Second order differentiation formula on RCD(K, N) spaces

Transcription

1 Secon orer ifferentiation formula on RCD(K, N) spaces Nicola Gigli Luca Tamanini February 8, 018 Abstract We prove the secon orer ifferentiation formula along geoesics in finite-imensional RCD(K, N) spaces. Our approach strongly relies on the approximation of W -geoesics by entropic interpolations an, in orer to implement this approximation proceure, on the proof of new (even in the smooth setting) estimates for such interpolations. Contents 1 Main result an comments 1 Strategy of the proof 4.1 The nee of an approximation proceure Entropic interpolation: efinition Entropic interpolations: uniform control an convergence Main result an comments This work is about the evelopment of calculus tools in the setting of RCD(K, N) spaces (X,, m) with K R an N [1, ) (see [] for the original efinition with N = an [9] for the case N < ). The proofs of the announce results are containe in [1] an, up to technical ifficulties, they rely on [11], where the same results are obtaine for compact RCD(K, N) spaces. SISSA, Via Bonomea 65, Trieste (Italy). ngigli@sissa.it Institut für Angewante Mathematik, Universität Bonn, Enenicher Allee 60, Bonn (Germany). tamanini@iam.uni-bonn.e 1

2 Recall that an optimal geoesic test plan π on X is a probability measure on C([0, 1], X) such that (e t ) π Cm for every t [0, 1] an some C > 0 an satisfying 1 0 γ t t π(γ) = W ( (e0 ) π, (e 1 ) π ). Here e t : C([0, 1], X) X is the evaluation map sening γ to γ t. Any such π is concentrate on constant spee geoesics an for any couple of measures µ 0, µ 1 P(X) with boune ensities an supports, there is a unique optimal geoesic test plan such that (e 0 ) π = µ 0, (e 1 ) π = µ 1. From the point of view of calculus on metric measure spaces as evelope in [1], the relation between optimal geoesic test plans an stanar geoesics is in some sense the same that there is between Sobolev functions an Lipschitz ones. An example of this phenomenon is the following result (a minor variant of a statement in [7]), which says that we can safely take one erivative of a W 1, (X) function along an optimal geoesic test plan: Theorem 1. Let (X,, m) be a RCD(K, ) space, π an optimal geoesic test plan with boune support (equivalently: such that {γ t : t [0, 1], γ supp(π)} X is boune) an h W 1, (X). Then the map [0, 1] t h e t L (π) is in C 1 ([0, 1], L (π)) an we have ( ) h et = h, φt e t, t for every t [0, 1], where φ t is any function such that for some s t, s [0, 1], the function (s t)φ t is a Kantorovich potential from (e t ) π to (e s ) π. Our main result here is the extension of the above to secon orer erivatives. Recalling that the secon orer Sobolev space H, (X) an the corresponing Hessian are efine in [8], we have: Theorem. Let (X,, m) be a RCD(K, N) space, N <, π an optimal geoesic test plan with boune support an h H, (X). Then the map [0, 1] t h e t L (π) is in C ([0, 1], L (π)) an we have t ( h et ) = Hess(h)( φt, φ t ) e t, (1) for every t [0, 1], where φ t is as in Theorem 1. Notice that by Theorem 1 we have that such result is really a statement about the C 1 regularity of t h, φ t e t. Let us collect a couple of equivalent formulations of Theorem. For the first recall that the space of Sobolev vector fiels H 1, C (T X) as well as the covariant erivative have been efine in [8]. Then we have:

3 Theorem 3. Let (X,, m) be a RCD(K, N) space, N <, π an optimal geoesic test plan with boune support an X H 1, C (X). Then the map [0, 1] t X, φ t e t L (π) is in C 1 ([0, 1], L (π)) an we have ( ) X, φt e t = X( φt, φ t ) e t, () t for every t [0, 1], where φ t is as in Theorem 1. From the ientity ( h) = Hess(h) (assuming to ientify tangent an cotangent vector fiels) we see that Theorem 3 implies Theorem. For the converse implication notice that Theorem an the Leibniz rule easily provie the correct formula for the erivative of t X, φ t e t for X = h i i h i, with ( h i ) L W 1, (X) an (h i ) H, (X), then conclue by the closure of the covariant erivative. Another equivalent formulation of Theorem, which is the one we shall actually prove, is: Theorem 4. Let (X,, m) be a RCD(K, N) space, N <, µ 0, µ 1 P (X) be such that µ 0, µ 1 Cm for some C > 0, with compact supports an let (µ t ) be the unique W -geoesic connecting µ 0 to µ 1. Also, let h H, (X). Then the map [0, 1] t h µ t R belongs to C ([0, 1]) an it hols t h µ t = Hess(h)( φ t, φ t ) µ t, (3) for every t [0, 1], where φ t is any function such that for some s t, s [0, 1], the function (s t)φ t is a Kantorovich potential from µ t to µ s. Since for any W -geoesic as in the statement there is a (unique) optimal geoesic test plan π such that µ t = (e t ) π for any t, we see that Theorem 4 follows from Theorem by integration w.r.t. π. For the converse implication one notices that for any optimal geoesic test plan π with boune support an Γ C([0, 1], X) Borel with π(γ) > 0, the curve t π(γ) 1 (e t ) (π Γ ) fulfils the assumptions of Theorem 4 with the same φ t s as in Theorem. The conclusion then follows by the arbitrariness of Γ observing that L (π)-erivatives exist for every t if an only if the ifference quotients converge in the weak L (π)-topology for every t. Let us comment about the assumptions in Theorems, 3, 4: - The first orer ifferentiation formula is vali on general RCD(K, ) spaces, while for the secon orer one we nee to assume finite imensionality. This is ue to the strategy of our proof, which among other things uses the Li-Yau inequality. 3

4 - There exist optimal geoesic test plans without boune support (if K = 0 or the ensities of the initial an final marginals ecay sufficiently fast) but in this case the functions φ t appearing in the statement(s) are not Lipschitz. As such it seems har to have Hess(h)( φ t, φ t ) e t L 1 (π) an thus we can not really hope for anything like (1), (), (3) to hol: this explains the nee of the assumption on boune supports. Having at isposal the secon orer ifferentiation formula is interesting not only at the theoretical level, but also for applications to the stuy of the geometry of RCD spaces. For instance, the proofs of both the splitting theorem [7] an of the volume cone implies metric cone [5] in this setting can be greatly simplifie by using such formula (in this irection, see [14] for comments about the splitting). Also, one aspect of the theory of RCD spaces which is not yet clear is whether they have constant imension: for Ricci-limit spaces this is known to be true by a result of Coling-Naber [4] which uses secon orer erivatives along geoesics in a crucial way. Thus our result is necessary to replicate Coling-Naber argument in the non-smooth setting (but not sufficient: they also use a calculus with Jacobi fiels which as of toay oes not have a non-smooth counterpart). Strategy of the proof.1 The nee of an approximation proceure Let us recall that a secon orer ifferentiation formula, vali for sufficiently regular curves, has been prove in [8]: Theorem 5. Let (µ t ) be a W -absolutely continuous curve solving the continuity equation t µ t + iv(x t µ t ) = 0, (4) for some vector fiels (X t ) L (T X) in the following sense: for every f W 1, (X) the map t f µ t is absolutely continuous an it hols Assume that t f µ t = f, X t µ t. (i) t X t L (T X) is absolutely continuous, (ii) sup t { X t L + X t L + X t L } < +. Then for f H, (X) the map t f µ t is C 1,1 an the formula t fµ t = Hess(f)(X t, X t ) + f, t X t + Xt X t µt (5) hols for a.e. t [0, 1]. 4

5 If the vector fiels X t are of graient type, so that X t = φ t for every t an the acceleration a t is efine as t φ t + φ t =: a t then (5) reas as t fµ t = Hess(f)( φ t, φ t ) µ t + f, a t µ t. (6) In the case of geoesics it is well-known that (4) hols exactly with X t = ϕ t for appropriate choices of Kantorovich potentials ϕ t (see also [10] in this irection) an moreover the functions ϕ t solve (in a sense which we will not make precise here) the Hamilton-Jacobi equation t ϕ t = ϕ t, (7) thus in this case the acceleration a t is ientically 0. Hence if the vector fiels ( ϕ t ) satisfie the regularity requirements (i), (ii) in the last theorem, we woul easily be able to establish Theorem. However in general this is not the case; informally speaking this has to o with the fact that for solutions of the Hamilton- Jacobi equations we o not have sufficiently strong secon orer estimates. In orer to establish Theorem it is therefore natural to look for suitable smooth approximations of geoesics for which we can apply Theorem 5 above an then pass to the limit in formula (5). Given that the source of non-smoothness is in the Hamilton-Jacobi equation it is natural to think at viscous approximation as smoothing proceure: all in all viscous limit is the way of approximating the correct solution of Hamilton-Jacobi an the Laplacian is well behave uner lower Ricci curvature bouns. However, this oes not really work: shortly sai, the problem is that not every solution of Hamilton-Jacobi is linke to W -geoesics, but only those for which shocks o not occur in the time interval [0, 1]. Since the conclusion of Theorem can only hol along geoesics, we see that we cannot simply use viscous approximation an PDE estimates to conclue (one shoul incorporate in the estimates the fact that the starting function is c-concave, but this seems har to o). We shall instea use entropic interpolation, which we now introuce.. Entropic interpolation: efinition Fix two probability measures µ 0 = ρ 0 m, µ 1 = ρ 1 m on X. The Schröinger functional equations are ρ 0 = f h 1 g ρ 1 = g h 1 f, (8) 5

6 the unknown being the Borel functions f, g : X [0, ), where h t f is the heat flow starting at f evaluate at time t. It turns out that in great generality these equations amit a solution which is unique up to the trivial transformation (f, g) (cf, g/c) for some constant c > 0. Such solution can be foun in the following way: let R be the measure on X whose ensity w.r.t. m m is given by the heat kernel r t (x, y) at time t = 1 an minimize the Boltzmann-Shannon entropy H(γ R) among all transport plans γ from µ 0 to µ 1. The Euler equation for the minimizer forces it to be of the form f g R for some Borel functions f, g : X [0, ), where f g(x, y) := f(x)g(y). Then the fact that f g R is a transport plan from µ 0 to µ 1 is equivalent to (f, g) solving (8). Once we have foun the solution of (8) we can use it in conjunction with the heat flow to interpolate from ρ 0 to ρ 1 by efining ρ t := h t f h 1 t g. This is calle entropic interpolation. Now we slow own the heat flow: fix ε > 0 an by mimicking the above fin f ε, g ε such that ρ 0 = f ε h ε/ g ε ρ 1 = g ε h ε/ f ε, (9) (the factor 1/ plays no special role, but is convenient in computations). Then efine ρ ε t := h tε/ f ε h (1 t)ε/ g ε. The remarkable an non-trivial fact here is that as ε 0 the curves of measures (ρ ε t m) converge to the W -geoesic from µ 0 to µ 1. In orer to state our results, it is convenient to introuce the (interpolate) Schröinger potentials ϕ ε t, ψ ε t as ϕ ε t := ε log h tε/ f ε ψ ε t := ε log h (1 t)ε/ g ε. In the limit ε 0 these will converge to forwar an backwar Kantorovich potentials along the limit geoesic (µ t ) (see below). In this irection, it is worth to notice that while for ε > 0 there is a tight link between potentials an ensities, as we trivially have ϕ ε t + ψ ε t = ε log ρ ε t, in the limit this becomes the well known (weaker) relation that is in place between forwar/backwar Kantorovich potentials an measures (µ t ): ϕ t + ψ t = 0 on supp(µ t ), ϕ t + ψ t 0 on X, see e.g. Remark 7.37 in [15] (paying attention to the ifferent sign convention). By irect computation one can verify that (ϕ ε t ), (ψ ε t ) solve the Hamilton-Jacobi- Bellman equations t ϕε t = 1 ϕε t + ε ϕε t t ψε t = 1 ψε t + ε ψε t, (10) 6

7 thus introucing the functions it is not har to check that it hols an t ϑε t + ϑε t ϑ ε t := ψε t ϕ ε t t ρε t + iv( ϑ ε t ρ ε t) = 0 (11) ( = a ε t, where a ε t := ε log ρ ε t + log ρ ε t ). 8.3 Entropic interpolations: uniform control an convergence With this sai, our main results about entropic interpolations can be summarize as follows. Uner the assumptions that the metric measure space (X,, m) is RCD(K, N), N <, an that ρ 0, ρ 1 belong to L (X) with boune supports it hols: - Zeroth orer boun For some C > 0 we have ρ ε t C for every ε (0, 1) an t [0, 1]. convergence The curves (ρ ε t m) W -uniformly converge to the unique W -geoesic (µ t ) from µ 0 to µ 1 an setting ρ t := µt m it hols ρε t ρ t in L (X) for all t [0, 1]. - First orer boun For any t (0, 1] the functions {ϕ ε t } ε (0,1) are locally equi- Lipschitz. Similarly for the ψ s. convergence For every sequence ε n 0 there is a subsequence - not relabele - such that for any t (0, 1] the functions ϕ ε t converge both locally uniformly an in W 1, loc (X) to a function ϕ t such that tϕ t is a Kantorovich potential from µ t to µ 0. Similarly for the ψ s. - Secon orer For every δ (0, 1/) we have boun sup 1 δ ε (0,1) δ 1 δ sup ε (0,1) δ ( Hess(ϑ ε t ) HS + ε Hess(log ρ ε t ) HS ( ϑ ε t + ε log ρ ε t ) ρ ε t t m <. ) ρ ε t t m <, (1) Notice that since in general the Laplacian is not the trace of the Hessian, there is no irect link between these two bouns. 7

8 convergence For every function h W 1, (X) with h L (X) it hols lim ε 0 1 δ δ h, a ε t ρ ε t t m = 0. (13) With the exception of the convergence ρ ε t m µ t, all these results are new even on compact smooth manifols (in fact, even on R ). The zeroth an first orer bouns are obtaine via a combination of Hamilton s graient estimate an Li-Yau s Laplacian estimate. Similar bouns can also be obtaine for the viscous approximation. The fact that the limit curve (µ t ) is the W -geoesic an that the limit potentials are Kantorovich potentials are consequence of the fact that we can pass to the limit in the continuity equation (11) an that the limit potentials satisfy the Hamilton-Jacobi equation. Notice that these zeroth an first orer convergences are sufficient to pass to the limit in the term with the Hessian in (6). The crucial avantage of ealing with entropic interpolations (which has no counterpart in viscous approximation) is in the secon orer bouns an convergence results. The key ingreient that allows to obtain these is a formula ue to Léonar [13], who realize that there is a connection between entropic interpolation an lower Ricci bouns; our contribution is the rigorous proof in the RCD framework of his formal computations: Proposition 6. For any ε > 0 the map t H(µ ε t m) belongs to C([0, 1]) C (0, 1) an for every t (0, 1) it hols t H(µε t m) = ρ ε t, ϑ ε t m = 1 ( ψ ε t ϕ ε t ) ρ ε t m, (14a) ε t H(µε t m) = ρ ε t ( Γ (ϑ ε t ) + ε 4 Γ (log(ρ ε t )) ) = 1 ρ ε t ( Γ (ϕ ε t ) + Γ (ψt ε ) ). (14b) Let us see how to use (14b) in the simplifie case K = 0 an m(x) = 1 to obtain (1). Observe that if h : [0, 1] R + is a convex function, then h(0) t h (t) h(1) 1 t for any t (0, 1) an thus 1 δ δ h (t) t = h (1 δ) h (δ) h(1) 1 δ + h(0) δ. (15) If K = 0 we have Γ 0, so that (14b) tells in particular that t H(µ ε t m) is convex for any ε > 0, an if m(x) = 1 such function is non-negative. Therefore (15) gives that for any δ (0, 1/) it hols 1 δ sup ε (0,1) δ ρ ε t ( Γ (ϑ ε t ) + ε 4 Γ (log(ρ ε t )) ) t H(µ 1 m) 1 δ + H(µ 0 m) δ <. (16) 8

9 Recalling the Bochner inequalities ([6], [3],[8]) Γ (η) Hess(η) HS m, Γ (η) ( η) N m, we see that (1) follows from (16). Then with some work (see [11] for the etails) starting from (16) we can euce (13) which in turn ensures that the term with the acceleration in (6) vanishes in the limit ε 0, thus leaing to our main result Theorem 4. References [1] L. Ambrosio, N. Gigli, an G. Savaré, Calculus an heat flow in metric measure spaces an applications to spaces with Ricci bouns from below, Invent. Math., 195 (014), pp [], Metric measure spaces with Riemannian Ricci curvature boune from below, Duke Math. J., 163 (014), pp [3] L. Ambrosio, A. Monino, an G. Savaré, Nonlinear iffusion equations an curvature conitions in metric measure spaces. Preprint, arxiv: , 015. [4] T. H. Coling an A. Naber, Sharp Höler continuity of tangent cones for spaces with a lower Ricci curvature boun an applications, Ann. of Math. (), 176 (01), pp [5] G. De Philippis an N. Gigli, From volume cone to metric cone in the nonsmooth setting, Geometric an Functional Analysis, 6 (016), pp [6] M. Erbar, K. Kuwaa, an K.-T. Sturm, On the equivalence of the entropic curvature-imension conition an Bochner s inequality on metric measure spaces, Inventiones mathematicae, 01 (014), pp [7] N. Gigli, The splitting theorem in non-smooth context. Preprint, arxiv: , 013. [8], Nonsmooth ifferential geometry - an approach tailore for spaces with Ricci curvature boune from below. Accepte at Mem. Amer. Math. Soc., arxiv: , 014. [9], On the ifferential structure of metric measure spaces an applications, Mem. Amer. Math. Soc., 36 (015), pp. vi+91. [10] N. Gigli an B. Han, The continuity equation on metric measure spaces, Calc. Var. Partial Differential Equations, 53 (013), pp [11] N. Gigli an L. Tamanini, Secon orer ifferentiation formula on compact RCD (K, N) spaces. Preprint, arxiv: , (017). [1] N. Gigli an L. Tamanini, Secon orer ifferentiation formula on RCD (K, N) spaces. Preprint, arxiv: , (018). [13] C. Léonar, On the convexity of the entropy along entropic interpolations. To appear in Measure Theory in Non-Smooth Spaces, Partial Differential Equations an Measure Theory. De Gruyter Open, 017. ArXiv: v. 9

10 [14] L. Tamanini, Analysis an geometry of RCD spaces via the Schröinger problem, PhD thesis, Université Paris Nanterre an SISSA, 017. [15] C. Villani, Optimal transport. Ol an new, vol. 338 of Grunlehren er Mathematischen Wissenschaften, Springer-Verlag, Berlin,