Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential

We represent minimal upper gradients of Newtonian functions, in the range $1\le p<\infty$, 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 $p$-weak charts 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 computed curvewise, is linear, and satisfies the usual Leibniz and chain rules. The arising $p$-weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger's structure in the presence of a Poincar\'e inequality. It is moreover compatible with, and gives a geometric interpretation of, Gigli's abstract differentiable structure, whenever it exists. The $p$-weak charts give rise to a finite dimensional $p$-weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm.

. Shanmugalingam [2000] developed Newtonian spaces N 1, p (X ) using the modulus perspective of [Heinonen and Koskela 1998] and proved that they coincide with the Sobolev space defined by Cheeger up to modification of its elements on a set of measure zero.Further notions of Sobolev spaces, based on test plans, were developed by Ambrosio, Gigli and Savaré [Ambrosio et al. 2014], with a corresponding 456 SYLVESTER ERIKSSON-BIQUE AND ELEFTERIOS SOULTANIS notion of minimal gradient.Earlier, Hajłasz [1996] had introduced a Sobolev space whose associated minimal gradient, however, lacks suitable locality properties.While the various Sobolev spaces (with the exception of Hajłasz's definition) are equivalent for generic metric measure spaces, Newtonian spaces consist of representatives which are absolutely continuous along generic curves, a property central to the results in this paper.
The minimal p-weak upper gradient g f ∈ L p (X ) of a Newtonian function f ∈ N 1, p (X ) on a metric measure space X is a Borel function characterized (up to a null-set) as the minimal function satisfying for all absolutely continuous γ : I → X outside a curve family of zero p-modulus.Here |γ ′ t | denotes the metric derivative of γ for a.e.t; see Section 2. When X = ‫ޒ‬ n and f ∈ C ∞ c ‫ޒ(‬ n ), g f is given by g f = ∥∇ f ∥; in this case, for each x ∈ X , there exists a (smooth) gradient curve γ : (−ε, ε) → X , with In general, however, despite the minimality of g f , the equality in (1-2) is not always attained.For example the fat Sierpiński carpet (with the Hausdorff 2-measure and Euclidean metric) constructed in [Mackay et al. 2013] with a sequence in ℓ 2 \ ℓ 1 , as pointed out in the introduction of that work, gives zero p-modulus ( p > 1) to the family of curves parallel to the x-axis, and thus to the family of gradient curves of the function f (x, y) = x.We remark that the example above is measure doubling and supports a Poincaré inequality; in this context an approximate form of (1-2) for Lipschitz functions was proven in [Cheeger and Kleiner 2009, Theorem 4.2].
Towards a positive answer for generic spaces, an "integral formulation" of (1-2) given by [Gigli 2015, Theorem 3.14] states that, when p > 1 and f ∈ N 1, p (X ), there exist probability measures η on C(I ; X ) (known as test plans representing the gradient of f ) such that In this paper we obtain a "pointwise" variant of (1-2) for general metric measure spaces using a combination of plan-modulus duality, developed in [Ambrosio et al. 2015b;Durand-Cartagena et al. 2021;Honzlová Exnerová et al. 2021], and disintegration techniques.(For p > 1, a pointwise variant also follows from Gigli's integral formulation; see Section 3C.) Theorem 1.1 below expresses the minimal weak upper gradient of a Newtonian function as the supremum of directional derivatives along generic curves passing through a given point.Here, it is crucial to use Newtonian functions, which are absolutely continuous along almost every curve.
This curvewise characterization of minimal upper gradients yields the existence of an abundance of curves in a given region of the space, provided the region supports nontrivial Newtonian functions.The idea of constructing an abundance of curves goes back to [Semmes 1996] in the presence of a Poincaré inequality.Under this assumption Cheeger showed that g f = Lip f, where Lip f denotes 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 [Cheeger 1999, pp. 432-433].
The work of Cheeger lead to many developments, including [Cheeger and Kleiner 2009] pioneering the idea of using directional derivatives along curves (and the early version of Theorem 1.1, appearing as Theorem 4.2 in that work) as well as the development and detailed analysis of Lipschitz differentiability spaces; see [Keith 2004a;Bate 2015;Bate and Speight 2013;Cheeger et al. 2016;Schioppa 2016a;2016b].In the latter, curves are replaced by curve fragments whose abundance is expressed using Alberti representations.Alberti representations are similar to plans used in this paper.The connection between such representations and the ideas in [Cheeger and Kleiner 2009] was first observed by Preiss, see [Bate 2015, p. 2], and can be used to prove the self-improvement of the Lip-lip inequality to the Lip-lip equality, see [Bate 2015;Schioppa 2016a;Cheeger et al. 2016].
Similarly the abundance of curves, obtained here using duality, yields geometric information on Sobolev functions on general metric measure spaces.(Indeed, duality, in the disguise of a minimax principle, was previously used to find Alberti representations in Lipschitz differentiability spaces; see [Bate 2015, Theorem 5.1] which uses [Rudin 1980, Lemma 9.4.3].)As an important first application, we use curvewise directional derivatives to define p-weak charts and a differential for Newtonian functions with respect to such charts.The arising p-weak differentiable structure, i.e., a covering by p-weak charts, exists far more generally than for Lipschitz differentiability spaces -indeed metric doubling and finite Hausdorff measure suffice; see Proposition 5.4.This existence result involves a new and crucial dimension bound for the charts and the induced differential structure; see Theorem 1.11(c) or Proposition 4.13.With the aid of Theorem 1.1 we adapt Cheeger's construction to produce a measurable L ∞ -bundle, called the p-weak cotangent bundle, over spaces admitting a p-weak differentiable structure; differentials of Newtonian functions are sections over this bundle.While the Cheeger differential d C f yields a linearization of a Lipschitz function f, our p-weak differential d f is given by a linearization along p-almost every curve, and the pointwise norm of d f recovers the minimal weak upper gradient; see Theorem 1.7.This definition of a weak differentiable structure seems to be the natural extension of the seminal work [Cheeger 1999] to settings without a Poincaré inequality and yields a "partial differentiable structure", which has been the aim of many authors previously; see [Lučić et al. 2021;Alberti and Marchese 2016;Schioppa 2016a;Cheeger et al. 2016].Namely, the p-weak cotangent 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 a concrete geometric meaning.A sequence of recent work has sought such concrete descriptions; see, e.g., [Ikonen et al. 2022;Lučić et al. 2021].Our approach yields a new unification of the concrete and abstract cotangent modules of [Cheeger 1999] and [Gigli 2018], respectively; the p-weak cotangent bundle is compatible with Gigli's cotangent module when the latter is locally finitely generated, and with Cheeger's cotangent bundle when the space satisfies a Poincaré 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 [Ambrosio et al. 2014;2015c;Lučić et al. 2021], and the identification of abstract tangent bundles with geometric tangent cones; see [Alberti and Marchese 2016;Lučić et al. 2021].Our methods give tools to generalize and refine the results mentioned above, and moreover enable a blow-up analysis to study analogues of generalized linearity considered for example in [Cheeger 1999;Cheeger et al. 2016].Indeed, blow-ups of plans that define the pointwise norm on a p-weak chart (see Lemmas 4.1,4.3 and 4.2) along a sequence of rescaled spaces yield curves in the limiting space along which limiting maps of rescaled Newtonian maps behave linearly.In this context we highlight [Schioppa 2016a], which gives a similar geometric and blow-up analysis in the context of abstract Weaver derivations.We leave the detailed exploration and development of these ideas for future work.
1B. Curvewise characterization of minimal upper gradients.Throughout the paper, we fix a metric measure space X = (X, d, µ), that is, a complete separable metric space (X, d) together with 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 evaluation map e : C(I ; X ) × I → X , (γ , t) → γ t , gives rise to the barycenter dη # := e * (|γ ′ t | dt dη) of η.If dη # = ρ dµ 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 [Ambrosio et al. 2015b]).Every finite measure π on C(I ; X ) × I admits a disintegration with respect to e: for e * π -almost every x ∈ X , there exists a (unique) measure π x ∈ P(C(I ; X ) × I ), concentrated on e −1 (x), such that the collection for all Borel sets B ⊂ C(I ; X ) × I.We refer to [Ambrosio et al. 2008;Bogachev 2007] for more details.
We use these notions to define a "generic curve": if η is a q-plan on X and {π x } the disintegration of dπ := |γ ′ t | dt dη with respect to e, then π x -a.e.-curve passes through x, for e * π -a.e.x ∈ X .In the forthcoming discussion, we omit the reference to e in the disintegration.We now formulate our first result, in which the equality in (1-2) is obtained as an essential supremum with respect to the disintegration for almost every point.In the statement below we write Theorem 1.1.Let 1 ≤ p < ∞, and let 1 < q ≤ ∞ satisfy 1/ p + 1/q = 1.Suppose f ∈ N 1, p (X ), g f is a Borel representative of the minimal p-weak upper gradient of f, and D := {g f > 0}.There exists a q-plan η with µ| D ≪ η # so that the disintegration Remark 1.2.The statement also holds when for each ball B(x, r ) ⊂ X .Indeed, a localization argument, replacing f by f η n with η n a sequence of Lipschitz functions with bounded support and η n | B(x 0 ,n−1) = 1 for some x 0 , reduces the statement for f ∈ N 1, p loc (X ) to Theorem 1.1.Similarly, other notions in this paper, such as charts, could use a local Sobolev space, but to avoid technicalities we do not discuss this point further.A reader can see Lemma 4.5 and its proof for a prototypical form of such a localization argument.
In particular, we have the following corollary.
Theorem 1.1 notably covers the case p = 1.In Section 3C we also prove a variant (Theorem 3.6) when p > 1, using test plans representing a gradient instead of plan-modulus duality.
1C. Application: p-weak differentiable structure.Cheeger [1999] showed that PI-spaces (metric measure spaces with a doubling measure supporting some Poincaré inequality) admit a countable cover by Cheeger charts, also called a Lipschitz differentiable structure (see [Keith 2004b]).Let LIP(X ) denote the collection of Lipschitz functions on X , and let LIP b (X ) consist those Lipschitz functions with 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 → ‫ޒ‬ n such that, for every f ∈ LIP(X ) and µ-a.e.x ∈ U, there exists a unique linear map d C,x f : (1-4) Not every space admits Lipschitz differentiable structure, as shown by the so called Rickman's rug X := [0, 1] 2 equipped with the metric d((x 1 , y 1 ), (x 2 , y 2 )) = |x 1 − x 2 | + |y 1 − y 2 | α , where α ∈ (0, 1) and µ = L 2 | X .Indeed, a Weierstrass-type function in the y-variable combined with [Schioppa 2016a, Theorem 1.14] would yield nonhorizontal rectifiable curves if the space were a differentiability space, contradicting the fact that all rectifiable curves in X are horizontal.
Here, we introduce p-weak differentiable structures, which exist in much more generality (including Rickman's rug, see the discussion after Definition 1.4), adapting Cheeger's construction by substituting (1-4) for a weaker curvewise control.To accomplish this, we replace the pointwise Lipschitz constant by the minimal p-weak upper gradient in the definition of "infinitesimal linear independence" (1-5) and use Theorem 1.1 to circumvent the difficulties arising from the fact that the latter is defined only up to a null-set.
In the remainder of the introduction, we use the notation |D f | p for the minimal p-weak upper gradient of f ∈ N 1, p loc (X ) and refer to Section 2 for more discussion on this notation.Given p ≥ 1 and N ∈ ‫,ގ‬ we say that a Sobolev map ϕ (1-5) and p-maximal in U if no Lipschitz map into a higher-dimensional target is p-independent in a positive 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 −1 ; see Section A2.Note that p-maximality does not depend on the particular map ϕ but rather the dimension of its target space.
Definition 1.4.An N -dimensional p-weak chart (U, ϕ) of X consists of a Borel set U ⊂ X with positive measure and a Lipschitz function ϕ : X → ‫ޒ‬ 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 by countably many p-weak charts.
By convention, zero-dimensional p-weak charts satisfy ϕ ≡ 0 and (1-5) is a vacuous condition, while maximality means that |D f | p = 0 µ-a.e. on U for every f ∈ LIP b (X ) (see also Proposition 4.4).In Section 4F we briefly discuss a lower-regularity requirement in Definition 1.4 and the fact the resulting notion yields essentially the same p-weak differentiable structure.We also show that an N -dimensional p-weak chart (U, ϕ) satisfies N ≤ dim H U, where dim H U denotes the Hausdorff dimension 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 differentiable structure for any p ≥ 1.In particular, this holds if the space is metrically doubling.
We refer to Proposition 5.4 for a more technical statement, which immediately implies the theorem.Next, we give an analogue of the Cheeger differential (1-4) using p-weak charts. (1-6) for p-a.e.absolutely continuous curve γ in X .We say that a function f ∈ N 1, p (X ) has a p-weak differential with respect to ϕ, if such a d f exists.
If the curve γ does not enter U, or only spends zero length in the set, then condition (1-6) becomes vacuously satisfied with both sides vanishing.The p-weak differential is uniquely determined up to almost everywhere equivalence by (1-6); see Lemma 4.3.Further, it is also local, i.e., if g ∈ N 1, p (X ) and f | A = g| A on a positive measure subset A ⊂ U, then d f | A = dg| A .The differential satisfies various natural computation rules; see Propositions 4.10 and 5.7 for the most important ones.
Theorem 1.7.Suppose p ≥ 1, and ϕ : X → ‫ޒ‬ N is a p-weak chart on U. Then any f ∈ N 1, p (X ) has a p-weak differential d f : U → ‫ޒ(‬ N ) * with respect to ϕ, which is µ-a.e.unique, and the map f → d f is linear.
Moreover, for µ-a.e.x ∈ U, there is a norm Whereas Lipschitz functions are differentiable with respect to Cheeger charts, (1-5) yields only the curvewise control (1-6).Indeed, if there are very few or no rectifiable curves, or if the curves only point into certain directions, then the p-weak differential vanishes, or measures only these directions, respectively.For example, given a fat Cantor set Lipschitz differentiability space but the minimal weak upper gradient of every Lipschitz function is zero.
On the other hand, Rickman's rug admits nontrivial p-weak charts ϕ(x, y) = x.The p-weak differential in this case can be identified with the x-derivative, d f ≡ ∂ x f, and the only curves with positive p-modulus are those which are horizontal.These examples demonstrate that p-weak differentiable structures might exist for spaces not admitting a Cheeger structure, but the two need not coincide even if both exist.However, if a Poincaré inequality is present, the two structures coincide.
Theorem 1.8.Suppose X is a p-PI space for p ≥ 1.Then any p-weak chart (U, ϕ) of X is a Cheeger chart.
It follows from the discussion after Definition 1.4 that a p-PI space admits p-weak charts.In Section 1D, we obtain a precise statement on the relationship between the p-weak and Lipschitz differentiable structure, as well as a characterization of the existence of p-weak differentiable structures in terms of Gigli's cotangent module [2018].Here we mention a noteworthy corollary of the existence of a p-weak differentiable structure.
Theorem 1.9.Let p ≥ 1.If X admits a p-weak differential structure, then LIP b (X ) is norm-dense in N 1, p (X ).
Theorem 1.9 has been obtained by other methods for p > 1 in [Ambrosio et al. 2013] but is new in the case p = 1.In particular, we highlight that the density holds if X has finite Hausdorff dimension.
1D. Connections to Cheeger's and Gigli's differentiable structures.Together with the pointwise norm from Theorem 1.7, a p-weak differentiable structure gives rise to a p-weak cotangent bundle T * p X over X , analogous to the measurable L ∞ -cotangent bundle T * C X arising from the Lipschitz differentiable structure [Cheeger 1999;Keith 2004b], which is equipped with the pointwise norm for µ-a.e.x ∈ U, where (U, ϕ) is an N -dimensional Cheeger chart.For any f ∈ LIP b (X ), the differentials d f and d C f are sections of the cotangent bundles T * p X and T * C X , respectively.We refer 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 a condition under which the bundle map is an isometric isomorphism.See Section 5 for the definition of 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 a surjective linear map L : Theorem 1.10.Suppose X admits a Cheeger structure and let p ≥ 1.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 for every f ∈ LIP b (X ).If there exists a collection {ω x } x∈X of moduli of continuity satisfying for 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 [Ikonen et al. 2022, Theorem 1.1] and the following theorem, which identifies the space p (T * p X ) of p-integrable sections of the p-weak cotangent bundle T * p X with Gigli's cotangent module L p (T * X ).We refer to Section 6 for the relevant definitions, and remark here that Gigli's construction is the most general in the sense that L p (T * X ) can be defined for any metric measure space.It is a priori defined only as an abstract L p -normed L ∞ -module in the sense of [Gigli 2015;2018].
We say that L p (T * X ) is locally finitely generated if X has a countable Borel partition B so that each B ∈ B admits a finite generating set in B. Here, a collection for every v ∈ V .Gigli's cotangent modules admit a dimensional decomposition, i.e., a Borel partition {A N } N ‫}∞{∪ގ∈‬ 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 .The dimensional decomposition is uniquely determined up to µ-negligible sets.
Below we denote by d G f and | • | G the abstract differential and pointwise norm 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 pointwise norm and has an inverse that is a morphism.
Theorem 1.11.Let X be a metric measure space and p ≥ 1.Then X admits a p-weak differentiable structure if and only if L p (T * X ) is locally finitely generated.In this case, (a) there exists an isometric isomorphism ι : p (T * p X ) → L p (T * X ) of normed modules satisfying ι(d f ) = d G f for every f ∈ N 1, p (X ) and uniquely determined by this property, (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, Theorem 1.11 gives a concrete interpretation of Gigli's cotangent module, and bounds the Hausdorff dimension of the sets in the dimensional decomposition.As corollaries we obtain the reflexivity of N 1, p (X ) when p > 1, and a characterization of infinitesimal Hilbertianity in terms of the pointwise norm of Theorem 1.7 when p = 2, for spaces admitting a p-weak differentiable structure; see Corollary 6.7.Reflexivity could also be obtained directly from Theorem 1.7 following the argument in [Cheeger 1999, Section 4].

Preliminaries
Throughout this paper X = (X, d, µ) will be a complete separable metric measure space equipped with 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 of absolutely continuous curves in X , where I ⊂ ‫ޒ‬ is an interval.Mostly, we will be concerned with statements independent of parametrization; thus the choice of the interval I is immaterial.However, when we need to refer to the end points of the curve, then we will take If γ is a curve, its value at t ∈ I is denoted by The metric derivative of the curve, in the sense of say [Ambrosio et al. 2008, Section I.1], is defined as |γ ′ t | = lim h→0 d(γ t+h , γ t )/ h, when it exists.The metric derivative is defined almost everywhere on I for γ ∈ AC(I ; X ).
2A.Plans and modulus.A finite measure η on C(I ; X ) is called a plan if it is concentrated on AC(I ; X ), and a q-plan if the barycenter dη # := e * (|γ ′ t | dt dη) satisfies dη # = ρ dµ for some ρ ∈ L q (µ).We denote by AC q (I ; X ) the space of curves γ ∈ AC(I ; X ) satisfying 1 0 |γ ′ t | q dt < ∞, 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 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 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 ⊂ AC(I ; X ), one can construct associated q-test plans supported on reparametrized curves, which are subcurves of curves in .
The argument for this is a combination of two observations in [Ambrosio et al. 2015b].First, for each q-plan one can reparametrize curves to get a plan with a good "parametric barycenter" [loc.cit., Definition 8.1 and Theorem 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 subsegments of curves and average these over shifts to get a compression bound, which is explained as part of the proof of [loc. cit.,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 ⊂ C(I ; X ) is a family of curves, then a Borel function ρ : X → [0, ∞] is called admissible if γ ρ ds ≥ 1 for each rectifiable γ ∈ .In particular, if there are no rectifiable curves, then this condition is vacuous.We define, for p ∈ [1, ∞), where the infimum is over all admissible ρ.We remark, that due to Vitali-Carathéodory, such an infimum can always be taken with respect to lower semicontinuous functions.Notice that the modulus is supported on rectifiable curves and is independent of the parametrization of such curves.We say that a property holds for p-almost every curve if there is a family of curves B so that Mod p ( B ) = 0 and the property holds for all γ ∈ C(I ; X ) \ B .Modulus is invariant of the parametrization of curves, but some of our statements depend on a parametrization.In those 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 )\ B with Mod p ( B ) = 0.The reader may consult [Heinonen et al. 2015, Sections 4-7] for a more in-depth treatment of modulus, upper gradients and Vitali-Carathéodory.
Remark 2.2.A crucial fact we will use is that if satisfies Mod p ( ) = 0, then for any q-plan η we have η( ) = 0 (which holds for p ∈ [1, ∞) and q its dual exponent).The converse is also true for p ∈ (1, ∞).
See the arguments and discussion in [Ambrosio et al. 2015b, Sections 4 and 9].One point here is that if we used q-test plans, this relationship would be more complex, and we would need to consider "stable" families of curves; see [loc. cit.,Theorem 9.4].The case of p = 1 is also somewhat subtle, and we will deal with a special case of this issue in Section 3. The argument of Proposition 2.3 would give the converse for compact families of curves and p = 1.See also, [Honzlová Exnerová et al. 2021] for a much more detailed exploration of this borderline case.
The previous remark concerns an inequality relating modulus and q-plans.However, there is a closer connection, and in a sense these are dual to each other.Previously, this has been explored in [Ambrosio et al. 2015b, Theorem 5.1] for p > 1, and in [Honzlová Exnerová et al. 2021, Theorem 6.3] for p = 1.Due to its importance for us, we summarize one main consequence of these results.We further briefly describe the main steps of a direct proof from [David and Eriksson-Bique 2020, Proposition 4.5].A similar argument appeared previously in a more specific context in [Durand-Cartagena et al. 2021, Theorem 3.7].
Proof.A power of the modulus Mod p (K ) 1/ p arises from a convex optimization problem on ρ with a constraint for every curve γ ∈ K .A dual formulation of this corresponds to a variable for each constraint, i.e., a measure ν supported on K .Thus, it is reasonable to consider a modified Lagrangian defined by where ρ : X → [0, ∞] is a function and ν is a probability measure supported on K .Let P(K ) be the collection of these probability measures supported on K equipped with the topology of weak* convergence.In order to obtain the required continuity, we will restrict to ρ ∈ G, with G := {ρ : X → [0, 1] : ρ compactly supported and continuous in X }.
The set G is equipped with the topology of uniform convergence.Then : G × P(K ) → ‫ޒ‬ is a functional with two properties: ( • , ν) is convex and continuous for each ν ∈ P(K ), and (ρ, • ) is concave and upper semicontinuous for each ρ : X → [0, 1].Further P(K ) is compact and convex in the weak* topology and G is a convex subset.By Sion's minimax theorem, see, e.g., statement in [David and Eriksson-Bique 2020, Theorem 4.7], we have sup We can compute inf ρ∈G sup ν∈P(K ) (ρ, ν) ≥ 0. Indeed, given any ρ ∈ G, we can use the definition of modulus to find a γ ∈ K with γ ρ ds ≤ ∥g∥ p /Mod p (K ) −1/ 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 (ρ, ν) ≥ 0. But, up to showing that this supremum is attained, there must be some η ∈ P(K ) for which we get inf ρ∈G (ρ, η) ≥ 0. After unwinding the definition of a q-plan, and an application of Radon-Nikodym on X , the measure η is our desired q-plan.□

2B. Sobolev spaces and functions
Let LIP b (X ) be the collection of Lipschitz maps f : X → ‫ޒ‬ with bounded support.
Definition 2.4.Let f : X → ‫ޒ‬ ∪ {±∞} 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 γ , if γ g ds < ∞ and We say that g is an upper gradient of f if it is an upper gradient along every rectifiable curve, and a p-weak upper gradient if g is an upper gradient of f along p-a.e.rectifiable curve.
The space N 1, p (X ) is defined as all µ-measurable functions f ∈ L p (X ) which have an upper gradient g in L p (X ).The (semi-)norm on this space is defined as where the infimum is taken over all L p -integrable upper gradients g of f.The theory of these spaces was largely developed in [Shanmugalingam 2000]; see also [Heinonen et al. 2015] for most of the classical theory.By the results there combined with an observation of [Hajłasz 2003] in the case of p = 1, one can show that there always exists a unique minimal g f , which is an upper gradient along p-almost every path, and for which 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 p loc (X ).We denote by N 1, p (X ; ‫ޒ‬ N ) ≃ N 1, p (X ) N the space of functions ϕ : Another notion of Sobolev space can be defined using q-test plans and we denote it by W 1, p (X ), with for every q-test plan η on X .The space has a norm , where the infimum is over all such functions g.We refer to [Di Marino and Squassina 2019] for details.
Note that any representative of an element of W 1, p (X ) still belongs to W 1, p (X ), whilst a representative of an element in N 1, p (X ) belongs to N 1, p (X ) if and only if they agree outside a p-exceptional set.The next theorem says that up to this ambiguity of representatives, the two approaches produce the same object.The measurability conclusion is also a corollary of [Eriksson-Bique 2023].We refer to [Ambrosio et al. 2015a] for a proof.
3. Curvewise (almost) optimality of minimal upper gradients 3A.Upper gradients with respect to plans.Given a plan η, we can speak of a gradient along its curves.
Definition 3.1.If η is a q-plan and f ∈ N 1, p (X ), then a Borel function g is an η-upper gradient if g is an upper gradient of f along γ for η-almost every γ .
The following lemma gives a notion of a minimal η-upper gradient and shows how to compute it by using derivatives along curves.
Lemma 3.2.Suppose g f is a minimal upper gradient and η is any q-plan and dπ = dη|γ ′ t | dt, with disintegration π x .Then: (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 .
Proof.Let g f be the minimal p-upper gradient for f.By Lemma A.2 there is a Borel family 0 ⊂ C(I ; X ), so that f is absolutely continuous on each curve γ ̸ ∈ c 0 with upper gradient g f and so that η( 0 ) = 0.By Corollary A.3 and Lemma A.1 there is a set N ⊂ C(I ; X ) × I so that for π (N ) = 0, and for each (γ , t) ̸ ∈ N, both ( f • γ ) ′ (t) and |γ ′ t | are defined and measurable.Let M 0 = 0 × I ∪ N. We get π (M 0 ) = 0.For each curve γ ̸ ∈ 0 the function f is absolutely continuous with upper gradient ) 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 c 0 , and thus the claim follows again from curvewise absolute continuity and by showing that the set of (γ , t) with The final claim follows since g η ′ must be a η-upper gradient for f. □ 3B.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 be less than the minimal upper gradient, and thus the task is to show that by varying over different plans η we can obtain the minimal upper gradient through maximization.In order to show equality of the result of this maximization, we argue by contradiction, that if it were not a minimal upper gradient, then we could witness this by a given plan.This is the core of the following result.It should be compared to [Ambrosio et al. 2015b, Sections 9-11], where a similar analysis is done, but with different terminology and only for p > 1.In the following statement we will need to refer to end points of curves, and thus choose the domain of curves as ), and q be its dual exponent.Let f ∈ N 1, p (X ).Suppose g is any nonnegative 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, 1] → X we have Proof.By Vitali-Carathéodory we may find a lower semicontinuous g ≥ g which is integrable and so that Ã = { g < g f } has positive measure.We will suppress the tildes below to simplify notation and thus only consider the case of g lower semicontinuous.Since g < g f on a positive measure subset, g cannot be a minimal upper gradient, and thus there must exist a family ⊂ C(I ; X ) of curves with Mod p ( ) > 0, so that (3-1) holds for each γ ∈ .Modulus is invariant under reparametrization of curves and so we may consider the subset of those γ ∈ which are Lipschitz.We want to find a plan supported on .However, the issue with this is that since p = 1 is allowed the family may not be compact, the duality of modulus and q-plans may fail.So, we seek to "cover" , 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 R of those curves in , which are contained in a ball B(x 0 , R) for some fixed x 0 ∈ 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 By passing to a subsequence of n, we may assume that n √ ε n < 1. Define as the collection of γ ∈ R so that f is absolutely continuous on γ and H 1 γ \ ∞ n=1 K n = 0.This holds for Mod p -almost every curve, since f ∈ N 1, p (X ) and since p-almost every curve spends measure zero in the null set X \ ∞ n=1 K n .Thus, Mod p ( ) > 0. Next, let m be those curves γ : I → X , which are m-Lipschitz, so that Len(γ ) ≤ m|b − a|, diam(γ ) ≥ 1/m, γ 0 , γ 1 ∈ K m and (3-1) holds.We will show that every γ ∈ contains a subcurve, up to reparametrization, in ∞ m=1 m .From this, and [Björn and Björn 2011, Lemma 1.34], it follows that Mod p ∞ m=1 m > 0, and thus there is some M > 0 so that Mod p ( M ) > 0. It is easy to show that 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., [Keith 2003, Proposition 4]).
To obtain the previous fact, consider a nonconstant curve γ ∈ .We have We may also parametrize γ by constant speed as the claim is invariant under reparametrizations.Since γ has constant speed, we know For n large enough we also have Len )) ≥ 1/n.Since the curves are parametrized by constant speed, they are n-Lipschitz.So γ ′ = γ [a n ,b n ] is, up to a reparametrization, in n for n large enough, and the claim follows.
Fix M > 0 so that Mod p ( M ) > 0. Next, choose δ < min(Mod p ( M ), 1).Define Thus, by subadditivity of modulus, see, e.g., [Fuglede 1957, Theorem 1], By Lemma 3.4, since M is closed, the family M t ⊂ M is a compact family of curves in a complete space.Then, by Proposition 2.3 there exists a q-plan η supported on M t .Each curve γ ∈ M t satisfies (3-1), and thus the claim follows. □ 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 ε (A) := a∈A B(a, ε) for ε > 0. Lemma 3.4.Suppose that K n are compact sets, η n > 0 constants with lim n→∞ η n = 0, L > 0 and let ⊂ C(I ; X ) be a closed family of curves in a complete space X .Let n,L be the family of curves γ ∈ for which Len(γ ) ≤ L, diam(γ ) ≥ 1/L and which are L-Lipschitz, with γ 1 X \K n ds ≤ η n for each n ∈ ‫.ގ‬ Then n,L is compact.Let γ ∈ n,L .We may suppose that η n < 1/(2L) by restricting to large enough n.Then, we have for each Thus γ ∩ K n ̸ = ∅.Moreover, if t ∈ I, and d(γ t , K n ) = s, then there will be a subsegment of length at least min(s, diam(γ k )/2) in X \ K n .This gives min(s, diam(γ To run the usual proof of Arzelà-Ascoli, since we have equicontinuity with the Lipschitz bound, we only need to show that for each fixed t ∈ I the set A t = {γ t : γ ∈ n,L } is precompact.However, since X is complete, it suffices to show that A t is totally bounded.Fix ε > 0. We concluded that d(γ , K n ) ≤ η n for all n ∈ ‫.ގ‬ Thus, we have for some large n that η n ≤ ε/4 and that compact, it is totally bounded, and the claim follows by covering K n by finitely many ε/4 balls and noting that ε > 0 is arbitrary.□ Proof of Theorem 1.1.Let q be the set of all q-plans, and for each η ∈ q , with its disintegration being given by π x , define .
Finally, define 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 where η # is the barycenter of η, which is absolutely continuous with respect to µ.Let η = ∞ n=1 a −1 n 2 −n η n .This will be a plan with g η ≥ g η n for each n by Lemma 3.2.For µ-almost every x, we have Since g f is a p-weak upper gradient, Lemma 3.2 gives |D π f | ≤ g f .Suppose for the sake of contradiction then that |D π f | < g f on a positive measure subset.Then, by Lemma 3.2, there exists a plan η ′ so that However, by the definition of a plan upper gradient, we have for η ′ almost every curve that 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 which contradicts the above inequalities.Finally, since |D π f | = g η = g f , we must have µ| D ≪ η# .Indeed, otherwise there would be a non-null Borel set E ⊂ D for which µ(E) > 0 and η# (E) = 0.However, then g η| E = 0, contradicting the equality µ-almost everywhere. □ We now prove Corollary 1.3.
Proof.Let f ∈ N 1, 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 ε > 0, and define x is the disintegration for η ′ , we have π x (B) > 0 for 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 support of η ′ we include also its reversal, and r preserves η. □ 3C.Alternative curvewise characterizations of upper gradients when p > 1.In this subsection we assume that p, q ∈ (1, ∞) satisfy 1/ p + 1/q = 1 and prove a variant Theorem 1.1 using test plans representing gradients, introduced by Gigli.Given f ∈ N 1, p (X ), a q-test plan η represents g f if where A test plan η representing the gradient of a Sobolev map f ∈ N 1, p (X ) is concentrated on "gradient curves" of f in an asymptotic and integrated sense.We refer to [Gigli 2015;Pasqualetto 2022] for discussion of the definition we are using here.The following result of Gigli states that Sobolev functions always possess test plans representing their gradient.In the statement, P q (X ) denotes probability measures ν on X with d(x 0 , x) q dν(x) < ∞ for some and thus any x 0 ∈ X .
We now state the main result of this subsection.
Theorem 3.6.Let f ∈ N 1, p (X ) and g f be a Borel representative of the minimal p-weak upper gradient of f, with D := {g f > 0} of positive µ-measure.Let η be a q-test plan representing g f with µ| D ≪ e 0 * η ≪ µ| D .
For the proof, we present the following three elementary lemmas.Define and +∞ otherwise.The following observation is essentially made in [Pasqualetto 2022, Lemma 1.19] (we are using different notation for our purposes).See Lemma A.1(3) for the Borel measurability of the functionals in the claim.
Lemma 3.10.Let E ⊂ X be a Borel set, t > 0, and let Then D E,t → (χ E g p f ) • e 0 in L 1 (η). Proof.Define This estimate, together with the Hölder inequality and Lemma 3.7, yields lim sup Since s → h • e s is continuous in L p (η) whenever h ∈ L p (µ) (see [Gigli and Pasqualetto 2020, Proposition 2.1.4])all terms above tend to zero, proving the claimed convergence.□ Proof of Theorem 3.6.Let N be the negligible set is as in Corollary A.3.The function is Borel.Let η represent g f and satisfy µ| D ≪ e 0 * η ≪ µ| D .Fix ε > 0, let δ > 0 be as in Lemma 3.8, and set δ 0 = min{ε, δ}.We define the Borel function Note that see (3-2) and Lemma 3.8.Once we show that dπ g(γ t ) p/q 1 − 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 | dt dη ≤ (1 + ε)g(γ t ) p/q dt dη and thus It remains to prove that µ| D ≪ e * π .Let E ⊂ D be a Borel set with µ(E) > 0. Then e 0 * η(E) = η({γ : , see Lemmas 3.7 and 3.10 respectively, there exists a sequence T n → 0 such that for η-almost every γ ∈ e −1 0 (E) the functions h γ (s) ) satisfy the hypotheses of Lemma 3.9.It follows that for η-almost every γ ∈ e −1 0 (E) the sets have positive measure for all n.Notice that, for η-almost every γ , if Having positive length in U is more restrictive than assuming that γ −1 (U ) has positive measure.We will also discuss p-weak differentials and covector fields of the form d f : U → ‫ޒ(‬ N ) * or ξ : U → ‫ޒ(‬ N ) * for measurable subsets U ⊂ X .The values of such a map at x ∈ U are denoted by d x f, ξ x , respectively.
4B. Canonical minimal gradients.Let p ≥ 1 and N ≥ 0 be given.For the next three lemmas we fix 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 Here {π x } is the disintegration of dπ := |γ ′ t | dη dt with respect to the evaluation map e.
We prove the claim in the statement for arbitrary ξ ∈ ‫ޒ(‬ N ) * .Let (ξ n l ) l ⊂ A be a sequence with On the other hand, for p-a.e.curve γ , we have Since η is a q-plan with µ| D ≪ η # , this implies lim sup for µ-a.e.x ∈ D. Thus ξ (x) = lim l→∞ ξ n l (x) for µ-a.e.x ∈ D. Since ξ = 0 = |D(ξ • ϕ)| p µ-a.e. on X \ D, the proof is completed.□ In the next two lemmas we collect the properties of the Borel function constructed above.
Fix a countable dense set A ⊂ ‫ޒ(‬ N ) * and one ξ ∈ A. We have that (ξ, x) is a weak upper gradient for ξ • ϕ, so there is family of curves ξ so that ξ • ϕ is absolutely continuous with upper gradient |D(ξ • ϕ)| p on each γ ∈ i , and so that Mod p ( \ i ) = 0. Let ′ = ξ ∈A ξ , whose complement B = AC(I ; X ) \ ′ has null p-modulus.
Since ζ • ϕ has as upper gradient ζ (x) on γ for each ζ ∈ A, by considering a sequence ξ l in A converging to ξ ̸ ∈ A we obtain the same conclusion.
Finally, fixing an absolutely continuous curve γ ̸ ∈ B there is a full measure set F 1 γ , where the components of ϕ • γ t are differentiable at t ∈ F 1 γ .Both sides of ( 4) are continuous and defined in ξ on the set F 1 γ .Since ξ (x) is an upper gradient for ξ • ϕ along γ , there is a full measure subset F γ ⊂ F 1 γ , where the inequality holds for ξ ∈ A. Continuity then extends it for all ξ ∈ ‫ޒ(‬ N ) * and t ∈ F γ and the claim follows by setting □ Next, we collect some basic properties of the canonical minimal gradient.Let be the map given by (4-1).
(3) If ϕ is p-independent on U and f ∈ N 1, p (X ), then the p-weak differential d f with respect to (U, ϕ), if it exists, must be unique.On the other hand, if ξ n , for n ∈ ‫,ގ‬ is a countably dense collection in the unit sphere of ‫ޒ(‬ N ) * , then we have almost everywhere.By intersecting the sets where this holds for different ξ n and since the collection is countable, we have that these hold simultaneously on a full-measure set.Specifically, inf n∈‫ގ‬ ξ n (x) ≥ ess inf ∥ξ ∥ * =1 |D(ξ • ϕ)| p .By Lemma 4.2, we have that ξ → ξ (x) is Lipschitz.Thus, almost everywhere, which gives the claim.
The final claim about uniqueness follows since, if d i f were two p-weak differentials for i = 1, 2, then we could define f and otherwise ξ x = 0. We then get immediately from the definition and the second part that x (ξ ) = 0 for µ-a.e.x ∈ U.This would contradict independence.□ 4C.Charts.The presentation here should be compared to [Cheeger 1999, Section 4], and specifically to the proof of Theorem 4.38 there, 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.
In the converse direction, (4-2) implies, for any f e. γ ∈ AC(I ; X ) and a.e.t ∈ γ −1 (U ).Then, by Theorem 1.1, together with measurability considerations from Corollary A.3, this gives |D f | p = 0 µ-a.e. on U for every f ∈ LIP b (X ), showing that (U, 0) is a 0-dimensional p-weak chart.□ For the remainder of this subsection we assume that N ≥ 1 and that (U, ϕ) is an N -dimensional chart of X .Denote by the canonical minimal gradient of ϕ (see Lemma 4.1).
Lemma 4.5.The function ξ → x (ξ ) is a norm on ‫ޒ(‬ 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 : for a.e.t ∈ γ −1 (U ), for p-a.e.absolutely continuous curves γ in X .The map d f is uniquely determined a.e. in U and satisfies 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, and conversely, the first follows by Lebesgue differentiation.Further, it would be enough to consider only p-a.e.curve 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 −1 (U ) and both sides of the equation vanish.
Observe that, if V j := U ∩ {I (ϕ) ≥ 1/j}, the sets U j := V j \ i< j V i partition U up to a null-set and we have an isometric identification ) is a Banach space.Recall, that an ℓ p -direct sum of Banach spaces B i with norms ∥ • ∥ B i with countable index set I is defined by for some f ∈ N 1, p (X ) and ξ ∈ p (T * U ). Then ξ is the (uniquely defined) differential of f in U, and In particular, (ξ , Proof.By Lemma 4.5 and Fuglede's theorem [1957, Theorem 3(f)] (applied to the sequence of functions 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), This shows that ξ is the differential of f, and uniqueness follows from Lemma 4.3(3).The identity for every f ∈ F and if for every ε > 0, there is an δ > 0 and a positive measure subset ε , so that for any measurable set E with µ(E) ≤ δ, we have c ε ∪E | f | p dµ ≤ ε for each f ∈ F. By the Dunford-Pettis theorem a set of L 1 functions is equi-integrable if and only if it is sequentially compact; see for example [Dunford and Schwartz 1958, Theorem IV.8.9].
Remark 4.8.It follows from (4-4) that, if (ξ n ) n ⊂ p (T * U ) is equi-integrable, then there exists ξ ∈ p (T * U ) such that ξ n ⇀ ξ weakly in p (T * U ) up to a subsequence and, by Mazur's lemma, that a convex combination of ξ n 's converges to ξ in p (T * U ). Indeed, the p > 1 case is direct and the p = 1 case uses the Dunford-Pettis argument above.
Next, we show that any Sobolev function f ∈ N 1, p has a uniquely defined differential with respect to 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 1, p (X ).Lemma 4.3(3) implies that d f, if it exists, is a.e.uniquely determined on U. The proof above also yields the following corollary.Note that, while the claim initially holds only after passing to a subsequence, since the limit is unique, the convergence holds along the full sequence.
Corollary 4.9.Let (U, ϕ) be a p-weak chart of X .Suppose that f ∈ N 1, p (X ) and Using Lemma 4.3 we prove that the differential satisfies natural rules of calculation.The following properties are stated for f, g ∈ N 1, p (X ), but they would equivalently hold if we assumed only we have the local assumption f, g ∈ N 1, p loc (X ).For the following, recall that if A ⊂ ‫ޒ‬ is a measurable set, then t is a density point of A if Here | • | denotes the Lebesgue measure of the set.
Proposition 4.10.Let (U, ϕ) be an N -dimensional p-weak chart of X , f, g ∈ N 1, p (X ), and F : X → Y be a Lipschitz map into a metric measure space (Y, d, ν) with F * µ ≤ Cν for some C > 0: (1) If (V, ψ) is a p-weak chart with ϕ| U ∩V = ψ| U ∩V then the p-weak differentials of f with respect to both charts agree µ-a.e. on U ∩ V.
(2) If f | A = g| A for some A, then d f = dg µ-a.e. on A ∩ U. ( holds µ-a.e. on U. (5) Let (V, ψ) be an M-dimensional p-weak chart of Y with µ(U ∩F −1 (V )) > 0. For µ-a.e.U ∩F −1 (V ) there exists a unique linear map D x F : ‫ޒ‬ N → ‫ޒ‬ M satisfying the following: if h ∈ N 1, p (Y ) and E is the set of y ∈ V where the differential d y h does not exist, then µ(U ∩ F −1 (E)) = 0 and . By the third part in Theorem 4.11 we have that V (ν ′ ) x = V (ν) x for ν ′ -a.e.x ∈ A. Further ϕ −1 (A) ⊂ U, so ϕ is still p-independent on ϕ −1 (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 by dropping 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 ‫ޒ‬ N , and thus there are vectors perpendicular to these.Since x → V (ν) ϕ(x) is Borel, we can choose a Borel map x → ξ x ∈ ‫ޒ(‬ N ) * so that ξ x is a unit vector that vanishes on V (ν) ϕ(x) for µ-a.e.x ∈ U (see, e.g., [Bogachev 2007, Theorem 6.9.1], which is an instance of a Borel selection theorem).Let U ⊂ U be the full measure subset where these properties hold for every x ∈ U. Now, by Lemma 4.12 we have for p-a.e.curve γ that (ϕ•γ ) ′ t ∈ V (ν) ϕ(γ t ) for almost every t ∈ γ −1 (U ).The set U \ U has null measure, and thus + U \ U has null modulus.Thus, for p-a.e.curve γ ∈ AC(I ; X ) and a.e.t ∈ γ −1 (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 −1 (U ) and such curves γ .By part (2) of Lemma 4.3, we have that I (ϕ) ≤ x (ξ x ) = 0 for µ-a.e.x ∈ U.This contradicts p-independence and proves the claim.□ 4F.Sobolev charts.By definition, a p-weak chart is a Lipschitz map which has target of maximal dimension with respect to Lipschitz maps.The notions of p-independence and maximality, however, are 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, the existence of maximal p-independent Sobolev maps also guarantees the existence of p-weak chart of the same dimension.This follows from the energy density of Lipschitz functions, see [Eriksson-Bique 2023], together with results of the previous subsection.
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 ; ‫ޒ‬ 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 [Keith 2004a, Proposition 3.1], there is a partition of U up to a null-set by p-weak charts V i , i ∈ ‫,ގ‬ of dimension ≤ N. By [Eriksson-Bique 2023, Theorem 1.1], Corollary 4.9 (with a diagonal argument) and Mazur's lemma we have that, for each component If and n denote the canonical minimal gradients associated to ϕ and ψ n := (ψ n 1 , . . ., ψ n N ), we have

It follows that lim
completing the proof, since ψ n is p-independent and maximal on the set {I (ψ n ) > 0}.□ Another condition in this context is strong maximality: a map ϕ ∈ N 1, p (X ; ‫ޒ‬ N ) is strongly maximal in U ⊂ X if no positive measure subset V ⊂ U admits a p-independent Sobolev map into a higherdimensional Euclidean space.This condition excludes not only Lipschitz, but also Sobolev functions into higher-dimensional targets, and is thus a priori stronger than maximality.However, it follows 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.
4G. p-weak charts in Poincaré spaces.Recall that a metric measure space X = (X, d, µ) is said to be a p-PI space if µ is doubling, and X supports a weak (1, p)-Poincaré inequality: there exist constants C, σ > 0 so that, for any f ∈ L 1 (X ) with upper gradient g, we have for all balls B ⊂ X of radius r .Here h B = − B h dµ = (1/µ(B)) B h dµ for a ball B ⊂ X and h ∈ L 1 (B).The celebrated result from [Cheeger 1999] states that a PI-space admits a Lipschitz differentiable structure.We will return to this structure in Section 6B, but here recall the constructions from [Cheeger 1999, Section 4].Cheeger's paper does not employ the following terminology, but it simplifies and clarifies our presentation.
Given a Lipschitz map ϕ : X → ‫ޒ‬ N and a positive measure subset U ⊂ X the pair (U, ϕ) is called a Cheeger chart if for every Lipschitz map f : X → ‫ޒ‬ and a.e.x ∈ U there is a unique element d This equality is equivalent to (1-4).
Proof of Theorem 1.8.Let (U, ϕ) be a p-weak chart of dimension N and let f ∈ LIP(X ).Denote by the canonical minimal gradient of (ϕ, f ) : X → ‫ޒ‬ 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 ⊂ ‫ޒ(‬ N ) * .From this, and the continuity of both sides in ξ , we obtain that for µ-a.e.x ∈ U, the equality holds simultaneously for all ξ ∈ ‫ޒ(‬ N ) * .Since the p-weak differential d f is characterized by the property x (d f, −1) = 0 for µ-a.e.x ∈ U, it follows that, for µ-a.e.x ∈ U, d x f ∈ ‫ޒ(‬ N ) * satisfies (4-6).Thus (U, ϕ) is a Cheeger chart.The uniqueness follows from the equality in a similar way.□ 5.The p-weak differentiable structure 5A.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.
) is a finite-dimensional normed space so that x → |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 linear map 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 by 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 )} forms a measurable bundle over X , the transformations given by the collection {D i, j,x } constructed in Lemma 5.2.
First, we construct the transformation maps.
In the proof, we denote by ϕ i 1 , . . ., ϕ i N i the components of ϕ i .Proof.For µ-a.e.x ∈ U 1 ∩ U 2 , define for p-a.e.γ ∈ + U 1 ∩U 2 .Note that, by the uniqueness of differentials, D is the unique linear map satisfying (5-1) for p-a.e.curve.By Lemma 4.3(2) it follows that for µ-a.e.x ∈ U 1 ∩U 2 .Thus D * is an isometric embedding and in particular N 1 ≤ N 2 .Reversing the roles of ϕ 1 and ϕ 2 we obtain that N 1 = N 2 and consequently D 1,2,x := D * For any f ∈ N 1, p (X ), the identity d 1 x f = d 2 x f • D 1,2,x for µ-a.e.x ∈ U 1 ∩ U 2 follows from (5-1) and (1-6).□ Proof of Proposition 5.1.Conditions (1) and ( 2) are satisfied by Lemma 4.2.The cocycle condition follows from Lemma 5.2.□ Definition 5.3.We call the measurable bundle given by Proposition 5.1 the p-weak cotangent bundle and denote it by T * p X .We define T ) for almost every x ∈ U, where (U, ϕ) is an N -dimensional p-weak chart and | • | x the norm given by the canonical minimal gradient ; see Lemmas 4.1 and 4.5.The spaces T p,x are here defined pointwise almost everywhere.By considering the adjoints of transition maps in the definition above, one can patch these together to form a measurable L ∞ tangent bundle, which is dual to T * p X , whose fibers are T p,x X .The next proposition establishes the existence of a p-weak differentiable structure under a mild finite dimensionality condition.
Proposition 5.4.Suppose X is a metric measure space and {X i } i∈‫ގ‬ a covering of X with dim H X i < ∞.Then, for any p ≥ 1, 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 ) > 0.
Proof.For any Borel set U ⊂ X with µ(U ) > 0 there exists i ∈ ‫ގ‬ such that µ(U ∩ X i ) > 0. By Proposition 4.13 we have that N ≤ dim H (U ∩ X i ) whenever ϕ ∈ LIP b (X ; ‫ޒ‬ N ) is p-independent in a positive measure subset of U ∩ X i .Using [Keith 2004b, Proposition 3.1] we can cover X up to a null-set by Borel sets U k for which there exist ϕ k ∈ LIP b (X ; ‫ޒ‬ N k ) that are p-independent and p-maximal on U k .The collection {(U k , ϕ k )} k∈‫ގ‬ is a p-weak differentiable structure on X .The last claim follows by the argument above.□ 5B.Sections of measurable bundles.A measurable bundle T over X comes with a projection map π : T → X , (x, v) → x, and a section of T is a collection ω = {ω i : U i → V i } of Borel 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 almost every x ∈ U i ∩ U j .Observe that the map x → |ω(x)| x given by 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 p (T ) be the space of sections ω of T with We call p (T ) the space of p-integrable sections of T .The space p (T * p X ) is called the p-weak cotangent module.
Note that p (T ), equipped with the pointwise norm (5-2) and the natural addition and multiplication operations, is a normed module in the sense of [Gigli 2015] for all m ∈ M and g ∈ L ∞ (X ).We refer to [Gigli 2015;2018] for a detailed account of the theory of normed modules.
Next we consider the p-weak cotangent module p (T * p X ).For a p-weak chart (U, ϕ) of X and f ∈ N 1, p (X ), denote by d (U,ϕ) f the differential of f with respect to (U, ϕ).Lemma 5.2 implies that the collection of differentials with respect to different charts satisfies the compatibility condition above.
Definition 5.6.Let p ≥ 1, and suppose A is a p-weak differentiable atlas of X .For any f ∈ N 1, p (X ), the differential d f ∈ p (T * p X ) is the element in the p-weak cotangent module defined by the collection {d (U,ϕ) f : U → ‫ޒ(‬ N ) * } (U,ϕ)∈A .We record the following properties of the differential.
Proposition 5.7.Let A ⊂ X be a Borel set and F : X → Y a Lipschitz map to a metric measure space (Y, d, ν) admitting a p-weak differentiable structure, with F * µ ≤ Cν.
(3) If E is the set of y ∈ Y for which T * p,y Y does not exist, then µ(F −1 (E)) = 0 and, for µ-a.e.x ∈ X \ F −1 (E) there exists a unique linear map D x F : T p,x X → T p,F(x) Y such that for µ-a.e.x, for every h ∈ N 1, p (Y ).
(5) If f i ∈ N 1, p (X ) and there is a function f ∈ L p and a w ∈ p (T * p X ) so that lim i→∞ f i = f (x) converges in L p (X ) and d f i → w converges in p (T * p X ), then, there is a function f ∈ N 1, p (X ) so that f = f almost everywhere with d f = w.
Proof.The proofs of the first four claims follow directly from Proposition 4.10 together with the compatibility condition of sections.Indeed, one can verify the identities for each chart (U, ϕ), from which the identities follows for everywhere.
Consider now f i ∈ N 1, p (X ) which converge in L p (X ) to f ∈ L p (X ) and so that d f i converge in p (T * p X ) to w ∈ p (T * p X ).We have therefore that g i = |D f i | p = |d f i | converges in L p to g = |w|.By Fuglede's theorem [1957, Theorem 3(f)] we can pass to a subsequence so that f i → f converges pointwise and so that ) is q-maximal in U ⊂ X , then it is p-maximal.
It follows (using this dimension upper bound and [Keith 2004b, Proposition 3.1]) that if X admits a q-weak differentiable structure then X also admits a p-weak differentiable structure.We remark that the structures may be different.
For the following statement we say that a bundle map π : T → T ′ between two measurable bundles 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 of linear 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 compatibility condition 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 the family {π i,x := π i,i,x }, since these determine a unique bundle map.
Proposition 5.8.Suppose q > p ≥ 1 and 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 a linear 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 = i∈‫ގ‬ U i ∪ N, with µ(N ) = 0, and ϕ q,i ∈ N 1, p (X ; ‫ޒ‬ m i ) is Lipschitz.Assume that U i are chosen to be pairwise disjoint.As |D f | p ≤ |D f | q (a.e.) for any f ∈ LIP b (X ), any p-independent map is also q-independent.Any map ϕ ∈ N 1, p (X ; ‫ޒ‬ n ) which is p-independent on some positive-measure subset of U i must have n ≤ m i ; see Proposition 4.14.By [Keith 2004b, Proposition 3.1] and this dimension bound we can cover X by maximal p-independent maps, i.e., charts, (V j , ϕ p, j ).By considering the countable collection of sets V i ∩ U j , and reindexing, we may assume that (U i , ϕ q,i ) and (U i , ϕ p,i ) are qand pcharts, respectively.
We define the matrix A x for x ∈ U i by taking as rows the vectors d i, p ϕ k q,i for each component k = 1, . . ., m i .We define the bundle map π p,q by setting π x p,q (ξ ) = ξ • A x for µ-a.e.x ∈ U i .For each ξ we get d p (ξ • ϕ q,i ) = ξ • A x .Thus, for p-a.e.curve γ ∈ AC(I ; X ) and a.e.t ∈ γ −1 (U ) we have 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 isometric isomorphism ι : p (T * p X ) → L p (T * X ).Note that ι(d f ) = d G f for every f ∈ N 1, p (X ), establishing (6-2).To prove uniqueness, note that if A : 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. □ 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 ‫}∞{∪ގ∈‬ is the dimensional decomposition of X and µ(A ∞ ) = 0.
Let N ∈ ‫ގ‬ be such that µ(A N ) ≥ µ(V ) > 0 for some Borel set V, and v 1 , . . ., v N ∈ L p (T * X ) is a basis of L p (T * X ) on V.By possibly passing to a smaller subset of V, we may assume that there exists C > 0 for which |g| p dµ for all g = (g 1 , . . ., g N ) ∈ L ∞ . (6-3) For each k = 1, . . ., N there are sequences with {A n j,k } j a Borel partition of X and ( f n j ) ⊂ N 1, p (X ) such that v n k → v k in L p (T * X ) as n → ∞, by the definition of L p (T * X ).We set J n = {1, . . ., M n 1 } × • • • × {1, . . ., M n N } and define new partitions A n ȷ := A n j 1 ,1 ∩ • • • ∩ A n j N ,N indexed by ȷ = ( j 1 , . . ., j N ) ∈ J n .Then for all n and k = 1, . . ., N. We claim that there exists n so that ϕ n, ȷ := ( f n j 1 ,1 , . . ., f n j N ,N ) ∈ N 1, p (X ; ‫ޒ‬ N ) is p-independent on a positive measure subset of A n ȷ ∩ V for some ȷ ∈ J n .By (6-3) we have the inequality for all g = (g 1 , . . ., g N ) ∈ L ∞ , where n, ȷ is the canonical minimal gradient of ϕ n, ȷ (see Lemma 6.3).By (6-4) there exists n ∈ ‫,ގ‬ ȷ ∈ J n and a Borel set U ⊂ A n ȷ ∩ V with 0 < µ(U ) ≤ µ(A n ȷ ∩ V ) such that Proof.Let I = [a, b].Since and n,L are closed, it suffices to show precompactness.
e.) for each n, m ∈ ‫.ގ‬ Thus, we can define d f (x) = d f n (x) for x ∈ B(x 0 , n −1) with only an ambiguity on a null set.It is easy to check that d f is a differential.□ 4D.Differential and pointwise norm.Let | • | x := x and define p (T * U ) = {ξ : U → ‫ޒ(‬ N ) * Borel : ∥ξ ∥ p (T * U ) < ∞}, ∥ξ ∥ p (T * U ) := U as n → ∞, which exists by [Eriksson-Bique 2023, Theorem 1.1].By Lemma 4.5, (d f n ) n ⊂ p (T * U ) is equi-integrable.It follows that there exists ξ ∈ p (T * U ) such that d f n ⇀ ξ weakly in L p (T * U ); see 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 dg n → ξ in 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; see Lemma 4.3(3).□ Here, we use that |D(ϕ k − ψ n k )| p ≤ |Dϕ k | p + |Dψ n k | p and the L p -convergence of the right-hand side from [Eriksson-Bique 2023].
Remark 4.15.The proof of Theorem 1.8 also yields the claim under the weaker assumption Lip f ≤ ω(|D f | p ) for some collection of moduli of continuity ω (compare Theorem 1.10) since the equality Lip f = |D f | p follows from this by [Ikonen et al. 2022, Theorem 1.1].

γ|
f i − f | ds → 0 and γ |g i − g| ds → 0 for p-a.e.absolutely continuous curves γ : [0, 1] → X .Then, for all such curves, we have that| f i (γ (0)) − f i (γ (1))| ≤ γ g i ds, which converges to | f (γ (0)) − f (γ (1))| ≤ γ g ds.Thus f ∈ N 1, p .Finally, one only needs to show that w = d f.This follows by another diagonal argument and computing d f in charts using the argument from Lemma 4.7.□We finish the subsection with a proof of the density of Lipschitz functions in Newtonian spaces.Proof of Theorem 1.9.Let f ∈ N 1, p (X ).By [Eriksson-Bique 2023], there exists a sequence( f n ) ⊂ LIP b (X ) with f n → f and |D f n | p → |D f | p in L p (µ).It follows that (d f n ) ⊂ p (T * X ) is equi-integrable,and Remark 4.8 and Lemma 4.7, together with a diagonalization argument over a union of charts covering X , show that d fn → d f in p (T * X ) for convex combinations fn ∈ LIP b (X ) of f n 's.Consequently|D( fn − f )| p → 0 in L p (µ). □ 5C.Dependence of the p-weak differentiable structures on p. Suppose 1 ≤ p < q.We have that|D f | p ≤ |D f | q µ-a.e. for every f ∈ LIP b (X ), and the inequality may be strict; see [Di Marino and Speight 2015].As a consequence, if ϕ ∈ LIP b (X ; ‫ޒ‬ N Overview.Minimal weak upper gradients of Sobolev-type functions on metric measure spaces were first introduced by Cheeger [1999], building on the notion of upper gradients from [Heinonen and Koskela