Tensorization of $p$-weak differentiable structures

We consider $p$-weak differentiable structures that were recently introduced by the first and last named authors, and prove that the product of $p$-weak charts is a $p$-weak chart. This implies that the product of two spaces with a $p$-weak differentiable structure also admits a $p$-weak differentiable structure. We make partial progress on the tensorization problem of Sobolev spaces by showing an isometric embedding result. Further, we establish tensorization when one of the factors is PI.

1. Introduction 1.1.Background.A weak notion of differentiable charts in singular metric spaces first arose in Cheeger's seminal paper [7].Lipschitz differentiability charts, or Cheeger charts, have since become ubiquitous in analysis in metric spaces, with connections to rectifiability, (non-)embedding results and other topics in geometric measure theory.A Cheeger chart (U, ϕ) consists of a Borel set U ⊂ X and a Lipschitz map ϕ : X → R n such that every f ∈ LIP(X) admits a differential x → d x f : U → (R n ) * , uniquely determined for µ-a.e.x ∈ U , satisfying (1.1) f (y) − f (x) = d x f ((ϕ(y) − ϕ(x)) + o(d(x, y)).
Cheeger charts describe the infinitesimal behaviour of Lipschitz functions but are not well suited for studying Sobolev functions in the absence of additional assumptions.In [9] the first and third named authors developed p-weak charts, a further weakening of Cheeger charts.We refer to Section 2 for their definition and mention here that p-weak charts control the behaviour of Sobolev functions curvewise and exist under very mild assumptions, e.g. when the underlying space has finite Hausdorff dimension.While weaker than the notion of Cheeger charts, the existence of non-trivial p-weak charts guarantees the existence of non-negligible families of curves, and induces a pointwise norm given as the essential supremum of directional derivatives along these curves.Indeed, Sobolev functions admit a p-weak differential with respect to p-weak charts, and the minimal upper gradient is recovered as the pointwise norm of the differential.
In this paper we consider products of spaces admitting a p-weak differentiable structure and prove that they likewise admit a p-weak differentiable structure.
1.2.Statement of main results.Let X = (X, d X , µ) and Y = (Y, d Y , ν) be two metric measure spaces, i.e. complete separable metric spaces equipped with Radon measures that are finite on balls.Throughout this paper we equip the product space X × Y with the product measure µ × ν and metric (1.2) d((x, y), (x ′ , y ′ )) := (d X (x, x ′ ), d Y (y, y ′ )) , (x, y), where • is a norm on R 2 .Note that the behaviour of the norm on the first quadrant determines the metric.In the theorem below we denote by (x, y) ′ := max{|ax + by| : (a, b) = 1, a, b ≥ 0}, (x, y) ∈ R 2 , the partial dual norm of a given planar norm • .While • ′ ≤ • * in general, the equality • ′ = • * holds for l p -norms (and more generally norms satisfying (a, b) = (|a|, |b|) for (a, b) ∈ R 2 ).Roughly speaking the need for the partial (rather than the "full") dual norm comes from the fact that the metric speed of product curves does not distinguish between the direction in which each of the component curves is traversed, see estimate (4.3) in the proof of Proposition 4.2.
Theorem 1.1 yields the following immediate corollary.
Corollary 1.2.Let X and Y be two metric measure spaces which admit a p-weak differentiable structure.Then X × Y with the product metric (1.2) admits a p-weak differentiable structure.
In proving Theorem 1.1 we will use a characterization of p-weak charts in terms of existence and uniqueness of differentials in the spirit of (1.1).In the following definition, let U ⊂ X be a Borel set with µ(U ) > 0 and ϕ ∈ N 1,p loc (X; R N ).Definition 1.3.A Borel map ξ : U → R N is a p-weak differential of a function f ∈ N 1,p (X) with respect to (U, ϕ), if for p-a.e.curve γ in X.
Note that there is no uniqueness condition imposed in the definition above.We say that f ∈ N 1,p (X) admits a unique p-weak differential with respect to (U, ϕ), if the map ξ in (1.3) is unique in the following sense: if ξ ′ : U → (R N ) * is another Borel map satisfying (1.3) for p-a.e.γ, then ξ = ξ ′ µ-a.e. on U .Theorem 1.4.Let U ⊂ X be a Borel set with µ(U ) > 0 and ϕ : X → R N a Lipschitz map.The following are equivalent.
1.3.Application: Tensorization problem of Sobolev spaces.We use Theorem 1.1 and Corollary 1.2 to make partial progress on the "tensorization problem" for Sobolev spaces.The tensorization problem first appeared in [4] and was formulated using Sobolev spaces defined via test plans, which we denote here by W 1,p , see Definition 2.4.It has later been investigated e.g. in [5].Given p ≥ 1, the Beppo-Levi space J 1,p (X, Y ) consists of Borel functions f ∈ L p (X × Y ), which satisfy the following: (a1) For µ X -a.e.x ∈ X, We refer to the Appendix for the measurability of the integrand in (a3).Observe that, although the Sobolev space W 1,p and N 1,p are isometrically isomorphic, it is not trivial that one can replace W 1,p by N 1,p in (a1) and (a2), a point we address in Section 5.The space J 1,p (X, Y ) equipped with the norm The "tensorization problem" of Sobolev spaces asks whether or not the equality holds.Roughly speaking this amounts to asking whether knowledge of the directional derivatives in the X and Y directions is enough to ensure Sobolev regularity in X × Y .While the inclusion W 1,p (X × Y ) ⊂ J 1,p (X, Y ) is elementary not much else is known without additional assumptions.
Theorem 1.5.Suppose X and Y admit a p-weak differentiable structure and let f ∈ N 1,p (X × Y ).Then for µ × ν-a.e.(x, y).In particular, the embedding W Theorem 1.5 leaves open the question of equality in (1.5) but shows that the elementary inclusion W 1,p (X × Y ) ⊂ J 1,p (X, Y ) is isometric (e.g. when the spaces are finite dimensional), providing partial evidence in favour of the tensorization property.
While a full solution to the tensorization problem remains open, we are able to establish (1.5) under the additional assumption that one of the factors is a PI-space.Recall that a p-PI space is a complete doubling metric measure space supporting a weak p-Poincaré inequality, cf.[11].
Theorem 1.6.Suppose that X is a p-PI-space and Y admits a p-weak differential structure.Then W 1,p (X × Y ) = J 1,p (X, Y ) and the norms coincide.
Previously the same conclusion was (essentially) known to hold if both factors are PI-spaces, see [5,Theorem 3.4] for a proof in the case p = 2.
1.4.Acknowledgements.The first author was partially supported by the Finnish Academy under Research postdoctoral Grant No. 330048.The second author was partially supported by the Finnish Academy, Grant No. 314789.The third author was supported by the Swiss National Science Foundation Grant 182423.The authors thank Jeff Cheeger, Nicola Gigli, Enrico Pasqualetto, and Nageswari Shanmugalingam for many enlightening discussions.

Preliminaries
2.1.Product spaces.Let X and Y be complete separable metric spaces, and • a norm on R 2 .Throughout this paper we will use the product metric on X × Y defined by Horizontal curves are curves whose Y -component is constant, and their collection is denoted H([0, 1]; X × Y ).Similarly, vertical curves have constant X-component, and their collection is denoted V ([0, 1]; X × Y ).In Sections 5 and 6 we will need the notion of horizontal-vertical curves which are obtained as concatenations of horizontal and vertical curves.The formal definition is given below.

Given an absolutely continuous curve
Given a HV-curve γ = (α, β) its constant speed parametrization γ = (ᾱ, β) is also a HV-curve with the additional property that γ| I is non-constant on any open interval I ⊂ [0, 1] unless γ is a constant curve.For (non-constant) γ we may find a decomposition 0 < t 1 . . .< t n < 1, with either α or β constant on each [t i−1 , t i ], which is maximal in the following sense: for any i = 1, . . ., n and open I ⊃ [t i−1 , t i ], both ᾱ| I and β| I are non-constant (i.e.none of the intervals [t i−1 , t i ] can be enlarged while keeping one of the component curves constant).By convention constant curves have an empty decomposition.Definition 2.2.Given a curve γ, we call the unique decomposition t(γ) = {t 1 < . . .< t n } as above the collection of turning times of γ.For n ∈ N, we denote by HV n the subset of HV consisting of curves γ such that the constant speed parametrization γ has exactly n turning times.
2.2.Partial upper gradients.We refer the reader to [11,6] for a good account of modulus, line integrals along curves, weak upper gradients, and Newton-Sobolev spaces, and omit their definitions here.
Let Γ ⊂ AC([0, 1]; X) be a family of curves in a metric measure space X, and let p ≥ 1.We say that a Borel function g : X → [0, ∞] is an upper gradient of a Borel function f : X → R along Γ, if the upper gradient inequality gds, 0 ≤ s < t ≤ 1 holds for every γ ∈ Γ.We moreover say that g is a p-weak upper gradient along Γ if (2.1) holds for p-a.e.curve γ ∈ Γ.If Γ = AC([0, 1]; X) we say that g is an upper gradient (resp.p-weak upper gradient) of f .We record the following result which establishes the existence of minimal weak partial upper gradients.
Proposition 2.3.Let Γ ⊂ AC([0, 1]; X) be a family of curves and f : X → R a locally integrable function which admits a p-weak upper gradient g ∈ L p loc (µ) along Γ.Then there exists a minimal p-weak upper gradient |Df | p,Γ of f along Γ.
Minimality in the claim above is intended in the sense that (a) |Df | p,Γ is a p-weak upper gradient of f along Γ, and (b) |Df | p,Γ ≤ g for any locally p-integrable p-weak upper gradient g of f .
Proof.By Fuglede's Lemma, the family of partial weak upper gradients of f is closed under (local) L p -convergence.The collection of partial weak upper gradients of f has the lattice property by argument in the proof of [11,Lemma 6.3.14].The existence of a minimal element in the lattice follows as in [11,Theorem 6.3.20].

Plans.
A plan on X is a finite measure η on C([0, 1]; X) concentrated on absolutely continuous curves.The barycenter η # of η is the measure on X defined by If η # is an L q -function -i.e.η # = ρµ for some ρ ∈ L p (µ) -we say η is a q-plan.To define test plans, denote for fixed t ∈ [0, 1].We say that η is a q-test plan, if (2.2) and there exists C > 0 such that e t * η ≤ Cµ for each t ∈ [0, 1].If q = ∞, we replace (2.2) by the requirement that η is concentrated on a family of L-Lipschitz curves, for some L. Test plans appear in the definition of the Sobolev spaces W 1,p .
Definition 2.4.Let p ≥ 1 and let q be the dual exponent of p.A function f ∈ L p (µ) belongs to the Sobolev space W 1,p (X) if there exists g ∈ L p (µ) such that for every q-test plan η on X.
We remark that the equality N 1,p (X) = W 1,p (X) holds if properly interpreted, see e.g.[9, Theorem 2.5].Next we define the restriction operation on plans.Definition 2.5 (Restriction of a plan).Let η be a plan on X and t, s ∈ [0, 1], t ≤ s.The restriction of η to [0, 1] is defined as the plan where e [t,s] : C(I; X) → C(I; X) is the restriction map satisfying for all γ ∈ C(I; X) the equality e [t,s] (γ) = γ with γ(r) = γ((1 − r)t + rs).
We record the following lemma which can be established by elementary arguments.We omit the proof.
Lemma 2.6.Let η be a q-plan and t, s ∈ [0, 1], t ≤ s.Then η| [t,s] is a q-plan.If η is a q-test plan then η| [s,t] is a q-test plan.
In Section 5 we define the concatenation of plans which is, in a sense, an opposite operation to restricting plans.
2.4.Disintegration.Disintegration of measures with respect to a Borel map is a far reaching generalization of Fubini's theorem.Although we will mostly apply it to plans and with respect to the evaluation map, we present a more general formulation below.The following theorem can be found in [2,Theorem 5.3.1].
2.5.p-Weak differentiable structure.Given a Borel set U ⊂ X of positive measure and ϕ ∈ N 1,p loc (X; R N ), we say that (U, ϕ) for some (and thus any) countable dense subset D ⊂ S N −1 .The pair (U, ϕ) is said to be p-maximal if, for all Lipschitz maps ψ ∈ LIP(X; R M ) with M > N and Borel sets V ⊂ U of positive measure, the pair (V, ψ) is not p-independent.
Definition 2.8.A pair (U, ϕ) is a p-weak chart if it is both p-independent and p-maximal.
To describe the pointwise norm associated to p-weak charts we record the following result which will be useful in the sequel.In the statement U ⊂ X is Borel and ϕ ∈ N 1,p loc (X; R N ).Theorem 2.9 (Lemmas 4.1-4.3 in [9]).There exists a q-plan η on X and a Borel set D ⊂ X with µ D ≪ η # , and {π x } the disintegration of dπ := |γ ′ t |dtdη, such that (c) For any Borel map ξ : U → (R N ) * and Borel set V ⊂ X we have that Φ x (ξ x ) = 0 µ-a.e.
x ∈ V if and only We will refer to a map Φ so that (a) in Theorem 2.9 holds as a canonical representative for the gradient of ϕ.

2.6.
From plans to test plans.In the sequel we want to consider canonical representations of gradients of Sobolev functions arising from test plans rather than plans.To achieve this we adapt arguments in [1, Theorem 8.5 and Theorem 9.4], which we present in detail here for the readers' convenience.
Proposition 2.10.Let η be a q-plan on X, q ∈ [1, ∞], and {π x } the disintegration of dπ := |γ ′ t |dtdη.Then there exists a q-test plan η on X such that Proof.First we obtain a plan with parametric barycenter in L ∞ by a suitable parametrization of η-a.e.curve (see also [1,Theorem 8.5]).Note that if q = ∞ the q-plan η already has parametric barycenter in L ∞ and thus we assume that q < ∞.

Existence of p-weak differentials
We lay some groundwork for the proof of Theorem 1.4.Fix a pair (U, ϕ), and let Φ : X ×(R N ) * → [0, ∞] canonically represent the gradient of ϕ.For µ-a.e.x ∈ U we denote by the pointwise seminorm, and write W x := (R N ) * /{ξ : |ξ| x = 0}.We also let L Define the vector space Γ p (T * U ) as the set of Borel maps ξ : is finite, with the usual identification of elements that agree µ-a.e.It is a standard exercise to show that • Γp is a norm making Γ p (T * U ) a Banach space.Moreover we let Γ p,loc (T * U ) the collection of all Borel maps ξ : for p-a.e.curve.By the definition of Φ this yields |ξ(x)| x ≤ g f (x) µ-a.e.x ∈ U .Thus g ≤ g f .The identity (1.3) also yields that for p-a.e.curve γ we have for a.e.t ∈ γ −1 (U ).Thus g is a p-weak upper gradient, and g ≥ g f and (2) follows.To prove (1) note that, since |ξ ′ − ξ| = 0 µ-a.e. on U by definition, it follows that ξ((ϕ Lemma 4.3(2)].The claim in (1) follows directly from this.Lemma 3.2.Let f ∈ N 1,p (X).Assume that there exist f i ∈ N 1,p (X), C i ⊂ X, for each i ∈ I in a countable index set I such that (1) f i admits a p-weak differential with respect to (U, ϕ) and Then f admits a p-weak differential with respect to (U, ϕ).
Proof.Let ξ i be a p-weak differential of f i for each i ∈ I. Identify I with a subset of N and define the sets Let J be the set of j ∈ I for which µ(W j ) > 0. Then {W j } j∈J is a partition of U up to a null-set and f | W j = f j | W j for all j ∈ J.We claim that is a p-weak differential of f with respect to (U, ϕ).Indeed, note that p-a.e.curve γ in X has the following properties: . For any such γ we have that for some j, for almost every t ∈ γ −1 (U ).This proves the claim.
Remark 3.3.Lemma 3.2 shows in particular the local nature of p-weak differentials: if V ⊂ U and f 1 = f 2 µ-a.e. on V , then a p-weak differential ξ of f 1 with respect to (U, ϕ) is a p-weak differential of f 2 with respect to (V, ϕ).
Lemma 3.4.Suppose (f j ) ⊂ N 1,p loc (X) is a sequence such that f j → f in L p loc (µ) and (|Df j | p ) j is equi-integrable.If f j has a p-weak differential with respect to (U, ϕ) for each j, then f has a p-weak differential with respect to (U, ϕ).
Proof.For each j, let ξ j be a p-weak differential of f j .By Lemma 3.1 we may assume that (ξ j ) ⊂ Γ p,loc (T * U ). Since the sequence (|ξ j | p ) is equi-integrable.A subsequence of (ξ j ) converges weakly (by reflexivity for p > 1 and by Dunford-Pettis for p = 1) to an element ξ ∈ Γ p,loc (T * U ). Denote by ξ j and f j the sequence of convex combinations (granted by Mazur's lemma) converging to ξ and f in Γ p,loc (T * U ) and L p loc (µ), respectively, in norm.Now we may argue as in the proof of [9, Lemma 4.7] (using Fuglede's lemma) to conclude that the identities ( for p-a.e.γ and a.e.t ∈ γ −1 (U ) pass to the limit and yield a.e.t ∈ γ −1 (U ) for p-a.e.γ.This proves that ξ is a p-weak differential of f with respect to (U, ϕ).
Remark 3.5.The proof above shows that any weak limit in Γ p (T * U ) of a sequence of p-weak differentials of the f j 's is a p-weak differential of f .We close this section by proving Theorem 1.4.
Assume that every f ∈ LIP(X) admits a unique p-weak differential with respect to (U, ϕ).It follows that (U, ϕ) is p-independent.Indeed, let Φ represent the gradient of ϕ canonically, cf.Theorem 2.9.Since any Borel map ξ : U → (R N ) * with ξ x ∈ ker Φ x is a p-weak differential of the zero function, the uniqueness of p-weak differentials with respect to (U, ϕ) implies that ker Φ x = {0} µ-a.e.x ∈ U .Theorem 2.9(d) implies that (U, ϕ) is p-independent.
It remains to show that (U, ϕ) is p-maximal.Suppose that V ⊂ U has positive measure and that ψ ∈ LIP(X; R M ) is p-independent on V .We will show that M ≤ N .Let dψ i ∈ (R N ) * be the unique p-weak differentials of the components ψ i of ψ for i = 1, . . .M .To reach a contradiction assume M > N .Then dψ i are linearly dependent, and there are Borel functions a i ∈ L ∞ (V ) so that M i=1 a i dψ i = 0, with a := (a 1 , . . ., a M ) = 0 µ-a.e. on V .But, then for p-a.e.absolutely continuous γ and a.e.t ∈ γ −1 (V ) we have By Theorem 2.9(c) this implies that Ψ x (a) = 0 for a.e.x ∈ V , where Ψ canonically represents the gradient of ψ.By Theorem 2.9(d) this is a contradiction to p-independence.

Products of charts and tensorization
4.1.Tensorization of charts.Throughout this section we fix p-weak charts (U, ϕ) and (V, ψ) of dimensions N and M in X and Y , respectively.To prove Theorem 1.1, the following two propositions will be used.
is a p-weak differential of f with respect to (U × V, ϕ × ψ), and We present the proof of Theorem 1.1 assuming Propositions 4.1 and 4.2 above, and after this prove the propositions.
Proof of Theorem 1.1.Proposition 4.2 implies that (U × V, ϕ × ψ) is p-independent and thus any pweak differentials are necessarily unique.By Theorem 1.4 it suffices to show that every f ∈ LIP(X) admits a p-weak differential with respect to (U × V, ϕ×, ψ).We do this in two steps using Lemma 3.4.
be a sequence of smooth functions with uniformly bounded Lipschitz constant converging to h pointwise.By Proposition 4.1 the functions f j := h j • (u, v) admit a p-weak differential.The sequence (f j ) is moreover uniformly Lipschitz and thus (|Df j | p ) is equi-integrable.Since h j → h locally uniformly we have that f j → f uniformly on bounded sets.By Lemma 3.4 it follows that f admits a p-weak differential, as claimed.
Note that sup N LIP(f N ) < ∞ and f N → f pointwise.By Lemma 3.4 f has a p-weak differential if f N has a p-weak differential with respect to (U × V, ϕ × ψ) for each N ∈ N. To see this, observe that there exists a Borel partition B 1 , . . ., B N of X such that for each 1 ≤ j ≤ N , and that It follows from Lemma 3.2 that f N has a p-weak differential for each N .The claim about the pointwise norm follows from Proposition 4.2, completing the proof.
The remainder of this subsection is devoted to proving Propositions 4.1 and 4.2.We start with a technical lemma.do not exist along the diagonal.In [2] this is avoided by using an upper derivative, but we need the actual derivative to find the differential.The slightly odd assumption on the existence guarantees the identity in the claim.
Proof.The absolute continuity of δ follows from the estimate (4.1) In particular, the distributional and classical derivatives of δ agree a.e.. Suppose ζ ∈ C ∞ (R), sptζ ⊂ (0, 1), and ε is small.On one hand, denoting C := A ∩ B we have On the other hand, and the right hand side tends to zero as ε → 0. Using this, dominated convergence and the fact that ∂ 1 h(t, t), ∂ 2 h(t, t) exist for every t ∈ C we may take the limit ε → 0 to obtain Since ζ is arbitrary the claim follows.
Proof of Proposition 4.1.By Lemma 3.1 the functions |du| and |dv| are p-weak upper gradients for u and v respectively.By using this observation, find curve families Γ X ⊂ AC([0, 1]; X) and Γ Y ⊂ AC([0, 1]; Y ) of zero p-modulus such that u • α and v • β are absolutely continuous and where the identities above fail for u and v, respectively.We have that and In particular, g is a p-weak upper gradient.
Proof of Proposition 4.2.We will show that Ξ((x, y), (ξ, ζ)) is a minimal p-weak upper gradient for ξ for µ-a.e. point (x, y).To show this, it suffices to consider any upper gradient Let η and η ′ be q-plans on X and Y , respectively, and D ⊂ X, D ′ ⊂ Y Borel sets such that , cf.Theorem 2.9.By Proposition 2.10 we may assume that η and η ′ are q-test plans.If and is µ × ν-a.e.defined in D × D ′ .Fix such disintegrations.
First, for a.e.(x, y) ∈ D × D ′ we have Ξ((x, y), (ξ, ζ)) = 0, and we have the trivial bound In what follows, we will concentrate on µ-a.e.(x, y) (notice that η × η ′ -almost every γ satisfies this.)A Fubini-type argument yields that for a.e.(t, s) for (a, b) ∈ G.By continuity we obtain the estimate for all (a, b) ∈ R 2 .By using Theorem 2.7, it follows that for µ × ν-a.e.(x, y) ∈ D × D ′ we have , where esssup τ is the essential supremum with respect to a measure τ .
This establishes (4.2) and consequently implies completing the proof of the proposition.

4.2.
Isometric inclusion W 1,p ⊂ J 1,p (X, Y ).With the results of Subsection 4.1 we can prove the isometric inclusion of N 1,p (X × Y ) in J 1,p (X, Y ).
Proof of Theorem 1.5.The product X × Y admits a p-weak differentiable structure with charts given by products of charts of X and Y , cf.Corollary 1.2.Thus every f ∈ N 1,p (X × Y ) has a differential df satisfying df = |Df | p .We will show that |df | (x,y) = (|d Given a p-weak chart (U, ϕ) of X and (V, ψ) of Y it suffices to prove the identity for almost every (x, y) ∈ U × V .Since (U × V, (ϕ, ψ)) is a p-weak chart, we have a local representation of the differential d (x,y) f = (a (x,y) , b (x,y) ).Since f ∈ N 1,p (X × Y ), for almost every x ∈ X, we have f x ∈ N 1,p (Y ), and for almost every y ∈ Y we have f y ∈ N 1,p (X).Thus for p-a.e.horizontal curve, i.e. for a.e.y ∈ V and p-a.e.γ ∈ AC([0, 1]; X) we have that However, since this holds for a.e.fixed y, and p-a.e.γ ∈ AC([0, 1]; X), by [9, Lemma 4.5], we have a (x,y) = d x f y for a.e.x ∈ U .In particular, this means that the map (x, y) → d x f y is measurable.Similarly, we get b (x,y) = d y f x for a.e.x ∈ X and a.e.y ∈ Y .We have obtained that It follows from Lemma 3.1(3) and Proposition 4.2 that for µ-a.e.(x, y) ∈ U × V .This completes the proof.

Properties of Beppo-Levi functions
In this section we establish a characterization of the Beppo-Levi space J 1,p in terms of Newtonian spaces.Note that the isomorphism N 1,p = W 1,p does not automatically allow one to replace W 1,p with N 1,p in the definition of J 1,p (X, Y ).The main result in this section achieves this by providing a good representative.
Recall the definition of HV-curves in Definition 2.1.In particular one obtains an equivalent definition if in (a1) and (a2) one replaces W 1,p by N 1,p .To prove Theorem 5.1 we develop a notion of concatenation of plans and apply it to plans concentrated on horizontal and vertical curves, which will henceforth be called horizontal and vertical plans, respectively.5.1.Concatenation of plans.The next definition will allow us to pass from horizontal and vertical plans to plans concentrated on HV-curves (HV plans).Here we give the definition and establish the basic properties of concatenation of plans.Definition 5.2 (Concatenation of plans).Let η and η ′ be plans on X with e 1 * η = e 0 * η ′ =: ν.Let {η x } and {η ′ x } be the disintegrations of η and η ′ with respect to the maps e 1 and e 0 , respectively.For ν-a.e.x ∈ X, set , where a : e −1 1 (x) × e −1 0 (x) → X is the concatenation map (α, β) → αβ, and define the concatenation η * η ′ of η and η ′ by Remark 5.3.Notice that in the definition of concatenation the choice of η x × η ′ x is somewhat arbitrary.One could produce new plans by choosing measurably other couplings of η x and η ′ x .In particular, for a given plan η the concatenation of the restrictions: η| [0,t] * η| [t,1] does not usually give back the original plan η.Lemma 5.4.Let η be a plan on X and s ∈ [0, 1].If both η| [0,s] and η| [s,1] are q-plans, then so is η.Moreover, we have Proof.For any bounded Borel function g : X → [0, ∞) we have 2).This proves the claim.
As a corollary we have the following.
Corollary 5.5.The concatenation of two q-plans η and η ′ , whenever defined, is a q-plan.Moreover, we have that Lemma 5.6.Suppose that η is a q-plan, s ∈ [0, 1] and that f ∈ L p (µ) and g ∈ L p (µ) satisfy the inequality Proof.By the argument in the proof of Lemma 5.4, we have that This proves the claim.

Concatenation of horizontal and vertical curves.
We now apply the lemmas from the previous subsection to HV-plans.Recall that HV ([0, 1]; X × Y ) denotes the set of all HV curves, and HV n ([0, 1; X ×Y ) the subset of HV ([0, 1]; X ×Y ) consisting of HV curves with exactly n turning times.We remark that HV 0 ([0, 1]; X × Y ) consists of the union of all horizontal and vertical curves.Our first step is to decompose plans concentrated on HV n ([0, 1]; X ×Y ).For the proof we denote by l : AC([0, 1]; X × Y ) → LIP([0, 1]; X × Y ) the map sending γ to its constant speed parametrization γ.
Lemma 5.8.Let η be a horizontal or vertical plan in Proof.We prove the claim for horizontal plans.The vertical case follows similarly.If we replace η by l * η, both sides of the inequality remaining unchanged.Thus we may assume that η-a.e.γ is constant speed parametrized.Further we may assume that η-a.e.γ is non-constant.Since η is a horizontal plan, for η-a.e.γ = (α, β) we have that β is a constant y β .Denote by h the map γ → y β and let {η y } be the disintegration of η with respect to h.Set ν := h * η.Observe that h −1 (y) = AC([0, 1]; X × {y}) ≃ AC([0, 1]; X) and we regard η y as a plan living on the metric space X × {y} whose metric is a constant multiple of the metric of X (constant independent of y).
It is not difficult to see that ν ≪ µ Y .We claim that η # y = ρ(•, y)µ X for ν-a.e.y, where ρ ∈ L q (µ X × µ Y ) is the density of η # with respect to µ X × µ Y .It follows from this that η y is a q-plan for ν-a.e.y.
For each Borel E ⊂ X and bounded Borel function Since E and g are arbitrary it follows that for ν-a.e.y the identity η # y = ρ(•, y)µ X holds.Now we have the estimate which proves the claim.
Notice that a plan η concentrated on HV 0 ([0, 1] : X × Y ) has a decomposition η = η H + η V , where η H is a horizontal and η V a vertical plan.This can be shown e.g. by disintegrating η with respect to the map P : HV 0 → {0, 1} which sends horizontal curves to 0 and vertical curves to 1 (we may by convention send constant curves to 0).Lemma 5.8 directly implies that, for f ∈ J 1,p (X, y) and g as in the claim, the inequality holds for all q-plans concentrated on HV 0 .
Let f ∈ J 1,p (X, Y ) be bounded and with bounded support.Let (f i ) be a sequence of Lipschitz functions satisfying the conclusion in (3).By reflexivity and Mazur's Lemma, a convex combination fi of a suitable subsequence of (f i ) converges in the norm of W 1,p (X × Y ) and in L p (X × Y ).Since the inclusion W 1,p (X × Y ) ֒→ L p (X × Y ) is injective, and since the sequence also converges in To construct Lipschitz approximants in the proof of Theorem 1.6 we use a so-called discrete convolution in the X-direction.First, define a Lipschitz partition on unity.For n ∈ N fix an 2 −nnet N n ⊂ X.That is, fix a set N n ⊂ X so that for each x ∈ X there is a a ∈ N n with d(a, x) ≤ 2 −n , and for each a, b ∈ N n we have d(a, b) > 2 −n .Let {ψ n a : a ∈ N n } be a Lipschitz partition of unity subordinate to the cover {B(a, 2 1−n )} so that supp(ψ n a ) ⊂ B(a, 2 2−n ).The functions ψ n a can be chosen to be C(D)2 n -Lipschitz with a constant C(D) depending on doubling of X.We will fix this partition of unity and the nets N n in the proofs below.
For a function f ∈ W 1,p (X) define the approximation (6.1) We have the following lemma.
X) is bounded, with norm bounded independent of n, and T n f → f in L p (X) for every f ∈ W 1,p (X).
Proof.That T n : L p (X) → L p (X) is bounded, and that T n f → f for all f ∈ L p (X) follows from [10, Lemma 5.2].Further, [10, Lemma 5.2] also implies that there is a constant C > 0 independent of n so that T n f is locally Lipschitz with By the p-Poincaré inequality, we have for all x ∈ B(a, 5λ2 2−n ).The balls B(a, 5λ2 2−n ) have bounded overlap by the doubling condition and thus we get for a constant C ′ = C(C, D, λ, c P I ), where D is the doubling constant of µ.
Remark 6.3.Indeed, our proof of Theorem 1.6 will use the Poincaré inequality only through the previous Lemma.One may conjecture that linear approximating operators T n exist more generally.
The crucial properties that we need are that the expression is given by a partition of unity, averages of the function, and that it is a bounded linear operator.PI-spaces are the only context where such a discrete convolution operators are known to exist.One plausible approach to tensorization of Sobolev spaces would involve constructing such convolutions more generally.
Proof of Theorem 1.6.We will verify (3) from Proposition 6.1.Let f ∈ J 1,p (X, Y ) be bounded and with bounded support.We assume that f is the good representative given by Theorem 5. Let η be any q-test plan in Y .
Since f has bounded support, the sum here is in fact finite.It follows that f n ∈ N 1,p (X × Y ).The doubling condition and p-Poincaré inequality on X easily implies that Integrating the p-th power of this estimate over Y using that the balls B(a, By the triangle inequality, and since the sum has at most C(D)-nonzero terms for any given x, we get Integrating over X, noting that ψ n a has support in B(a, 2 2−n ), and using the doubling condition yields that |d Next, consider the X-derivative.Let b ∈ N n .We may write using the linearity of the differential and the fact that ψ a n is a partition of unity.The doubling condition implies the existence of a constant C D such that, for each x ∈ B(b, 2 Appendix A. A.1.Elementary properties of disintegration.We record some elementary properties of disintegrations of plans.In the following statement, the space ACB([0, 1]) consists of absolutely continuous bijections σ : [0, 1] → [0, 1] with absolutely continuous inverse, and {π x } is the disintegration of the measure π given by dπ := |γ ′ t |dtdη with respect to e.The following properties are easy to verify from the definition by direct calculation using the uniqueness of disintegration.Thus we omit the proofs.Proof.Lemma A.1 implies that η # ≪ η # F,H ≪ η # (since f > 0 η # -a.e.) and moreover F (H σ −1 ) * π x η # − a.e.x ∈ X.
Proposition A.3.Let f ∈ J 1,p (X, Y ).For µ-a.e.x ∈ X there exists a representative of df x ∈ Γ p (T * Y ) so that (x, y) → d y f x is Borel measurable.Similarly for ν-a.e.y ∈ Y there exists a representative of df y ∈ Γ p (T * X) so that (x, y) → d x f y is Borel.
Proof.Let (U, ϕ) and (V, ϕ) be p-weak charts of dimension N and M , respectively.Let E ⊂ U , F ⊂ V be null-sets such that the pointwise norms Φ x and Ψ y are well-defined and f y ∈ W 1,p (X), f x ∈ W 1,p (Y ) whenever x / ∈ E and y / ∈ V .We may define d x f y as the unique vector ξ ∈ (R N ) * for which ξ((ϕ = 0, whenever this exists, and 0 otherwise.Here f y α is the absolutely continuous representative of f y •α if this exists and 0 otherwise.It follows that U × V ∋ (x, y) → d x f y thus defined is µ × ν-measurable.A similar argument gives the claim for d y f x .Since µ × ν is Borel regular it follows that d x f y and d y f x have Borel representatives.By the arbitrariness of (U, ϕ) and (V, ψ) the claim follows.We remark that the measurability of |Df y | p (x) and |Df x | p (y) can also be proven without using the p-weak differentiable structure.

Corollary A. 4 .
Let f ∈ J 1,p (X, Y ).For µ-a.e.x ∈ X there exists a representative of|Df x | p ∈ L p (ν) so that (x, y) → |Df x | p (y) is Borel measurable.Similarly,for ν-a.e.y ∈ Y there exists a representative of |Df y | p so that (x, y) → |Df y | p (x) is Borel.Proof.The claim follows from Proposition A.3 since Φ x (d x f y ) and Ψ y (d y f x ) are Borel representatives of |Df y | p and |Df x | p .