maximal in abounded Borel set U ⊂ X . For any ε > V ⊂ U with µ ( U \ V ) < ε , and a Lipschitzfunction ψ : X → R N such that ( V, ψ ) is an N -dimensional p -weak chart. Proof. For any V ⊂ U with µ ( V ) > 0, let n V be the supremum of numbers n so that there exists ψ ∈ LIP b ( X ; R n ) which is p -independent on a positive measure subset of V . By the maximality of N we have that n V ≤ N . Thus n V is attained for every such V and, by [34, Proposition 3.1], there isa partition of U up to a null-set by p -weak charts V i , i ∈ N of dimension ≤ N . By [22, Theorem 1.1],Corollary 4.9 (with a diagonal argument) and Mazur’s lemma we have that, for each component ϕ k ∈ N ,p ( X ) of ϕ , there exists a sequence ( ψ nk ) ⊂ LIP b ( X ) with | D ( ϕ k − ψ nk ) | p → L p ( V i ).Thus, | D ( ϕ k − ψ nk ) | p → L p ( U ). Here, we use that | D ( ϕ k − ψ nk ) | p ≤ | Dϕ k | p + | Dψ nk | p and the L p -convergence of the right hand side from [22].If Φ and Ψ n denote the canonical minimal gradients associated to ϕ and ψ n := ( ψ n , . . . , ψ nN ) wehave thatsup k ξ k ∗ =1 | Φ( ξ, · ) − Ψ n ( ξ, · ) | ≤ ess sup k ξ k ∗ =1 | D ( ξ ◦ ( ϕ − ψ n )) | p ≤ N X k =1 | D ( ϕ k − ψ nk ) | p µ − a.e. in U. It follows that lim n →∞ µ ( U \ { I ( ψ n ) > } ) = 0 , completing the proof, since ψ n is p -independent and maximal on the set { I ( ψ n ) > } . (cid:3) Another condition in this context is strong maximality: a map ϕ ∈ N ,p ( X ; R N ) is stronglymaximal in U ⊂ X if no positive measure subset V ⊂ U admits a p -independent Sobolev map intoa higher dimensional Euclidean space. This condition excludes not only Lipschitz, but also Sobolevfunctions into higher dimensional targets, and is thus a priori stronger than maximality. However, itfollows from Proposition 4.14 that a maximal p -independent Sobolev map is also strongly maximal.Conversely, if one has a Lipschitz chart, then the Lipschitz chart is also strongly maximal.4.7. p -weak charts in Poincar´e spaces. Recall that a metric measure space X = ( X, d, µ ) issaid to be a p -PI space if µ is doubling, and X supports a weak (1 , p )-Poincar´e inequality: thereexist constants C, σ > f ∈ L ( X ) with upper gradient g , we have − Z B | f − f B | d µ ≤ Cr (cid:18) − Z σB g p d µ (cid:19) /p for all balls B ⊂ X of radius r . Here h B = − R B h d µ = µ ( B ) R B h d µ for a ball B ⊂ X and h ∈ L ( B ).Cheeger’s celebrated result, from [12], states that a PI-space admits a Lipschitz differentiablestructure. We will return to this structure in Section 6.2, but here recall the constructions from [12,Section 4]. Cheeger’s paper does not employ the following terminology, but it simplifies and clarifiesour presentation.Given a Lipschitz map ϕ : X → R N and a positive measure subset U ⊂ X the pair ( U, ϕ ) iscalled a Cheeger chart if for every Lipschitz map f : X → R and a.e. x ∈ U there is a uniqueelement d C,x f ∈ ( R N ) ∗ satisfying(4.6) Lip(d C,x f ◦ ϕ − f )( x ) = 0 . URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 29 This equality is equivalent to Equation (1.4). Proof of Theorem 1.8. Let ( U, ϕ ) be a p -weak chart of dimension N and let f ∈ LIP( X ). Denoteby Φ the canonical minimal gradient of ( ϕ, f ) : X → R N +1 , cf. Lemma 4.1. Since X is a p -PIspace, it follows that Lip h = | Dh | p µ -a.e for any h ∈ LIP( X ), see [12, Theorem 6.1]. (In fact,the slightly easier comparability from [12, Lemma 4.35] suffices for the following.) Then, for any ξ ∈ ( R N ) ∗ and for µ -a.e. x ∈ U , we haveLip( ξ ◦ ϕ − f )( x ) = Φ x ( ξ, − , ξ ∈ ( R N ) ∗ . Arguing using in the proof of Lemmas 4.1 and 4.5 we obtain this equality, simultaneously, for a.e. x ∈ U and for any ξ ∈ A for a dense subset of A ⊂ ( R N ) ∗ . From this, and the continuity of bothsides in ξ , we obtain that for µ -a.e. x ∈ U , the equality holds simultaneously for all ξ ∈ ( R N ) ∗ .Since the p -weak differential d f is characterized by the property Φ x (d f, − 1) = 0 for µ -a.e. x ∈ U ,it follows that for µ -a.e. x ∈ U , d x f ∈ ( R N ) ∗ satisfies Equation (4.6). Thus ( U, ϕ ) is a Cheegerchart. The uniqueness follows from the equality in a similar way. (cid:3) Remark 4.15. The proof of Theorem 1.8 also yields the claim under the weaker assumptionLip f ≤ ω ( | Df | p ) for some collection of moduli of continuity ω (compare Theorem 1.10) since theequality Lip f = | Df | p follows from this by [32, Theorem 1.1].5. The p -weak differentiable structure The p -weak cotangent bundle. A measurable L ∞ -bundle T over X consists of a collection( { U i , V i,x } ) i ∈ I together with a collection ( { φ i,j,x } ) of transformations with a countable index set I ,where,(1) U i ⊂ X are Borel sets for each i ∈ I , and cover X up to a µ -null set;(2) for any i ∈ I and µ -a.e. x ∈ U i , V i,x = ( V i , | · | i,x ) is a finite dimensional normed space sothat x 7→ | v | i,x is Borel for any v ∈ V i ;(3) for any i, j ∈ I and µ -a.e. x ∈ U i ∩ U j , φ i,j,x : V i,x → V j,x is an isometric bijective linearmap satisfying the cocycle condition : for any i, j, k ∈ I and µ -a.e. x ∈ U i ∩ U j ∩ U k , we have φ j,k,x ◦ φ i,j,x = φ i,k,x .For each i ∈ I and µ -a.e. x ∈ U i , we denote T x the equivalence class of the normed vector space V i,x under identification by isometric isomorphisms. By (3), T x is well-defined for µ -a.e. x ∈ X .We now show that a p -weak differentiable structure A on X gives rise to a measurable bundle. Proposition 5.1. Let p ≥ 1, and let { ( U i , ϕ i ) } be an atlas of p -weak charts on X . The collection { ( U i , ( R N i ) ∗ , | · | i,x ) } forms a measurable bundle over X , the transformations given by the collection { D Φ i,j,x } constructed in Lemma 5.2.First, we construct the transformation maps. Lemma 5.2. Let ( U i , ϕ i ) be N i -dimensional p -weak charts on X , with corresponding differentials d i and norms | · | i,x , for i = 1 , . If µ ( U ∩ U ) > , then N = N := N and, for µ -a.e. x ∈ U ∩ U ,there exists a unique bijective isometric isomorphism D Φ , ,x : (( R N ) ∗ , | · | ,x ) → (( R N ) ∗ , | · | ,x ) such that d f = d f ◦ D Φ , ,x . Further D Φ , ,x satisfies the measurability constraint (2) for In the proof, we denote by ϕ i , . . . , ϕ iN i the components of ϕ i . Proof. For µ -a.e. x ∈ U ∩ U , define D x = D = (d ϕ , . . . , d ϕ N ) : R N → R N .D is a linear map satisfying, for all ξ ∈ ( R N ) ∗ ξ ◦ D (( ϕ ◦ γ ) ′ t ) = ξ (( ϕ ◦ γ ) ′ t ) a.e. t ∈ γ − ( U ∩ U )(5.1)for p -a.e. γ ∈ Γ + U ∩ U . Note that, by the uniqueness of differentials, D is the unique linear mapsatisfying (5.1) for p -a.e. curve. By Lemma 4.3(2) it follows that | ξ ◦ D | ,x = | ξ | ,x , ξ ∈ ( R N ) ∗ for µ -a.e. x ∈ U ∩ U . Thus D ∗ is an isometric embedding and in particular N ≤ N . Reversingthe roles of ϕ and ϕ we obtain that N = N and consequently D Φ , ,x := D ∗ x : (( R N ) ∗ , | · | ,x ) → (( R N ) ∗ , | · | ,x ) is an isometric isomorphism for µ -a.e. x ∈ U i ∩ U j .For any f ∈ N ,p ( X ), the identity d x f = d x f ◦ D Φ , ,x for µ -a.e. x ∈ U ∩ U follows from (5.1)and (1.6). (cid:3) Proof of Proposition 5.1. Conditions (1) and (2) are satisfied by Lemma 4.2. The cocycle conditionfollows from Lemma 5.2. (cid:3) Definition 5.3. We call the measurable bundle given by Proposition 5.1 the p -weak cotangentbundle and denote it by T ∗ p X . We denote T ∗ p,x X = (( R N ) ∗ , | · | x ) and T p,x X = ( R N , | · | ∗ ,x ) foralmost every x ∈ U where ( U, ϕ ) is an N -dimensional p -weak chart and | · | x the norm given by thecanonical minimal gradient Φ, cf. Lemmas 4.1 and 4.5. The spaces T p,x are here defined pointwisealmost everywhere. By considering the adjoints of transition maps in the definition above, one canpatch these together to form a measurable L ∞ tangent bundle, which is dual to T ∗ p X , whose fibersare T p,x X .The next proposition establishes the existence of a p -weak differentiable structure under a mildfinite dimensionality condition. Proposition 5.4. Suppose X is a metric measure space and { X i } i ∈ N a covering of X withdim H X i < ∞ . Then, for any p ≥ X admits a p -weak differentiable structure. Moreover, N ≤ dim H X i whenever ( U, ϕ ) is an N -dimensional p -weak chart with µ ( U ∩ X i ) > Proof. For any Borel set U ⊂ X with µ ( U ) > i ∈ N such that µ ( U ∩ X i ) > 0. ByProposition 4.13 we have that N ≤ dim H ( U ∩ X i ) whenever ϕ ∈ LIP b ( X ; R N ) is p -independent ina positive measure subset of U ∩ X i . Using [35, Proposition 3.1] we can cover X up to a null-set byBorel sets U k for which there exists ϕ k ∈ LIP b ( X ; R N k ) that are p -independent and p -maximal on U k . The collection { ( U k , ϕ k ) } k ∈ N is a p -weak differentiable structure on X . The last claim followsby the argument above. (cid:3) Sections of measurable bundles. A measurable bundle T over X comes with a projectionmap π : T → X, ( x, v ) x , and a section of T is a collection ω = { ω i : U i → V i } of Borel URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 31 measurable maps satisfying π ◦ ω i = id U i µ -a.e. and φ i,j,x ( ω i ) = ω j for each i, j ∈ I and almostevery x ∈ U i ∩ U j . Observe that the map x 7→ | ω ( x ) | x given by | ω ( x ) | x := | ω i ( x ) | i,x µ − a.e. x ∈ U i (5.2)is well-defined up to negligible sets by the cocycle condition and the fact that φ i,j,x is isometric. Definition 5.5. For p ∈ [1 , ∞ ], let Γ p ( T ) be the space of sections ω of T with k ω k p := k x 7→ | ω ( x ) | x k L p ( µ ) < ∞ . We call Γ p ( T ) the space of p -integrable sections of T . The space Γ p ( T ∗ p X ) is called the p -weakcotangent module.Note that Γ p ( T ), equipped with the pointwise norm (5.2) and the natural additition and multi-plication operations, is a normed module in the sense of [25]. Recall that an L p -normed L ∞ -moduleover X is a Banach module ( M , k · k ) over L ∞ ( X ), equipped with a pointwise norm | · | : X → R that satisfies | gm | = | g || m | and k m k = (cid:18)Z X | m | px d µ ( x ) (cid:19) /p for all m ∈ M and g ∈ L ∞ ( X ). We refer to [25, 26] for a detailed account of the theory of normedmodules.Next we consider the p -weak cotangent module Γ p ( T ∗ p X ). For a p -weak chart ( U, ϕ ) of X and f ∈ N ,p ( X ), denote by d ( U,ϕ ) f the differential of f with respect to ( U, ϕ ). Lemma 5.2 implies thatthe collection of differentials with respect to different charts satisfies the compatibility conditionabove. Definition 5.6. Let p ≥ 1, and suppose A is a p -weak differentiable atlas of X . For any f ∈ N ,p ( X ), the differential d f ∈ Γ p ( T ∗ p X ) is the element in the p -weak cotangent module defined bythe collection { d ( U,ϕ ) f : U → ( R N ) ∗ } ( U,ϕ ) ∈ A .We record the following properties of the differential. The claims follow directly from Proposition4.10 together with the compatibility condition of sections. We omit the proof. Proposition 5.7. Let A ⊂ X be a Borel set and F : X → Y a Lipschitz map to a metric measurespace ( Y, d, ν ) admitting a p -weak differentiable structure, with F ∗ µ ≤ Cν .(1) If f, g ∈ N ,p ( X ) agree on A ⊂ X , then d f = d g µ -a.e. on A .(2) If f, g ∈ N ,p ( X ) ∩ L ∞ ( X ), then d( f g ) = g d f + f d g µ -a.e.(3) If E is the set of y ∈ Y for which T ∗ p,y Y does not exist, then µ ( F − ( E )) = 0 and, for µ -a.e. x ∈ X \ F − ( E ) there exists a unique linear map D x F : T p,x X → T p,F ( x ) Y such thatd x ( h ◦ F ) = d F ( x ) h ◦ D x F µ − a.e. x for every h ∈ N ,p ( Y ).We finish the subsection with a proof of the density of Lipschitz functions in Newtonian spaces. Proof of Theorem 1.9. Let f ∈ N ,p ( X ). By [22], there exists a sequence ( f n ) ⊂ LIP b ( X ) with f n → f and | Df n | p → | Df | p in L p ( µ ). It follows that (d f n ) ⊂ Γ p ( T ∗ X ) is equi-integrable, andRemark 4.8, Lemma 4.7 together with a diagonalization argument over a union of charts covering X shows that d ˜ f n → d f in Γ p ( T ∗ X ) for convex combinations ˜ f n ∈ LIP b ( X ) of f n ’s. Consequently | D ( ˜ f n − f ) | p → L p ( µ ). (cid:3) Dependence of the p -weak differentiable structures on p . Suppose 1 ≤ p < q . We havethat | Df | p ≤ | Df | q µ -a.e. for every f ∈ LIP b ( X ), and the inequality may be strict, see [18]. As aconsequence, if ϕ ∈ LIP b ( X ; R N ) is q -maximal in U ⊂ X , then it is p -maximal. It follows (usingthis dimension upper bound and [35, Proposition 3.1]) that if X admits a q -weak differentiablestructure then X also admits a p -weak differentiable structure. We remark that the structures maybe different.For the following statement we say that a bundle map π : T → T ′ between two measurablebundles T = ( { U i , V i,x } , { φ i,l,x } ) i ∈ I and T ′ = ( { U ′ j , V ′ j,x } , { ψ j,k,x } ) j ∈ J over X is a collection oflinear maps { π i,j,x : V i → V ′ j } for µ -a.e. x ∈ U i ∩ U ′ j such that(a) for each i ∈ I , j ∈ J , the map x π i,j,x ( v ) : U i ∩ U ′ j → V ′ j is Borel for any v ∈ V i ;(b) for each i, l ∈ I , j, k ∈ J , and µ -a.e. x ∈ U i ∩ U l ∩ U ′ j ∩ U ′ k , we have the compatibilitycondition: ψ j,k,x ◦ π i,j,x = φ l,j,x ◦ π i,l,x .When the underlying index sets agree and U i = V i for all i ∈ I , it is sufficient to consider thefamily { π i,x := π i,i,x } , since these determine a unique bundle map. Proposition 5.8. Suppose q > p ≥ X admits a q -weak differentiable structure. Then X admits p -weak differentiable structure and there is a bundle map π p,q : T ∗ q X → T ∗ p X which is alinear 1-Lipschitz surjection µ -a.e. Moreover, this map satisfies π p,q = π p,s ◦ π s,q for q > s > p , and π p,q (d q f ) = d p f for any f ∈ LIP b ( X ) where d q f, d p f are the p - and q -weak differentials respectively. Proof. Since X admits a q -differential structure, we can find q -charts ( U i , ϕ q,i ) so that X = S i ∈ N U i ∪ N with µ ( N ) = 0, and ϕ q,i ∈ N ,p ( X ; R m i ) is Lipschitz. Assume that U i are chosento be pairwise disjoint. As | Df | p ≤ | Df | q (a.e.) for any f ∈ LIP b ( X ), any p -independent map isalso q -independent. Any map ϕ ∈ N ,p ( X ; R n ) which is p -independent on some positive measuresubset of U i must have n ≤ m i , see Proposition 4.14. By [35, Proposition 3.1] and this dimensionbound we can cover X by maximal p -independent maps, i.e. charts, ( V j , ϕ p,j ). By considering thecountable collection of sets V i ∩ U j , and re-indexing, we may assume that ( U i , ϕ q,i ) and ( U i , ϕ p,i )are q - and p - charts, respectively.We define the matrix A x for x ∈ U i , by taking as rows the vectors d i,p ϕ kq,i for each component k = 1 , . . . , m i . We define the bundle map π p,q by setting π xp,q ( ξ ) = ξ ◦ A x for µ -a.e. x ∈ U i . Foreach ξ we get that d p ( ξ ◦ ϕ q,i ) = ξ ◦ A x . Thus, for p -a.e. curve γ ∈ AC ( I ; X ) and a.e. t ∈ γ − ( U )we have ξ ( ϕ q,i ◦ γ ) ′ t = ( ξ ◦ A x )( ϕ p,i ◦ γ ) ′ t . By the definition of the differential, we get immediately that π p,q (d q f ) = d p f for every f ∈ LIP b ( X ). Thus, the 1-Lipschitz property follows immediately from the definition of norms combinedwith | Df | p ≤ | Df | q . The map is clearly a surjective bundle map as well, and by uniqueness of the p -differential, we automatically get π p,s ◦ π s,q = π p,q . (cid:3) URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 33 Relationship Cheeger’s and Gigli’s differentiable structures Gigli’s cotangent module. Fix p ≥ 1. Gigli’s cotangent module is the L p -normed L ∞ -module given by the following theorem. Theorem 6.1. There exists an L p -normed L ∞ -module L p ( T ∗ X ) , with pointwise norm denoted | · | G , and a bounded linear map d G : N ,p ( X ) → L p ( T ∗ X ) satisfying | d G f | G = | Df | p , f ∈ N ,p ( X ) , (6.1) such that the subspace V defined by V := M X j χ A j d G f j : ( A j ) j Borel partition of X, f j ∈ N ,p ( X ) is dense in L p ( T ∗ X ) . The module L p ( T ∗ X ) is uniquely determined up to isometric isomorphismof normed modules by these properties. Following [26, Definition 1.4.1] we say that a collection { v , . . . , v N } ⊂ L p ( T ∗ X ) is linearlyindependent in a Borel set U ⊂ X if, whenever g , . . . , g N ∈ L ∞ ( X ) satisfy (cid:12)(cid:12)(cid:12)P Nj g j v j (cid:12)(cid:12)(cid:12) G = 0 µ -a.e.on U , we have that g = · · · = g N = 0 µ -a.e. in U . A linearly independent collection { v , . . . , v N } in U is a basis of L p ( T ∗ X ) in U if, for any v ∈ L p ( T ∗ X ) there exists a Borel partition { U i } i ∈ N of U and g i , . . . , g iN ∈ L ∞ ( X ) such that (cid:12)(cid:12)(cid:12) v − P Nj g ij v j (cid:12)(cid:12)(cid:12) G = 0 µ -a.e. on U i , for every i ∈ N . Definition 6.2. Let p ≥ 1. The cotangent module L p ( T ∗ X ) is locally finitely generated if thereexists a Borel partition such that L p ( T ∗ X ) has a finite basis in each set of the partition.By [26, Proposition 1.4.5], there exists a Borel partition { A N } N ∈ N ∪{∞} of of X such that L p ( T ∗ X )has a basis of N elements on A N , for each N ∈ N ∪{∞} . We call the partition { A N } the dimensionaldecomposition of X . Notice that L p ( T ∗ X ) is locally finitely generated if and only if µ ( A ∞ ) = 0.In the forthcoming discussion we identify vectors (and vector fields) ξ ∈ R N with their dualelement v v · ξ where necessary. Lemma 6.3. Let p ≥ , N ≥ , ϕ = ( ϕ , . . . , ϕ N ) ∈ N ,p ( X ) N , and Φ the canonical minimalgradient associated to ϕ . If g = ( g , . . . , g N ) ∈ L ∞ ( X ; ( R N ) ∗ ) , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 g k d G ϕ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G,x = Φ x ( g ) µ − a.e. x ∈ X. In particular, ϕ is p -independent on U ⊂ X if and only if d G ϕ , . . . , d G ϕ N ∈ L p ( T ∗ X ) are linearlyindependent on U .Proof. If g , . . . , g N are simple functions, then g = P Mj χ A j ξ j for disjoint Borel A j and some ξ j ∈ ( R N ) ∗ . It follows that P Nk =1 g k d G ϕ k = P Mj χ A j d G ( ξ j ◦ ϕ ) as elements of L p ( T ∗ X ). Thus (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 g k d G ϕ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X j χ A j d G ( ξ j ◦ ϕ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = M X j χ A j | D ( ξ j ◦ ϕ ) | p = Φ x ( g ) for µ -a.e. x ∈ X .The estimate Φ x ( g ) ≤ N X k | g k | q ! /q N X k | Dϕ k | pp ! /p ≤ C | g | N X k | Dϕ k | p , valid for all simple vector valued g , implies that the equality in the claim is stable under local L ∞ -convergence of g . Since simple functions are dense in L ∞ , the claim follows. The remainingclaim follows in a straightforward way from the equality. (cid:3) Remark 6.4. If ϕ ∈ LIP( X ; R N ) is a chart in U , and f ∈ N ,p ( X ), then for the canonical minimalupper gradient Φ x ( a, ξ )) of ( f, ϕ ) ∈ N ,ploc ( X ; R N +1 ) we have by Lemma 4.3(2) that Φ x (1 , − d f ) = 0.Thus, by the previous lemma, we get that d G f − P Nk =1 g k d G ϕ k = 0, where g k are the componentsof d f and ϕ k are the components of ϕ . Indeed, this follows by considering this first on the sets U M = { x ∈ U : | g k ( x ) | ≤ M, k = 1 , . . . , N } and sending M → ∞ combined with locality. Lemma 6.5. If ( U, ϕ ) is an N -dimensional p -weak chart in X , then the differentials of the com-ponent functions d G ϕ , . . . , d G ϕ N form a basis of L p ( T ∗ X ) in U .Proof. By Lemma 6.3, d G ϕ , . . . , d G ϕ N ∈ L p ( T ∗ X ) are linearly independent on U . To see thatthey span L p ( T ∗ X ) in U , let f ∈ N ,p ( X ), and set g k := d f ( e k ) for each k = 1 , . . . , N , where e k is the standard basis of R N . Then, since d ϕ k = e k , where e k is the dual basis of ( R N ) ∗ , weget d f = P Nk =1 g k d ϕ k . Thus, by Remark 6.4 we have d G f = P Nk =1 g k d G ϕ k . Since the abstractdifferentials d G f span L p ( T ∗ X ), this completes the proof. (cid:3) Lemma 6.6. Suppose p ≥ and X admits a p -weak differentiable structure. There exists anisometric isomorphism ι : Γ p ( T ∗ p X ) → L p ( T ∗ X ) of normed modules satisfying ι (d f ) = d G f, f ∈ N ,p ( X ) . (6.2) The map ι is uniquely determined by (6.2) . Uniqueness here means that if A : Γ p ( T ∗ p X ) → L p ( T ∗ X ) is L ∞ -linear and satisfies (6.2) then A = ι . Proof. The set W = M X j χ A j d f j : ( A j ) j Borel partition of X , f j ∈ N ,p ( X ) is dense in Γ p ( T ∗ p X ), since it contains all the simple Borel sections of T ∗ p X . We set ι ( v ) := M X j χ A j d G f j , v = M X j χ A j d f j ∈ W , We have that | ι ( v ) | G = M X j χ A j | Df j | p = M X j χ A j | d f j | = | v | µ − a.e., URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 35 for v ∈ W . This implies that ι is well-defined and preserves the pointwise norm on the dense set W .By Remark 6.4 we have that ι is linear. Since ι ( W ) = V , it follows that ι extends to an isometricisomorphism ι : Γ p ( T ∗ p X ) → L p ( T ∗ X ). Note that ι (d f ) = d G f for every f ∈ N ,p ( X ), establishing(6.2).To prove uniqueness, note that if A : Γ p ( T ∗ p X ) → L p ( T ∗ X ) is linear and satisfies (6.2), then A ( v ) = ι ( v ) for all v ∈ W which implies that A = ι by the density of W . (cid:3) Proof of Theorem 1.11. If X admits a p -weak differentiable structure, Lemma 6.5 implies that L p ( T ∗ X ) is locally finitely generated. To prove the converse implication, suppose { A N } N ∈ N ∪{∞} isthe dimensional decomposition of X and µ ( A ∞ ) = 0.Let N ∈ N be such that µ ( A N ) ≥ µ ( V ) > V , and v , . . . , v N ∈ L p ( T ∗ X ) isa basis of L p ( T ∗ X ) on V . By possibly passing to a smaller subset of V , we may assume that thereexists C > Z V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k g k v k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) pG d µ ≥ C Z V | g | p d µ, g = ( g , . . . , g N ) ∈ L ∞ . For each k = 1 , . . . , N there are sequences v nk = M nk X j χ A nj,k d G f nj,k with { A nj,k } j a Borel partition of X and ( f nj ) ⊂ N ,p ( X ) such that v nk → v k in L p ( T ∗ X ) as n → ∞ ,by the definition of L p ( T ∗ X ).We set J n = { , . . . , M n } × · · · × { , . . . , M nN } and define new partitions A n ¯ := A nj , ∩ · · · ∩ A nj N ,N indexed by ¯ = ( j , . . . , j N ) ∈ J n . Then v nk = X ¯ ∈ J n χ A n ¯ d G ( f nj k ,k ) , µ ( V ) = X ¯ ∈ J n µ ( A n ¯ ∩ V ) , andlim n →∞ Z X | v nk − v k | pG d µ = lim n →∞ X ¯ ∈ J n Z A n ¯ | d G ϕ n, ¯ k − v k | pG d µ = 0(6.4)for all n and k = 1 , . . . , N . We claim that there exists n so that ϕ n, ¯ := ( f nj , , . . . , f nj N ,N ) ∈ N ,p ( X ; R N ) is p -independent on a positive measure subset of A n ¯ ∩ V , for some ¯ ∈ J n .By (6.3) we have the inequality1 C Z A n ¯ ∩ V | g | p d µ ≤ Z A n ¯ ∩ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k g k v k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p d µ ≤ C ′ Z A n ¯ ∩ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k g k d G ϕ n, ¯ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) pG d µ + C ′ Z A n ¯ ∩ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k g k (d G ϕ n, ¯ k − v k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) pG d µ ≤ C ′ Z A n ¯ ∩ V Φ n, ¯ ( g ( x ) , x )d µ + C ′′ Z A n ¯ ∩ V | g | p N X k | d G ϕ n, ¯ k − v k | pG ! d µ for all g = ( g , . . . , g N ) ∈ L ∞ , where Φ n, ¯ is the canonical minimal gradient of ϕ n, ¯ (cf. Lemma6.3). By (6.4) there exists n ∈ N , ¯ ∈ J n and a Borel set U ⊂ A n ¯ ∩ V with 0 < µ ( U ) ≤ µ ( A n ¯ ∩ V )such that P Nk | d G ϕ n, ¯ k − v k | pG < ε on U , where C ′′ ε < C . Thus1 C Z U | g | p d µ ≤ C ′ Z U Φ n, ¯ ( g ( x ) , x )d µ + 12 C Z U | g | p d µ for all g = ( g , . . . , g N ) ∈ L ∞ ( U ; R N ) by extending g by zero to V \ U . This readily implies that I ( ϕ n, ¯ ) > U , proving the p -independence of ϕ n, ¯ in U . Note that ϕ n, ¯ is also maximal,since the existence of a Lipschitz map on a positive measure subset of U with a higher dimensionaltarget would imply that the local dimension of L p ( T ∗ X ) in V would be > N , cf. Lemma 6.3. ByProposition 4.14, U contains a N -dimensional p -weak chart, and [35, Proposition 3.1] implies that X admits a differentiable structure.The argument above shows that each A N with µ ( A N ) > N -dimensional p -weak charts, proving (b), while (a) follows directly from Lemma 6.6. Finally, (c)is implied by Proposition 4.13. (cid:3) Theorem 1.11 and [26, Chapter 2] immediately yield the following corollary. Corollary 6.7. Let p ≥ and suppose X admits a p -weak differentiable structure. (i) If p > , then N ,p ( X ) is reflexive. (ii) If p = 2 , then N , ( X ) is infinitesimally Hilbertian if and only if, for µ -a.e. x ∈ X , thepointwise norm | · | x (cf. Theorem 1.7) is induced by an inner product. (cid:3) Lipschtiz differentiability spaces. A space X is said to be a Lipschitz differentiability spaceif it admits a Cheeger structure. Recall that a Cheeger structure is a countable collection of Cheegercharts ( U i , ϕ i ), see Section 4.7, so that µ ( X \ S i U i ) = 0. Following [12, Section 4, p. 458], we notethat the differentials d C,i f of a Lipschitz function f with respect to overlapping charts satisfy aco-cycle condition almost everywhere and the transition maps preserve the pointwise norm. Thus,they define a measurable L ∞ -bundle T ∗ C X called the measurable cotangent bundle.Suppose now that X admits a Cheeger structure. Denote by T ∗ C X the associated measurablecotangent bundle, and by | ξ | C,x := Lip( ξ ◦ ϕ )( x ) , ξ ∈ ( R N ) ∗ the pointwise norm for µ -a.e. x ∈ U , where ( U, ϕ ) is an N -dimensional Cheeger chart of X .Fix p ≥ 1. Any Lipschitz differentiability space X admits a p -weak differentiable structure.Indeed, the asymptotic doubling property of the measure (cf. [9]) implies, by [8, Lemma 8.3], that X decomposes into finite-dimensional pieces. The existence of the p -weak differentiable structurenow follows from Proposition 5.4, and the associated measurable cotangent bundle is denoted T ∗ p X .We have the following result from [32, Theorem 3.4]: Theorem 6.8. Let p ≥ . There exists morphism P : Γ p ( T ∗ C X ) → L p ( T ∗ X ) of normed modulessuch that (a) P (d C f ) = d G f for every f ∈ LIP( X ) ; (b) | P ( ω ) | G ≤ | ω | C for every ω ∈ Γ p ( T ∗ C X ) ; and URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 37 (c) for every w ∈ L p ( T ∗ X ) there exists ω ∈ P − ( w ) with | w | G = | ω | c . Remark 6.9. The proof of [32, Theorem 3.4] can be modified to cover the case p = 1: the energydensity of Lipschitz functions holds for p = 1 by [22], and equicontinuity can be used instead of L p -boundedness to obtain the weakly convergent subsequence in the proof. Proof of Theorem 1.10. Arguing as in the proof of Proposition 5.8 we may assume that X has aBorel partition { U i } and Lipschitz maps ϕ ip = ( ϕ ip, , . . . , ϕ ip,N i ) , ϕ iC = ( ϕ iC, , . . . , ϕ iC,M i ) such that( U i , ϕ ip ) is a p -weak chart and ( U i , ϕ iC ) is a Cheeger chart on X (of possibly different dimensions N i and M i ) for each i ∈ N . For each i and µ -a.e. x ∈ U i define σ i,x = (d p,x ϕ iC, , . . . , d p,x ϕ iC,M i ) : R N i → R M i . It is easy to see that the collection { π i,x = σ ∗ i,x } defines a bundle map T ∗ C X → T ∗ p X satisfyingd p,x f = d C,x f ◦ σ x µ − a.e. x ∈ X. for every f ∈ N ,p ( X ). This proves Equation (1.7). In particular, for each i ∈ N and ξ ∈ ( R M i ) ∗ we have π i,x ( ξ ) = ξ ◦ σ i,x = d p,x ( ξ ◦ ϕ iC ) , and consequently | π i,x ( ξ ) | x = | D ( ξ ◦ ϕ iC ) | p ( x ) ≤ Lip( ξ ◦ ϕ iC )( x ) = | ξ | C,x , for µ -a.e. x ∈ U i . Moreover, for any ζ ∈ ( R N i ) ∗ setting ξ := d C,x ( ζ ◦ ϕ ip ), we have that π i,x ( ξ ) = d C,x ( ζ ◦ ϕ ip ) ◦ σ x = d p,x ( ζ ◦ ϕ ip ) = ζ, proving that π i,x is surjective for µ -a.e. x ∈ U i .To prove that π i,x is a submetry for µ -a.e. x ∈ U i , suppose to the contrary that there exist aBorel set B ⊂ U i with 0 < µ ( B ) < ∞ such that π i,x is not a submetry for x ∈ B . Then there existsa Borel map ζ : B → ( R N i ) ∗ with | ζ x | x = 1 and | ζ x | x = 1 and inf ξ ∈ π − i,x ( ζ x ) | ξ | C,x > µ -a.e. x ∈ B. (6.5)We derive a contradiction using Theorem 6.8 and the isometric isomorphism ι : Γ p ( T ∗ p X ) → L p ( T ∗ X ) from Theorem 1.11(a). We may view ζ as an element of Γ p ( T ∗ p X ) by extending it byzero outside B . Set w := ι ( ζ ) ∈ L p ( T ∗ X ). Then | w | G = χ B . By Theorem 6.8(c) there exists ω ∈ Γ p ( T ∗ C X ) with P ( ω ) = w and | ω | C = | w | G = χ B µ -a.e. However, since ω x ∈ π − i,x ( ζ x ) for µ -a.e. x ∈ B , we have | ω | C,x ≥ inf ξ ∈ π − i,x ( ζ x ) | ξ | C,x > µ -a.e. x ∈ B by (6.5), which is a contradiction.This completes the proof that π i,x is a submetry for µ -a.e. x ∈ U i .If Lip f ≤ ω ( | Df | p ) holds for every f ∈ LIP b ( X ), then by [32, Theorem 1.1] we have that | Df | p = Lip f µ -a.e. for every f ∈ LIP b ( X ). It follows that p -weak charts are Cheeger charts (cf.Theorem 1.8 and Remark 4.15) and that the pointwise norms agree µ -almost everywhere. Thisimplies that the maps π i,x are isometric bijections for µ -a.e. x . (cid:3) Appendix A. General measure theory A.1. Measurability questions. Here we record a host of measurability statements that areneeded throughout the paper. See [4, 11, 27] for more details. Given f ∈ N ,p ( X ) and a Borelrepresentative g of p -weak upper gradient of f , we denote Γ( f ) := { γ ∈ AC ( I ; X ) : f ◦ γ ∈ AC ( I ; R ) } , Γ( f, g ) := { γ ∈ AC ( I ; X ) : g upper gradient of f along γ } ⊂ Γ( f )and M D = { ( γ, t ) ∈ AC ( I ; X ) × I : | γ ′ t | exists } , Diff( f ) = { ( γ, t ) ∈ AC ( I ; X ) × I : γ ∈ Γ( f ) , ( f ◦ γ ) ′ t and | γ ′ t | > } , Diff( f, g ) = { ( γ, t ) ∈ Diff( f ) : γ ∈ Γ( f, g ) , | ( f ◦ γ ) ′ t | ≤ g f ( γ t ) | γ ′ t |} . Also, let Len( γ ) be the length of a curve γ , if the curve is rectifiable, and otherwise infinity. Thefunction der is defined by der( γ, t ) := | γ ′ t | = lim h → d ( γ t + h ,γ t ) | h | , when the limit exists, and otherwiseis infinity. Lemma A.1. (1) The functions Len : C ( I ; X ) → [0 , ∞ ] and der : AC ( I ; X ) × I → [0 , ∞ ] areBorel measurable. (2) If g : X → [0 , ∞ ] is a Borel function, then I : AC ( I ; X ) → R , given by γ R γ g d s or ∞ ifthe curve is not rectifiable, is Borel. (3) If H : AC ( I ; X ) × I → [0 , ∞ ] is Borel, then I H ( γ ) := R H ( γ, s )d s : AC ( I ; X ) → [0 , ∞ ] isBorel. (4) The set M D is Borel, and the map M D → R defined by ( γ, t ) → | γ ′ t | is Borel.Proof. (1) The length function is a lower semicontinuous function with respect to uniform con-vergence, and thus is Borel. Fix r, p ∈ Q positive. Then define A p,r = ∪ n ∈ N ∩ q ∈ Q ∩ ( − n , n ) { ( γ, t ) : | d ( γ t + q , γ t ) − qp | < r | q |} , and thus Borel. The set M where the metric derivative ex-ists is of the form ∩ r ∈ Q ∩ (0 , ∞ ) ∪ p ∈ Q ∩ (0 , ∞ ) A p,r . On this set we have M ∩ A p,r = der − ( B ( p, r ))and thus der( γ, t ) is Borel.(2) The claims for the integral function being Borel follow from a monotone family argument,and considering g first a characteristic function of an open set and using lower semi-continuity of the integral in that case.(3) If H is a characteristic function of a product set A × B , where A and B are open sets suchthat A ⊂ C ( I ; X ) , B ⊂ I , then the claim follows just as in Statement (2). Again, by amonotone family argument, we obtain the claim for all Borel measurable functions.(4) Define for every q ∈ Q and ε, h > A ( ε, q, h ) and B ( ε, q ) by A ( ε, q, h ) := (cid:26) ( γ, t ) ∈ C ( I ; X ) × I : (cid:12)(cid:12)(cid:12)(cid:12) d ( γ t + h , γ t ) | h | − q (cid:12)(cid:12)(cid:12)(cid:12) < ε (cid:27) B ( ε, q ) := [ δ ∈ Q + \ h ∈ (0 ,δ ) ∩ Q A ( ε, q, h ) . We note that | γ ′ t | exists if and only if ( γ, t ) ∈ \ j ∈ N [ q ∈ Q B (2 − j , q ) = M D . On the set M D ,where the limit exists, we can write | γ ′ t | = lim n →∞ n ( d ( γ t + n − , γ t )), which shows measura-bility. (cid:3) URVEWISE CHARACTERIZATIONS AND A SOBOLEV DIFFERENTIAL 39 Lemma A.2. Let g be a Borel p -weak upper gradient of f ∈ N ,p ( X ) . There exists a Borel set Γ ⊂ AC ( I ; X ) with Mod p (Γ ) = 0 such that AC \ Γ ⊂ Γ( f, g ) .Suppose moreover that f is Borel. Then the set A := Γ c × I ∩ Diff( f, g ) is Borel, and π ( A c ) = 0 whenever π = L × η and η is a q -test plan.If f is Lipschitz, and g = Lip[ f ] , then we can choose Γ = ∅ , and Diff( f, g ) = Diff( f ) is Borel. Note that we make no claims about the Borel measurability of the set Γ( f, g ). Proof. We model the argument after [37, Lemma 1.9]. Since Mod p (Γ( f, g ) c ) = 0 there existsan L p -integrable Borel function ρ : X → [0 , ∞ ] with R γ ρ d s = ∞ for every γ / ∈ Γ f,g . ThenΓ := { γ ∈ AC ( I ; X ) : R γ ρ d s = ∞} ⊃ Γ cf,g is a Borel set, by Lemma A.1 and η (Γ ) = 0 for every q -plan η (see Remark 2.2). If f is Lipschitz, then Γ( f, g ) = AC ( I ; X ). Thus, we can choose Γ = ∅ .For the second part assume f ∈ N ,p ( X ) is Borel, and set A ( ε, q, h ) = (cid:26) ( γ, t ) ∈ Γ c × I : (cid:12)(cid:12)(cid:12)(cid:12) f ( γ t + h ) − f ( γ t ) h − q (cid:12)(cid:12)(cid:12)(cid:12) < ε (cid:27) B ( ε, q ) = [ δ ∈ Q + \ h ∈ (0 ,δ ) ∩ Q A ( ε, q, h )for each q ∈ Q and ε, h > 0. It is easy to see that for each γ / ∈ Γ , ( f ◦ γ ) ′ t exists if and only if( γ, t ) ∈ \ j ∈ N [ q ∈ Q B (2 − j , q ) =: A. Note that A is a Borel set with A ∩ M D ⊂ Diff( f ). Moreover, ( γ, t ) ( f ◦ γ ) ′ t is Borel whenrestricted to A ∩ M D Define the Borel function H ( γ, t ) = ( f ◦ γ ) ′ t if ( γ, t ) ∈ A ∩ M D and H = + ∞ otherwise, and G ( γ, t ) = | H | − g ( γ t ) | γ ′ t | (here we use the convention ∞ − ∞ = ∞ ). Then the set { G ≤ } = Γ c × I ∩ Diff( f, g )is Borel.Set N := { G > } , suppose η is a q -test plan and π := L × η . Note that N ⊂ Γ × I ∪ { ( γ, t ) ∈ Γ c × I : G ( γ, t ) > } . But for all γ / ∈ Γ , we have that G ( γ, t ) ≤ L -a.e. t ∈ I . Thus π ( N ) ≤ η (Γ ) + Z Γ c Z χ { G ( γ, · ) > } ( t )d t d η ( γ ) = 0 , finishing the proof of the second part. (cid:3) Corollary A.3. Every pointwise defined function f ∈ N ,p ( X ) has a Borel representative ¯ f ∈ N ,p ( X ) . Moreover, if f ∈ N ,p ( X ) and g is a Borel p -weak upper gradient of f , there exists aBorel set N ⊂ C ( I ; X ) × I with N c ⊂ Diff( f, g ) and π ( N ) = 0 whenever π = L × η , η a q -testplan. The map ( γ, t ) ( f ◦ γ ) ′ t if ( γ, t ) / ∈ N and + ∞ otherwise is Borel. If f is Lipschitz therepresentative can be chosen as the same function. Proof. The first claim follows directly from [22, Theorem 1.1]. To see the second, let ¯ f ∈ N ,p ( X )be a Borel representative of f . The set E := { f = ¯ f } is p -exceptional, i.e. Γ E := { γ : γ − ( E ) = ∅ } has zero p -modulus. Note that, if f is Lipschitz, then f is automatically Borel and we do not needto change representatives, and we can set Γ E = ∅ .If ¯ A is the set in Lemma A.2 for ¯ f , g , then A := ¯ A \ (Γ E × I ) ⊂ Diff( f, g ) and N := A c satisfiesthe claim since it is Borel and N ⊂ Γ E × I ∪ ¯ A c .The last claim follows since N c is Borel and, if ( γ, t ) / ∈ N , we have that( f ◦ γ ) ′ t = lim n →∞ n ( f ( γ t +1 /n ) − f ( γ t )) . (cid:3) A.2. Essential supremum.Definition A.4. Let X be a σ -finite measure space and F a collection of measurable functions on X , then there exists a function g : X → R ∪ {∞ , −∞} which is measurable, andA For each f ∈ F , f ≤ g almost everywhere.B For each g ′ that satisfies [A], will satisfy g ≤ g ′ almost everywhere.We call g = ess sup f ∈F f . Similarly, we define g = ess inf f ∈F f , by switching the directions ofthe inequalities and assuming g : X → R ∪ {∞ , −∞} . We will need the following standard lemma. While its proof is standard, we provide it for thesake of completeness. Lemma A.5. If X is any σ -finite measure space and F is any collection of measurable func-tions, then ess sup f ∈F f and ess inf f ∈F f exists, and further, there are sequences f n , g n ∈ F so that ess sup f ∈F f = sup n f n and ess inf f ∈F f = inf n g n almost everywhere.Proof. By considering { arctan( f ) : f ∈ F } , we can assume that the collection is bounded. Further,by σ -finiteness, and after exhausting the space by finite measure sets, it suffices to consider abounded measure. Define G to be the collection of all functions of the form max( f , . . . , f k ) forsome f i ∈ F . By construction, if g, g ′ ∈ G , then max( g, g ′ ) ∈ G ′ .Consider U = sup g ∈G R g d µ . There is a sequence g n so that lim n →∞ R g n d µ = U . 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