Sobolev-Kantorovich Inequalities

Abstract In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.


Introduction
In the recent contribution [6], E. Cinti and F. Otto established a number of interpolation inequalities which 15 arise in the analysis of pattern formation in physics (more precisely in the study of branching in superconductors). Within this framework, they showed in particular that for any positive periodic smooth function f : [ , ] n → R such that´[ , ] n fdx = , where r = n + n , θ = n+ n and C > only depends on n. Here (f − C)+ = max(f − C, ) and W (f , ) is the Kantorovich-Wasserstein distance between the probability measures fdx and dx (see below). The proof of this inequality 20 in [6] relies on a speci c geometric construction of cut-o functions rst put forward in [5]. The exponent r is optimal as veri ed on peak functions (and, as explained in [6], from this point of view the Kantorovich distance behaves like a negative fractional Sobolev norm). Note furthermore that (1.1) is invariant under the change´[ , ] n dx → Λ n´[ ,Λ] n dx for any Λ > .
The purpose of this work is to show that the inequality (1.1) is actually one specimen in a all family of 25 interpolation inequalities that we call Sobolev-Kantorovich inequalities and that actually hold in a wide generality.
One suitable framework for such Sobolev-Kantorovich inequalities is the one of (weighted) Riemannian manifolds with non-negative Ricci curvature, covering in particular the previous model example of the torus. Let M = (M, g) be a complete connected n-dimensional Riemannian manifold with Riemannian measure dx, 30 and let, as reference measure, µ be a probability measure on M (with smooth density with respect to dx). The counterpart of the Euclidean geometric decomposition in [6] will be achieved via non-negative curvature conditions, speci cally curvature-dimension CD( , N) of the underlying di usion operator invariant with respect to µ, or metric measure space (M, g, µ), in the sense of [1,10,13] (see Section 2). In case of an n-dimensional Riemannian manifold (with nite, normalized, volume element dx = dµ), the CD( , N) condition with N = n amounts to a non-negative lower bound on the Ricci curvature.
Given two probability measures µ and ν on the Borel sets of M, and p ≥ , the Kantorovich-Wasserstein distance Wp(ν, µ) between ν and µ is de ned by where the in mum is taken over all couplings π on the product space M × M with respective marginals ν and µ and where d(x, y) denotes the Riemannian distance between x and y in M (cf. [13]).
Given the probability µ, the norm of the Lebesgue space L r (µ), r ≥ , is denoted by · r . 5 With these notation, the general family of Sobolev-Kantorovich interpolation inequalities emphasized in this work take the following form. Theorem 1.1. Let (M, g, µ) be a weighted Riemannian manifold with weighted probability measure µ satisfying the curvature-dimension condition CD( , N) for some N ≥ . Given p, q ≥ , there is a constant C > only depending on p, q, N such that for any probability dν = fdµ with smooth density f with respect to µ, 10 ( and θ = r p + q = + p + N . The Cinti-Otto inequality (1.1) on the at n-dimensional torus corresponds to q = , p = and N = n for which therefore r = n+ n . Other values of interests are q = p = for which r = N+ N . In this case, the inequality may be compared to the Poincaré-Sobolev inequality (see Section 2) In a somewhat weaker form, (1.2) may also be expressed by It is natural to expect that C = =´M fdµ on the left-hand side of (1.2) but the proof below is far from such a claim. The exponents in Theorem 1.1 are optimal on peak functions (in dimension one).
The proof of Theorem 1.1 will be based on heat ow arguments on the semigroup (P t ) t≥ with invariant measure µ underlying the weighted manifold (M, g, µ), and slicing methods in the spirit of [6] and [8]. The 20 CD( , N) condition will allow for the use of heat kernel and Harnack inequalities, consequences of the Li-Yau parabolic inequality in this framework. These tools are actually also available in the more general context of Markov Triples (E, µ, Γ) in the sense of [1]. For the speci c torus example, it is plausible that the Euclidean arguments of [6] also allow for the full range of parameters.
In nite dimensional examples of curvature-dimension CD( , ∞) may be addressed similarly. For CD( , ∞) 25 models, the classical Li-Yau parabolic inequality is however no more available and has to be supplemented by Wang's Harnack inequality (cf. [1,15]). The resulting Sobolev-Kantorovich inequalities (Theorem 4.1) are then dimension free, and the values q = p = read in particular for some numerical C > . Under a diameter bound, this result turns into a Nash-type inequality of possible independent interest, In the form (1.4), Sobolev-Kantorovich interpolation inequalities are also in the spirit of the Otto-Villani HWI inequality [1,12,13] stating in this context that for any probability dν = fdµ, To outline the content of the article, we present in the next section the framework of investigation and the basic geometric tools which will be used in the proof of Theorem 1.1. These rely on the curvature-dimension condition CD( , N) in the form of Harnack and pseudo-Poincaré inequalities. Section 3 is then devoted to the proof of Theorem 1.1 which adapts, with the preceding geometric tools and a slicing argument, the scheme 5 emphasized in [6]. The subsequent section addresses the in nite-dimensional case under CD( , ∞) with the tool of Wang's Harnack inequality.

Framework and geometric tools
For simplicity in the exposition, the results of this work are presented in the weighted Riemannian setting, that is a complete connected Riemannian manifold (M, g) equipped with a weighted probability measure 10 dµ = e −V dx where V : M → R is a smooth potential, to which is naturally attached the di usion operator L = ∆ − ∇V · ∇ with invariant and reversible measure µ and associated heat semigroup (P t ) t≥ .
The curvature-dimension CD( , N), for some N ≥ , condition on L, or the weighted Riemannian manifold (M, g, µ), is described by the Bochner-type inequality holding for all smooth f : M → R. For example, the Laplace-Beltrami operator ∆ on an n-dimensional Riemannian manifold with non-negative Ricci curvature satis es CD( , n). The curvature condition CD( , ∞) covers in nite dimensional models in the sense of the operator L, such as for example log-concave measures 15 on R n . We refer to [1,10,11,13] for standard references on such a framework. As mentioned in the introduction, the Sobolev-Kantorovich interpolation inequalities emphasized here may actually be developed in the more general setting of a Markov di usion Triple (E, µ, Γ) in the sense of [1], consisting of a state space E equipped with a di usion semigroup (P t ) t≥ with in nitesimal generator L and carré du champ operator Γ, and invariant and reversible probability measure µ. In the weighted Riemannian 20 context, Γ(f , f ) = |∇f | for smooth functions f . The curvature condition CD( , N) in this context stems from the abstract form of the Bochner identity in terms of the Γ operator. The state space E may be endowed with an intrinsic distance d for which Lipschitz functions f are such that Γ(f ) is bounded (µ-almost everywhere), which coincides with the Riemannian distance in the manifold case.
One basic tool in the heat kernel estimates necessary in this investigation will be the Li-Yau parabolic 25 inequality [9] in the form of its following consequences. These results, established rst on Riemannian manifolds with non-negative curvature, extend to the case of weighted Riemannian manifolds and more generally Markov Triples under the curvature-dimension condition CD( , N) (cf. [1]).
About the proof of this proposition (see e.g. Lemma 3 in [8]), it should be pointed out that the case ≤ q ≤ indeed does not require the dimensional Li-Yau inequality and rather relies on Poincaré inequalities for heat kernel measures valid under the only curvature condition CD( , ∞). Speci cally, the argument If ≤ q ≤ ∞, by Jensen's inequality on (2.1), To complete this section, we provide a direct proof of the Poincaré-Sobolev inequality (1.3) relying on the preceding tools as we could not locate a suitable reference. For a given f : M → R, by Proposition 2.2 with q = , for any t ≥ , f ≤ Hence, after integration in the x variable, P t f ∞ ≤ + D t N/ f and, by interpolation, Combining the preceding, for any (smooth) f and every t ≥ , Optimizing in t > yields the Nash-type inequality where C > only depends on N. By a standard slicing procedure (see [1,Chap. 6]), the latter amounts to the Sobolev inequality . Now, it is also classical in this context (cf. [1,3]) that a Poincaré inequality for mean-zero functions f holds true with constant proportional to the diameter, namely

Proof of Theorem 1.1
This section is devoted to the proof of the main result, Theorem 1.1. The general scheme follows the approach developed by E. Cinti and F. Otto [6]. To get a better feeling about the argument, we start with the simpler weak-type version of the statement. Actually, as put forward and achieved in [6], a main step will be to pass from the weak formulation to the strong one. Proof. Let thus f be a smooth probability density on M, and recall dν = fdµ. By Proposition 2.2, for every u > and t > , Now, by Markov's inequality and symmetry of (P t ) t≥ with respect to µ, By the Kantorovich duality (cf. [13]), for every ε > , where Qε is the sup-convolution Clearly, for any x ∈ M, the supremum may be restricted to those y for which d(x, y) p ≤ ε. Use next the Harnack inequality from Proposition 2.1 with s replaced by ts to get that for every s > and x, y ∈ M, P t (1 F )(y) ≤ P t(s+ ) (1 F )(x)( + s) N/ e d(x,y) / ts .
In particular, for every y such that d(x, y) p ≤ ε with ε = (ts) p/ , so that, with this choice of ε, Therefore, after integration with respect to µ, Choose s = − u /N so that u ( + s) N/ ≤ provided that u ≥ C = C N large enough. Hence, together with (3.1), Optimizing in t > yields the conclusion.
On the basis of the preceding weak-type estimate, we address the proof of Theorem 1.1 together with a standard 10 slicing argument as extensively presented in [1] for related functional inequalities.
Starting as above, for every u > and t > , by Proposition 2.2, Apply the preceding (which holds true for any smooth -Lipschitz -positive function, not only probability densities) to f k = min((f − k )+, k ), k ∈ Z, and u = k− , t = t k > , so to get and where we used that f k ≤ f in the last step. Note that µ(f ≥ k+ ) = µ(f k ≥ k ), and sum the preceding inequalities, after multiplication by rk , over the set of integers I = {k , k + , . . . , k } for some k ≥ to be determined below (and where k is arbitrarily large, tending to in nity at the end of the argument). Hence, By the Kantorovich duality, for any ε > , where It is an elementary, yet crucial, observation emphasized in Claim B in the proof of Proposition 1.3 of [6], that for some c = cr > only depending on r > . Therefore, for every xed x ∈ M, where M k (x) = {y ∈ E; d(x, y) p ≤ cε (r− )k }, k ∈ I. 10 Apply next the Harnack inequality from Proposition 2.1 with t = t k > to be speci ed and s = t k s k = (cε) /p (r− )k/p . Recalling that F k = {P t k f k ≥ k− }, for every y ∈ M k (x), 1 F k (y) ≤ −k+ P t k f k (y) ≤ −k+ ( + s k ) N/ P t k (s k + ) f k (x).
As a consequence of (3.5), Integrating with respect to µ,ˆM Choose t k = λ − (r−q)k/q and replace ε by λ p/ ε where λ > . Thereforê On the other hand The nal step will be to absorb the sum over k ∈ I on the right-hand side by the one on the left-hand side. To this task, note that r = (r − ) + N r by the very de nition of r. For any η > , there exist ε = ε(η) > and k = k (η) ≥ large enough (with further dependence only on p, r, N) so that (r− )k + (cε) /p r k N/ ≤ η rk for every k ≥ k . Hence, If η > is chosen so that r+ cη ≤ , then To conclude, optimize in λ > and make use of the weak-type Proposition 3.1 to control the last term on the right-hand side of this inequality. It follows that, for k large enough, there is a constant C > depending only on p, q, N such that Finally, it is an easy exercise to check that with for example C = k + . Therefore, letting k → ∞ in (3.6) then yields the conclusion. The proof of Theorem 1.1 is complete.

The case of the CD( , ∞) condition
As discussed in the introduction, the dimensional Harnack inequality from Proposition 2.1 is no more available under the in nite-dimensional non-negative curvature CD( , ∞) condition and has to be substituted by Wang's Harnack inequality [1,14,15]. The latter expresses, under thus CD( , ∞), that for any positive (measurable) f on M, and all t > , x, y ∈ M, With this tool in hand, we may state the main result in the CD( , ∞) case. Only p = is considered. Besides, values of q have to be restricted to the interval [ , ] by Proposition 2.2. In addition, since (3.4) does not hold for r = , the value q = has also to be excluded (although a weak-type estimate holds in this case).
Theorem 4.1. Let (M, g, µ) be a weighted Riemannian manifold with weighted probability measure µ satisfying the curvature condition CD( , ∞). Given < q ≤ , there is a constant C > only depending on q such that for 10 any probability dν = fdµ with smooth density f with respect to µ, where r = q q + .
As in the preceding section, we start with a weak-type estimate, actually valid for any q ∈ [ , ]. The main argument relies here on a somewhat di erent use of the Harnack inequality, namely a reverse transportation cost inequality along the ow (cf. [1,2]).

Proposition 4.2. In the setting of Theorem 4.1, there is a constant C > only depending on q ∈ [ , ] such that
Proof. As in the proof of Proposition 3.1, for every u > and t > , Recall now the entropic inequality stating that for any (say bounded) measurable h on M, M hP t f dµ ≤ˆM P t f log P t f dµ + logˆM e h dµ.
Hence, provided that u ≥ (for example), Now, as a consequence of Wang's Harnack inequality (4.1), the entropy of P t f is bounded from above by the Kantorovich distance W (ν, µ) as (see e.g. (1.11) in [2]) Therefore, for any u ≥ , As a conclusion, which is the result of the proposition after optimization in t > .
Proof of Theorem 4.1. We follow the proof of Theorem 1.1 until (3.5) at which point we make use of (4.1). Recall Hence,ˆM where we used that´M f k dµ ≤ k µ(f ≥ k ). Now, going back to (3.2), the value of r ensures that (r−q)k t q/ k = (cε) q/ so that For k large enough (only depending on r), The last term on the right-hand of this inequality may then be handled by the weak-type estimate of Proposition 4.2, concluding the proof of Theorem 4.1 after optimization in ε > .
In addition to the preceding statements, there is a version of Proposition 4.2 for p = q = . This result shares 5 some similarities with the recent [4]. The proof of this proposition follows the standard pattern. For any u > and t > , Now, by (2.1), √ t P t (1 F )f is -Lipschitz (where as usual F = {P t f ≥ u}) so that, for the W metric, Hence µ(P t f ≥ u) ≤ uˆM 1 F P t fdµ = uˆM P t (1 F )fdµ ≤ u µ(P t f ≥ u) + √ t u W (ν, µ) and for u ≥ , µ(P t f ≥ u) ≤ √ t u W (ν, µ). Therefore from which the claim follows after optimization in t > . is the Minkowski content (surface measure) of A and, to ease the notation, W (A) = W (ν, µ) where dν = fdµ = 1 A µ(A) dµ. The inequality (4.3) may be seen as a local version of the links between concentration and isoperimetric-type inequalities under non-negative curvature put forward by E. Milman in [10,11]. Indeed, 5 concentration at some scale amounts to an upper bound on W (A) in terms of a function of µ(A) which in turn, by (4.3), ensures that for sets with small measure the surface measure µ + (A) is (uniformly) bounded away from zero. Due to the concavity of the isoperimetric pro le under non-negative curvature, the latter always yields a comparison to the exponential isoperimetric pro le (see [10,11] for details).