Korevaar-Schoen's directional energy and Ambrosio's regular Lagrangian flows

We develop Korevaar-Schoen's theory of directional energies for metric-valued Sobolev maps in the case of ${\sf RCD}$ source spaces; to do so we crucially rely on Ambrosio's concept of Regular Lagrangian Flow. Our review of Korevaar-Schoen's spaces brings new (even in the smooth category) insights on some aspects of the theory, in particular concerning the notion of `differential of a map along a vector field' and about the parallelogram identity for ${\sf CAT}(0)$ targets. To achieve these, one of the ingredients we use is a new (even in the Euclidean setting) stability result for Regular Lagrangian Flows.


Introduction
In the seminal paper [15], Eells-Sampson proved Lipschitz regularity of harmonic maps from a manifold M to a simply connected manifold N with non-positive sectional curvature, the estimate being in term of a bound from below on the Ricci curvature and an upper bound on the dimension of the source manifold. A crucial step in their argument is the proof of the now-called Bochner-Eells-Sampson inequality, namely: and then a Moser's iteration argument gives that |du| HS is locally bounded from above (the upper dimension bound comes into play in the constants appearing in this process), which was the claim.
Given that the final estimate does not depend on the smoothness of M, N but only in the stated curvature bounds, it is natural to wonder whether such smoothness can be removed. This problem attracted the attention of several mathematicians, see in particular [22], [24], [28] and the survey [23] for an overview on this and related topics. We remark that given the kind of assumptions in Eells-Sampson work, the natural non-smooth class of spaces for which such Lipschitz regularity is expected to hold is that of RCD(K, N ) spaces as source and CAT(0) ones as target; so far this generality has been out of reach.
This paper is part of a bigger project aiming at reproducing (1.1) in such fully synthetic setting, see also [21] and [14] for other contributions in this direction. The purpose of the current manuscript is to generalize part of Korevaar-Schoen's theory in [24] to the case of source spaces which are RCD. Specifically, one of the definitions proposed in [24] is that of 'map from a smooth manifold to a metric space which is Sobolev along a given direction': we adapt this construction to the case of RCD source and postpone to a future contribution the study of what in [24] has been called 'total energy functional'. Our main results here are: i) We obtain new stability results for Regular Lagrangian Flows both on RCD spaces and in the Euclidean setting, see Theorems 3.4, 3.8. ii) We reproduce the theory of what we call Korevaar-Schoen (Sobolev) space relying on the aforementioned concept of Regular Lagrangian Flow. In particular we introduce the Korevaar-Schoen space KS p Z (Ω, Y) of maps from Ω ⊂ X to Y which are Sobolev along the vector field Z, and for u ∈ KS p Z (Ω, Y) we define the quantity |du(Z)| which plays the role of the modulus of the differential of u applied to Z (and corresponding to the p-th root of the directional energy density in [24]). See Section 4.2. iii) Using our stability result for RLF we prove the 'triangle inequality' |du(α 1 Z 1 + α 2 Z 2 )| ≤ |α 1 | |du(Z 1 )| + |α 2 | |du(Z 2 )| iv) We show that for u ∈ KS p Z (Ω, Y) not only the quantity |du(Z)| is well defined, but also the differential du(Z) of u applied to Z makes sense, see Definition 4.14. v) Using the previous point and a duality argument we show that under some kind of Sobolev condition on the target space Y, we also have the parallelogram identity (1.3) |du(Z 1 + Z 2 )| 2 + |du(Z 1 − Z 2 )| 2 = 2 |du(Z 1 )| 2 + |du(Z 2 )| 2 , see Theorem 4.19. According to [14], CAT(0) spaces have the required condition.
Let us briefly comment the above results. In (i) the relevant notion of convergence of the underlying vector fields is that of 'weak convergence in time and strong in space', see Definition 3.3. Previous results in this direction (see [2,Remark 5.11]) required quantitative estimates on the regularity of the flows which are not available neither in the RCD setting (but see [12], [11]) nor in the Euclidean one for BV vector fields (but see [10]). More recent contributions [1] avoid the use of such quantitative estimates, but still our setting was not covered.
To the expert's eye, what claimed in (ii) is perhaps not so surprising, as it is by now clear that the concept of Regular Lagrangian Flow provides the correct replacement for the notion of flow of a vector field in a non-smooth environment. Even so, let us mention that our presentation offers some (marginal) improvement w.r.t. the original one in [24], see in particular Definition 4.6 and compare to the original proof of the absolute continuity of the directional energy densities.
For what concerns the triangle inequality mentioned in (iii), we can obtain it under the only assumption that the map u is in KS p Z1 (Ω, Y) ∩ KS p Z2 (Ω, Y), without needing a control of the total energy as in [24] (and indeed we won't mention total energy at all in this manuscript). In particular, even in the case of smooth source space, our result strengthen previously existing ones. This is possible thanks to a kind of Trotter-Kato formula for Regular Lagrangian Flows that we obtain as a corollary of the stability results in (i), see Proposition 3.5.
The definition in (iv) makes use of the theory of L 0 -normed modules as tools to develop firstorder calculus on metric measure spaces as proposed in [17]. More in detail, our approach for defining du(Z) should be seen as an adaptation to the current framework of the recent construction of differential of a metric-valued Sobolev map proposed in [21].
Finally, the proof of the parallelogram identity (1.3) is perhaps what conceptually differs the most from the approach in [24]. Indeed, in [24] it is observed that CAT(0) spaces have a sort of metric parallelogram (in)equality and this information is directly exploited to obtain (1.3); here, instead, in some sense we decouple the study of the geometry of the target from the one of Sobolev maps valued in it. More precisely, thanks to the existence of a sort of linear differential (point (iv)) we can easily prove that if the target space (Y, d Y ) is so that 'for any Radon measure µ on it the Sobolev space W 1,2 (Y, d Y , µ) is Hilbert' (see Definition 4.18), then necessarily (1.3) holds. Thus here the discussion is fully at the 'Sobolev' level. The question is then evidently whether there are spaces Y as above, and in particular if CAT(0) spaces have this property: the affirmative answer (valid more generally for locally CAT(k) spaces) has been obtained in [14].
We conclude this introduction remarking that it seems impossible to obtain the desired Lipschitz regularity of harmonic maps from RCD to CAT(0) spaces fully mimicking the approach in [24]. The problem is that Regular Lagrangian Flows are not Lipschitz in general (because typically there are not Lipschitz vector fields on RCD spaces) and as such they cannot be used in the same spirit as in [24] to provide any kind of Euler's equation for our minimizers of the energy functional. This is one of the reasons that led to the attempt of establishing the 'full' inequality (1.1) rather than focussing 'only' on its version for harmonic maps (1.2).
Acknowledgments This research has been supported by the MIUR SIR-grant 'Nonsmooth Differential Geometry' (RBSI147UG4).

Sobolev calculus.
To keep the presentation short we assume that the reader is familiar with the concept of Sobolev functions on a metric measure space ( [13], [27], [5], [4]), with that of L 0 -normed modules and differentials of real valued Sobolev maps and with second order calculus on RCD spaces ( [17], [16]).
Here we only recall those concepts we shall use most frequently. For a generic metric space (X, d) we shall denote by Lip(X), Lip bs (X) the spaces of real valued Lipschitz functions and Lipschitz functions with bounded support, respectively. The Lipschitz constant of f ∈ Lip(X) will be denoted Lip(f ) ∈ [0, +∞); the local Lipschitz constant lipf : X → [0, ∞) is defined as We shall most often work with a metric measure space (X, d, m) which is complete and separable as metric space and equipped with a non-negative and non-zero Radon measure giving finite mass to bounded sets.
For f ∈ W 1,p (X) we recall that there is a minimal, in the m-a.e. sense, non-negative function G ∈ L p (m) for which the situation in Definition 2.1 occurs: it will be denoted |Df | and called minimal weak upper gradient. One can check that Now suppose that m ′ is another Radon measure on X giving finite mass to bounded sets and such that m ≤ m ′ . Then, given p ∈ (1, ∞), it is clear that L p (m ′ ) ⊂ L p (m) with continuous inclusion, thus a direct consequence of Definition 2.1 above and of minimal weak upper gradient is that and where with |D m f |, |D m ′ f | we denoted the minimal weak upper gradients in W 1,p (X, d, m), The concept of L 0 (m)-normed module is introduced in order to 'extract' a notion of differential from that of minimal weak upper gradient: Theorem 2.2 (Cotangent module and differential). With the above notation and assumptions, there is a unique (up to unique isomorphism) couple (L 0 (T * X), d) with L 0 (T * X) being a L 0 (m) normed module, d : W 1,2 (X) → L 0 (T * X) linear and such that: |df | = |Df | m-a.e. for every f ∈ W 1,2 (X) and {df : f ∈ W 1,2 (X)} generates L 0 (T * X).
Among the various constructions related to L 0 -normed modules, we shall make use of the one of pullback: The module u * M is called the pullback module and [u * ] the pullback map. These can also be characterized by the following universal property: Proposition 2.4 (Universal property of the pullback). With the same notation and assumptions as in Theorem 2.3 above, let V ⊂ M a generating subspace, N a L 0 (m X )-normed module and . Then there exists a unique L 0 (m X )-linear and continuous map These properties of pullbacks have been studied in [17], [16] for maps satisfying u * m X ≤ Cm Y ; the generalization to the case of L 0 -normed modules has been considered in [20] and [9].
Finally, let us present the construction of the 'extention functor'. Informally speaking, it might happen that one deals with a measure µ ≪ m and with a L 0 (µ)-normed module M and would like to think M as L 0 (m)-normed module, where its elements are 0 on those regions which µ does not see. The extension functor formalizes this construction.
Thus let M be a L 0 (µ)-normed module with µ ≪ m. Notice that we have a natural projection/restriction operator proj : L 0 (m) → L 0 (µ) given by passage to the quotient up to equality µ-a.e. and a natural right inverse of it, namely an 'extension' operator ext : L 0 (µ) → L 0 (m) which sends f ∈ L 0 (µ) to the function equal to f m-a.e. on { dµ dm > 0} and to 0 on { dµ dm = 0}. Then we put Ext(M ) := M as set, define the multiplication of v ∈ Ext(M ) by f ∈ L 0 (m) as proj(f )v ∈ M = Ext(M ) and the pointwise norm as ext(|v|) ∈ L 0 (m). We shall denote by ext : M → Ext(M ) the identity map and notice that in a rather trivial way we have In what follows we shall always implicitly make this identification.
For the definition of RCD(K, ∞) space see [6] and for the second order calculus see [17]. Recall that the space Test(X) of test functions (introduced in [26]) is defined as and that for f ∈ Test(X) we have ∇f ∈ L ∞ ∩ H 1,2 C (T X) with ∇(∇f ) = Hess(f ) (having freely identified tangent and cotangent modules via the Riesz isomorphism).
Finally recall the following form of Leibniz rule: for v ∈ L ∞ ∩H 1,2 C (T X) and w ∈ L ∞ ∩W 1,2 C (T X) we have 2.2. Sobolev and absolutely continuous curves. We recall here some basic properties of Sobolev and absolutely continuous curves with values in a metric space. Throughout this section, (Y, d Y ) will be a complete metric space.
Definition 2.5 (Absolutely continuous curves). A curve γ : [0, T ] → Y is said to be absolutely continuous provided there is f ∈ L 1 ((0, T )) such that it defines a function in L p ((0, T )) and is the least -in the a.e. sense -function f for which (2.5) holds.
We now turn to the definition of Sobolev curves and in order to do so we begin by spending few words on metric-valued L p spaces. Let (X, d, m) be a metric measure spaces as before (i.e. complete, separable and with m finite on bounded sets) and (Y, d Y ) a complete space.
For p ≥ 1 the space L p (X, Y) consists of those (equivalence class up to m-a.e. equality) Borel maps u : X → Y which are essentially separably valued, i.e. for some negligible set N ⊂ X we have that u(X \ N ) ⊂ Y is separable, and satisfying´d p Y (u(x),ȳ) dm < ∞ for some, and thus any, y ∈ Y.
The space L p (X, Y) is equipped with the distance It is easy to see that L p (X, Y) is complete w.r.t. this distance and separable if (Y, d Y ) is so. Moreover, arguing as for the study of so-called 'strong measurability' of Banach-valued functions, it is not hard to check that where 'simple' means 'attaining a finite number of values.
We now come back to the study of Sobolev curves and consider the above construction for X := [0, T ], T > 0 equipped with the canonical distance and measure. For In order to study the properties of Sobolev curves, let us first study the functionals E p,ε : ii) Let ε ∈ (0, T ) and λ i ∈ [0, 1], i = 1, . . . , n, n ∈ N, with i λ i = 1. Then Noticing that the triangle inequality on Y gives |d for every a, b, c, d ∈ Y and using again the triangle inequality in L p ([0, T − ε], R) we then have . This shows that the maps T ε are equiLipschitz, hence to conclude the proof it is sufficient to find a dense subset of L p ([0, T ], Y) such that for any γ in this subset it holds T ε (γ) → 0 as ε ↓ 0. It is readily checked that simple curves have this property, hence the proof is complete.
As in the real-valued case, there is a tight connection between the notions of absolutely continuous and Sobolev curves: Theorem 2.11 (Sobolev and AC curves). Let (Y, d Y ) be a complete space and p ∈ (1, ∞). Then: . Then (the equivalence class up to a.e. equality of ) γ belongs to ii-a) γ admits a continuous representative and such representative belongs to ii-c) |∂ t γ| coincides for a.e. t with the metric speed of the continuous representative of γ and given that this holds for every h ∈ (0, T ), the claim is proved.
T ] (which are uniformly bounded in L p ((0, T )) by assumption) weakly converge to some limit function g. Also, let {x n : n ∈ N} ⊂ Y be countable and dense set and define f n (t) := d Y (γ t , x n ). Then the triangle inequality ensures that f n ∈ L p ((0, T )) and |f n (t + h) − f n (t)| ≤ d Y (γ t+h , γ t ) and hence up to pass to a subsequence we can assume that for every n ∈ N the functions t → fn(t+h k )−fn(t) h k converge to some limit g n in the weak convergence of L p ((0, T )) as h k ↓ 0. The construction ensures that |g n | ≤ g and for any ϕ ∈ C ∞ c ((0, T )) we have which shows that f n ∈ W 1,p ((0, T )) with ∂ t f n = g n . It turn, it is well known -and easy to prove -that this implies that f n admits a continuous representativef n satisfying Thus for N ⊂ [0, T ] Borel, negligible and such that f n (t) =f n (t) for any t / ∈ N and n ∈ N we have which, e.g. by the absolute continuity of the integral, shows that the restriction of γ to [0, T ] \ N is uniformly continuous and thus it can be extended to a continuous curveγ which clearly satisfies By the very definition, this meansγ ∈ AC p ([0, T ], Y) and Theorem 2.6 also tells that therefore the chain of inequalities ≤ g p p is actually made of equalities. In particular, the functions dY(γ ·+h k ,γ·) h k converge to g also in norm (because L p -spaces are uniformly convex when p ∈ (1, ∞)), thus strongly in L p ((0, T )). Also, the equality |γ| p p = g p p and (2.14) force g = |γ| showing at once that the limit of dY(γ ·+h ,γ·) h as h ↓ 0 does not depend on the particular subsequence chosen and that it coincides with the metric speed of the continuous representative of γ. Final statements It is clear that the functionals E p,ε : For the Borel regularity of the 'continuous representative' map it is sufficient to show that for any c > 0 such map is continuous from As for the classical L p spaces -and with the same proof -up to pass to a subsequence we can assume that γ n t → γ t for a.e. t. Now notice that the bound grants that the curves γ n are uniformly continuous, thus from pointwise a.e. convergence we deduce uniform convergence to a limit curveγ. It is then clear thatγ is the continuous representative of γ, thus concluding the proof of the claim. Finally, the Borel regularity of the 'distributional derivative' map follows easily noticing that for any n ∈ N the map γ → dY(γ t+1/n ,γt) is continuous, hence Borel. The conclusion follows noticing that W 1,p ([0, T ], Y) coincides with the class of γ's such that the maps have limit in L p ((0, T )) as n → ∞, the distributional derivative being such limit.
A direct and simple corollary of the above is the following chain rule: and any such family has the following property: For any p ∈ (1, ∞) and γ ∈ L p ([0, 1], X) we have that γ ∈ W 1,p ([0, 1], X) if and only if f n • γ ∈ W 1,p ((0, 1)) for every n ∈ N with sup n ∂ t (f n • γ) ∈ L p ((0, 1)). Moreover, if these holds we also Proof. To prove that a countable family for which (2.15) holds exists, simply pick f k,n (x) : For the second part of the claim, let (f n ) be an arbitrary sequence of 1-Lipschitz functions for which (2.15) holds and notice that for every absolutely continuous curve γ the function f n • γ is absolutely continuous with 1)). Then the same arguments used in the proof of point (ii) of Theorem 2.11 give that γ admits a continuous representative in AC p ([0, 1], X) and the conclusion follows.
We now study the particular case of curves with values in some L p space.
Then the following are equivalent: Moreover, if these holds h is the least function g ≥ 0, in the m × L 1 -a.e. sense, for which (2.16) holds and where the incremental ratios are defined to be Proof. We shall deal with the absolutely continuous case, as the Sobolev one can be obtained through very similar arguments taking also into account Theorem 2.11. (i) ⇒ (ii) Recall that L p (X) has the Radon-Nikodym property and let (g t ) ∈ L p ([0, 1], L p (X)) be the derivative of (f t x ∈ X. It is then easy to see that the same formula holds also m-a.e., so that the conclusion holds with g t := |g t |. (ii) ⇒ (i) For any t, s ∈ [0, 1], t < s from (2.16) we have and since the identity´1 0 g t p L p (X) dt =˜1 0 |g t | p dt dm < ∞ shows that ( g t L p (X) ) ∈ L p ((0, 1)), this is sufficient to conclude.
(ii) ⇒ (iii) Continuity follows from the already proved implication (ii) ⇒ (i). The assumption is equivalent to asking that for a.e. t, ε ∈ [0, 1] with t + ε ∈ [0, 1] it holds Notice also that Fubini's theorem and the assumption g Also, by a simple approximation argument it is sufficient to check the above for ϕ running in a countable set D, dense in the C 1 -topology in the class of admissible ϕ's.
With this said, for any ϕ ∈ D we havê and therefore (2.18) gives that m-a.e. the bound According to what previously said, this is sufficient to conclude.
and thus Fubini's theorem gives that for a.e. t, s ∈ [0, 1] with t < s it holds To conclude that the same holds for every t, s, notice that the continuity assumption on (f t ) grants that the left hand side of the above is continuous in t, s with values in L p (X). The same holds for the right hand side due to the assumption ∂ t f t ∈ L p (X × [0, 1]).
Last statements The fact that h is the least g ≥ 0 for which (2.16) holds follows by the arguments given. To prove (2.17) notice that by standard results about W 1,p ((0, 1)) functions we know that for m-a.e. x ∈ X the given incremental ratios converge to t → h t (x) in L p ((0, 1)). Hence by dominate converence and Fubini's theorem the conclusion follows if we show that the incremental ratios are dominated in L p (X × [0, 1]). This is a direct consequence of (2.16).

Regular Lagrangian Flows.
Here we very briefly recall the main definitions and results of the metric theory of Regular Lagrangian Flows as developed in [7]. The concept of Regular Lagrangian Flow provides the correct substitute, in this setting, for the concept of solution of the ODE γ ′ t = Z t (γ t ), see also [2] and [8] for overviews of the subject and historical remarks on R d and RCD spaces respectively.
We begin with: ii) For every f ∈ W 1,2 (X) it holds: for m-a.e. x the map Notice that part of the role of (i) is to ensure that (ii) makes sense, as the function df (Z t ) is only defined up to m-a.e. equality, so that its composition with Fl Z t makes sense because (Fl Z t ) * m ≪ m. The concept of Regular Lagrangian Flow is tightly linked to the continuity equation: Let be a weakly * continuous curve of probability densities and [0, T ] ∋ t → v t ⊂ L 0 (T X) be a Borel map. We say that (ρ t , v t ) solves the continuity equation provided: ii) For any f ∈ W 1,2 (X) the map t →´f ρ t dm is absolutely continuous and its derivative is Albeit the last two definitions make sense on arbitrary metric measure spaces, to develop a good theory it seems necessary to impose a lower bound on the Ricci curvature. In particular, the following notion of regularity for vector fields is important: Definition 2.17 (Regular vector fields). We say that a vector field Z over a RCD(K, ∞) space X is regular provided Z ∈ L ∞ ∩ W 1,2 C (T X) and moreover it is in the domain of the divergence with (divZ) − ∈ L ∞ (X). For a Borel time-dependent vector field (Z t ) defined for t ∈ [0, T ] we say that it is regular provided Z t is regular in the previous sense for a.e. t and The main/basic result of the theory of Regular Lagrangian Flows on RCD spaces is: Theorem 2.18. Let (X, d, m) be a RCD(K, ∞) space and (Z t ) a regular vector field parametrized on t ∈ I ⊂ R. Then: i) There is a unique Regular Lagrangian flow (Fl Z t ) of (Z t ) (uniqueness is intended at the level of curves, i.e.: if (Fl t ) is another flow, then for m-a.e. x it holds Fl Z t (x) =Fl t (x) for any t ∈ I). ii) For any bounded probability density ρ 0 with bounded support there is a unique family (ρ t ), t ∈ I, such that ρ t ≤ C for some C > 0 and every t ∈ I and (ρ t , Z t ) solves the continuity equation. Moreover, ρ t is the density w.r.t. m of (Fl Z t ) * (ρ 0 m) and the Regular Lagrangian Flow is the only flow with this property. iii) For m-a.e. x the curve t → Fl Z t (x) is absolutely continuous and its metric speed ms(Fl Z · (x), t) at time t is given by e. x a.e. t. Also, for (ρ t ) as above we have Proof.
(ii) for the existence see for instance Theorem 6.1 in [8] and notice that Example 4.1 ensures that in our setting such theorem is applicable. For uniqueness see Theorem 6.4 in [8] (and recall that Corollary 6.3 in [7] grants that a L 4 − Γ estimate holds).
(iii) the fact that the flow is concentrated on a family of absolutely continuous curves and inequality ≤ in (2.19) follows from the superposition principle, Definition 7.3 in [7] and our Lemma 2.14. For the opposite inequality follows from Lemma 2.13 and the definition of norm of a vector field (equivalently, notice that for every probability measure µ ≤ Cm on X the plan π := (Fl · ) * µ is a test plan, that the superposition principle tells that π ′ t = e * t Z t and conclude with Theorem 2.3.18 in [17] converge to Fl Z1+Z2 t as n → ∞, see Theorem 3.4 and Proposition 3.5 for the precise statement.
We start with the following very general fact: a complete and separable metric space and T n : X → Y, n ∈ N ∪ {∞}, be such that: for any bounded probability density ρ with bounded support the sequence n → (T n ) * (ρm) weakly converges to (T ∞ ) * (ρm) in duality with C b (Y). Then T n → T ∞ locally in measure.
Proof. Suppose not. Then there are ε > 0 and E ⊂ X bounded such that  (because if T ∞ (x) ∈ Bī and d Y (T n (x), T ∞ (x)) ≥ ε then since the diameter of Bī is ≤ ε 2 we necessarily have that T n (x) is not in the ε 2 -neighbourhood of Bī). Finally, put ρ := χ Cī m(Cī) −1 (notice that (3.2) ensures that m(Cī) > 0 so that ρ is well defined), observe that ρ is a bounded probability density with bounded support and that putting µ := ρm by construction we have´ϕ d(T ∞ ) * µ = 1 and for every n for which (3.2) holds. Hence lim n→∞´ϕ d(T n ) * µ <´ϕ d(T ∞ ) * µ violating the weak convergence of ((T n ) * µ) to (T ∞ ) * µ and thus concluding the proof.
We can use such abstract result in conjunction with the theory of Regular Lagrangian Flows to deduce the following stability result, which links stability of solutions of the continuity equation to stability of the associated flows: Proposition 3.2. Let (X, d, m) be a RCD(K, ∞) space and (Z n ), n ∈ N ∪ {∞} regular time dependent vector fields such that S t := sup n∈N∪{∞} |Z n,t | L ∞ (B) ∈ L 1 ((0, 1)). Assume that for any probability density ρ 0 ∈ L ∞ (X) with bounded support, letting (ρ n,t ) be the solution of the continuity equation for Z n starting from ρ 0 , we have weakly in duality with C b (X). Then Fl Zn → Fl Z∞ locally in measure as maps from X to C([0, 1], X). In particular, for every t ∈ [0, 1] we have Fl Zn t → Fl Z∞ t locally in measure as maps from X to X.
Proof. Recall from (ii) of Theorem 2.18 that ρ n,t is the density w.r.t. m of (Fl Zn t ) * (ρ 0 m). Hence our assumptions and Lemma 3.1 above grant that Fl Zn t → Fl Z∞ t locally in measure as maps from X to X for every t ∈ [0, 1]. Now let m ′ ∈ P(X) be such that m ≪ m ′ ≪ m and recall that the local convergence in measure of maps from X to C([0, 1], X) is metrized by the distancē Also, by (2.19) and our assumption we have d(Fl Zn t (x), Fl Zn s (x)) ≤´s t S r dr for m-a.e. x and n ∈ N ∪ {∞}.
Thus for ε > 0 let k ∈ N be such that´i +1 k i k S t dt ≤ ε for every i = 0, . . . , k and notice that having used the local convergence in measure of (Fl Zn Our question is now to find appropriate conditions on a sequence of vector fields which ensures convergence of solutions of the continuity equation. We shall work with: Definition 3.3. Let (Z n ) ⊂ L 1 ([0, 1], L 1 (T X)), n ∈ N ∪ {∞}. We say that Z n → Z ∞ weakly in time and strongly in space provided for any ϕ ∈ C c (R) we have Z ϕ n → Z ϕ ∞ strongly in L 1 ([0, 1], L 1 (T X)), where we put and it is intended that Z n,s = 0 for s / ∈ [0, 1].
The main result of this section is the following theorem: Theorem 3.4. Assume that (Z n ) ⊂ L 1 ([0, 1], L 1 (T X)) converges weakly in time and strongly in space to the regular vector field Z ∞ . Assume also that Then the Regular Lagrangian Flows (Fl Zn ) converge locally in measure to the Regular Lagrangian Flow Fl Z∞ and, for every t ∈ [0, 1], the maps (Fl Zn t ) converge in measure to Fl Z∞ t . Proof. Let ρ 0 ∈ L ∞ be a probability density with bounded support and let (ρ n,t ) be the solution of the continuity equation for Z n starting from ρ 0 . According to Proposition 3.2, to conclude it is sufficient to prove that ρ n,t ⇀ ρ ∞,t in duality with C b (X) for any t ∈ [0, 1].
Compactness To this aim start observing that by the assumption (3.4) and the bound (2. 20) we have that ρ n,t ≤ Cm for some C > 0 independent on n, t. Analogously, from the bound (2.19) it easily follow that supp(ρ n,t ) ⊂ B for some bounded closed set B ⊂ X independent on n, t. Thus and since m | B is a finite Radon measure we can conclude that the family {ρ n,t } n,t is tight. Finally, again the bound (2.19) gives that the curves t → ρ n,t m are W 1 -equiLipschitz. This and the previous observations imply that such sequence of curves is precompact in C([0, 1], (P 1 (X), W 1 )) and thus up to pass to a non-relabeled subsequence we can assume that it converges to a limit curve (µ t ) ∈ C([0, 1], (P 1 (X), W 1 )). Since the bound (3.5) passes to the limit, we have that µ t = η t m for some η t ≤ C χ B for every t ∈ [0, 1]. Identification of the limit To conclude it is sufficient to show that η t = ρ t for every t ∈ [0, 1] and since clearly η 0 = ρ 0 , this will follow if we show that (η t ) solves the continuity equation for Z ∞ . Thus let f ∈ Test(X), recall that t →´f ρ n,t dm is absolutely continuous with (3.6) d dtˆf ρ n,t dm =ˆdf (Z n,t )ρ n,t dm a.e. t and notice that what already proved grants that´f ρ n,t dm →´f η t dm as n → ∞ for every t ∈ [0, 1]. Hence to conclude it is sufficient to show that´df (Z n,t )ρ n,t dm converges tó df (Z ∞,t )η t dm in the sense of distributions. Here and below we shall put Z n,t ≡ 0 for t / ∈ [0, 1], ρ n,t = ρ 0 for t ≤ 0 and ρ n,t = ρ n,1 for t ≥ 1, similarly for η t . Now fix ϕ ∈ C ∞ c (R) and let (ψ k ) ⊂ C c (R) be a sequence of functions such that´ψ k = 1 and ψ k ≥ 0 and supp(ψ k ) ⊂ [− 1 k , 1 k ] for every k ∈ N. Then for every k ∈ N we have ¨1 Now notice that what previously proved ensures that (ρ n,t ) converges to (η t ) in the weak * topology of L ∞ ([0, 1] × X), while the assumption and the fact that |df | ∈ L ∞ (X) grant that df (Z ψ k n,t ) → df (Z ψ k ∞,t ) in L 1 ([0, 1] × X) as n → ∞ for every k ∈ N. Since moreover it is clear that df (Z ψ k ∞,t ) → df (Z ∞,t ) in L 1 ([0, 1] × X) as k → ∞, by letting first n → ∞ and then k → ∞ in the above we obtain: Thus to conclude the proof it is sufficient to show that the right hand side in the above is 0. Put for brevity I n (t, s) :=´df (Z n,t )ρ n,s dm and notice that since f ∈ Test(X), from (3.4) we have that t → I t := sup n |I n (t, t)| ∈ L 1 (0, 1) and recalling also (2.4) we see that df (Z n,t ) ∈ W 1,2 (X) for a.e. t, with |d(df (Z n,t ))| ≤ C(f )(|Z n,t | + |∇Z n,t | HS ). Thus from (3.6) (which is valid for functions f ∈ W 1,2 (X)) we obtain (3.7) |I n (t, s) − I n (t, t)| ≤¨s t d(df (Z n,t ))(Z n,r )ρ n,r dr dm ≤ |s − t|C(f )g t where g t := sup n´B |Z n,t | 2 + |∇Z n,t ||Z n,t | dm. The assumption (3.4) give g ∈ L 1 (0, 1). Now observe that˜1 0 ϕ t df (Z n,t )ρ n,t dt dm =˜1 0 ϕ t ψ t−s I n (t, t) ds dt and thaẗ having used the fact that ψ k is even in the last step. Therefore using also (3.7) we get Recalling that by construction we have ψ k t−s = 0 for |s − t| > 2 k we obtain and the conclusion follows letting k → ∞.
For later use, we notice the following: Proposition 3.6. Let Z 1 , Z 2 be two regular vector fields, define Z n as in (3.8) and let ρ 0 be a bounded probability density with bounded support. Let ρ t be the density of (Fl Z 1 +Z 2 t ) * (ρ 0 m) and define ρ 1 n,t , ρ 2 n,t by Then for every t ∈ [0, 1] both the sequences (ρ 1 n,t ), (ρ 1 n,t ) converge to ρ t in the weak * topology of L ∞ (X).
Proof. We shall prove the result for ρ 1 n,t only, as the study of ρ 2 n,t follows along similar lines. Fix t ∈ [0, 1] and let i n ∈ N be such that t ∈ [ 2in 2 n , 2(in+1) 2 n ). The estimates (2.20) and (2.19) grant that the densities {ρ 1 n,t } n are equibounded, hence to conclude it is sufficient to prove that´ϕρ 1 n,t dm →´ϕρ t dm for a set ϕ's dense in L 1 (X). We shall consider ϕ bounded and Lipschitz and notice that Theorem 3.4 ensures that B n → 0 as n → ∞, for A n we put µ n := (Fl Zn 2in 2 n ) * (ρ 0 m) ∈ P(X), 2 n ) * µ n ∈ P(X 2 ) and notice that Thus recalling, from (2.19), that for m-a.e. x we have the conclusion follows from 3.2. The Euclidean case. The arguments used in the previous section can also be used in the Euclidean context to extend known stability results for vector fields converging weakly in time and strongly in space, to the BV case. Compare with [2,Remark 5.11].
The following is a rather trivial observation: Then t →´f ρ t dL d is absolutely continuous and for its derivative it holds Proof. For f ∈ C ∞ c (R d ) the claim is a direct consequence of the distributional formulation of the continuity equation, which ensures that Then the conclusion follows recalling that Df TV = |df | L 1 . The general case follows from a standard approximation procedure; we omit the details.
With this last lemma and adapting the arguments used for Theorem 3.4 we deduce the following result: Theorem 3.8. Assume that (Z n ) ⊂ L 1 ([0, 1], L 1 (R d , R d ; L d )) converges weakly in time and strongly in space to Z ∞ . Assume also that for every bounded set B ⊂ R d it holds Then the Regular Lagrangian Flows (Fl Zn ) converge locally in measure to the Regular Lagrangian Flow Fl Z∞ and, for every t ∈ [0, 1], the maps (Fl Zn t ) converge in measure to Fl Z∞ t . Proof. The argument is verbatim the same used in the proof of Theorem 3.4, with the following differences: the function f is taken in C ∞ c (R d ), so that the assumption (3.10) yields that df (Z n,t ) ∈ BV (R d ) and taking into account Lemma 3.7 above we obtain the estimate f (Z n,t )ρ n,r dL d dr ≤ D(df (Z n,t )) TVˆs t ρ n,r L ∞ Z n,r L ∞ dr.
Using this estimate in place of (3.7), the conclusion is obtained arguing as in Theorem 3.4.

4.1.
Basic considerations about the space L p (X, Y). In this short section we collect some basic simple properties of the space L p (X, Y). Let us fix a complete and separable metric space (X, d) equipped with a non-negative Radon measure m giving finite mass to bounded sets and a complete space (Y, d Y ) (which often, but not always, will be separable).
The behaviour of AC p curves with values in L p (X, Y) is described in the following lemma (compare with Lemma 2.14): Then the following are equivalent: iii) For m-a.e. x ∈ X we have f · (x) ∈ W 1,p ([0, 1], Y) and the function (t, x) → |∂ t f t (x)| =: H t (x) belongs to L p (X × [0, 1]) (resp. and moreover (f t ) ∈ C([0, 1], L p (X, Y))). Moreover, if these holds H is the least function G ≥ 0 in the m × L 1 -a.e. sense, for which (4.1) holds and where the incremental ratios are defined to be Proof. We shall deal with the absolutely continuous case, as the Sobolev one can be obtained through very similar arguments taking also into account Theorem 2.11. Moreover, since (f t ) ∈ L p (X × [0, 1], Y), by definition there is a separable subset of Y containing, up to negligible sets, the image of (f t ); thus up to replacing Y with the closure of such separable subset we can assume that Y is separable. Hence, without loss of generality we may assume that L p (X, Y) is separable. Y) be countable and dense, put F n,t := d Y (f t , f n ) ∈ L p (X) and notice that the triangle inequality in Y gives |F n,s − F n,t | ≤ d Y (f t , f s ) m-a.e.. On the other hand, the triangle inequality in L p (X) gives F n, In particular, for every n ∈ N we have (F n,t ) ∈ AC p ([0, 1], L p (X)) so that by the Radon-Nikodym property of L p we obtain that G n,t := ∂ t F n,t is a well defined function in L p (X) for a.e. t ∈ [0, 1]. Now observe that the assumption (f t ) ∈ AC p ([0, 1], L p (X, Y)) ensures (by arguing as in (2.13)) that given a sequence h i ↓ 0 the incremental ratios dY(f t+h i ,ft)(x) hi are bounded in L p (X × [0, 1]) as h i → 0, hence up to subsequences they must converge to a limitG weakly in L p (X × [0, 1]). Thus (4.3) forces |G n,t |(x) ≤G t (x) for m × L 1 -a.e. (x, t) and in turn this grants that G := sup n G n belongs to L p (X × [0, 1]). The conclusion follows noticing that for any t, s ∈ [0, 1], t < s it holds as desired.
(ii) ⇒ (i) Directly from (4.1) we obtain for any t, s ∈ [0, 1], t < s, and since the identity´1 0 G t p L p (X) dt =˜1 0 |G t | p dt dm < ∞ shows that ( G t L p ) ∈ L p (0, 1), this is sufficient to conclude. (ii) ⇒ (iii) Continuity is obvious from the implication (ii) ⇒ (i) already proved. For any 1-Lipschitz function ϕ : Y → R the function ϕ • f satisfies (i) of Lemma 2.14 with g := G and thus Lemma (2.14) ensures that for m-a.e. x ∈ X the function t → ϕ • f t (x) belongs to W 1,p ([0, 1]) and its distributional derivative is bounded above by G t (x). Letting ϕ running over the countable set given by Lemma 2.13 we conclude that t → f t (x) belongs to W 1,p ([0, 1], Y) with distributional derivative bounded above by G t (x) for m-a.e. x. (iii) ⇒ (ii) It is trivial to notice that for any 1-Lipschitz function ϕ : Y → R the function ϕ • f belongs to W 1,p ((0, 1)) and ∂ t (ϕ(f t (x))) =: h ϕ t (x) ≤ H t (x). Thus Lemma (2.14) ensures that for every t, s and m-a.e. x ∈ X it holds hence taking the supremum as ϕ varies over the countable family given by Lemma 2.13 we conclude. Final statements. The fact that H is the minimal G for which (4.1) holds follows directly from the proof given. For what concerns (4.2), notice that (4.1) and the choice G := H give where the claimed convergence is an easy consequence of the definition of Bochner integral. Hence lim h→0 dY(f ·+h (·),f·(·)) |h| L p ≤ H L p . Now letH ∈ L p be any L p -weak limit of dY(f ·+h (·),f·(·)) |h| along some sequence h n → 0, so that H L p ≤ H L p , and notice that to conclude it is sufficient to prove thatH = H. For any ϕ : Y → R 1-Lipschitz, the function ϕ • f satisfies (2.16) with g := H, hence putting h ϕ (x) := ∂ t (ϕ(f t (x))) as before, from the trivial bound Hence letting ϕ run in the countable set given by Lemma 2.13 we deduce that H ≥ H and then the conclusion.
We now want to prove a continuity result for L p functions valued in Y and to this aim it is convenient to first analyze the case Y := ℓ ∞ .
Proof. Let E ⊂ X Borel, f ∈ ℓ ∞ and (g n ) ⊂ C b (X, R) be converging to χ E in the L p (X, R)topology. Then (g n f ) ⊂ C b (X, ℓ ∞ ) converges to χ E f in the topology of L p (X, ℓ ∞ ). Since linear combinations of functions of the form χ E f with E, f as above are dense in L p (X, ℓ ∞ ) (recall (2.6)), the proof is completed.
Proof. Up to a left composition with a (Kuratowski) isometric embedding of Y in ℓ ∞ we can assume that Y = ℓ ∞ . Then observe that the trivial bound shows that the right composition with Fl Z t is a Lipschitz map from L p (X, ℓ ∞ ) to L p (X, ℓ ∞ ), thus to conclude it is sufficient to prove that there is a dense subset of L p (X, ℓ ∞ ) made of functions u such that t → u • Fl Z t ∈ L p (X, ℓ ∞ ) is continuous. An application of the dominated convergence theorem shows that this is the case for u ∈ C b (X, ℓ ∞ ), thus the conclusion follows from Lemma 4.2.
4.2. The Korevaar-Shoen space KS p Z (Ω, Y). Let us fix some regular vector field Z not depending on time on the RCD(K, ∞) space (X, d, m) and p ∈ (1, ∞) and denote by Fl Z the unique regular Lagrangian flow associated to Z.

Definition 4.4. We say that a Borel map u belongs to Korevaar-Schoen space KS
Then the following are equivalent: iii) There exists G ∈ L p (Ω) such that the following holds. For every closed set C ⊂ Ω with T C > 0 we have (and in particular the map t → u • Fl Z t belongs to AC p loc ([0, T C ), L p (C, Y))). iv) For m-a.e. x ∈ Ω the map t → u(Fl Z t (x)) belongs to W 1,p ([0, T x ], Y) and for some H ∈ L p (Ω) the distributional derivative |∂ t u(Fl Z t (x))| satisfies the identity (4.7) Moreover if these hold the functions (e Z p,ε [u, Ω]) 1/p converge to nonnegative H in L p (Ω) as ε ↓ 0, we have Proof.
(iv) ⇒ (iii) By Proposition 4.3 we know that t → u • Fl Z t ∈ L p (C, Y) is continuous. Then the conclusion follows from Lemma 4.1.
and in particular is independent on C, as desired.
(i) ⇒ (iv) Let α > β > 0 be two parameters and consider the closed set C α ⊂ Ω defined as if Ω = X we pick C α = X as well -if Z = 0 the set C α might be not closed but in this case the claim is trivial). We start claiming that (4.9) To check the first, notice that Denote its distributional derivative by t → F α,β,t (x) and notice that by the last part of Theorem 2.11 the m × L 1 -a.e. defined function F α,β : [0, α − β] × C α → R is Borel and by the second in (4.9) for a.e. s, t. Thus letting µ α,β : there is a µ α,β -a.e. uniquely defined Borel functionF α,β : Ω → R such that for a.
Now observe that The 'easy' implication that m(E) = 0 implies µ α,β (E) = 0 for every α, β obviously follows from definitions. To prove the converse implication we proceed as follows. Let ρ t be the density of (F t ) * m w.r.t. m, so that the functions ρ t are uniformly bounded in L ∞ for t ∈ [0, 1] (for this we only need a bound on the negative part of the divergence). The measures (F t ) * m converge to m weakly in duality with continuous functions with bounded support on X as t ↓ 0 (by the dominated convergence theorem and because the flow is concentrated on continuous curves). This weak convergence plus the uniform L ∞ bound imply that ρ t converge to 1 in the weak * topology of L ∞ . Therefore for any E of finite measure we have This proves that if m(E) > 0, then for t sufficiently small it holds (F t ) * m(E) > 0 as well. Then the conclusion follows from the definition of µ α,β .
Thus from (4.11) it follows that there exists and is m | Ω -a.e. uniquely determined a Borel function H such that H =F α,β µ α,β − a.e. ∀α > β > 0. We claim that such H has the required properties. We start by proving that H ∈ L p (Ω) and to this aim we start noticing that (4.13) lim This can be easily proved for f bounded and Lipschitz, then the case of f ∈ L 1 (Ω) follows by a density argument based on the bound (Fl Z t ) * m ≤ e t divZ L ∞ m and finally the case of non-negative f 's comes by monotone approximation. Now notice that by construction (and (2.12)) it holds´C = lim (4.14) Now we prove (4.7). Letting β ↓ 0 in (4.10) we see that for every α > 0 it holds: m-a.e. x ∈ C α the curve t → u(Fl Z t (x)) belongs to W 1,p ([0, α], Y) (e.g. by recalling the relation between Sobolev and AC curves stated in Theorem 2.11) and, by definition, its distributional derivative is given by H • Fl Z t . Thus for m-a.e. x ∈ Ω we have that: for every α ∈ Q with α < T x the curve t → u(Fl Z t (x)) belongs to W 1,p ([0, α], Y) and its distributional derivative is given by H • Fl Z t . Arguing as before by calling into play Theorem 2.11 we conclude that for m-a.e. x t → u(Fl Z t (x)) belongs to W 1,p ([0, T x ], Y) and its distributional derivative is given by H • Fl Z t , as desired. Last statements The fact that the choice G := H is the least for which (4.6) holds is a direct consequence of the analogous statement in Lemma 4.1. Inequality ≥ in (4.8) is proved in (4.14) while the opposite comes with the proofs (iii) ⇒ (ii) and (ii) ⇒ (i).
It remains to prove L p (Ω)-convergence of (e Z p,ε [u, Ω]) 1/p to H. Extend H to the whole X by putting it 0 outside Ω and notice that what we already proved gives where the claimed convergence can be proved along the same lines used to show (4.13). Now notice that for α > β > 0, Lemma 4.1 applied to X := C α and f t : . This is the same as to say that (e Z p,ε [u, Ω]) 1/p → H in L p (µ α,β ) and in particular any L p (Ω)-weak limit of (e Z p,ε [u, Ω]) 1/p must coincide with H µ α,β -a.e.. Thus by (4.12) we deduce that (e Z p,ε [u, Ω]) 1/p ⇀ H in L p (Ω), which together with (4.15) gives the conclusion.  In the smooth category, the quantity |du(Z)| is the norm of the differential of u applied to Z, whence the notation chosen. Notice that for the moment we only defined |du(Z)|, not the underlying object du(Z), so the notation chosen is purely formal. We will define du(Z) in Section 4.4.
We conclude this section with a the following kind of regularity result which will be useful in what comes next. Proposition 4.7. Let p ∈ (1, ∞), Z a regular vector field on X, Ω ⊂ X open and u ∈ KS p Z (Ω, Y). Then for every C ⊂ Ω closed for which T C > 0 (recall the definition (4.5)) and f ∈ Lip bs (Y), the is C 1 and for its derivative we have for every t ∈ [0, T C ) Proof. For any t, s ∈ [0, T C ) we have e. on C and thus (4.6) yields that t → f • u • Fl Z t ∈ L p (C) is Lipschitz. Since L p (C) has the Radon-Nikodym property, we deduce that such curve is differentiable for a.e. t. Then from the identity for every differentiability points s > t. Since Proposition 4.3 grants continuity in s with values in L p (C) of the right hand side, C 1 regularity follows. Then the bound (4.16) follows from the definition of |du(Z)|, Corollary 2.12 and Lemma 2.14.
The study of the above will be divided in two parts: a first (easy) one where we study the effect of multiplication of vector fields by constants and a second (more delicate) where we study sums of vector fields.
We start with the following simple lemma: Lemma 4.8. Let (X, d, m) be a RCD(K, ∞) space and Z a regular vector field. Then for every α ∈ R and t ∈ R we have Fl αZ t = Fl Z αt m-a.e.. Proof. Start noticing that αZ is also a regular vector field, so that the statement makes sense. To conclude, according to Theorem 2.18 it is sufficient to show that if t → ρ t solves the continuity equation for v t ≡ Z, then t → ρ αt solves the continuity equation for v t ≡ αZ. But this is obvious, whence the conclusion follows.
As a direct consequence of the above we obtain: Proposition 4.9 (Multiplication of the vector field by a constant). Let K ∈ R, (X, d, m) be RCD(K, ∞) space, Ω ⊂ X open and Z a regular vector field on it. Let (Y, d Y ) be a complete metric space and u ∈ KS p Z (Ω, Y). Then for every α ∈ R we also have u ∈ KS p αZ (Ω, Y) and |du(αZ)| = |α||du(Z)|. Proof. It is clear that if t → γ t is absolutely continuous then so is t → γ αt and with metric speed which changes by a factor |α|. Then by Theorem 2.11 the same holds for Sobolev curves and distributional derivatives. Then conclusion easily follows from Theorem 4.5.
We now turn to the study of the effect of the sum of vector fields on Regular Lagrangian Flows and start with a simple result about stability of convergence in measure under left composition: Lemma 4.10. Let T n : X → X, n ∈ N ∪ {∞} be Borel and such that T n → T ∞ locally in measure as n → ∞. Assume also that the measures (T n ) * m are locally equi-absolutely continuous w.r.t. m, i.e. that: for every ε > 0 and B ⊂ X bounded there is δ > 0 such that for every E ⊂ B Borel with m(E) < δ we have (T n ) * m(E) ≤ ε for every n ∈ N.
Then for every complete metric space Y and every Borel map u : X → Y which is essentially separably valued we have that (u • T n ) converges locally in measure to u • T ∞ .
Proof. Replacing Y with a closed separable subset containing, up to negligible sets, the image of u we can assume that Y is separable. Then up to a (Kuratowski) isometric embedding of Y in ℓ ∞ we can replace the former with the latter. Then Lemma 4.2 and a simple cut-off argument shows that C b (X, ℓ ∞ ) is dense in the space of essentially separably valued Borel maps from X to ℓ ∞ w.r.t. to local convergence in measure. Now let m ′ ∈ P(X) be such that m ≪ m ′ ≪ m and notice that the assumption on equi-absolute continuity of (T n ) * m implies (4.17) ∀ε > 0 ∃δ > 0 s.t. ∀E ⊂ X Borel the bound m ′ (E) ≤ δ implies (T n ) * m ′ (E) ≤ ε ∀n ∈ N ∪ {∞} and recall that the distance d 0 (u, v) :=´1 ∧ d ℓ ∞ (u, v) dm ′ metrizes the local convergence in measure. Then for u : X → ℓ ∞ Borel and essentially separably valued and ε > 0 let first δ be given by (4.17) and then v ∈ C b (X, ℓ ∞ ) be such that for E := {d ℓ ∞ (u, v) > ε} it holds m ′ (E) < δ. We have (4.18) ∀n ∈ N and for every n ∈ N ∪ {∞} it holds ≤ ε + ε.
Notice that if f : Y → R is defined up to µ-a.e. equality, then f • u is defined up to m-a.e. equality on {|du(Z)| > 0}; hence the function f • u|du(Z)| is defined up to m-a.e. equality on Ω. Then the trivial identitŷ shows that (4.23) the map L p (µ) ∋ f → f • u|du(Z)| ∈ L p (m | Ω ) is linear and continuous.