Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups
aa r X i v : . [ m a t h . M G ] A ug GENERALIZED BAKRY–ÉMERY CURVATURE CONDITION ANDEQUIVALENT ENTROPIC INEQUALITIES IN GROUPS
GIORGIO STEFANI
Abstract.
We study a generalization of the Bakry–Émery pointwise gradient estimatefor the heat semigroup and its equivalence with some entropic inequalities along the heatflow and Wasserstein geodesics for metric-measure spaces with a suitable group structure.Our main result applies to Carnot groups of any step and to the SU (2) group. Introduction
The Riemannian framework.
Let ( M , g) be a (complete and connected) N -di-mensional smooth Riemannian manifold with Laplace–Beltrami operator ∆. The cele-brated Bochner formula states that12 ∆ |∇ f | = h∇ ∆ f, ∇ f i g + || Hess f || + Ric( ∇ f, ∇ f ) (1.1)for all f ∈ C ∞ ( M ). DefiningΓ( f, g ) = 12 (cid:16) ∆( f g ) − f ∆ g − g ∆ f (cid:17) = h∇ f, ∇ g i g , Γ ( f, g ) = 12 (cid:16) ∆Γ( f, g ) − Γ( f, ∆ g ) − Γ(∆ f, g ) (cid:17) (1.2)for all f, g ∈ C ∞ ( M ), we can rewrite (1.1) as12 ∆Γ( f ) = Γ(∆ f, f ) + || Hess f || + Ric( ∇ f, ∇ f ) , so that Γ ( f ) = || Hess f || + Ric( ∇ f, ∇ f )for all f ∈ C ∞ ( M ). Here and in the following, we write Γ( f ) = Γ( f, f ) and Γ ( f ) =Γ ( f, f ) for simplicity. Using Cauchy–Schwartz inequality, we can estimate || Hess f || ≥ N (∆ f ) , thus the geometric information Ric ≥ K for some K ∈ R (1.3)implies the analytical informationΓ ( f ) ≥ N (∆ f ) + K Γ( f ) (1.4) Date : September 1, 2020.2010
Mathematics Subject Classification.
Primary 47D07, 28D20. Secondary 53C17.
Key words and phrases.
Bakry–Émery curvature condition, metric-measure space, Carnot group, SU (2)group, topological groups, Gamma Calculus, entropy, entropic inequalities, Wasserstein distance. Acknowledgements . The author thanks Luigi Ambrosio and Giuseppe Savaré for helpful discussionsand many valuable suggestions about the subject. for all f ∈ C ∞ ( M ). Nowadays, (1.4) is the so-called and well-known Bakry–Émerycurvature-dimension inequality CD ( K, N ). Remarkably, it is also possible to prove theconverse implication, see [22, Proposition 6.2]: if a Riemannian manifold M satisfies CD ( K, N ) for some K ∈ R and N ∈ (0 , + ∞ ), then dim M ≤ N and Ric ≥ K .Let us now drop the role of the dimension of M (which formally corresponds to thechoice N = + ∞ in (1.4)) and focus on the lower bound on the Ricci tensor encoded by the CD ( K, ∞ ) condition. After the works of Bakry–Émery [27], Otto–Villani [119], Cordero-Erausquin–McCann–Schmuckenschläger [63] and von Renesse–Sturm [123], the analyticalcondition (1.4) for N = + ∞ on a Riemannian manifold can be equivalently formulatedin other three ways (at least, see [123] for other equivalent statements): via the pointwisegradient estimate for the heat flow, via the Wasserstein contractivity property of the dualheat flow and via the K -convexity of the entropy along geodesics in the Wasserstein space.The heat kernel p t ( x, y ) of the Riemannian manifold ( M , g) is the fundamental solutionof the heat differential operator ∂ t − ∆. The function p t ( x, y ) is smooth in ( t, x, y ) ∈ (0 , + ∞ ) × M × M , symmetric in ( x, y ) and naturally defines the associated heat semigroup P t : C ∞ ( M ) → C ∞ ( M ) as P t f ( x ) = Z M f ( y ) p t ( x, y ) dVol g ( y ) , x ∈ M , for all f ∈ C ∞ ( M ). Inequality (1.4) describes the beaviour of the commutation betweenthe gradient ∇ and the heat semigroup ( P t ) t> . More precisely, the CD ( K, ∞ ) conditionis equivalent to the Bakry–Émery pointwise gradient estimate Γ( P t f ) ≤ e − Kt P t Γ( f ) (1.5)for all t > f ∈ C ∞ ( M ).The dual heat semigroup H t : P ( M ) → P ( M ) is nothing but the extension of the heatsemigroup to the space P ( M ) of probability measures on M and can be defined by setting Z M f d H t µ = Z M P t f d µ for all f ∈ C b ( M ), whenever µ ∈ P ( M ). The subset of P ( M ) of probability measureswith finite second moment P ( M ) = (cid:26) µ ∈ P ( M ) : Z M d ( x, x ) d µ ( x ) < + ∞ for some x ∈ M (cid:27) endowed with the 2 -Wasserstein distance W , given by12 W ( µ, ν ) = sup (cid:26)Z M ϕ d µ + Z M ψ d ν : ϕ ( x ) + ψ ( y ) ≤ d ( x, y ) for all x, y ∈ M (cid:27) for all µ, ν ∈ P ( X ), is a complete and separable geodesic space. The lower bound (1.3)on the Ricci tensor can be equivalently stated as a contractivity property of the dual heatsemigroup with respect to the 2-Wasserstein distance, in the sense that W ( H t µ, H t ν ) ≤ e − Kt W ( µ, ν ) (1.6)for all t > µ, ν ∈ P ( X ).The (Boltzmann) entropy with respect to the volume measure Vol g is defined as Ent ( µ ) = Z M f log f dVol g if µ = f Vol g , + ∞ otherwise , ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 3 whenever µ ∈ P ( X ). Note that, by the Bishop Volume Comparison Theorem, it actuallyholds that Ent ( µ ) > −∞ for all µ ∈ P ( M ), see [70, Lemma 4.1] for the proof. The CD ( K, ∞ ) condition can be equivalently reformulated as a convexity property of theentropy along all (constant speed, as usual) geodesics joining two measures in P ( M ).More precisely, if [0 , ∈ s µ s ∈ P ( M ) is a geodesic joining µ , µ ∈ P ( M ), then thefollowing displacement K -convexity inequality Ent ( µ s ) ≤ (1 − s ) Ent ( µ ) + s Ent ( µ ) − K s (1 − s ) W ( µ , µ ) (1.7)holds for all s ∈ [0 , The non-smooth framework: CD ( K, ∞ ) spaces. The peculiar feature of inequal-ity (1.7) is that it can be stated uniquely in terms of the distance and the volume measure,no matter they come from the underlying smooth structure of the Riemannian manifold,and thus can be considered as a metric-measure definition of the lower bound on the Riccitensor.This observation has led Lott–Villani and Sturm in their groundbreaking works [107,133, 134] to study the properties of very general metric-measure spaces ( X, d , m ) satisfy-ing the displacement K -convexity for some K ∈ R . Besides the many powerful conse-quences successively derived from their ideas, see [78, 106, 122] and the monograph [137]for example, a key feature of the Lott–Sturm–Villani approach is that the displacement K -convexity of the entropy actually provides a metric-measure definition of the CD ( K, ∞ )condition that is stable under Gromov–Hausdorff convergence.As pointed out by Gigli [78], starting from the CD ( K, ∞ ) condition, it is possible toprove that the metric gradient flow (see the monograph [12]) ( S t ) t> of the entropy func-tional in ( P ( X ) , W ) is an evolution semigroup on the convex subset of P ( X ) given byprobability measures with finite entropy. However, since also Finsler geometries (as in theflat case of R N endowed with a non-Euclidean norm) can satisfy the CD ( K, ∞ ) condition,the semigroup ( S t ) t> can be non-linear in such a general setting. Nevertheless, ( S t ) t> can be extended to a continuous semigroup of contractions in L ( X, m ) (and actually inany L p -space) which can be also characterized as the gradient flow in L ( X, m ) of theconvex and 2-homogeneous functional Ch ( f ) = inf (cid:26) lim inf n → + ∞ Z X | D f n | d m : f n ∈ Lip b ( X ) , f n → f in L ( X, m ) (cid:27) , (1.8)the Cheeger energy of f ∈ L ( X, m ), see the celebrated work [62], where | D f | ( x ) = lim sup y → x | f ( y ) − f ( x ) | d ( y, x ) (1.9)is the slope at x ∈ X of the bounded Lipschitz f ∈ Lip b ( X ). Since the slope (1.9) playsthe same role of the absolute value of the gradient in the smooth framework, it is naturalto consider the gradient flow of the Cheeger energy as a metric-measure definition of theheat flow ( P t ) t> in the non-smooth context. The identification of the entropic semigroupand the heat flow has been proved in Euclidean spaces in [97] by Jordan–Kinderleher–Otto(see also [118]) and then extended to Riemannian manifolds [70, 137], Hilbert spaces [19],Finsler spaces [116], Alexandrov spaces [81] and eventually to CD ( K, ∞ ) spaces in thefundamental work [14] of Ambrosio–Gigli–Savaré. We refer the reader also to the worksof Kuwada [94, 95] for their key role in the understanding of the equivalence between G. STEFANI the gradient estimate (1.5) and the W -contraction inequality (1.6) in the non-smoothframework.Having a metric-measure notion of heat flow ( P t ) t> at hand, it is then natural to see ifthe displacement K -convexity is still equivalent to suitable analogues of the Bakry–Émeryinequality (1.5) and the W -contractivity property (1.6) in this abstract setting. Buildingupon the non-smooth Calculus developed in [14], Ambrosio–Gigli–Savaré in [15, 16] andAmbrosio–Gigli–Mondino–Rajala in [11] proved this equivalence under the additional as-sumption that the heat flow ( P t ) t> is linear or, equivalently, that the Cheeger energy (1.8)is a Dirichlet (and thus quadratic ) form on L ( X, m ), in order to naturally rule out Finslergeometries. For this reason, such metric-measure spaces, forming a smaller family of CD ( K, ∞ ) spaces remarkably still stable under Gromov–Hausdorff convergence, are saidto have Riemannian
Ricci curvature bounded from below by K ∈ R , or infinitesimallyHilbertian CD ( K, ∞ ) spaces, or RCD ( K, ∞ ) spaces for short. We refer the reader to [117]and to [81, 142] for strictly related results in Finsler and Alexandrov spaces respectively.One of the most important results of [11, 15] is that RCD ( K, ∞ ) spaces can be equiv-alently characterized as those metric-measure spaces for which the gradient flow ( S t ) t> of the entropy in the Wasserstein space ( P ( X ) , W ) satisfies the following EvolutionVariational Inequality with parameter K ∈ R , EVI K for short,dd t W ( S t µ, ν )2 + K W ( S t µ, ν ) + Ent ( S t µ ) ≤ Ent ( ν ) for a.e. t > µ, ν ∈ P ( X ). Thus, in the infinitesimally Hilbertian case (and so in theparticular case of smooth Riemannian manifolds), inequality (1.10) provides an alternativeequivalent metric-measure formulation of lower bound on the Ricci curvature.The above analysis has been extended also to the finite dimensional case N ∈ (0 , + ∞ ),where however the equivalent metric-measure formulations of the lower bound on theRicci curvature and the upper bound on the dimension become more involved. We referthe reader to the seminal works of Erbar–Kuwada–Sturm [71] and Kuwada [96], and tothe more recent developments obtained in [53, 54, 77]. The theory of CD ( K, N ) spaces hasbeen extended also to the case of negative dimension N ∈ ( −∞ , The sub-Riemannian framework.
Although the class of CD ( K, N ) spaces is verybroad, a large and widely-studied family of spaces is left out, the sub-Riemannian mani-folds. For an introduction on the subject, we refer the reader to the papers [101, 129, 130]and to the monographs [1, 55, 112].A sub-Riemannian manifold is a triple ( M , H , h· , ·i H ), where M is a (connected) smoothmanifold, H ⊂ T M is a sub-bundle of the tangent bundle T M and h· , ·i H is a smoothlyvarying positive definite quadratic form on H . Typically, the sub-bundle H is assumedto be bracket generating and equiregular , that is, at each point x ∈ M the directionsin H x together with all their Lie brackets generate the full tangent space T x M , and thedimensions of the intermediate sub-bundles of commutators obtained at each step do notdepend on the choice of x ∈ M .A sub-Riemannian manifold ( M , H , h· , ·i H ) naturally carries a metric notion, the so-called Carnot–Carathéodory (CC for short) distance , defined as d cc ( x, y ) = inf (cid:26)Z | ˙ γ t | H d t : γ : [0 , → M is horizontal , γ = x, γ = y (cid:27) (1.11) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 5 for all x, y ∈ M . A Lipschitz curve γ : [0 , → M is horizontal if ˙ γ t ∈ H γ t for a.e. t ∈ [0 , H ensures that the function in (1.11) is a finitedistance — this is the celebrated Chow–Rashevskii Theorem, see [1, Theorem 3.31] forexample.A sub-Riemannian manifold ( M , H , h· , ·i H ) can be endowed with the Hausdorff mea-sure associated to d cc . However, differently from the Riemannian case, the Hausdorffmeasure does not coincide in general with the volume induced by the distribution H , the Popp measure µ H . We refer the reader to [1, Chapter 20] for the precise definitions andconstructions.The CD ( K, N ) condition fails for the metric-measure space ( M , d cc , µ H ) and, for thisreason, sub-Riemannian manifolds are said to have Ricci curvature unbounded from below.The first result in this direction was obtained by Juillet in [98] for the Heisenberg group H N ,building upon some results on optimal transportation in H N established by Ambrosio–Rigot in the seminal paper [18]. Later, by exploiting a result of [40], Ambrosio and theauthor in [20] showed that any non-commutative Carnot group is not a CD ( K, ∞ ) space.Carnot groups, of which R N and H N are the simplest examples, are nilpotent Lie groupsthat, in some sense, capture the local infinitesimal behavior of sub-Riemannian manifolds.Precisely, by a famous result of Mitchell [111], Carnot groups are the tangent metriccones to sub-Riemannian manifolds, see [101, Section 4.1] and the references therein.Recently, among other non-embedding results, Huang–Sun [92] proved that equiregularsub-Riemannian manifolds do not satisfy the CD ( K, N ) condition, and Juillet [100] treatedthe case of rank-varying distributions.Despite their intrinsic wild nature of non- CD spaces, in the last fifteen years sub-Riemanninan manifolds have become an active and promising research topic for the studyof Optimal Transport, heat and entropy flows and generalized curvature notions beyondthe well-established Riemannian and CD frameworks.After the pioneering work [18] of Ambrosio–Rigot, the well-posedness of optimal trans-portation was studied in Heisenberg and H -type groups [67, 72, 124], for non-holonomicdistributions [4, 93] and in more general sub-Riemannian manifolds [21, 73].The identification between the heat and the entropy semigroups was established first inthe Heisenberg groups by Juillet in [99] and then in all Carnot groups by Ambrosio andthe author in [20].Several notions of curvature in sub-Riemannian manifolds have been introduced inrecent years, following either the Lagrangian or the
Eulerian approach.The
Lagrangian point of view has its roots in the study of Jacobi vector fields initi-ated in the fundamental works of Agrachev–Li–Zelenko [6, 7, 141] (ALZ for short) andlater developed in [3, 32, 33, 83, 84, 113]. Besides the numerous applications inspired bysome classical results of Riemannian Geometry, see [2, 5, 31, 32, 103, 126] for example,a deep and powerful byproduct of the Lagrangian approach — in the original spirit ofCordero-Erausquin–McCann–Schmuckenschläger [63] — is a precise control of the dis-torsion coefficients in the sub-Riemannian interpolation inequalities . These results wereobtained for the first time by Balogh–Kristály–Sipos in Heisenberg and corank-1 Carnotgroups [29, 30] via direct methods based on the special structure of these spaces. The linkwith the ALZ theory of sub-Riemannian Jacobi fields was made manifest shortly after byBarilari–Rizzi in the more general context of ideal structures [35, 36].Sub-Riemannian interpolation inequalities have two interesting consequences: the
Mea-sure Contraction Property , MCP for short, and the distorted displacement convexity ofthe entropy along Wasserstein geodesics.
G. STEFANI
The
MCP ( K, N ) condition, introduced for the first time by Ohta [114], keeps track ofthe distortion of the volume of a set when it is transported to a Dirac delta. Althoughfor a Riemannian manifold the
MCP ( K, N ) and the CD ( K, N ) conditions are equivalent(with N the topological dimension of the manifold), the MCP ( K, N ) condition is in generalweaker than the CD ( K, N ) condition. The first result in this direction was obtainedby Juillet [98] for the Heisenberg group H N (see also [60]). The same property wasthen proved for other Carnot groups [34, 125] and later established for more general sub-Riemannian manifolds in [35, 36] .The entropy Ent H with respect to the Popp measure µ H of the sub-Riemannian mani-folds ( M , d cc , µ H ) considered in [29, 30, 35, 36] satisfies a distorted displacement convexityinequality in the following sense: if [0 , ∈ s µ s ∈ P ( M ) is the geodesic connectingtwo probability measures µ , µ ∈ P ( M ) with compact support, then Ent H ( µ s ) ≤ (1 − s ) Ent H ( µ ) + s Ent H ( µ ) + w ( s ) (1.12)for all s ∈ [0 , w : [0 , → [0 , + ∞ ) is a function, concave and such that w (0) = w (1) = 1, depending only on the lower bounds on the distorsion coefficients of ( M , d cc , µ H )and compensating the lack of K -convexity of the function s Ent H ( µ s ).Although not strictly related to the present work, for the sake of completeness wewarn the reader that there are other lines of research in the Lagrangian direction for thedefinition of curvature in the sub-Riemannian context besides the ALZ approach. Werefer the interested reader to [48, 49, 88] for generalizations of the notion of connectionand to [50] for the so-called Solov’ev method .The
Eulerian point of view arises from the fundamental work of Baudoin–Garofalo [44](BG for short) and relies on a clever adaptation of the Bakry-Émery curvature-dimensioninequality (1.4) to sub-Riemannian manifolds with transverse symmetries . Roughly said,the tangent space of the sub-Riemannian manifolds considered in [44] splits into theaforementioned subspace of horizontal directions H and a subspace of vertical directions V .To this splitting, it is possible to associate two Γ-operators, the usual horizontal one Γ H associated to the CC distance and the horizontal Laplacian ∆ H , and a new vertical one Γ V which satisfies the commutation propertyΓ H ( f, Γ V ( f )) = Γ V ( f, Γ H ( f )) (1.13)for all f ∈ C ∞ ( M ). Property (1.13) is typical of step 2 distributions H , where V = [ H , H ]and [ V , H ] = 0. DefiningΓ H ( f, g ) = 12 (cid:16) ∆ H Γ H ( f, g ) − Γ H ( f, ∆ H g ) − Γ H (∆ H f, g ) (cid:17) , Γ V ( f, g ) = 12 (cid:16) ∆ H Γ V ( f, g ) − Γ V ( f, ∆ H g ) − Γ V (∆ H f, g ) (cid:17) , for all f, g ∈ C ∞ ( M ), as in the Riemannian case (1.2), the generalized BG curvature-dimension inequality , CD ( K H , K V , κ, N ) for short, with K H ∈ R , K V > κ ≥ N ∈ (0 , + ∞ ], amounts to say thatΓ H ( f ) + ε Γ V ( f ) ≥ N (∆ H f ) + (cid:18) K H − κε (cid:19) Γ H ( f ) + K V Γ V ( f ) (1.14)holds for all f ∈ C ∞ ( M ) and all ε >
0. In (1.14), the parameter K H ∈ R plays the role ofthe lower bound on the ‘generalized Ricci tensor’ and, if κ >
0, then (cid:16) K H − κε (cid:17) → −∞ as ε → + , coherently with the non- CD nature of sub-Riemannian spaces. The usual CD ( K, N ) condition is thus recovered when Γ V = 0 and κ = 0, with K = K H . ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 7
Th BG theory has been developed in several directions, see the numerous applicationsobtained in [37, 39, 41–43, 46, 47] and the generalizations made in [85, 86]. A simple butinteresting consequence of (1.14) is the following pointwise gradient bound for the heatflow ( P t ) t> associated to the horizontal Laplacian ∆ H , in analogy with (1.5): there exists α ∈ R such that Γ H ( P t f ) + Γ V ( P t f ) ≤ e − αt (cid:16) P t Γ H ( f ) + P t Γ V ( f ) (cid:17) (1.15)for all f ∈ C ∞ ( M ) and t ≥
0, see [44, Corollary 4.6].Pointwise gradient bounds for the heat flow ( P t ) t> associated to the horizontal Lapla-cian ∆ H , similar to (1.15) but closer to the Riemannian one (1.5), were proved for thefirst time by Driver–Melcher for the Heisenberg groups [68] and later generalized to allCarnot groups by Melcher [109] (see also [87] for a different proof). Baudoin–Bonnefontobtained similar inequalities for the SU (2) group in [38]. Stronger inequalities have beenproved for the Heisenberg groups [26, 104], H -type groups [91] and the Grushin plane [140]with different techniques, and very recently Baudoin–Kelleher treated the case of metricgraphs via the theory of differential forms on Dirichlet spaces [45] (concerning Dirichletspaces, we also refer the reader to [64] for a strictly related, although slightly weaker,pointwise gradient bound).In all the spaces quoted above, the heat flow ( P t ) t> satisfies an inequality of the formΓ( P t f ) ≤ Ce − Kt P t Γ( f ) (1.16)for all f ∈ C ∞ ( M ) and t ≥
0, for some constants C ≥ K ∈ R (with K = 0 for Carnotgroups, coherently with their homogeneous nature). Since (1.16) reduces to (1.5) when C = 1, by analogy with the CD framework the (optimal) parameter K ∈ R in (1.16) can bethought as a lower bound on the ‘generalized Ricci tensor’. Accordingly to this interpreta-tion, thanks to the celebrated work [94] of Kuwada, the pointwise gradient bound (1.16)is equivalent to the following contractivity property of the dual heat flow ( H t ) t> withrespect to the 2-Wasserstein distance: if µ, ν ∈ P ( M ), then W ( H t µ, H t ν ) ≤ √ C e − Kt W ( µ, ν ) (1.17)for all t ≥
0. In view of the equivalence between (1.16) and (1.17), and in analogy withthe CD framework, the study of (1.16), or equivalently of (1.17), belongs to the Eulerianside of the approach to the definition of curvature in the sub-Riemannian setting.1.4. Sub-Riemannian groups as weak
RCD spaces.
At the present moment, no linkis known between the Lagrangian and the Eulerian approach presented above, in the sensethat no relation has been shown between the distorted convexity of the entropy (1.12) andthe inequalities (1.14) and (1.16) satisfied by the Γ operator and the heat flow ( P t ) t> , inthe same manner of the CD framework.The main contribution of this work is to make a partial step towards the connectionbetween the Langrangian and the Eulerian approach in the sub-Riemannian context byshowing that the pointwise gradient bound (1.16) is equivalent to a heated version of thedisplacement convexity inequality (1.7) and an almost-integrated form of the EvolutionVariational Inequality (1.10) in the context of metric-measure spaces with a group struc-ture, extending to this non- CD setting the dimension-free results obtained by Ambrosio–Gigli–Savaré [13, 15] and Ambrosio–Gigli–Mondino–Rajala [11]. Since these inequalitiesnaturally embeds the corresponding ones of the CD framework, our main equivalence resultcan be seen as an attempt to understand the problem of the grande unification synthétique G. STEFANI proposed by Villani [138] for the special case of metric-measure groups. The present workwas also motivated by some questions raised by Balogh–Kristály–Sipos in [29, Section 5].Let us give a sketch of our idea. We start by assuming that, in a metric-measurespace ( X, d , m ), the metric heat flow ( P t ) t> , i.e., the gradient flow of the Cheeger energyassociated to the distance d (recall (1.8) for the definition) is linear and satisfies thepointwise gradient bound Γ( P t f ) ≤ c ( t ) P t Γ( f ) m -a.e. in X (1.18)for all t ≥ f ∈ Dom( Ch ), for some function c : [0 , + ∞ ) → (0 , + ∞ ) locallypositively bounded from above and below.In this non-smooth context, we precisely have Γ( f ) = |∇ f | w for all f ∈ Dom( Ch ),where |∇ f | w ∈ L ( X, m ) is the so-called minimal relaxed gradient of f in the sense of [14,Definition 4.2] and represents the Cheeger energy (1.8) as Ch ( f ) = 12 Z X |∇ f | w d m for all f ∈ Dom( Ch ). However, to avoid technicalities, in what follows we simply con-sider X as a sub-Riemannian manifold and Γ as the squared modulus of the gradient,Γ( f ) = |∇ f | .We can think of the function c in (1.18) as the curvature function of the space ( X, d , m )replacing the function t e − Kt of the standard pointwise gradient (1.5). Actually, thanksto the Fekete Lemma for sub-additive functions, the optimal curvature function c in (1.18)does satisfy c ( t ) ≤ Ce − Kt for all t ≥ C ≥ K ∈ R , as for the pointwisegradient bound (1.16), provided that lim sup t → + c ( t ) < + ∞ . The (optimal) constant K ∈ R plays the role of the lower bound on the ‘generalized Ricci tensor’ in this situation. We callinequality (1.18) the weak Bakry–Émery curvature condition with respect to the curvaturefunction c , BE w ( c , ∞ ) for short.In this general framework, the equivalence between the pointwise gradient bound (1.18)and the W -contractivity property of the dual heat flow ( H t ) t> , W ( H t µ, H t ν ) ≤ c ( t ) W ( µ, ν ) (1.19)for all µ, ν ∈ P ( X ) and t ≥
0, has already been addressed by Ambrosio–Gigli–Savaréin [13, Section 3.2] adapting the original idea of Kuwada [94]. Actually, in [13] only theimplication (1.18) ⇒ (1.19) is proved in detail, while the other implication (1.19) ⇒ (1.18)— of no need for the scopes of [13] — is only stated with a sketch of its proof. However,the line suggested in [13] for the proof of this implication is not completely correct, seeRemark 3.20 below for the technical details. Our first task is thus to amend the strategyof [13] and to give a self-contained and complete proof of the equivalence between (1.18)and (1.19).Having the correspondence between the pointwise gradient bound (1.18) and the W -contractivity property (1.19) of the dual heat semigroup at hand, we can focus on theproof of the almost-integrated form of the EVI . The following (formal) computations area sketch of the action estimates performed in [14, Section 4.3] for the dimension-freecase N = + ∞ . Actually, our approach takes advantage of the more general point of viewassumed in [71, Section 4.2] for the finite dimensional case N < + ∞ . For the presentation,we also took inspiration from [25, Section 6]. ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 9
Let s µ s = f s m , s ∈ [0 , µ , µ ∈ P ( X ) and let us define a new curve s ˜ µ s = ˜ f s m , s ∈ [0 , µ s = H η ( s ) µ ϑ ( s ) , so that ˜ f s = P η ( s ) f ϑ ( s ) , for all s ∈ [0 , , where η ∈ C ([0 , , + ∞ )) and ϑ ∈ C ([0 , , ϑ (0) = 0 and ϑ (1) = 1. Atleast formally, we can computedd s ˜ f s = ˙ η ( s ) ∆ P η ( s ) f ϑ ( s ) + ˙ ϑ ( s ) P η ( s ) ˙ f ϑ ( s ) for s ∈ (0 , s Ent m (˜ µ s ) = dd s Z X ˜ f s log ˜ f s d m = Z X (1 + log ˜ f s ) dd s ˜ f s d m = − ˙ η ( s ) Z X p ′ ( ˜ f s ) Γ( ˜ f s ) d m + ˙ ϑ ( s ) Z X p ( ˜ f s ) P η ( s ) ˙ f ϑ ( s ) d m for s ∈ (0 , p ( r ) = 1 + log r for all r >
0. Observing that p ′ ( r ) = r ( p ′ ( r )) forall r >
0, by the chain rule Γ( ϕ ( f )) = ( ϕ ′ ( f )) Γ( f ) valid for all ϕ : R → R sufficientlysmooth and all f ∈ Dom( Ch ), we can writedd s Ent m (˜ µ s ) = − ˙ η ( s ) Z X Γ( g s ) d˜ µ s + ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) g s d m (1.20)for s ∈ (0 , g s = p ( ˜ f s ) for brevity.On the other hand, by Kantorovich duality, we can write12 W ( µ, ν ) = sup (cid:26)Z X Q ϕ d µ − Z X ϕ d ν : ϕ ∈ Lip( X ) with bounded support (cid:27) , (1.21)where Q s ϕ = inf y ∈ X ϕ ( y ) + d ( y, x )2 for x ∈ X and s > , is the Hopf–Lax infimum-convolution semigroup . Recalling that ϕ s = Q s ϕ solves theHamilton–Jacobi equation ∂ s ϕ s + |∇ ϕ s | = 0, again integrating by parts we can computedd s Z X ϕ s ˜ f s d m = Z X ∂ s ϕ s d˜ µ s + Z X ϕ s dd s ˜ f s d m = − Z X Γ( ϕ s ) d˜ µ s − ˙ η ( s ) Z X Γ( ϕ s , ˜ f s ) d m + ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ϕ s d m (1.22)for s ∈ (0 , s Z X ϕ s ˜ f s d m + ˙ η ( s ) dd s Ent m (˜ µ s ) ≤ − Z X (cid:16) Γ( ϕ s ) + ˙ η ( s ) Γ( g s ) (cid:17) d˜ µ s − ˙ η ( s ) Z X Γ( ϕ s , ˜ f s ) d m + ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ( ϕ s + ˙ η ( s ) g s ) d m (1.23) for s ∈ (0 , − ˙ η ( s ) R X Γ( g s ) d˜ µ s ≤ ϕ s + ˙ η ( s ) g s ) = Γ( ϕ s ) + 2 ˙ η ( s ) Γ( ϕ s , g s ) + ˙ η ( s ) Γ( g s )and, by the chain rule, Γ( ϕ s , g s ) = Γ( ϕ s , p ( ˜ f s )) = p ′ ( ˜ f s ) Γ( ϕ s , ˜ f s ) . Since r p ′ ( r ) = 1, we have Z X Γ( ϕ s , g s ) d˜ µ s = Z X ˜ f s p ′ ( ˜ f s ) Γ( ϕ s , ˜ f s ) d m = Z X Γ( ϕ s , ˜ f s ) d m , and thus (1.23) simplifies todd s Z X ϕ s ˜ f s d m + ˙ η ( s ) dd s Ent m (˜ µ s ) ≤ − Z X Γ( ϕ s + ˙ η ( s ) g s ) d˜ µ s + ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ( ϕ s + ˙ η ( s ) g s ) d m (1.24)for s ∈ (0 , s µ s = f s m is that Z X ˙ f s ψ d m ≤ | ˙ µ s | (cid:18)Z X Γ( ψ ) d µ s (cid:19) (1.25)for all sufficiently ‘nice’ functions ψ ∈ Dom( Ch ), where | ˙ µ s | = lim h → W ( µ s + h ,µ s ) h , s ∈ (0 , metric velocity of the curve s µ s with respect to the 2-Wasserstein distance.With (1.25) at disposal, we may choose ψ = P η ( s ) ( ϕ s + ˙ η ( s ) g s ) and estimate˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ( ϕ s + ˙ η ( s ) g s ) d m = Z X d d s f ϑ ( s ) ! P η ( s ) ( ϕ s + ˙ η ( s ) g s ) d m ≤ | ˙ ϑ ( s ) | | ˙ µ ϑ ( s ) | (cid:18)Z X Γ( P η ( s ) ( ϕ s + ˙ η ( s ) g s )) d µ s (cid:19) ≤ c ( η ( s ))2 ˙ ϑ ( s ) | ˙ µ ϑ ( s ) | + c − ( η ( s ))2 Z X Γ( P η ( s ) ( ϕ s + ˙ η ( s ) g s )) d µ s ≤ c ( η ( s ))2 ˙ ϑ ( s ) | ˙ µ ϑ ( s ) | + 12 Z X Γ( ϕ s + ˙ η ( s ) g s ) d˜ µ s (1.26)by Young inequality, in virtue of (1.18). By combining (1.24) with (1.26), we concludethat dd s Z X ϕ s ˜ f s d m + ˙ η ( s ) dd s Ent m (˜ µ s ) ≤ c ( η ( s ))2 ˙ ϑ ( s ) | ˙ µ ϑ ( s ) | for s ∈ (0 , ϑ ( s ) = c − ( η ( s )), then we can integrate in s ∈ (0 ,
1) so that,by Kantorovich duality (1.21), we finally get12 W ( H η (1) µ , H η (0) µ ) −
12 R( η ) W ( µ , µ ) + ˙ η (1) Ent m ( H η (1) µ ) ≤ ˙ η (0) Ent m ( H η (0) µ ) + Z ¨ η ( s ) Ent m ( H η ( s ) µ ϑ ( s ) ) d s, (1.27)where R( η ) = Z c − ( η ( s )) d s . ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 11
Note that (1.27) is actually equivalent to the pointwise gradient bound (1.18). Indeed,the choice of the constant function η ( s ) = t for all s ∈ [0 ,
1] immediately gives (1.19).Moreover, if c ( t ) = e − Kt for some K ∈ R , then we recover (1.10) by choosing η ( s ) = st for all s ∈ [0 , W ( H t µ , µ ) − Kte Kt − W ( µ , µ ) + t (cid:16) Ent m ( H t µ ) − Ent m ( µ ) (cid:17) ≤ t = 0, which is enough thanks to the semigroup property) byobserving that Kte Kt − = (1 − Kt + o ( t )) as t → + . For this reason, and adopting thesame terminology of [66, Proposition 3.1], we may think of (1.27) as an almost-integrated form of the EVI (1.10).Since we have no information about the behavior of the function s Ent m ( H η ( s ) µ ϑ ( s ) ),to simplify (1.27) it is convenient to choose η ( s ) = (1 − s ) t + st for s ∈ [0 , ≤ t ≤ t are fixed. With this choice, inequality (1.27) reduces to W ( H t µ , H t µ ) + 2( t − t ) (cid:16) Ent m ( H t µ ) − Ent m ( H t µ ) (cid:17) ≤ A [ t ; t ] − W ( µ , µ ) , (1.28)where A [ t ; t ] = − Z t t c − ( s ) d s . In analogy with (1.10), we call the above inequality (1.28)the weak Evolution Variational Inequality with respect to the curvature function c , EVI w ( c )for short.Arguing exactly as in the proof of [66, Theorem 3.2], from (1.28) we deduce that, if s µ s is a 2-Wasserstein (constant speed) geodesic, then Ent m ( H t + h µ s ) ≤ (1 − s ) Ent m ( H t µ ) + s Ent m ( H t µ )+ s (1 − s )2 h (cid:16) A [ t ; t + h ] − W ( µ , µ ) − W ( H t µ , H t µ ) (cid:17) (1.29)for all t ≥ h >
0. Note that (1.29) is still equivalent to the pointwise gradientbound (1.18) since, by multiplying both of its sides by h > h → + ,we again recover (1.19). Moreover, if we choose t = 0 in (1.29), then we obtain Ent m ( H h µ s ) ≤ (1 − s ) Ent m ( µ ) + s Ent m ( µ ) + B [ h ]2 s (1 − s ) W ( µ , µ ) (1.30)for all h >
0, where B [ h ] = A [0 , h ] − − h . In particular, if c ( t ) = e − Kt then B [ h ] = − K + o (1) as h → + , so that we immediately recover the displacement K -convexity (1.7).For this reason, we may think of (1.29) as a heated version of the displacement convexityof the entropy and we call it the (dimension-free) weak Riemannian Curvature-Dimensioncondition with respect to the curvature function c , RCD w ( c , ∞ ) for short.Inequality (1.30) is very close to the distorted convexity inequality (1.12) but, atthe same time, it reflects the idea behind the generalized Baudoin–Garofalo CD con-dition (1.14), in the sense that B [ h ] → + ∞ as h → + in the non- CD setting, coherentlywith the fact that sub-Riemannian spaces have Ricci curvature unbounded from below.The explosion of the right-hand side of (1.30) as h → + can be interpreted also in thelight of the singularity problem of 2-Wasserstein geodesics, still open for a general sub-Riemannian manifold: if µ ≪ m , then do any 2-Wasserstein geodesic s µ s , s ∈ [0 , µ , µ ∈ P ( X ) still satisfy µ s ≪ m ? This problem was posed for the first timeby Ambrosio–Rigot [18] for the Heisenberg group and positively answered by Figalli–Juillet [72]. Later Figalli–Rifford [73] gave an affirmative answer also for more general sub-Riemannian manifolds (see also [35]). As pointed out by Cavaletti–Mondino [61],the answer is still positive if the ambient metric-measure space satisfies the MCP ( K, N )condition and is essentially non-branching , a condition roughly saying that branching geodesics, i.e., geodesics splitting at intermediate times, are not too many. Note thatsome sub-Riemannian spaces do have branching geodesics, see [110]. Thus, in this sense,considering the heated version of the 2-Wasserstein geodesic in (1.29) can be seen as away to bypass its possible singularity.All in all, apart from technicalities, if we can construct sufficiently good 2-Wassersteincurves s µ s = f s m satisfying (1.25), then, under the linearity of the heat flow, for themetric-measure space ( X, d , m ) we have the equivalences BE w ( c , ∞ ) ⇐⇒ EVI w ( c ) ⇐⇒ RCD w ( c , ∞ ) . Therefore, in analogy with the CD setting, we may call such a metric-measure space( X, d , m ) a weak RCD -space .Property (1.25) can be obtained from a celebrated result of Lisini [105], so that thecentral problem we need to face for the construction of the curve s µ s is the absolutecontinuity property µ s ≪ m . Due to the aforementioned singularity problem in thisgeneral framework, we cannot choose s µ s to be a geodesic. Nevertheless, we maychoose s µ s to be a suitable regularization s µ εs , for all ε >
0, of a geodesic (orof any other probability curve realizing the 2-Wasserstein distance between µ and µ up to a smaller and smaller error). However, since we need the Lisini inequality (1.25),the regularized curve s µ εs has to have 2-Wasserstein metric velocity controlled by thevelocity of the original curve.In [13, 71], the regularized procedure takes advantage of the smoothing property of theheat flow and, precisely, leads to the choice µ εs = H ε µ s . Indeed, on the one hand, thepointwise gradient bound (1.5) implies the instantaneous diffusion property H t µ ≪ m for all t > µ ∈ P ( X ). On the other hand, the W -contractivity property (1.6)immediately gives | ˙ µ εs | ≤ e − Kε | µ s | .Under the weaker BE ( c , ∞ ) condition (1.18), it is still possible to prove the instan-taneous diffusion property. However, for the choice µ εs = H ε µ s , the W -contractivityproperty (1.19) only gives the weaker estimate | ˙ µ εs | ≤ c ( ε ) | µ s | , which is of no use unlesslim t → + c ( t ) = 1, a property the curvature function does not satisfy for the sub-Riemannianmanifolds under consideration.Since we cannot rely on the sole properties the heat flow, it is at this point thatwe assume that the ambient space X has a group operation left-compatible with themetric-measure structure and exploit the property of convolution. In fact, under thisadditional assumption, we may choose the regularized curve as the left-convoluted curve µ εs = ( ̺ ε ⋆ µ s ) m , where ( ̺ ε ) ε> ⊂ L ( X, m ) is a suitable family of convolution kernels.Since the group operation is left-compatible with the metric, it is not difficult to provethat the 2-Wasserstein metric velocity of the left-convoluted curve does not increase, i.e. | ˙ µ εs | ≤ | ˙ µ s | (a property not expected for right-convoluted curves, see Remark 5.14 below).We can thus perform all the above computations on the left-convoluted curve s µ εs andobtain the desired entropic inequalities by passing to the limit as ε → + at the end.Although the present work is focused only on (metric-measure) groups, we believe thatour results may be valid also for other non-group spaces, such as the Grushin plane [140]and metric graphs [45], where the regularization of 2-Wasserstein curves could possibly beperformed by exploiting the particular structure of the underlying space. Another interest-ing problem is whether some sub-Riemmanian manifolds (possibly, with a group structure) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 13 may satisfy a more precise form of the pointwise gradient bound (1.16) also taking intoaccount a dimensional parameter N ∈ (0 , + ∞ ). Finally, from a purely metric-measurepoint of view, on the one side we do not know if the weak RCD condition (1.29) may imply(a weaker form of) the
MCP condition and, on the other side, if the weak
EVI (1.28) maybe useful for the definition of a weaker notion of metric gradient flow and/or for provingstability properties of weak
RCD spaces under Gromov–Hausdorff convergence. We willhopefully come again over these and related topics in a future work.1.5.
Organization of the paper.
The paper is organized as follows.In Section 2 we recall all the known definitions and results in the metric-measure settingwe will use in the sequel, in order to keep the paper the most self-contained as possible.Almost all the theorems are stated without proofs, but we give the reader precise referencesto the existing literature where to find the needed technical details.In Section 3 we introduce the BE w ( c , ∞ ) condition in the metric-measure framework andstudy its consequences, such as Poincaré inequalities and the definition and the propertiesof the dual and the pointwise version of the heat semigroup. The main result of this part isthe equivalence between the BE w ( c , ∞ ) condition and the weak W -contractivity propertyof the dual heat flow, the so-called Kuwada duality.In Section 4 we deal with the Fisher information and the L log L-regularization prop-erty. The results, which will be frequently used in the remaining part of the paper, aretechnical and provide a generalization of the known theory to the context of the BE w ( c , ∞ )condition.Finally, in Section 5, we prove our main equivalence result for metric-measure groups,see Theorem 5.16 for the precise statement. In the last part of this section, we show howthis theorem applies to Carnot groups and to the SU (2) group. Also, we briefly comparethe heated displacement convexity (1.30) we obtain with the distorted displacement con-vexity (1.12) in the case of 2-Wasserstein geodesics induced by right-translation optimaltransport maps. 2. Preliminaries
In this section, we recall the main technical tools we will use throughout the paper.For a more detailed exposition of the results presented below, we refer the reader to [8,10–17, 22, 24, 57, 62, 79, 80, 136, 137]. At the end of this section, we summarize the mainassumptions we will use in the rest of the present work. For the reader’s convenience, wewill try to keep the paper the most self-contained as possible.2.1.
AC curves.
Let ( X, d ) be a metric space. Let I ⊂ R be a closed interval and let p ∈ [1 , + ∞ ]. We say that a curve γ : I → X belongs to AC p ( I ; X ) if d ( γ s , γ t ) ≤ Z ts g ( r ) d r s, t ∈ I, s < t, (2.1)for some g ∈ L p ( I ). The space AC p loc ( I ; X ) is defined analogously. The exponent p = 1corresponds to absolutely continuous curves and is simply denoted by AC( I ; X ). It turnsout that, if γ ∈ AC p ( I ; X ), then there is a minimal function g ∈ L p ( I ) satisfying (2.1),called metric derivative of the curve γ , which is given by | ˙ γ t | = lim s → t d ( γ s , γ t ) | s − t | for L -a.e. t ∈ I, see [12, Theorem 1.1.2] for the simple proof. We thus say that an absolutely continuouscurve γ has constant speed if t
7→ | ˙ γ t | is (equivalent to) a constant.We say that ( X, d ) is a length (metric) space if for all x, y ∈ X we have d ( x, y ) = inf (cid:26)Z | ˙ γ t | d t : γ ∈ AC([0 , X ) , γ = x, γ = y (cid:27) . In addition, we call ( X, d ) a geodesic metric space if for every x, y ∈ X there exists acurve γ : [0 , → X such that γ = x , γ = y and d ( γ s , γ t ) = | s − t | d ( γ , γ ) ∀ s, t ∈ [0 , . In this case, we say that the curve γ : [0 , → X is a ( constant ) unit-speed geodesic andwe write s γ s ∈ Geo([0 , X ).2.2. Slopes.
Let ( X, d ) be a metric space. Let R = R ∪ {−∞ , + ∞} and let f : X → R be a function. We define the effective domain of f asDom( f ) = { x ∈ X : f ( x ) ∈ R } . Given x ∈ Dom( f ), we define the slope and the asymptotic Lipschitz constant of f at x by | D f | ( x ) = lim sup y → x | f ( y ) − f ( x ) | d ( x, y ) , | D ∗ f | ( x ) = lim sup y,z → xy = z | f ( y ) − f ( z ) | d ( y, z ) (2.2)The descending slope and the ascending slope of f at x are respectively given by | D − f | ( x ) = lim sup y → x [ f ( y ) − f ( x )] − d ( x, y ) , | D + f | ( x ) = lim sup y → x [ f ( y ) − f ( x )] + d ( x, y ) . Here and in the following, a + and a − denote the positive and negative part of a ∈ R respectively. When x ∈ Dom( f ) is an isolated point of X , we set | D f | ( x ) = | D ∗ f | ( x ) = | D − f | ( x ) = | D + f | ( x ) = 0. By convention, we set | D f | ( x ) = | D ∗ f | ( x ) = | D − f | ( x ) = | D + f | ( x ) = + ∞ for all x ∈ X \ Dom( f ). Clearly, | D f | ≤ | D ∗ f | on X and the asymptoticLipschitz constant | D ∗ f | : X → [0 , + ∞ ] is an upper semicontinuous function. Note thatthe slopes of a Borel function f : X → R are universally measurable, see [14, Lemma 2.6].According to [62] (see also [14, Section 2.3]), we say that a function g : X → [0 , + ∞ ] isan upper gradient of f : X → R if, for any curve γ ∈ AC([0 , f ) , d )), s g ( γ s ) | ˙ γ s | is measurable in [0 ,
1] (with the convention 0 · ∞ = 0) and | f ( γ ) − f ( γ ) | ≤ Z g ( γ s ) | ˙ γ s | d s. (2.3)If f ∈ Lip( X ), then | D f | , | D ∗ f | , | D − f | and | D + f | are upper gradients of f , see [14,Remark 2.8]. In addition, if ( X, d ) is a length space, then | D ∗ f | ( x ) = lim sup y → x | D f | ( y ) , Lip( f ) = sup x ∈ X | D f | ( x ) = sup x ∈ X | D ∗ f | ( x ) , see [16, Section 3.1]. In particular, | D ∗ f | is the upper semicontinuous envelope of | D f | . ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 15
Hopf–Lax semigroup.
Let ( X, d ) be a length space. For all s >
0, the
Hopf–Laxsemigroup Q s : C b ( X ) → C b ( X ) is given by Q s f ( x ) = inf y ∈ X f ( y ) + d ( y, x )2 s for all x ∈ X and f ∈ C b ( X ) . (2.4)By convention, we set Q f = f for all f ∈ C b ( X ). If f ∈ C b ( X ), thend + d s Q s f ( x ) + 12 | D Q s f | ( x ) = 0 (2.5)for all s > x ∈ X , see [14, Theorem 3.6]. If f ∈ Lip b ( X ), then we also haveLip( Q s f ) ≤ f ) and Lip( Q · f ( x )) ≤ f ) (2.6)for all s ≥ x ∈ X , see the discussion in [16, Section 3.1]. In addition, by [14, Proposi-ton 3.2 and Theorem 3.6], for all s > x
7→ | D Q s f | ( x ) is upper semicontinuous,so that | D Q s f | ( x ) = | D ∗ Q s f | ( x ) (2.7)for all s > x ∈ X .2.4. Wasserstein space.
Let (
X, d ) be a complete and separable (
Polish , for short)metric space. We denote by P ( X ) the set of probability Borel measures on X . Given p ∈ [1 , + ∞ ), the p -Wasserstein (extended) distance between µ, ν ∈ P ( X ) is given by W pp ( µ, ν ) = inf (cid:26)Z X × X d p ( x, y ) d π : π ∈ Plan ( µ, ν ) (cid:27) ∈ [0 , + ∞ ] , (2.8)where Plan ( µ, ν ) = { π ∈ P ( X × X ) : ( p ) ♯ π = µ, ( p ) ♯ π = ν } . (2.9)Here p i : X × X → X , i = 1 ,
2, denote the the canonical projections on the components. Asusual, if µ ∈ P ( X ) and T : X → Y is a µ -measurable map with values in the topologicalspace Y , the push-forward measure T ♯ ( µ ) ∈ P ( Y ) is defined by T ♯ ( µ )( B ) = µ ( T − ( B ))for every Borel set B ⊂ Y . The set Plan ( µ, ν ) introduced in (2.9) is call the set of admissible plans or couplings for the pair ( µ, ν ). Since the metric space ( X, d ) is completeand separable, there exist optimal couplings where the infimum in (2.8) is achieved.The function W p is a finite distance on the so-called p -Wasserstein space ( P p ( X ) , W p ),where P p ( X ) = (cid:26) µ ∈ P ( X ) : Z X d p ( x, x ) d µ ( x ) < + ∞ for some, and thus any, x ∈ X (cid:27) . The space ( P p ( X ) , W p ) is complete and separable. If ( X, d ) is geodesic, then ( P p ( X ) , W p )is geodesic as well. Moreover, µ n W p −−→ µ as n → + ∞ if and only if µ n ⇀ µ as n → + ∞ and lim n → + ∞ Z X d p ( x, x ) d µ n ( x ) = Z X d p ( x, x ) d µ ( x ) for some x ∈ X. As usual, we write µ n ⇀ µ as n → + ∞ , and we say that µ n weakly converges to µ as n → + ∞ , if we have lim n → + ∞ Z X ϕ d µ n = Z X ϕ d µ for all ϕ ∈ C b ( X ) . Given p ∈ [1 , + ∞ ), the p -Wasserstein distance can be equivalently obtained via the Kantorovich duality formula p W pp ( µ, ν ) = sup (cid:26)Z X ϕ c d µ − Z X ϕ d ν : ϕ ∈ Lip b ( X ) (cid:27) ∈ [0 , + ∞ ] (2.10) for all µ, ν ∈ P ( X ), where ϕ c ( x ) = inf y ∈ X ϕ ( y ) + d p ( y, x ) p , for all x ∈ X, (2.11)is the c -conjugate of ϕ ∈ Lip b ( X ) with respect to the cost function c = d p /p . In particular,if p = 1 then (2.11) immediately gives ϕ c = ϕ and thus we can rewrite (2.10) as W ( µ, ν ) = sup (cid:26)Z X ϕ d( µ − ν ) : ϕ ∈ Lip( X ) with Lip( ϕ ) ≤ (cid:27) (2.12)for all µ, ν ∈ P ( X ), the so-called Kantorovich–Rubinstein formula , see [137, ParticularCase 5.16]. If instead p = 2, then by (2.4) we have ϕ c = Q ϕ and thus we can rewrite (2.10)as 12 W ( µ, ν ) = sup (cid:26)Z X Q ϕ d µ − Z X ϕ d ν : ϕ ∈ Lip( X ) with bounded support (cid:27) (2.13)for all µ, ν ∈ P ( X ). Note that the integral expressions appearing in the right-hand sidesof (2.12) and (2.13) are invariant by adding constants to ϕ , so that we can additionallyassume ϕ ≥ m on X , finite on bounded setsand such that supp( m ) = X , for p ∈ [1 , + ∞ ) we let P ac ( X ) = { µ ∈ P ( X ) : µ ≪ m } , P ac p ( X ) = { µ ∈ P ac ( X ) : µ ∈ P p ( X ) } . Thanks to [137, Theorem 6.18], P ac p ( X ) is a W p -dense subset of P p ( X ).For a proof of the above results and as well as for an agile introduction to the Wassersteindistance, we refer the reader to [10, Section 3], [136, Chapter 1] and [137, Chapters 4–6].2.5. Entropy.
Let ( X, d , m ) be a metric-measure space, i.e. ( X, d ) is a Polish metricspace and m is a non-negative Borel-regular measure, finiteon bounded sets and such that supp m = X . (2.14)Note that, in particular, m is a Radon measure on X , see [89, Proposition 3.3.44]. Inaddition, assume that ∃ x ∈ X ∃ A, B > m ( B r ( x )) ≤ A exp( Br ) for all r > . (2.15)The functional Ent m : P ( X ) → ( −∞ , + ∞ ] given by Ent m ( µ ) = Z X f log f d m if µ = f m ∈ P ( X ) , + ∞ otherwise , (2.16)is called the ( relative ) entropy of µ ∈ P ( X ). According to our definition, µ ∈ Dom(
Ent m )implies that µ ∈ P ( X ) and that the effective domain Dom( Ent m ) is convex.As pointed out in [14, Section 7.1], the growth condition (2.15) guarantees that in fact Ent m ( µ ) > −∞ for all µ ∈ P ( X ), see [70, Lemma 4.1]. Hence, if µ = f m ∈ P ( X ) forsome f ∈ L p ( X, m ) with p ∈ (1 , + ∞ ], then f | log f | ∈ L ( X, m ) and µ ∈ Dom(
Ent m ). ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 17
When m ∈ P ( X ), the entropy functional Ent m naturally extends to P ( X ), is lowersemicontinuous with respect to the weak convergence in P ( X ) and positive by Jenseninequality. In addition, if F : X → Y is a Borel map, then Ent F ♯ m ( F ♯ µ ) ≤ Ent m ( µ ) for all µ ∈ P ( X ) , (2.17)with equality if F is injective, see [12, Lemma 9.4.5].When m ( X ) = + ∞ , if we set n = e − c d ( · ,x ) m , where x ∈ X is as in (2.15) and c > n ( X ) < + ∞ (the existence of such c > Ent m ( µ ) = Ent n ( µ ) − c Z X d ( x, x ) d µ for all µ ∈ P ( X ) . (2.18)This shows that Ent m is lower semicontinuous in ( P ( X ) , W ).2.6. Cheeger energy.
Let ( X, d , m ) be a metric-measure space with ( X, d ) a Polishmetric space and m as in (2.14). The functional Ch : L ( X, m ) → [0 , + ∞ ] given by Ch ( f ) = inf (cid:26) lim inf n Z X | D f n | d m : f n → f in L ( X, m ) , f n ∈ Lip( X ) (cid:27) , (2.19)for all f ∈ L ( X, m ), is called Cheeger energy . Here | D f | denotes the slope of f ∈ Lip( X )as defined in (2.2). We letW , ( X, d , m ) = Dom( Ch ) = n f ∈ L ( X, m ) : Ch ( f ) < + ∞ o be the Sobolev space naturally associated to Ch endowed with the norm given by k f k , ( X, d , m ) = k f k ( X, m ) + 2 Ch ( f ) . (2.20)The space (W , ( X, d , m ) , k · k W , ( X, d , m ) ) is a separable Banach space but can fail to be aHilbert space in general, see [14, Remark 4.6].2.7. Minimal weak gradient.
Let ( X, d , m ) be a metric-measure space with ( X, d ) aPolish metric space and m as in (2.14). If f ∈ L ( X, m ), then Grad ( f ) = ( G ∈ L ( X, m ) : " ∃ f n ∈ Lip b ( X, m ) such that f n → f and | D f n | ⇀ G in L ( X, m ) as n → + ∞ ) (2.21)is a convex set, possibly empty (see [14, Definition 4.2] or [127, Section 4.1]). If f ∈ W , ( X, d , m ), then it is possible to show that Grad( f ) = ∅ and thus, by the reflexivity ofL ( X, m ), Grad( f ) has a unique element of minimal L -norm, the minimal weak (upper orrelaxed) gradient of f , | D f | w ∈ L ( X, m ), that is also minimal with respect to the orderstructure, i.e. G ∈ Grad( f ) = ⇒ | D f | w ≤ G m -a.e. in X. (2.22)Thanks to [14, Theorems 6.2 and 6.3] (see also [14, Remark 4.7]), if f ∈ W , ( X, d , m ),then the minimal weak gradient | D f | w ∈ L ( X, m ) provides an integral representation ofthe Cheeger energy, so that Ch ( f ) = 12 Z X | D f | w d m for all f ∈ W , ( X, d , m ) . The minimal weak gradient is a local operator, i.e. f, g ∈ W , ( X, d , m ) = ⇒ | D f | w = | D g | w m -a.e. on { f − g = c } , (2.23) for all c ∈ R , obeys a Leibniz-rule estimate , in the sense that if f, g ∈ W , ( X, d , m ) ∩ L ∞ ( X, m ) then f g ∈ W , ( X, d , m ) with | D( f g ) | w ≤ | f | | D g | w + | D f | w | g | , (2.24)and satisfies the following chain rule f ∈ W , ( X, d , m ) = ⇒ ϕ ( f ) ∈ W , ( X, d , m ) with | D ϕ ( f ) | w ≤ | ϕ ′ ( f ) | | D f | w (2.25)for any Lipschitz function ϕ : I → R defined on an interval I ⊂ R containing the imageof f (with 0 ∈ I and ϕ (0) = 0 if m ( X ) = + ∞ ), with | D ϕ ( f ) | w = ϕ ′ ( f ) | D f | w if ϕ isnon-decreasing. Also, if f, g ∈ W , ( X, d , m ), then f ∧ g, f ∨ g ∈ W , ( X, d , m ) with | D( f ∧ g ) | w = | D f | w m -a.e. on { f ≤ g }| D g | w m -a.e. on { f ≥ g } (2.26)and | D( f ∨ g ) | w = | D f | w m -a.e. on { f ≥ g }| D g | w m -a.e. on { f ≤ g } , (2.27)see [14, Lemma 2.5 and Proposition 4.8(e)] and their proofs.In addition, by [14, Theorem 6.3 and Lemma 4.3(c)], bounded Lipschitz functions aredense in energy in W , ( X, d , m ), i.e. f ∈ W , ( X, d , m ) = ⇒ " ∃ f n ∈ Lip b ( X ) ∩ L ( X, m ) such that f n → f and | D f n | → | D f | w in L ( X, m ) as n → + ∞ . (2.28)In particular, we have that f ∈ Lip b ( X, d ) ∩ W , ( X, d , m ) = ⇒ | D f | w ≤ | D f | m -a.e. in X, (2.29)see also [14, Remark 5.5]. As observed in [13, Section 8.3], the approximation (2.28) canbe enforced by replacing slopes with asymptotic Lipschitz constants (recall (2.2) for thedefinition), so that f ∈ W , ( X, d , m ) = ⇒ " ∃ f n ∈ Lip b ( X ) ∩ L ( X, m ) such that f n → f and | D ∗ f n | → | D f | w in L ( X, m ) as n → + ∞ , (2.30)see [80, Theorem 2.8] and [8, Section 4] for a more detailed discussion.2.8. Heat semigroup.
Let ( X, d , m ) be a metric-measure space with ( X, d ) a Polishmetric space and m as in (2.14). By [14, Theorem 4.5], Ch is convex, lower semicontinuousand 2-homogeneous. The effective domain of the Cheeger energy, which we denote byW , ( X, d , m ), is dense in L ( X, m ). Thus, by the Hilbertian theory of gradient flows,see [12, 57] for the general theory and [10, Theorem 3.1] for a plain exposition of the mainresults, for each given f ∈ L ( X, m ) there exists a curve t f t = P t f ∈ AC loc ((0 , + ∞ ); L ( X, m )) , (2.31)called heat semigroup , such that dd t f t ∈ − ∂ − Ch ( f t ) for L -a.e. t ∈ (0 , + ∞ )lim t → + f t = f in L ( X, m ) . (2.32) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 19
Here and in the following, ∂ − Ch ( f ) ⊂ L ( X, m ) denotes the subdifferential of Ch at f ∈ W , ( X, d , m ) and is defined by ℓ ∈ ∂ − Ch ( f ) ⇐⇒ Ch ( g ) ≥ Ch ( f ) + Z X ℓ ( g − f ) d m for all g ∈ L ( X, m ) . The heat flow (2.31) is uniquely determined by (2.32), is 1-homogeneous, i.e. f ∈ L ( X, m ) , λ ∈ R = ⇒ P t ( λf ) = λ P t f for all t ≥ , and defines a strongly continuous semigroup of contractions in L ( X, m ), meaning that k f t k L ( X, m ) ≤ k f k L ( X, m ) for all t > f ∈ L ( X, m ) . (2.33)By [14, Theorem 4.16(a)], the heat semigroup preserves one-side essential bounds ( maxi-mum principle ). Precisely, for C ∈ R it holds f ≤ C (resp. f ≥ C ) = ⇒ f t ≤ C (resp. f t ≥ C ) for all t ≥ f ≤ g + C = ⇒ f t ≤ g t + C for all t ≥ , (2.35)whenever f, g ∈ L ( X, m ). By [14, Theorem 4.16(b)], the heat semigroup satisfies thecontraction property k f t − g t k L p ( X, m ) ≤ k f − g k L p ( X, m ) for all f, g ∈ L ( X, m ) ∩ L p ( X, m ) , (2.36)whenever p ∈ [1 , + ∞ ]. Since L p ( X, m ) ∩ L ( X, m ) is L p -dense in L p ( X, m ) for all p ∈ [1 , + ∞ ), we can uniquely extend the heat semigroup to a strongly continuous semigroupof contractions in L p ( X, m ), p ∈ [1 , + ∞ ), for which we retain the same notation. Theheat semigroup can thus be extended to a weakly*-continuous semigroup of contractionsin L ∞ ( X, m ) by duality, i.e. Z X ϕ P t f d m = Z X f P t ϕ d m for every f ∈ L ( X, m ) and ϕ ∈ L ∞ ( X, m ) . (2.37)By [14, Theorem 4.20], thanks to (2.36), if m satisfies the growth condition (2.15), thenthe heat semigroup satisfies the mass preservation property Z X f t d m = Z X f d m for all t ≥ f ∈ L ( X, m ) . (2.38)The heat semigroup is regularizing as stated in Lemma 2.1 below. This result is wellknown to experts, but we quickly prove it here for the reader’s convenience. Lemma 2.1 (Heat flow regularization) . Let f ∈ L ( X, m ) . Then t Ch ( f t ) ∈ AC loc ((0 , + ∞ ); [0 , + ∞ )) (2.39) with Ch ( f t ) ≤ inf (cid:26) Ch ( g ) + 12 t Z X | f − g | d m : g ∈ W , ( X, d , m ) (cid:27) (2.40) for all t > , and t
7→ | D f t | w ∈ C((0 , + ∞ ); L ( X, m )) . (2.41) Moreover, if f ∈ W , ( X, d , m ) , then the continuity of the maps in (2.39) and (2.41) extends to t = 0 . Proof.
We divide the proof in two steps.
Step 1: regularity of Cheeger energy along the heat flow . The AC loc -regularity of theChegeer energy along the heat flow in (2.39) and the inf-formula in (2.40) follow from thetheory of Hilbertian gradient flows, see [10, Theorem 3.1]. If moreover f ∈ W , ( X, d , m ),then Ch ( f t ) ≤ Ch ( f ) for all t > Ch ( f t ) → Ch ( f ) as t → + by thelower semicontinuity of the Cheeger energy. Step 2: regularity of the minimal weak gradient along the heat flow . Fix t > t n > n ∈ N , be such that t n → t as n → + ∞ . By (2.39), the sequence ( | D f t n | w ) n ∈ N is bounded in L ( X, m ). We can thus find a subsequence ( | D f t nk | w ) k ∈ N and a function G ∈ L ( X, m ) such that | D f t k | w ⇀ G in L ( X, m ) as k → + ∞ . By the weak lower semi-continuity of the L -norm and again by (2.39), we must have k G k L ( X, m ) ≤ k| D f t | w k L ( X, m ) .By definition of minimal weak gradient and [14, Lemma 4.3(b)], we must also have that | D f t | w ≤ G m -a.e. in X . Hence G = | D f t | w m -a.e. in X and thus | D f t nk | w → | D f t | w inL ( X, m ) as k → + ∞ by the uniform convexity of L ( X, m ) (see [58, Proposition 3.32]for example). Hence (2.41) readily follows. If moreover f ∈ W , ( X, d , m ), then we cantake t = 0 in the above argument and use the continuity in t = 0 of the map in (2.39) toextend the L -continuity of the minimal weak gradient along the heat flow to t = 0. (cid:3) Metric-measure Laplacian.
Let ( X, d , m ) be a metric-measure space with ( X, d )a Polish metric space and m as in (2.14). If f ∈ L ( X, m ) and ∂ − Ch ( f ) = ∅ , then theelement of minimal L -norm in − ∂ − Ch ( f ) is called the ( metric-measure ) Laplacian ofthe function f and is denoted by ∆ d , m f , see [14, Definition 4.13]. The effective domainDom(∆ d , m ) of the Laplacian is a L -dense subset of W , ( X, d , m ) (and thus, in particular,a L -dense subset of L ( X, m )), see [57, Proposition 2.11]. Note that the operator ∆ d , m isnot linear in general, but is 1-homogeneous, in the sense that f ∈ Dom(∆ d , m ) , λ ∈ R = ⇒ λf ∈ Dom(∆ d , m ) with ∆ d , m ( λf ) = λ ∆ d , m f. By the regularizing properties of gradient flows in Hilbert spaces (see [10, Theorem 3.1]),for every t > d + d t f t exists and it is actually the element with min-imal L -norm in − ∂ − Ch ( f ), so that f t ∈ Dom(∆ d , m ) for all t > d + d t f t = ∆ d , m f t for every t ∈ (0 , + ∞ )lim t → + f t = f in L ( X, m )whenever f ∈ L ( X, m ) is given. Moreover, by the integration-by-part formula providedby [14, Proposition 4.15], it holds thatd + d t Ch ( f t ) = −k ∆ d , m f t k L ( X, m ) for all t > , whenever f ∈ L ( x, m ).2.10. Quadratic Cheeger energy.
Let ( X, d , m ) be a metric-measure space with ( X, d )a Polish metric space and m as in (2.14) and (2.15). As in [15, Section4.3], we say thatthe Cheeger energy is quadratic if it satisfies the parallelogram identity Ch ( f + g ) + Ch ( f − g ) = 2 Ch ( f ) + 2 Ch ( g ) for all f, g ∈ W , ( X, d , m ) . (2.42) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 21
In this case, Ch is a quadratic form on W , ( X, d , m ), the functional E : W , ( X, d , m ) → [0 , + ∞ ) defined by the formula E ( f, g ) = Ch ( f + g ) − Ch ( f ) − Ch ( g ) for all f, g ∈ W , ( X, d , m ) (2.43)is a symmetric bilinear form on W , ( X, d , m ) and (W , ( X, d , m ) , k · k W , ( X, d , m ) ) is aHilbert space, see [80, Proposition 4.22]. In particular, thanks to (2.28), the set Lip b ( X ) ∩ W , ( X, d , m ) is W , -dense in W , ( X, d , m ), see [80, Corollary 2.9].For simplicity, we set E ( f ) = E ( f, f ) for all f ∈ W , ( X, d , m ). The chain rule (2.25)for the minimal weak gradient proves that E is Markovian , i.e. E ( ϕ ◦ f ) ≤ E ( f ) for all f ∈ W , ( X, m ) , whenever ϕ : R → R is 1-Lipschitz with ϕ (0) = 0. Since Ch is lower semicontinuous, theform E is also closed. Thus, thanks to the density of W , ( X, d , m ) in L ( X, m ), we canextend the form E given in (2.43) to a symmetric bilinear form on L ( X, m ), for which weretain the same notation. By the locality property of the minimal weak gradient (2.23),the form E is strongly local , meaning that f, g ∈ W , ( X, d , m ) , ( f + c ) g = 0 m -a.e. in X = ⇒ E ( f, g ) = 0 . By [80, Proposition 4.24], the Laplacian ∆ d , m coincides with the generator of E andhence satisfies the integration-by-part formula E ( f, g ) = − Z X g ∆ d , m f d m for all f ∈ Dom(∆ d , m ) , g ∈ W , ( X, d , m ) . (2.44)Thus, since E is symmetric, the Laplacian ∆ d , m is a self-adjoint operator in L ( X, m ). Inaddition, by [80, Proposition 4.23], the Laplacian is a linear operator.Therefore, if the Cheeger energy is quadratic, the heat semigroup ( P t ) t ≥ is a linear analytic Markov semigroup in L ( X, m ), is a self-adjoint operator in L ( X, m ) and themap (2.31) is the unique C map with values in Dom(∆ d , m ) satisfying dd t f t = ∆ d , m f t for t ∈ (0 , + ∞ ) , lim t → + f t = f in L ( X, m ) . Because of this, ∆ d , m can be equivalently characterized in terms of the strong convergence P t f − ft → ∆ d , m f in L ( X, m ) as t → + . (2.45)By [80, Proposition 4.21] (see also [15, Theorem 4.18]), if the Cheeger energy is qua-dratic, then the parallelogram identity (2.42) can be localized at the level of minimal weakgradient, in the sense that | D( f + g ) | w + | D( f − g ) | w = 2 | D f | w + 2 | D g | w m -a.e. in X for all f, g ∈ W , ( X, d , m ). Thus, the naturally associated Γ operator, given byΓ( f, g ) = | D( f + g ) | w − | D f | w − | D g | w for all f, g ∈ W , ( X, d , m ) , defines a strongly-continuous, symmetric and bilinear map from W , ( X, d , m ) to L ( X, m )which represents the form E , i.e. E ( f, g ) = Z X Γ( f, g ) d m for all f, g ∈ W , ( X, d , m ) . The Γ operator satisfies the pointwise estimateΓ( f, g ) ≤ | D f | w | D g | w for all f, g ∈ W , ( X, d , m ) , (2.46)the chain rule Γ( ϕ ( f ) , g ) = ϕ ′ ( f ) Γ( f, g ) for all f, g ∈ W , ( X, d , m ) , (2.47)whenever ϕ ∈ Lip( R ) with ϕ (0) = 0, and the Leibiniz ruleΓ( f g, h ) = g Γ( f, h ) + f Γ( g, h ) for all f, g, h ∈ W , ( X, d , m ) , f, g ∈ L ∞ ( x, m ) , see the discussion in [80, Chapters 3 and 4]. The operatorΓ( f ) = | D f | w , defined for f ∈ W , ( X, d , m ) , is therefore the carré du champ associated to E and obeys the rules of Γ -Calculus . Withthese notations, the Laplacian satisfies the following chain rule , see [80, Proposition 4.28]:if f ∈ Dom(∆ d , m ) ∩ W , ( X, d , m ) and ϕ ∈ C ( R ) with ϕ (0) = 0, then ϕ ( f ) ∈ Dom(∆ d , m )with ∆ d , m ( ϕ ◦ f ) = ϕ ′ ( f ) ∆ d , m f + ϕ ′′ ( f ) Γ( f ) . (2.48)For an account on Γ-Calculus in the present and related frameworks, we refer the readerto [16, 17, 22, 79, 80] and to the monograph [24].2.11. Main assumptions and length property.
We conclude this section summarizingthe main assumptions we are going to use throughout this paper. We assume that ( X, d , m )is a metric-measure space satisfying the following properties:(P.1) ( X, d ) is a complete and separable metric space;(P.2) m is a non-negative Borel-regular measure on X , finite on bounded sets and suchthat supp m = X ;(P.3) there exist x ∈ X and A, B > m ( B r ( x )) ≤ A exp( Br ) for all r > Ch is quadratic, i.e. Ch ( f + g ) + Ch ( f − g ) = 2 Ch ( f ) + 2 Ch ( g )for all f, g ∈ W , ( X, d , m );(P.5) if L ∈ [0 , + ∞ ) and f ∈ W , ( X, d , m ) satisfies | D f | w ≤ L m -a.e. in X , then f = ˜ f m -a.e. in X for some ˜ f ∈ Lip( X ) with Lip( ˜ f ) ≤ L .We say that a metric-measure space ( X, d , m ) is admissible if it satisfies the properties(P.1)-(P.5) listed above.Let us briefly comment on these assumptions. As we have already seen, assumptions(P.1)–(P.4) ensure that ( X, d , m ) satisfies all the properties we have recalled in this section.The additional assumption (P.5), instead, allows to identify the metric-measure structureof ( X, d , m ) with the energetic-measure structure of ( X, B , E , m ), where B is the Borel σ -algebra generate by the topology of open d -balls, thus making the metric-measure andthe energetic-measure approaches equivalent. More precisely, once the Dirichlet–Cheegerenergy E is available, it is relevant to check if the function d E ( x, y ) = sup n | f ( x ) − f ( y ) | : f ∈ W , ( X, d , m ) ∩ C( X ) with Γ( f ) ≤ m -a.e. o , defined for all x, y ∈ X , actually coincides with the starting distance function d . Thefunction d E is known as the Biroli–Mosco distance, see [51] and also [59, 128, 131, 132].From (2.29), we immediately get that d ≤ d E . The opposite inequality follows if anyfunction f ∈ W , ( X, d , m ) ∩ C( X ) with Γ( f ) ≤ m -a.e. in X is 1-Lipschitz, which isprecisely (P.5).In RCD ( K, ∞ ) spaces, assumption (P.5) is an important consequence of the BE ( K, ∞ )condition. Namely, since Γ( P t f ) ≤ e − Kt P t Γ( f ) for all f ∈ W , ( X, d , m ), the 1-Lipschitz ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 23 regularity of f ∈ W , ( X, d , m ) ∩ C( X ) can be obtained passing to the limit as t → + from the e − Kt -Lipschitz regularity of the (more regular) function P t f , see [16, Remark 3.8]and the proof of (v) ⇒ (ii) in [16, Theorem 3.17].In the more general situation in which the function t e − Kt is replaced by a func-tion t c ( t ) such that lim t → + c ( t ) > d E = d provided by (P.5) that will beemployed several times in the sequel is the length property of the metric space ( X, d ),see [16, Theorems 3.10] (and also [128]). Proposition 2.2 (Length property) . If ( X, d , m ) satisfies properties (P.1), (P.2), (P.4)and (P.5) (and thus, in particular, if it is admissible), then ( X, d ) is a length space. Weak Bakry–Émery curvature condition and Kuwada duality
In this section, we introduce and study a generalization of the
Bakry–Émery curva-ture condition for Sobolev functions and its equivalence with the Wasserstein contractionproperty of the dual heat semigroup. The presentation of the results will be close in spiritto that of [16, 94, 95]. For the reader’s ease, we adopt the notation of [16].If not otherwise stated, from now on we assume that ( X, d , m ) is an admissible metric-measure space as in Section 2.11.3.1. Semigroup mollification.
We begin this section by recalling an useful technicaltool that we will use in the following. Let κ ∈ C ∞ c ((0 , + ∞ )) be such that κ ≥ Z + ∞ κ ( r ) d r = 1 . (3.1)Let p ∈ [1 , + ∞ ]. For every f ∈ L p ( X, m ), let us set h ε f = 1 ε Z + ∞ P r f κ (cid:16) rε (cid:17) d r, for all ε > , (3.2)be the semigroup mollification operator , where the integral is intended in the Bochnersense if p < + ∞ and by duality with any function in L ( X, m ) if p = + ∞ .Obviously, by the semigroup property, we have P t ( h ε f ) = h ε ( P t f ) for all t, ε >
0. Since,by a simple change of variable, h ε f = Z + ∞ P εr f κ ( r ) d r, for all ε > , by (3.1) we immediately deduce that h ε f converges to f as ε → + strongly in L p ( X, m )if p < + ∞ and weakly* in L ∞ ( X, m ). The semigroup mollification operator satisfies thefollowing natural W , -approximation property. Lemma 3.1 (W , -approximation via ( h ε ) ε> ) . If f ∈ W , ( X, d , m ) , then h ε f → f in W , ( X, d , m ) as ε → + .Proof. From the definition in (2.21), it follows thatΓ( h ε f − f ) ≤ Z + ∞ Γ( P εr f − f ) κ ( r ) d r m -a.e. in X, so that the conclusion follows by Lemma 2.1 and the Dominated Convergence Theorem. (cid:3) For the reader’s convenience, we briefly prove the following regularity result for theLaplacian of the semigroup mollification operator.
Lemma 3.2 (Laplacian of ( h ε ) ε> ) . Let p ∈ [1 , + ∞ ] . If f ∈ L ( X, m ) ∩ L p ( X, m ) , then − ∆ d , m ( h ε f ) = 1 ε Z + ∞ P r f κ ′ (cid:16) rε (cid:17) d r ∈ L ( X, m ) ∩ L p ( X, m ) for all ε > .Proof. We argue as in the proof of [121, Theorem 2.7]. Without loss of generality, we canassume ε = 1. If t >
0, then P t ( h f ) − h ft = Z + ∞ P r + t f − P r ft κ ( r ) d r = Z + ∞ P r f κ ( r − t ) − κ ( r ) t d r. As t → + , the integrand in the last term converges to − P r f κ ′ ( r ) uniformly for r ∈ [0 , + ∞ ). Thus, in virtue of (2.45), we get that h f ∈ Dom(∆ d , m ) and∆ d , m ( h f ) = lim t → + P t ( h f ) − h ft = − Z + ∞ P r f κ ′ ( r ) d r and the conclusion follows. (cid:3) A differentiation formula.
We now prove Lemma 3.3 below. This result wasproved for the first time in [16, Lemma 2.1] to provide a very general formulation, in theweak sense and with minimal requirements on the regularity of the functions involved, ofthe simple differentiation formula ∂∂s P s ( P t − s f ) = P s n ∆( P t − s f ) − P t − s f )(∆ P t − s f ) o = P s |∇ P t − s f | (3.3)valid for all f ∈ C ∞ ( M ) on a Riemannian manifold ( M , g), see [23, 24, 28, 139] for anaccount.Note that the differentiation formula (3.3), as well as Lemma 3.3 below, does not requireany information about the curvature of the ambient space. For the reader’s convenienceand in order to keep this work the most self-contained as possible, we provide a proof ofthis result in our setting. Lemma 3.3 (Differentiation formula) . Let f ∈ L ( X, m ) and ϕ ∈ L ( X, d , m ) ∩ L ∞ ( X, m ) .If t > , then s A t [ f ; ϕ ]( s ) = 12 Z X ( P t − s f ) P s ϕ d m ∈ C([0 , t ]) ∩ C ([0 , t )) ,s B t [ f ; ϕ ]( s ) = Z X Γ( P t − s f ) P s ϕ d m ∈ C([0 , t )) (3.4) and ∂∂s A t [ f ; ϕ ]( s ) = B t [ f ; ϕ ]( s ) for all s ∈ [0 , t ) . (3.5) The regularity of the functions A and B in (3.4) and the differentiation formula in (3.5) extend to s = t if f ∈ W , ( X, d , m ) .Proof. We divide the proof in four steps.
Step 1: continuity of A . The function s P t − s f , s ∈ [0 , t ], is strongly continuous inL ( X, m ) by the definition of the heat flow. Thanks to the L -contraction property ofthe heat semigroup, by a simple approximation argument we easily get that s P s ϕ , s ∈ [0 , t ], is weakly* continuous in L ∞ ( X, m ). This prove that s A t [ f ; ϕ ]( s ) ∈ C([0 , t ]).
ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 25
Step 2: continuity of B . Since the function s Γ( P t − s f ), s ∈ [0 , t ), is stronglycontinuous in L ( X, m ) by (2.41), and since the function s P s ϕ , s ∈ [0 , t ], is weakly*continuous in L ∞ ( X, m ) by Step 1, we easily deduce that s B t [ f ; ϕ ]( s ) ∈ C([0 , t )).
Step 3: proof of the differentiation formula (3.5). Let us first assume that f ∈ L ( X, m ) ∩ L ∞ ( X, m ) , ϕ ∈ L ∞ ( X, m ) ∩ Dom(∆ d , m ) with ∆ d , m ϕ ∈ L ∞ ( X, m ) . (3.6)Then we have lim h → P t − ( s + h ) f − P t − s fh = − ∆ d , m P t − s f strongly in L ( X, m ) for all s ∈ [0 , t ) by (2.45) andlim h → P s + h ϕ − P s ϕh = ∆ d , m P s ϕ weakly* in L ∞ ( X, m ) for all s ∈ [0 , t ] by (2.37) and again by (2.45). Hence we get that ∂∂s A t [ f ; ϕ ]( s ) = Z X − P t − s f ∆ d , m P t − s f P s ϕ + 12 ( P t − s f ) ∆ d , m P s ϕ ! d m for all s ∈ [0 , t ). Since P t − s f ∈ L ∞ ( X, m ) by (3.6) according to (2.34), we have ( P t − s f ) ∈ W , ( X, d , m ) and thus, thanks to the integration-by-part formula (2.44) and the chainrule (2.48) for the Laplacian, we can compute Z X ( P t − s f ) ∆ d , m P s ϕ d m = Z X ∆ d , m ( P t − s f ) P s ϕ d m = − Z X P t − s f ∆ d , m P t − s f P s ϕ d m + 2 Z X Γ( P t − s f ) P s ϕ d m for all s ∈ [0 , t ) and (3.5) follows.Now let f ∈ L ( X, m ) and keep ϕ as in (3.6). Let f n = − n ∨ f ∧ n ∈ L ( X, m ) ∩ L ∞ ( X, m )for all n ∈ N and note that f n → f in L ( X, m ) as n → + ∞ . By (3.5) applied to f n and ϕ , we know that A t [ f n ; ϕ ]( s ) − A t [ f n ; ϕ ]( s ) = Z s s B t [ f n ; ϕ ]( s ) d s (3.7)for all 0 ≤ s < s < t . Since || D P t − s f n | w − | D P t − s f | w | ≤ Γ( P t − s ( f n − f ))for all n ∈ N and s ∈ [0 , t ) by (2.46), and since Z X Γ( P t − s ( f n − f )) d m = 2 Ch ( P t − s ( f n − f )) ≤ t − s Z X | f n − f | d m for all n ∈ N and s ∈ [0 , t ) by (2.40), we have that Γ( P t − s f n ) → Γ( P t − s f ) in L ( X, m )as n → + ∞ for all s ∈ [0 , t ). Since also P t − s f n → P t − s f in L ( X, m ) as n → + ∞ for all s ∈ [0 , t ), we can pass to the limit as n → + ∞ in (3.7) and prove (3.5) for all f ∈ L ( X, m )and ϕ as in (3.6).Finally, let f ∈ L ( X, m ) and ϕ ∈ L ( X, m ) ∩ L ∞ ( X, m ). For all ε >
0, we set ϕ ε = h ε ϕ .By Lemma 3.2, we know that ϕ ε is as in (3.6) for all ε > ϕ ε → ϕ weakly* inL ∞ ( X, m ) as ε → + . By applying (3.5) to f and ϕ ε in its integrated form and then passingto the limit as ε → + , we prove (3.5) for all f ∈ L ( X, m ) and ϕ ∈ L ( X, m ) ∩ L ∞ ( X, m ). Step 4: the limit case s = t . Let f ∈ W , ( X, d , m ). By the Mean Value Theorem, wejust need to prove that the continuity of the function B extends to s = t . This immediatelyfollows since Γ( P t − s f ) → Γ( f ) in L ( X, m ) as s → t − thanks to Lemma 2.1. (cid:3) BE w ( c , ∞ ) condition. We now come to the central definition of our paper. Hereand in the following, we let c : [0 , + ∞ ) → (0 , + ∞ ) be such that c , c − ∈ L ∞ ([0 , T ]) for all T > . (3.8) Definition 3.4 ( BE w ( c , ∞ ) condition) . We say that ( X, d , m ) satisfies the weak Bakry–Émery curvature condition with respect to the function c : [0 , + ∞ ) → (0 , + ∞ ) in (3.8), BE w ( c , ∞ ) for short, if for all f ∈ W , ( X, d , m ) and t ≥ P t f ∈ W , ( X, d , m )satisfies Γ( P t f ) ≤ c ( t ) P t Γ( f ) m -a.e. in X. (3.9)Although not strictly necessary, we always assume that c (0) = 1 for simplicity.Clearly, if c ( t ) = e − Kt for t ≥
0, then (3.9) states that Γ( P t f ) ≤ e − Kt P t Γ( f ) m -a.e.in X for all f ∈ W , ( X, d , m ), which is precisely the standard Bakry–Émery curvaturecondition BE ( K, ∞ ). We also observe that (3.9) naturally rephrases condition ( G ) in [94,Theorem 2.2(ii)] in our more general framework for ˜ d = c ( t ) d whenever t ≥ c : [0 , + ∞ ) → [0 , + ∞ ) (and so not necessarily locally positively bounded from above and below asin (3.8)), then we can replace it with another measurable function c ⋆ : [0 , + ∞ ) → [0 , + ∞ )which is optimal in the following sense: if t > c ⋆ ( t ) >
0, then for all ε > f ε ∈ W , ( X, d , m ) such that m (cid:16)n x ∈ X : Γ( P t f ε )( x ) ≥ ( c ⋆ ( t ) − ε ) P t Γ( f ε )( x ) o(cid:17) > . By (P.5), we immediately get that m ( { x ∈ X : Γ( f )( x ) > } > f ∈ W , ( X, d , m ) is not m -equivalent to a constant function, and so c ⋆ (0) = 1.In the following result we collect the elementary properties of c ⋆ . Lemma 3.5 (Properties of c ⋆ ) . The function c ⋆ : [0 , + ∞ ] → [0 , + ∞ ) satisfies:(i) c ⋆ ( s + t ) ≤ c ⋆ ( s ) c ⋆ ( t ) for all s, t ≥ ;(ii) c ⋆ is lower semicontinuous;(iii) c ⋆ ( t ) > for all t ≥ .Proof. Property (i) follows from the semigroup property of the heat flow and the opti-mality of c ⋆ . Property (ii) is a consequence of Lemma 2.1 and again of the optimalityof c ⋆ . By (i), if c ( t ) = 0 for some t >
0, then c ( t ′ ) = 0 for all t ′ > t . So let us set t = inf { t > c ⋆ ( t ) = 0 } . By (ii), we get that c ⋆ ( t ) = 0. Since c ⋆ (0) = 1, we musthave that t >
0. Now let f ∈ W , ( X, d , m ) ∩ L ( X, m ) be non-negative, non-constantand such that R X f d m = 1. Since c ⋆ ( t ) = 0, we must have Γ( P t f ) = 0 m -a.e. in X , sothat P t f ( x ) = a for all x ∈ X , for some a ∈ R , by (P.5). By (2.38), we thus get that a = 1 and so we must have that m ( X ) < + ∞ . Without loss of generality, we can assume m ( X ) = 1. Then, by (2.38) again and Jensen inequality, we get1 = (cid:18)Z X f d m (cid:19) = (cid:18)Z X P t / f d m (cid:19) ≤ Z X ( P t / f ) d m = Z X f P t f d m = 1 . By the strict convexity of the square function, we thus get that P t / f = 1 m -a.e. in X .Hence again R X f P t / f d m = 1, so that P t / n f = 1 m -a.e. in X for all n ∈ N by iteratingthe argument above. Thus f = 1 m -a.e. in X , contradicting the fact that the function f was taken non-constant. The proof is thus complete. (cid:3) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 27
The following result is a simple consequence of well-known properties of subadditivefunctions (see [90, Chapter VII] for example), but we briefly sketch its proof here for thereader’s convenience.
Lemma 3.6 (Local boundedness of c ⋆ ) . There exist t ⋆ ≥ and K ∈ R such that c ⋆ ( t ) ≤ e − Kt for all t ≥ t ⋆ . (3.10) In addition, c ⋆ ∈ L ∞ loc ((0 , + ∞ )) and c − ⋆ ∈ L ∞ loc ([0 , + ∞ )) .Proof. We define ϕ : [0 , + ∞ ) → R by setting ϕ ( t ) = log c ⋆ ( t ) for all t ≥
0. By Lemma 3.5, ϕ is well posed, lower semicontinuous and subadditive. By Fekete Lemma (see [90, The-orem 7.6.1] for example), we have that ∃ lim t → + ∞ ϕ ( t ) t = inf t> ϕ ( t ) t < + ∞ , (3.11)from which we immediately deduce (3.10). By [90, Theorem 7.4.1], we have that ϕ ∈ L ∞ loc ((0 , + ∞ )). Since lim inf t → + c ⋆ ( t ) ≥ c ⋆ ( t ) ≥ M forall t ∈ [0 , δ ] for some δ, M >
0, concluding the proof. (cid:3)
From Lemma 3.6, we easily deduce the following exponential upper bound for theoptimal function c ⋆ . Corollary 3.7 (Exponential bound for c ⋆ ) . If (3.9) holds for some everywhere finitemeasurable function c : [0 , + ∞ ) → [0 , + ∞ ) such that lim sup t → + c ( t ) < + ∞ , then the optimal function c ⋆ : [0 , + ∞ ) → (0 , + ∞ ) fulfills (3.8) and is such that c ⋆ ( t ) ≤ M e − Kt for all t ≥ for some M ≥ and K ∈ R . By Corollary 3.7, in analogy with the classical Bakry–Émery condition, we may thinkof the (best) constant K ∈ R appearing in (3.12) as a bound from below of the generalizedmetric-measure Ricci curvature of the space ( X, d , m ). In analogy with the usual RCD framework, we may say that ( X, d , m ) is negatively / zero / positively curved if we can choose K < K = 0/ K > BE w ( c ⋆ , + ∞ )condition behaves like BE ( K, + ∞ ) for some limit K ∈ R as t → + ∞ .3.4. Poincaré inequalities.
Exploiting the differentiation formula in (3.5), we can provethe following consequence of Definition 3.4 in analogy with [16, Corollary 2.3]. Seealso [139, Theorem 1.1(3) and (4)] for the same inequalities in the Riemannian setting.
Proposition 3.8 (Poincaré inequalities) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) .(i) If f ∈ L ( X, m ) and t > , then P t f ∈ W , ( X, d , m ) with − ( t ) Γ( P t f ) ≤ P t ( f ) − ( P t f ) m -a.e. in X. (3.13) (ii) If f ∈ W , ( X, d , m ) and t > , then P t ( f ) − ( P t f ) ≤ ( t ) P t Γ( f ) m -a.e. in X. (3.14) Here and in the following, we let I p ( t ) = Z t c p ( s ) d s (3.15) for all t ≥ and p ∈ R .Proof. Fix t > f ∈ W , ( X, d , m ). By Lemma 3.3, we have12 Z X ( P t ( f ) − ( P t f ) ) ϕ d m = 12 Z dd s Z X ( P t − s f ) P s ϕ d m d s = Z t Z X Γ( P t − s f ) P s ϕ d m d s = Z t Z X P s Γ( P t − s f ) ϕ d m d s for all ϕ ∈ L ( X, m ) ∩ L ∞ ( X, m ). Now assume ϕ ≥
0. By the weak Bakry-Émerycondition (3.9) and the semigroup property of the heat flow, we can estimate c − ( s ) Z X Γ( P t f ) ϕ d m ≤ Z X P s Γ( P t − s f ) ϕ d m ≤ c ( t − s ) Z X P t Γ( f ) ϕ d m for all s ∈ [0 , t ]. ThusI − ( t ) Z X Γ( P t f ) ϕ d m ≤ Z X ( P t ( f ) − ( P t f ) ) ϕ d m ≤ I ( t ) Z X P t Γ( f ) ϕ d m (3.16)for all ϕ ∈ L ( X, m ) ∩ L ∞ ( X, m ) such that ϕ ≥
0. In particular, we can choose ϕ = χ E for any set E ⊂ X with finite m -measure, so that inequalities (3.13) and (3.14) follow forall f ∈ W , ( X, d , m ).Now assume f ∈ L ( X, m ). By the density of W , ( X, d , m ) in L ( X, m ), there exists( f n ) n ∈ N ⊂ W , ( X, d , m ) such that f n → f in L ( X, m ) as n → + ∞ . As a consequence,we have that P t ( f n ) → P t ( f ) and ( P t f n ) → ( P t f ) in L ( X, m ) as n → + ∞ . Moreover,by (2.40) and (2.46), we can estimate Z X || D P t f n | w − | D P t f | w | d m ≤ Z X Γ( P t ( f n − f )) d m = 2 Ch ( P t ( f n − f )) ≤ t Z X | f n − f | d m for all n ∈ N , so that Γ( P t f n ) → Γ( P t f ) in L ( X, m ) as n → + ∞ . By the first inequalityin (3.16), we have thatI − ( t ) Z X Γ( P t f n ) ϕ d m ≤ Z X ( P t ( f n ) − ( P t f n ) ) ϕ d m (3.17)for all ϕ ∈ L ( X, m ) ∩ L ∞ ( X, m ) such that ϕ ≥
0. Passing to the limit as n → + ∞ in (3.17) and arguing as before, we get (3.13). (cid:3) BE w inequality for Lipschitz functions. Thanks to Proposition 3.8, we can provethat the heat flow of a bounded Lipschitz functions in L ( X, m ) has a Lipschitz represen-tative with controlled Lipschitz constant. Proposition 3.9 ( BE w for Lip-functions, I) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) . If f ∈ Lip b ( X, d ) ∩ L ( X, m ) , then P t f ∈ Lip b ( X, d ) ∩ L ( X, m ) with Lip( P t f ) ≤ c ( t ) Lip( f ) (3.18) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 29 for all t ≥ Proof.
Let t ≥ P t f ∈ W , ( X, d , m ) ∩ L ∞ ( X, m ) with 2 I − ( t ) Γ( P t f ) ≤ P t ( f ) ≤ k f k ∞ ( X, m ) m -a.e. in X. Thus, recalling property (P.5), P t f coincides m -a.e. in X with a (bounded) Lipschitzfunction. We now divide the proof in two steps. Step 1 . Assume that supp f is bounded. Then f ∈ Lip b ( X, d ) ∩ W , ( X, d , m ). Hence,by applying the weak Bakry–Émery condition (3.9) to f , we find thatΓ( P t f ) ≤ c ( t ) P t Γ( f ) ≤ c ( t ) Lip( f ) m -a.e. in X. Thus, again by property (P.5), we get (3.18).
Step 2 . Now fix x ∈ X and let R >
0. We let η x ,R : X → [0 ,
1] be such that η x ,R ( x ) = − d ( x, x ) R ! + for all x ∈ X. (3.19)Note that η x ,R ∈ Lip b ( X, d ) withsupp η x ,R ⊂ B R ( x ) , | D η x ,R | ≤ R χ B R ( x ) , Lip( η x ,R ) ≤ R for all R >
0. Let us set f n = f η x ,n for all n ∈ N . Then f n ∈ Lip b ( X, d ) ∩ L ( X, m ) hasbounded support and is such that k f n k L ∞ ( X, d ) ≤ k f k L ∞ ( X, d ) , Lip( f n ) ≤ Lip( f ) + 1 n k f k L ∞ ( X, d ) for all n ∈ N . By Step 1, we get thatLip( P t f n ) ≤ c ( t ) Lip( f n ) (3.20)for all n ∈ N . Since ( f n ) n ∈ N is equi-bounded and equi-Lipschitz, by (2.34) and (3.20)also ( P t f n ) n ∈ N is equi-bounded and equi-Lipschitz. Hence ( P t f n ) n ∈ N converges locallyuniformly to a (bounded) Lipschitz function g t with Lip( g t ) ≤ c ( t ) Lip( f ). Since f n → f in L ( X, m ) as n → + ∞ , then P t f n → P t f in L ( X, m ) as n → + ∞ and thus, up tosubsequences, P t f n → P t f m -a.e. in X . But then g t = P t f m -a.e. in X and thus (3.18)readily follows. (cid:3) Dual heat semigroup.
Thanks to (2.38), we can define the dual heat semigroup H t : P ac ( X ) → P ac ( X ) for all t ≥ H t µ = ( P t f ) m for all µ = f m ∈ P ac ( X ) . (3.21)Note that ( H t ) t ≥ is a linearly convex semigroup on P ac ( X ), in the sense that H s + t µ = H s ( H t µ ) (3.22)and H t ((1 − λ ) µ + λν ) = (1 − λ ) H t µ + λ H t ν for all λ ∈ [0 ,
1] (3.23)whenever s, t ≥ µ, ν ∈ P ac ( X ). Note that (3.21) is well posed without assumingthe BE w ( c , ∞ ) condition.The following result (which still does not require the BE w ( c , ∞ ) condition) proves thatthe dual heat semigroup preserves the finiteness of the second moments of the measuresin the domain of the entropy. Lemma 3.10 (Second moment estimate) . If µ = f m ∈ Dom(
Ent m ) , then µ t = H t µ ∈ P ac2 ( X ) with Z X d ( x, x ) d µ t ≤ e t Ent m ( µ ) + 2 Z X d ( x, x ) d µ ! (3.24) for all t ≥ , whenever x ∈ X is given. In particular, H t (Dom( Ent m )) ⊂ P ac p ( X ) for all p ∈ [1 , and t ≥ .Proof. Fix x ∈ X and set V ( x ) = d ( x, x ) for all x ∈ X . Then V ∈ Lip( X ) withLip( V ) ≤ f ∈ L ( X, m ). The conclusionthen follows by considering f n = q − n ( f ∧ n ) ∈ L ( X, m ), q n = R X f ∧ n d m , n ∈ N ,and passing to the limit as n → + ∞ by the Monotone Convergence Theorem. Since P ( X ) ⊂ P p ( X ) by Jensen inequality, the proof is complete. (cid:3) The following simple result provides a useful sufficient condition to extend the dualheat semigroup to a W p -Lipschitz map on the whole p -Wasserstein space for p ∈ [1 , Lemma 3.11 (Lip-extension of H t on P p ( X ) for p ∈ [1 , . Let p ∈ [1 , and let C : [0 , + ∞ ) → (0 , + ∞ ) be locally bounded. If the dual heat semigroup defined in (3.21) satisfies W p ( H t µ, H t ν ) ≤ C ( t ) W p ( µ, ν ) for all µ, ν ∈ D for some W p -dense subset D of P p ( X ) , then it uniquely extends to a W p -Lipschitz map(for which we retain the same notation) H t : P p ( X ) → P p ( X ) such that W p ( H t µ, H t ν ) ≤ C ( t ) W p ( µ, ν ) for all µ, ν ∈ P p ( X ) . In addition, the maps ( H t ) t ≥ : P p ( X ) → P p ( X ) still satisfy (3.22) and (3.23) for all µ, ν ∈ P p ( X ) .Proof. The extension of the dual heat semigroup readily follows from the W p -densityof D in P p ( X ) and the completeness of the p -Wasserstein space. The validity of (3.22)and (3.23) for all µ, ν ∈ P p ( X ) is a direct consequence of the joint convexity of the p -Wasserstein distance. (cid:3) In the following result, we prove that the BE w ( c , ∞ ) condition implies that the dualheat semigroup can be extended to a W -Lipschitz map on the whole 1-Wasserstein space.See [16, Proposition 3.2(i)], [94, Proposition 3.7] and [95, Theorem 2.2(ii)]. Proposition 3.12 ( BE w ( c , ∞ ) ⇒ H t is W -Lip) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) .For t ≥ , the dual heat semigroup (3.21) uniquely extends to a W -Lipschitz map with W ( H t µ, H t ν ) ≤ c ( t ) W ( µ, ν ) for all µ, ν ∈ P ( X ) . (3.25) Proof.
Let t ≥ W ( H t µ, H t ν ) ≤ c ( t ) W ( µ, ν ) for all µ, ν ∈ Dom(
Ent m ) . (3.26)So let µ, ν ∈ Dom(
Ent m ) with µ = f m and ν = g m . Let ϕ ∈ Lip( X ) with Lip( ϕ ) ≤ ϕ ∈ Lip b ( X ) ∩ L ( X, m ) and so ϕ t = P t ϕ ∈ Lip b ( X ) ∩ L ( X, m )with Lip( ϕ t ) ≤ c ( t ) for all t ≥ µ t = H t µ and ν t = H t ν ,we can estimate Z X ϕ d( µ t − ν t ) = Z X ϕ f t d m − Z X ϕ g t d m = Z X ϕ t f d m − Z X ϕ t g d m = Z X ϕ t d( µ − ν ) ≤ c ( t ) W ( µ, ν ) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 31 for all t ≥ ϕ ∈ Lip( X ) with Lip( ϕ ) ≤ (cid:3) Pointwise version of the heat semigroup.
Let ( H t ) t ≥ be the dual heat semi-group defined from P ac ( X ) to itself as in (3.21) and assume that, for some p ∈ [1 , H t ) t ≥ admits a unique W p -continuous extension from P p ( X ) to itself . (3.27)If (3.27) holds, then for all t ≥ pointwiseversion of the heat semigroup˜ P t f ( x ) = Z X f d H t δ x , x ∈ X, (3.28)whenever f : X → R is either a bounded or a non-negative Borel function. Note that,by the very definition (3.28), (˜ P t ) t ≥ defines a linear semigroup of L ∞ -contractions onbounded Borel functions, in the sense that˜ P s + t f = ˜ P s (˜ P t f )and k ˜ P t f k L ∞ ( X, m ) ≤ k f k L ∞ ( X, m ) (3.29)whenever f ∈ L ∞ ( X, m ) is Borel and s, t ≥
0. The following result lists the main proper-ties of (˜ P t ) t ≥ , see [16, Proposition 3.2(ii) and (iii)]. Proposition 3.13 (Properties of ˜ P t ) . Assume (3.27) holds for some p ∈ [1 , .(i) If f ∈ L ∞ ( X, m ) is lower (resp., upper) semicontinuous, then ˜ P t f is lower (resp.,upper) semicontinuous for all t ≥ . As a consequence, if f ∈ C b ( X ) , then ˜ P t f ∈ C b ( X ) for all t ≥ .(ii) If f ∈ L ∞ ( X, m ) is Borel, then ˜ P t f = P t f m -a.e. in X for all t ≥ .(iii) If f ∈ L ∞ ( X, m ) is Borel and µ ∈ P p ( X ) , then Z X ˜ P t f d µ = Z X f d H t µ (3.30) for all t ≥ .Proof. Let t ≥ Proof of (i) . By the linearity of ˜ P t and P t , we can assume that f ≥ f be lower semicontinuous. If x n → x in X as n → + ∞ , then δ x n → δ x in P p ( X ) as n → + ∞ and thus H t δ x n → H t δ x in P p ( X ) as n → + ∞ by (3.27). Hence H t δ x n ⇀ H t δ x in P ( X ) as n → + ∞ and thus H t δ x ( { f > t } ) ≤ lim inf n → + ∞ H t δ x n ( { f > t } ) . Thus˜ P t f ( x ) = Z + ∞ H t δ x ( { f > t } ) d t ≤ lim inf n → + ∞ Z + ∞ H t δ x n ( { f > t } ) d t = lim inf n → + ∞ ˜ P t f ( x n )by Fatou Lemma. The proof is similar in the case f is upper semicontinuous. Thisproves (i). Proof of (ii) . We divide the proof in two steps.
Step 1 . Let g ∈ L ( X, m ) have compact support K = supp g and be such that µ = g m ∈ P p ( X ). We claim that H t ( g m ) = Z X g ( x ) H t δ x d m ( x ) in P p ( X ) . (3.31) Indeed, given n ∈ N , we can find n points p n , . . . , p nn ∈ K such that K ⊂ n [ i =1 B (cid:16) p ni , n (cid:17) . Let us set A n = B (cid:16) p n , n (cid:17) , A ni = B (cid:16) x ni , n (cid:17) \ [ j
Step 1 . Assume f ∈ C b ( X ). By (ii), we thus know that ˜ P t f = P t f m -a.e. in X , so thatwe can compute Z X ˜ P t f d µ = Z X P t f g d m = Z X f P t g d m = Z X f d H t µ (3.32)for all µ = g m ∈ P ac ( X ). Now if µ ∈ P p ( X ), then we can find µ n ∈ P ac p ( X ) such that µ n W p −→ µ as n → + ∞ . By (3.27), we also have H t µ n W p −→ H t µ as n → + ∞ , and thusby (3.32) and (i) we get Z X ˜ P t f d µ = lim n → + ∞ Z X ˜ P t f d µ n = lim n → + ∞ Z X f d H t µ n = Z X f d H t µ, proving (3.30) whenever f ∈ C b ( X ). Step 2 . Let K ⊂ X be a non-empty bounded closed set. For each n ∈ N , let us set f n ( x ) = [1 − n dist( x, K )] + for all x ∈ X. Then f n ∈ C b ( X ) and χ K ≤ f n ≤ χ H for all n ∈ N , where H = { x ∈ X : dist( x, K ) ≤ } ,and f n ( x ) ↓ χ K ( x ) for all x ∈ X as n → + ∞ . Hence ˜ P t f n ( x ) ↓ ˜ P t χ K ( x ) for all x ∈ X as n → + ∞ and thus Z X ˜ P t χ K d m = lim n → + ∞ Z X ˜ P t f n d µ = lim n → + ∞ Z X f n d H t µ = Z X χ K d H t µ by Step 1 and the Monotone Convergence Theorem. Thus (iii) follows by the MonotoneClass Theorem (see [69, Theorem 5.2.2] for example). (cid:3) BE w inequality for Lipschitz functions, refined. We now refine Proposition 3.9to the following Proposition 3.14, where we prove an everywhere pointwise gradient boundfor the heat flow starting from Lipschitz functions.
Proposition 3.14 ( BE w for Lip-functions, II) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) . If f ∈ Lip b ( X, d ) ∩ L ( X, m ) with | D ∗ f | ∈ L ( X, m ) , then ˜ P t f ∈ Lip b ( X, d ) ∩ W , ( X, d , m ) with | D ∗ ˜ P t f | ( x ) ≤ c ( t ) ˜ P t ( | D ∗ f | )( x ) for all x ∈ X (3.33) for all t ≥ . In the proof of Proposition 3.14, we need the following technical result, see [16, Propo-sition 3.11]. For the reader’s convenience, we recall its short proof here.
Lemma 3.15 (Reverse slope estimate) . Let f ∈ C b ( X ) ∩ W , ( X, d , m ) . If Γ( f ) ≤ G m -a.e. in X for some upper semicontinuous G ∈ L ∞ ( X, m ) , then f ∈ Lip( X ) and | D ∗ f | ( x ) ≤ G ( x ) for all x ∈ X .Proof. Since G ∈ L ∞ ( X, m ), we immediately get f ∈ Lip( X ) from property (P.5). Let x ∈ X be fixed. For all ε >
0, set G ε = sup y ∈ B ε ( x ) ζ ( y ) and define ψ ε ( y ) = min n max {| f ( y ) − f ( x ) | , G ε d ( y, x ) } , G ε [ ε − d ( y, x )] + o for all y ∈ X. Then ψ ε ∈ Lip( X ) with supp ψ ⊂ B ε ( x ), so that ψ ε ∈ W , ( X, d , m ). By (2.26) and (2.27),we have | D ψ ε | w ≤ max { ζ , G ε } m -a.e. in X . Since ψ ε ( y ) = 0 for d ( y, x ) ≥ ε , we must havethat | D ψ ε | w ≤ G ε m -a.e. in X . Again by (P.5), we get Lip( ψ ε ) ≤ G ε . Since ψ ε ( x ) = 0, weconclude that ψ ε ( y ) ≤ G ε d ( y, x ) for all y ∈ X. (3.34)Now, if d ( y, x ) < ε , then [ ε − d ( y, x )] + > ε and ψ ε ( y ) < G ε ε by (3.34), so that | f ( y ) − f ( x ) | ≤ ψ ε ( y ) ≤ G ε d ( y, x ) . Hence | D f | ( x ) = lim sup y → x | f ( y ) − f ( x ) | d ( y, x ) ≤ lim sup y → x G ε d ( y, x ) d ( y, x ) = G ε . Since ζ is upper semicontinuous, we have lim ε → + G ε = ζ ( x ) and thus | D f | ( x ) ≤ ζ ( x ) when-ever x ∈ X . Since ( X, d ) is a length space by Proposition 2.2, again by the uppersemicontinuity of ζ we also get | D ∗ f | ≤ ζ . (cid:3) Proof of Proposition 3.14.
Let t ≥ P t f ∈ Lip b ( X, d ) ∩ L ( X, m ). By Proposition 3.13, we thus know that the continuous represen-tative of P t f coincides with ˜ P t f . Since | D ∗ f | ∈ L ( X, m ), we have f ∈ W , ( X, d , m ) andthus, by (2.29), | D f | w ≤ | D ∗ f | m -a.e. in X . Therefore, thanks to (2.35) and (3.18), wecan estimate Γ(˜ P t f ) ≤ c ( t ) P t Γ( f ) ≤ c ( t ) P t ( | D ∗ f | ) m -a.e. in X and hence, since ˜ P t ( | D ∗ f | ) = P t ( | D ∗ f | ) m -a.e. in X by Proposition 3.13(ii), we getΓ(˜ P t f ) ≤ c ( t ) ˜ P t ( | D ∗ f | ) m -a.e. in X. Since the function x ˜ P t ( | D ∗ f | ) is bounded and upper semicontinuous by Proposi-tion 3.13(i), inequality (3.33) immediately follows by Lemma 3.15. (cid:3) Kuwada duality.
We now come the the main result of this section, the equivalencebetween the weak Bakry–Émery inequality and the W -contractivity property of the (dual)heat semigroup. This duality property is well known for the BE ( K, ∞ ) condition and isdue to Kuwada, see the pioneering works [94–96]. This duality is also known for thestronger BE ( K, N ) condition (with
N < + ∞ ), see [71]. In a very general framework, thisequivalence has been obatined in [16, Theorem 3.5 and Corollary 3.18]. Theorem 3.16 (Kuwada duality) . The following are equivalent.(i) ( X, d , m ) satisfies BE w ( c , ∞ ) .(ii) There exists a W -dense subset D of P ac2 ( X ) such that H t ( D ) ⊂ P ac2 ( X ) and W (( P t f ) m , ( P t g ) m ) ≤ c ( t ) W ( f m , g m ) whenever f m , g m ∈ D (3.35) for all t ≥ .If either (i) or (ii) holds, then for all t ≥ the dual heat semigroup (3.21) uniquely extendsto a map H t : P ( X ) → P ( X ) such that W ( H t µ, H t ν ) ≤ c ( t ) W ( µ, ν ) for all µ, ν ∈ P ( X ) . (3.36)In the proof of Theorem 3.16, we will need the following two technical results.The first one will be use in the proof of the implication (i) ⇒ (ii) and was proved forthe first time in [94]. In the present framework, this result was proved in [16, Lemma 3.4]. ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 35
For the reader’s convenience, we provide a proof of it below. Here and in the following,we let Lip ⋆ ( X ) = { f ∈ Lip( X ) : supp f is bounded and f ≥ } . (3.37)Thanks to (2.29), we immediately see that Lip ⋆ ( X ) ⊂ W , ( X, d , m ). Moreover, from itsvery definition (2.4), we see that the Hopf–Lax semigroup satisfies Q s (Lip ⋆ ( X )) ⊂ Lip ⋆ ( X )for all s ≥
0, since if f ∈ Lip ⋆ ( X ) then supp( Q s f ) ⊂ supp f for all s ≥ Lemma 3.17 (Kuwada estimate) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) . If f ∈ Lip ⋆ ( X ) ,then ˜ P t Q f ( y ) − ˜ P t f ( x ) ≤ c ( t ) d ( y, x ) (3.38) for all x, y ∈ X and t ≥ . In the proof of Lemma 3.17, we will use the following generalization of Fatou Lemma.For its proof, we refer the reader to [16, Lemma 3.3].
Lemma 3.18 (Generalized Fatou Lemma) . If µ n ∈ P ( X ) weakly converges to µ ∈ P ( X ) and ( f n ) n ∈ N are Borel equi-bounded functions such that lim sup n → + ∞ f n ( x n ) ≤ f ( x ) whenever x n → x as n → + ∞ for some Borel function f , then lim sup n → + ∞ Z X f n d µ n ≤ Z X f d µ. Proof of Lemma 3.17.
Let t ≥ x, y ∈ X be fixed. Since Q s f ∈ Lip ⋆ ( X ), weclearly have Q s f ∈ Lip b ( X ) ∩ L ( X, m ) with | D ∗ Q s f | ∈ L ( X, m ) for all s ≥
0. ByProposition 3.13(i), Proposition 3.9 and Proposition 3.14, we thus have thatLip(˜ P t Q s f ) ≤ c ( t ) Lip( Q s f ) (3.39)for all s ≥ | D ∗ ˜ P t Q s f | ( x ) ≤ c ( t ) ˜ P t ( | D ∗ Q s f | )( x ) (3.40)for all s ≥ x ∈ X . Now let γ ∈ AC([0 , X ) be such that γ = x and γ = y . Weclaim that s ˜ P t Q s f ( γ s ) ∈ AC([0 , R ). Indeed, by (2.6) and (3.39), we can estimate | ˜ P t Q s f ( γ s ) − ˜ P t Q s f ( γ s ) | ≤ | ˜ P t Q s f ( γ s ) − ˜ P t Q s f ( γ s ) | + | ˜ P t Q s f ( γ s ) − ˜ P t Q s f ( γ s ) |≤ Lip(˜ P t Q s f ) d ( γ s , γ s ) + Z X | Q s f − Q s f | d H t γ s ≤ c ( t ) Lip( f ) Z s s | ˙ γ s | d s + 2 Lip( f ) ( s − s )for all 0 ≤ s < s ≤
1. We can now write˜ P t Q s + h f ( γ s + h ) − ˜ P t Q s f ( γ s ) h = Z X Q s + h f − Q s fh d H t γ s + h + ˜ P t Q s f ( γ s + h ) − ˜ P t Q s f ( γ s ) h for all 0 ≤ s < s + h ≤
1. On the one hand, we havelim sup h → + Z X Q s + h f − Q s fh d H t γ s + h ≤ Z X d + d s Q s f d H t γ s = − Z X | D Q s f | d H t γ s = −
12 ˜ P t ( | D Q s f | )( γ s ) by Lemma 3.18 and (2.5) for all s ∈ [0 , h → + | ˜ P t Q s f ( γ s + h ) − ˜ P t Q s f ( γ s ) | h ≤ lim sup h → + h Z s + hs | D ∗ ˜ P t Q s f | ( γ r ) | ˙ γ r | d r = | D ∗ ˜ P t Q s f | ( γ s ) | ˙ γ s | for L -a.e. s ∈ [0 , | D ∗ ˜ P t Q s f | ( γ s ) | ˙ γ s | ≤ c ( t ) | ˙ γ s | q ˜ P t ( | D ∗ Q s f | )( γ s ) ≤ c ( t ) | ˙ γ s | + 12 ˜ P t ( | D ∗ Q s f | )( γ s )for all s ∈ [0 , s ˜ P t Q s f ( γ s ) ≤ −
12 ˜ P t ( | D Q s f | )( γ s ) + 12 c ( t ) | ˙ γ s | + 12 ˜ P t ( | D ∗ Q s f | )( γ s )= 12 c ( t ) | ˙ γ s | for L -a.e. s ∈ [0 , P t Q f ( y ) − ˜ P t f ( x ) = Z dd s ˜ P t Q s f ( γ s ) d s ≤ c ( t ) Z | ˙ γ s | d s and (3.38) follows by minimizing with respect to all curves γ ∈ AC([0 , X ) such that γ = x and γ = y . This concludes the proof. (cid:3) The second preliminary result is a well known result proved for the first time by Lisiniin [105]. In this general framework, this result was proved in [16, Lemma 4.12] (se also [80,Theorem 2.1]). For the reader’s convenience, we provide a proof of it below.
Lemma 3.19 (Lisini Theorem for Lip-functions) . If s µ s ∈ AC ([0 , , P ( X )) , then s R X ϕ d µ s ∈ AC ([0 , R ) with (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z X ϕ d µ − Z X ϕ d µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Z X | D ϕ | d µ s ! / | ˙ µ s | d s (3.41) for all ϕ ∈ Lip b ( X ) . In particular, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dd s Z X ϕ d µ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ˙ µ s | Z X | D ϕ | d µ s for L -a.e. s ∈ [0 ,
1] (3.42) for all ϕ ∈ Lip b ( X ) .Proof. Let ϕ ∈ Lip b ( X ) be fixed. By Lisini Theorem, see [105, Theorem 5] or [80, The-orem 2.1], there exists η ∈ P ( C ), C = C([0 , , ( X, d )), concentrated on AC([0 , , X ),such that ( e s ) ♯ η = µ s for all s ∈ [0 , , (3.43)where e s : C → X is the evaluation map at time s ∈ [0 , Z C | ˙ γ s | d η ( γ ) = | ˙ µ s | for L -a.e. s ∈ [0 , . (3.44) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 37
By the upper gradient property of the slope (recall (2.3)), by (3.43), (3.44) and Hölderinequality, we can estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z X ϕ d µ − Z X ϕ d µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z C ( ϕ ( γ ) − ϕ ( γ )) d η ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Z C | D ϕ | ( γ s ) | ˙ γ s | d η ( γ ) d s ≤ Z (cid:18)Z C | D ϕ | ( γ s ) d η ( γ ) (cid:19) / (cid:18)Z C | ˙ γ s | d η ( γ ) (cid:19) / d s = Z (cid:18)Z X | D ϕ | d µ s (cid:19) / | ˙ µ s | d s, proving (3.41). Inequality (3.42) follows easily. (cid:3) We are now ready to prove the main result of this section.
Proof of Theorem 3.16.
We prove the two implications separately.
Proof of ( i ) ⇒ ( ii ). Fix t ≥
0. We divide the proof in three steps.
Step 1: definition of H t and ˜ P t . We define H t : P ac ( X ) → P ac ( X ) by setting H t µ = ( P t f ) m for all µ = f m ∈ P ac ( X ) , (3.45)as in (3.21). By Proposition 3.12, this map can be extended to a map H t : P ( X ) → P ( X ) which satisfies (3.25) with C ( t ) = c ( t ). Hence (3.27) holds with p = 1 and we canthus define ˜ P t f ( x ) = Z X f d H t δ x , x ∈ X, whenever f : X → R is either a bounded or a non-negative Borel function, as in (3.28). Step 2: W -estimate for H t on Dirac deltas . Let x, y ∈ X . By Lemma 3.17, we canestimate Z X Q ϕ d H t δ y − Z X ϕ d H t δ x = ˜ P t Q ϕ ( y ) − ˜ P t ϕ ( x ) ≤ c ( t ) d ( y, x ) (3.46)for all ϕ ∈ Lip ⋆ ( X ). Hence, by (2.13) and taking the supremum on all ϕ ∈ Lip ⋆ ( X )in (3.46), we get W ( H t δ y , H t δ x ) ≤ c ( t ) d ( y, x ) (3.47)whenever x, y ∈ X . Step 3: W estimate for H t on P ( X ). If µ ∈ P ( X ), then we can write µ = Z X δ x d µ ( x ) (3.48)and thus, by Proposition 3.13(iii), we we can also write H t µ = Z X H t δ x d µ ( x ) . (3.49)Now let µ, ν ∈ P ( X ). If π ∈ Plan ( µ, ν ), then we may use a Measurable SelectionTheorem (see [137, Corollary 5.22] or [52, Theorem 6.9.2] for example) to select in a π -measurable way an optimal plan ω tx,y ∈ OptPlan ( H t δ x , H t δ y ) for all x, y ∈ X. (3.50) By (3.48) and (3.49), we thus get thatΩ t = Z X × X ω tx,y d π ( x, y ) ∈ Plan ( H t µ, H t ν ) . Hence, by (2.8), the optimality of (3.50) and by (3.47), we can estimate W ( H t µ, H t ν ) ≤ Z X × X d ( u, v ) dΩ t ( u, v )= Z X × X Z X × X d ( u, v ) d ω tx,y ( u, v ) d π ( x, y )= Z X × X W ( H t δ x , H t δ y ) d π ( x, y ) ≤ c ( t ) Z X × X d ( x, y ) d π ( x, y )whenever π ∈ Plan ( µ, ν ). Again by (2.8), we thus get (3.36). By (3.45) and Lemma 3.10,this proves (ii) with D = Dom( Ent m ). Proof of ( ii ) ⇒ ( i ). Fix t ≥
0. We divide the proof in three steps.
Step 1: definition of H t and ˜ P t . We define H t : D → P ac2 ( X ) by setting H t ( f m ) = ( P t f ) m for all f m ∈ D (3.51)as in (3.21). Thanks to Lemma 3.11, by (3.35) we can extend the map (3.51) to a map H t : P ( X ) → P ( X ) (for which we retain the same notation) such that W ( H t µ, H t ν ) ≤ c ( t ) W ( µ, ν ) for all µ, ν ∈ P ( X ) . Hence (3.27) holds for p = 2 and we can thus define˜ P t f ( x ) = Z X f d H t δ x , x ∈ X, whenever f : X → R is either a bounded or a non-negative Borel function, as in (3.28). Step 2: BE w on Lip -functions via ˜ P t . Let f ∈ Lip b ( X ) ∩ L ( X, m ) with | D ∗ f | ∈ L ( X, m ). We claim that Γ( P t f ) ≤ c ( t ) P t | D ∗ f | m -a.e. in X. (3.52)Indeed, thanks to Proposition 3.13(ii), we have ˜ P t f = P t f m -a.e. in X , so that ˜ P t f ∈ L ( X, m ) ∩ L ∞ ( X, m ) in particular. Fix x, y ∈ X and let γ ∈ AC ([0 , X ) such that γ = x and γ = y . We thus have that s µ s = H t δ γ s ∈ AC ([0 , P ( X )) , since we can estimate W ( µ s , µ s ) ≤ c ( t ) W ( δ γ s , δ γ s ) ≤ c ( t ) d ( γ s , γ s ) ≤ c ( t ) Z s s | ˙ γ r | d r for all 0 ≤ s < s ≤
1, which immediately gives | ˙ µ s | ≤ c ( t ) | ˙ γ s | for L -a.e. s ∈ [0 , . (3.53)By Lemma 3.19 and (3.53), we thus get | ˜ P t f ( y ) − ˜ P t f ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z X f d µ − Z X f d µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z (cid:18)Z X | D f | d µ s (cid:19) / | ˙ µ s | d s ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 39 ≤ c ( t ) Z (cid:18)Z X | D f | d µ s (cid:19) / | ˙ γ s | d s = c ( t ) Z (cid:16) ˜ P t | D f | ( γ s ) (cid:17) / | ˙ γ s | d s. Thanks to (3.29), we thus get | ˜ P t f ( y ) − ˜ P t f ( x ) | ≤ c ( t ) Lip( f ) Z | ˙ γ s | d s, so that ˜ P t f ∈ Lip( X ) with Lip(˜ P t f ) ≤ c ( t ) Lip( f ) by the length property of ( X, d ) (recallProposition 2.2). In addition, again by the length property of ( X, d ), we have (cid:12)(cid:12)(cid:12) ˜ P t f ( y ) − ˜ P t f ( x ) (cid:12)(cid:12)(cid:12) ≤ c ( t ) d ( y, x ) sup (cid:26)(cid:16) ˜ P t | D ∗ f | ( z ) (cid:17) / : d ( z, y ) ≤ d ( y, x ) (cid:27) (3.54)for all x, y ∈ X . Since x
7→ | D ∗ f | ( x ) is upper semicontinuous and bounded, by Propo-sition 3.13(i) also x (cid:16) ˜ P t | D ∗ f | ( x ) (cid:17) / is upper semicontinuous. Therefore, taking thelim sup as y → x in (3.54), we get | D ∗ ˜ P t f | ( x ) ≤ c ( t ) (cid:16) ˜ P t | D ∗ f | ( x ) (cid:17) / for all x ∈ X. Since | D ∗ f | ∈ L ( X, m ) ∩ L ∞ ( X, m ) is Borel, we have that ˜ P t | D ∗ f | = P t | D ∗ f | m -a.e. in X by Proposition 3.13(i), and thus ˜ P t | D ∗ f | ∈ L ( X, m ). Hence ˜ P t f ∈ Lip b ( X ) ∩ L ( X, m )with | D ∗ ˜ P t f | ∈ L ( X, m ), so that ˜ P t f ∈ W , ( X, d , m ) by (2.22), with Γ(˜ P t f ) ≤ | D ∗ ˜ P t f | m -a.e. in X by (2.29). Since ˜ P t f = P t f m -a.e. in X , we must have Γ(˜ P t f ) = Γ( P t f )(recall (2.21) and again the definition of minimal weak gradient). Claim (3.52) is thusproved. Step 3: approximation . Let f ∈ W , ( X, d , m ). By (2.30), we can find f n ∈ Lip b ( X ) ∩ L ( X, m ) such that f n → f and | D ∗ f | → | D f | w in L ( X, m ) as n → + ∞ . By Step 2, wehave Γ( P t f n ) ≤ c ( t ) P t | D ∗ f n | m -a.e. in X (3.55)for all n ∈ N . Since P t | D ∗ f n | → P t Γ( P t f ) in L ( X, m ) as n → + ∞ by (2.33) andΓ( P t f n ) → Γ( P t f ) by (2.40) and again (2.33), up to possibly pass to a subsequence, wecan pass to the limit as n → + ∞ in (3.55) and get (3.9). This proves (i). (cid:3) Remark 3.20 (Errata to the proof of [16, Theorem 3.5]) . In [16, Section 3.2], insteadof (3.33), the authors consider the pointwise inequality (see [16, Equation (3.16)]) | D P t f | ( x ) ≤ c ( t ) ˜ P t ( | D f | )( x ) for all x ∈ X, (3.56)whenever f ∈ Lip b ( X ) ∩ L ( X, m ). In the proof of [16, Theorem 3.5], the authors thenstate that inequality (3.56), together with inequality (3.18) (which corresponds to [16,Equation (3.15)]) are implied by the W -contractivity property of the dual heat semigroup.Since they do not use this implication in their paper, the authors do not provide a proofof this statement and only refer to [15, Theorem 6.2] and to [94]. However, the proofof [15, Theorem 6.2] uses the fact that ˜ P t | D f | ∈ C( X ) for all t >
0, i.e. the L ∞ -to- C -regularization property of (˜ P t ) t> previously proved in [15, Theorem 6.1] thanks to the EVI K property of the gradient flow of the entropy. In the general framework consideredin [16, Section 3.2], as well as in the present one, the L ∞ -to-C-regularization property of (˜ P t ) t> is not available, and thus the continuity of the function x ˜ P t ( | D f | )( x )for t > W -contractivity of ( H t ) t ≥ = ⇒ (3.18) and (3.56)stated in [16, Theorem 3.5] is not completely justified. One can get rid of this problemby replacing [16, Equation (3.16)] with (3.33) and arguing as we have done in the proofof the implication (ii) ⇒ (i) of Theorem 3.16 above thanks to the upper semicontinuityof the asymptotic Lipschitz constant (recall its definition in (2.2)), without affecting thevalidity of all the other results of [16, Section 3.2]. We let the interested reader check thedetails.3.10. Strong Feller property and densities of the dual heat semigroup.
Thefollowing result deals with the regularization property of the pointwise heat semigroup(˜ P t ) t> on L ∩ L ∞ -functions, see the proof of the implication (i) ⇒ (v) in [16, Theo-rem 3.17]. We briefly provide its proof below for the reader’s convenience. Corollary 3.21 (Strong Feller property) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) . If f ∈ L ( X, m ) ∩ L ∞ ( X, m ) is Borel, then ˜ P t f ∈ Lip b ( X ) with q − ( t ) Lip(˜ P t f ) ≤ k f k L ∞ ( X, m ) (3.57) for all t > .Proof. Let t > f ∈ C b ( X ) ∩ L ( X, m ). Then P t f ∈ W , ( X, d , m )by Proposition 3.8(i) with 2 I − ( t ) Γ( P t f ) ≤ k f k ∞ ( X, m ) m -a.e. in X . Hence P t f has aLipschitz representative by (P.5). Thanks to Theorem 3.16 and Proposition 3.13(i), theLipschitz representative of P t f must coincide with ˜ P t f . We thus get that ˜ P t f ∈ Lip b ( X )satisfies (3.57). Hence, arguing as in Step 2 of the proof of Proposition 3.13(iii), we getthat ˜ P t χ K ∈ Lip b ( X ) satisfies (3.57) whenever K ⊂ X is a non-empty bounded closed set.The conclusion thus follows by the Monotone Class Theorem (see [69, Theorem 5.2.2] forexample), since (3.57) allows to convert monotone equibounded convergence of a sequence( f n ) n ∈ N into pointwise convergence on X of the sequence (˜ P t f n ) n ∈ N . (cid:3) An important consequence of Corollary 3.21 is the absolute continuity property of thedual heat semigroup ( H t ) t> on measures in P ( X ), see the proof of [16, Theorem 3.17].We provide a sketch of its proof below for the reader’s convenience. Corollary 3.22 ( H t ( P ( X )) ⊂ P ac2 ( X ) for t > . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) .If µ ∈ P ( X ) , then H t µ ≪ m for all t > .Proof. Let t > µ ∈ P ( X ) and let A ⊂ X be a Borel set with m ( A ) = 0.Then χ A ∈ L ( X, m ) ∩ L ∞ ( X, m ) and so ˜ P t χ A ∈ Lip b ( X ) by Corollary 3.21. By Proposi-tion 3.13(ii), we must have that ˜ P t f = P t f = 0 m -a.e. in X , and thus ˜ P t f ( x ) = 0 for all x ∈ X . Hence H t µ ( A ) = Z X χ A d H t µ = Z X ˜ P t χ A d µ = 0by Theorem 3.16 and Proposition 3.13(iii). The proof is complete. (cid:3) Remark 3.23 (Extension of ( H t ) t ≥ on P ( X )) . Although we do not need such a gen-erality for our purposes, it is possible to show that the dual heat semigroup can beextended to a weakly continuous map ( H t ) t ≥ : P ( X ) → P ( X ) such that (3.36) holds forall µ, ν ∈ P ( X ) with W ( µ, ν ) < + ∞ . Moreover, the validity of Proposition 3.13(iii) and ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 41 of Corollary 3.22 extends to any µ ∈ P ( X ). We refer the interested reader to [16, Sec-tion 3.2] for the details.By Corollary 3.22, for all x ∈ X there exists a non-negative density p t [ x ] ∈ L ( X, m )such that H t δ x = p t [ x ] m for all t > . (3.58)Therefore, accordingly with (3.28), if f : X → R is either a bounded or a non-negativeBorel function, we can then write˜ P t f ( x ) = Z X f ( y ) d H t δ x ( y ) = Z X f ( y ) p t [ x ]( y ) d m ( y ) , (3.59)for all t >
0, so that the definition of (˜ P t f ) t> does not depend on the particular choiceof the representative of f . By linearity, (˜ P t f ) t> is thus well defined whenever f : X → R is a one-side bounded measurable function. Lemma 3.24 (Properties of ( p t [ · ]) t> ) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) and let t > . The following hold.(i) ˜ P s ( p t [ x ]) = p s + t [ x ] m -a.e. in X for all x ∈ X and s ≥ .(ii) p t [ x ]( y ) = p t [ y ]( x ) for m -a.e. x, y ∈ X .Proof. Let t >
Proof of (i) . Let ϕ ∈ L ( X, m ) ∩ L ∞ ( X, m ) be a Borel non-negative function and set ϕ = ¯ ϕ k ϕ k L ( X, m ) . We can compute Z X ϕ p s + t [ y ] d m = Z X ϕ d H s + t δ y by (3.59), (3.22) and Lemma 3.11 = Z X ˜ P s ϕ d H t δ y by (3.28) = Z X P s ϕ p t [ y ] d m by Proposition 3.13(ii) and (3.59) = k ϕ k L ( X, m ) Z X p t [ y ] d H s ( ¯ ϕ m ) by (3.21) = Z X ˜ P s ( p t [ y ]) ϕ d m by Proposition 3.13(iii) and (3.59)for all s ≥ , so that (i) immediately follows. Proof of (ii) . Let ϕ, ψ ∈ L ( X, m ) ∩ L ∞ ( X, m ) be two Borel non-negative functions. ByTonelli Theorem, (3.59) and Proposition 3.13(ii), we can compute Z X Z X ϕ ( x ) ψ ( y ) p t [ x ]( y ) d m ( x ) d m ( y ) = Z X ϕ ( x ) Z X ψ ( y ) p t [ x ]( y ) d m ( y ) d m ( x )= Z X ϕ ( x ) ˜ P t ψ ( x ) d m ( x )= Z X ˜ P t ϕ ( y ) ψ ( y ) d m ( y )= Z X ψ ( y ) Z X ϕ ( x ) p t [ y ]( x ) d m ( x ) d m ( y )= Z X Z X ϕ ( x ) ψ ( y ) p t [ y ]( x ) d m ( x ) d m ( y ) , so that (ii) immediately follows. (cid:3) BE w inequality for Lipschitz functions, again. We conclude this section withthe following result, which provides a refined version of Proposition 3.9 and Proposi-tion 3.14, see the proof of [16, Theorem 3.17]. We give its proof below for the reader’sconvenience.
Proposition 3.25 ( BE w for Lip -functions, III) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) . If f ∈ Lip b ( X ) ∩ W , ( X, d , m ) , then P t f ∈ Lip b ( X ) ∩ W , ( X, d , m ) with | D ∗ P t f | ( x ) ≤ c ( t ) ˜ P t Γ( f )( x ) for all x ∈ X whenever t > .Proof. Let t > be fixed. By Proposition 3.9 we already know that P t f ∈ Lip b ( X ) ∩ W , ( X, d , m ) . From (3.9), (2.29), Proposition 3.13(ii) and (3.59) we get that Γ( P t f ) ≤ c ( t ) P t Γ( f ) = c ( t ) ˜ P t Γ( f ) m -a.e. in X. Since Γ( f ) ∈ L ( X, m ) ∩ L ∞ ( X, m ) , by Corollary 3.21 and again (3.59) we must have that ˜ P t Γ( f ) ∈ C b ( X ) . The conclusion thus follows from Lemma 3.15. (cid:3) Corollary 3.26 (Weak reverse slope estimate for ( P t ) t> ) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) . If f ∈ L ( X, m ) ∩ L ∞ ( X, m ) , then | D ∗ P t f | ≤ c (0 + ) Γ( P t f ) m -a.e. in X (3.60) for all t > , where c (0 + ) = lim inf s → + c ( s ) ≥ .Proof. Let t > be fixed and let ( ε n ) n ∈ N ⊂ (0 , t ) be such that ε n ↓ as n → + ∞ and c (0 + ) = lim n → + ∞ c ( ε n ) . By Corollary 3.21, we have f n = P t − ε n f ∈ Lip b ( X ) ∩ W , ( X, d , m ) for all n ∈ N and thus, by Proposition 3.25, Proposition 3.13(ii) and (3.59), we canestimate | D ∗ P t f | = | D ∗ P ε n f n | ≤ c ( ε n ) ˜ P ε n Γ( f n ) = c ( ε n ) P ε n Γ( f n ) m -a.e. in X for all n ∈ N . Since Γ( f n ) → Γ( f ) in L ( X, m ) as n → + ∞ by (2.41) in Lemma 2.1, theconclusion follows by passing to the limit as n → + ∞ . (cid:3) As a completely natural (although painful) drawback of the weakness of the BE w ( c , ∞ ) property, if the function c in (3.8) is such that c (0 + ) = lim inf t → + c ( t ) > , (3.61)then Corollary 3.26 provides no useful information, since | D ∗ P t f | ≥ | D P t f | ≥ | D P t f | w m -a.e. in X whenever t > by (2.29) and Corollary 3.21. In Section 5 (precisely, in theproof of Lemma 5.3), similarly to [16], we will need the following regularization propertyof the heat semigroup. Definition 3.27 (Heat-smoothing admissible space) . We say that an admissible metric-measure space is heat-smoothing if f ∈ L ∞ ( X, m ) ∩ W , ( X, d , m ) = ⇒ | D P t f | = | D P t f | w m -a.e. in X for all t > . (3.62)Note that, if c (0 + ) = 1 , then inequality (3.60) in Corollary 3.26 immediately implies (3.62)(actually, in the stronger form assuming f ∈ L ( X, m ) ∩ L ∞ ( X, m ) only). ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 43 Fisher information and
L log L -regolarization
In this section, we recall some useful properties of the Fisher information and theentropy functional in admissible metric-measure spaces. We only detail the proofs of theresults which rely on the BE w ( c , ∞ ) condition.4.1. Fisher information, entropy and Kuwada Lemma.
Let us set L ( X, m ) = n f ∈ L ( X, m ) : f ≥ m -a.e. in X o the convex cone of non-negative L -functions. As in [14, Definition 4.9], the Fisher infor-mation F : L ( X, m ) → [0 , + ∞ ] is defined for all f ∈ L ( X, m ) as F ( f ) = Ch ( √ f ) if q f ∈ W , ( X, d , m )+ ∞ otherwise . In particular, we have
Dom( F ) = (cid:26) f ∈ L ( X, m ) : q f ∈ W , ( X, d , m ) (cid:27) . Since f n → f in L ( X, m ) as n → + ∞ implies that q f n → q f in L ( X, m ) as n → + ∞ ,the Fisher information F is lower semicontinuous in L ( X, m ) . Thanks to the localityproperty (2.23) and the chain rule (2.25), if f ∈ Dom( F ) then f n = f ∧ n ∈ W , ( X, d , m ) with | D f n | w = 2 q f n | D q f | w χ { f ≤ n } ∈ L ( X, m ) for all n ∈ N and | D f n | w ↑ q f | D q f | w in L ( X, m ) as n → + ∞ . Hence we can write F ( f ) = lim n → + ∞ Z X | D q f | w χ {
Ent m ) and set f t = P t f and µ t = f t m for all t ≥ . (i) For all t ≥ , we have Ent m ( µ t ) ≤ Ent m ( µ ) .(ii) If x ∈ X and T > , then Z T F ( f t ) d t + 2 Z T Z X d ( x, x ) d µ t ( x ) d t ≤ e T (cid:18) Ent m ( µ ) + 2 Z X d ( x, x ) d µ ( x ) (cid:19) . (4.3)Thanks to Lemma 4.1, we have the following fundamental result. We refer the interestedreader to [14, Lemma 6.1] (see also [81, Proposition 3.7]) for its proof. Lemma 4.2 (Estimate on the W -velocity) . If µ = f m ∈ Dom(
Ent m ) with f ∈ L ( X, m ) ,then t µ t = f t m ∈ AC ([0 , + ∞ ); P ( X )) , where f t = P t f for all t ≥ , with | ˙ µ t | ≤ F ( f t ) = Z { f t > } | D f t | w f t d m for L -a.e. t > . As a simple consequence of Lemma 4.2 and the BE w ( c , ∞ ) condition, we can prove thefollowing W -continuity property of the dual heat flow. For the same result under thestandard BE ( K, ∞ ) condition, see the last part of [16, Lemma 4.2]. Lemma 4.3 ( W -continuity of t H t ) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) . If µ ∈ P ( X ) , then t H t µ ∈ C([0 , + ∞ ); P ( X )) . Equivalently, if µ ∈ P ( X ) then t H t µ is weakly continuous on [0 , + ∞ ) and t R X d ( x, x ) d H t µ ( x ) is continuous on [0 , + ∞ ) whenever x ∈ X is given.Proof. If µ = f m ∈ Dom(
Ent m ) with f ∈ L ( X, m ) , then the W -continuity of the map t H t µ follows immediately from Lemma 4.2. If µ ∈ P ( X ) , then we can find ( µ n ) n ∈ N ⊂ P ac2 ( X ) such that µ n W −→ µ as n → + ∞ . Possibly performing a truncation argument, wecan also assume that Ent ( µ n ) < + ∞ and µ n = f n m with f n ∈ L ( X, m ) for all n ∈ N . If t ≥ , then we can estimate lim sup s → t W ( H s µ, H t µ ) ≤ W ( H s µ n , H s µ ) + W ( H t µ n , H t µ ) + lim sup s → t W ( H s µ n , H t µ n ) ≤ M W ( µ n , µ ) , where we have set M = sup s ∈ [ t ,t +1] c ( s ) . The conclusion thus follows by passing to thelimit as n → + ∞ . (cid:3) Log-Harnack and
L log L estimates.
The rest of this section is dedicated to theproof of the following fundamental regularization property of the dual heat semigroup,see [16, Theorem 4.8] for the same result in the standard BE ( K, ∞ ) setting. Theorem 4.4 ( L log L regularization) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) . If µ ∈ P ( X ) , then Ent m ( H t µ ) ≤
12 I − ( t ) (cid:18) r + Z X d ( x, x ) d µ ( x ) (cid:19) − log m ( B r ( x )) (4.4) for all x ∈ X and r, t > . In particular, H t ( P ( X )) ⊂ Dom(
Ent m ) for all t > . To prove Theorem 4.4, we follow the same strategy adopted in [16, Section 4.2]. Beforethe proof of Theorem 4.4, we need two preliminary results.The first one is the following generalization of the differentiation formula proved inLemma 3.3. The proof goes as that of [16, Lemma 4.5] with minor modifications, so weomit it.
ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 45
Lemma 4.5 (General differentiation formula) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) andlet ω ∈ C ([0 , + ∞ )) . If f ∈ Lip b ( X ) ∩ W , ( X, d , m ) and µ ∈ P ( X ) , then for all t > we have s G ( s ) = Z X ω ( P t − s f ) d H s µ ∈ C([0 , t ]) ∩ C ((0 , t )) with G ′ ( s ) = Z X ω ′′ ( P t − s f ) Γ( P t − s f ) d H s µ for all s ∈ [0 , t ] . The second preliminary result is an adaptation to the abstract setting of an inequalityproved for the first time in the Riemannian framework by Wang, see [139, Theorem 1.1(6)].Lemma 4.6 below is the reformulation, under the more general BE w ( c , ∞ ) condition,of [16, Lemma 4.6]. Although its proof is very similar to that of [16, Lemma 4.6], wedetail it here for the reader’s convenience. Lemma 4.6 (Wang log-Harnack inequality) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) andlet ε > . If f ∈ L ( X, m ) is non-negative, then ˜ P t (log( f + ε ))( y ) ≤ log(˜ P t f ( x ) + ε ) + d ( x, y )4 I − ( t ) , (4.5) for all x, y ∈ X and t > , where I − is as in (3.15) .Proof. Let ε > and t > be fixed. We divide the proof in three steps. Step 1 . Assume f ∈ Lip b ( X ) ∩ W , ( X, d , m ) . Fix x, y ∈ X and let γ ∈ AC([0 , R ) besuch that γ = x and γ = y . We set ϑ ( r ) = I − ( r )I − ( t ) ∈ [0 , for all r ∈ [0 , t ] . Note that ϑ ∈ Lip([0 , t ]) . We also set ω ε ( r ) = log( r + ε ) − log ε for all r ≥ . Note that ω ∈ C ([0 , + ∞ )) with ω (0) = 0 . We claim that s Z X ω ε ( P t − s f ) d H s δ γ ϑ ( s ) ∈ AC([0 , t ]; R ) . (4.6)To this aim, we set G ε ( s, r ) = Z X ω ε ( P t − s f ) d H s δ γ ϑ ( r ) for all s, r ∈ [0 , t ] . On the one hand, by Lemma 4.5 applied with µ = δ γ ϑ ( r ) for each r ∈ [0 , t ] , we get that | ∂ s G ε ( s, r ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z X ω ′′ ε ( P t − s f ) Γ( P t − s f ) d H s δ γ ϑ ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( t − s ) ε Lip( f ) for all s ∈ (0 , t ) and r ∈ [0 , t ] by (2.29) and Proposition 3.9, so that s G ( s, r ) isLipschitz on [0 , t ] uniformly in r ∈ [0 , t ] . On the other hand, by (2.12), Proposition 3.9,Jensen inequality and Theorem 3.16, we can estimate | G ε ( s, r ) − G ε ( s, r ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z X ω ε ( P t − s f ) d (cid:16) H s δ γ ϑ ( r − H s δ γ ϑ ( r (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Lip( ω ε ( P t − s f )) W (cid:16) H s δ γ ϑ ( r , H s δ γ ϑ ( r (cid:17) ≤ c ( t − s ) ε Lip( f ) W (cid:16) H s δ γ ϑ ( r , H s δ γ ϑ ( r (cid:17) ≤ c ( t − s ) c ( s ) ε Lip( f ) W (cid:16) δ γ ϑ ( r , δ γ ϑ ( r (cid:17) ≤ c ( t − s ) c ( s ) ε Lip( f ) d ( γ ϑ ( r ) , γ ϑ ( r ) ) for all ≤ r ≤ r ≤ t and s ∈ [0 , t ] , so that r G ( s, r ) ∈ AC([0 , t ]; R ) uniformly in s ∈ [0 , t ] . This prove the claim in (4.6). Now write G ε ( s, r ) = F εs ( γ ϑ ( r ) ) , F εs ( x ) = ˜ P s (cid:16) ω ε ( P t − s f ) (cid:17) ( x ) for all x ∈ X. Then, whenever s ∈ [0 , t ] , we can estimate ∂ r G ε ( s, r ) ≤ | D F εs | ( γ ϑ ( r ) ) | ˙ γ ϑ ( r ) | | ϑ ′ ( r ) | for L -a.e. r ∈ [0 , t ] . By Proposition 3.25, we have | D F εs | ( γ ϑ ( r ) ) = (cid:12)(cid:12)(cid:12) D˜ P s (cid:16) ω ε ( P t − s f ) (cid:17)(cid:12)(cid:12)(cid:12) ( γ ϑ ( r ) ) ≤ c ( s ) ˜ P s Γ (cid:16) ω ε ( P t − s f ) (cid:17) ( γ ϑ ( r ) ) (4.7)for all r ∈ [0 , t ] . Recalling (3.58), by the chain rule (2.25) we can write ˜ P s Γ (cid:16) ω ε ( P t − s f ) (cid:17) ( γ ϑ ( r ) ) = Z X Γ (cid:16) ω ε ( P t − s f ) (cid:17) d H s γ ϑ ( r ) = Z X Γ (cid:16) ω ε ( P t − s f ) (cid:17) p s [ γ ϑ ( r ) ] d m = Z X ( ω ′ ε ( P t − s f )) Γ( P t − s f ) p s [ γ ϑ ( r ) ] d m = − Z X ω ′′ ε ( P t − s f ) Γ( P t − s f ) d H s γ ϑ ( r ) (4.8)for all s ∈ (0 , t ] and r ∈ [0 , t ] , since ( ω ′ ε ) = − ω ′′ ε . Therefore, by Young inequality, we getthat ∂ r G ε ( s, r ) ≤ c − ( s ) | D F εs | ( γ ϑ ( r ) ) + c ( s )4 | ˙ γ ϑ ( r ) | | ϑ ′ ( r ) | for all s ∈ (0 , t ] and L -a.e. r ∈ [0 , t ] . By combining (4.7) with (4.8), we conclude that dd s Z X ω ε ( P t − s f ) d H s δ γ ϑ ( s ) = ∂ s G ε ( s, s ) + ∂ r G ε ( s, s ) ≤ Z X ω ′′ ε ( P t − s f ) Γ( P t − s f ) d H s δ γ ϑ ( s ) + c − ( s ) | D F εs | ( γ ϑ ( s ) ) + c ( s )4 | ˙ γ ϑ ( s ) | | ϑ ′ ( s ) | ≤ c ( s )4 | ˙ γ ϑ ( s ) | | ϑ ′ ( s ) | for L -a.e. s ∈ [0 , t ] , so that Z X ω ε ( f ) d H t δ y − Z X ω ε ( P t f ) d δ x = Z t dd s Z X ω ε ( P t − s f ) d H s δ γ ϑ ( s ) d s ≤ Z t c ( s ) | ˙ γ ϑ ( r ) | | ϑ ′ ( s ) | d s = 14 I − ( t ) Z | ˙ γ τ | d τ. ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 47
Recalling Proposition 2.2, we immediately deduce that ˜ P t ( ω ε ( f ))( y ) ≤ ω ε (˜ P t f )( x ) + d ( x, y )4 I − ( t ) , (4.9)for all x, y ∈ X , whenever f ∈ Lip b ( X ) ∩ W , ( X, d , m ) . Step 2 . Assume f ∈ L ( X, m ) ∩ L ∞ ( X, m ) . Since W , ( X, d , m ) is dense in L ( X, m ) ,by (2.28) we can find ( f n ) n ∈ N ⊂ Lip b ( X ) ∩ W , ( X, d , m ) such that f n → f m -a.e. in X as n → + ∞ . Since f ∈ L ∞ ( X, m ) is non-negative, by (2.26) and (2.27) we can also assumethat ≤ f n ≤ k f k L ∞ ( X, m ) for all n ∈ N . By (3.59) and the Dominated ConvergenceTheorem, we thus get that ˜ P t f n ( x ) → ˜ P t f ( x ) for all x ∈ X as n → + ∞ . Hence, by FatouLemma and by (4.9) in Step 1, we get ˜ P t ( ω ε ( f ))( y ) ≤ lim inf n → + ∞ ˜ P t ( ω ε ( f n ))( y ) ≤ lim n → + ∞ ω ε (˜ P t f n )( x ) + d ( x, y )4 I − ( t )= ω ε (˜ P t f )( x ) + d ( x, y )4 I − ( t ) , for all x, y ∈ X , proving (4.9) whenever f ∈ L ( X, m ) ∩ L ∞ ( X, m ) . Step 3 . Assume f ∈ L ( X, m ) . Then f n = f ∧ n ∈ L ( X, m ) ∩ L ∞ ( X, m ) for all n ∈ N and thus the conclusion follows by Step 2 and the Monotone Convergence Theorem. (cid:3) An simple but interesting consequence of Lemma 4.6 is the following result, see [16,Corollary 4.7] for the same result in the standard BE ( K, ∞ ) setting. Corollary 4.7 (Wang inequality for p t [ · ] ) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) . Forevery ε, t > and every y ∈ X , we have Z X p t [ y ] log( p t [ y ] + ε ) d m ≤ log( p t [ y ]( x ) + ε ) + d ( x, y )4 I − ( t ) for m -a.e. x ∈ X . In particular, if m ∈ P ( X ) , then for all t > and y ∈ X we have p t [ y ]( x ) ≥ exp − d ( x, y )4 I − ( t ) ! (4.10) for m -a.e. x ∈ X .Proof. Thanks to Lemma 3.24(i), the result immediately follows by applying Lemma 4.6to f = p t [ y ] ∈ L ( X, m ) and passing to the limit as ε → + . (cid:3) We are now ready to prove the main result of this section.
Proof of Theorem 4.4.
Let x ∈ X and r, t > be fixed. We divide the proof in two steps. Step 1 . Assume µ = f m for some non-negative f ∈ L ( X, m ) . Let us set f t = P t f and ˜ f t = ˜ P t f . Note that ˜ f t = f t m -a.e. in X since, as in the proof of Corollary 4.7, we cancompute Z X ˜ f t ϕ d m = k ϕ k L ( X, m ) Z X f d H t ( ¯ ϕ m ) = Z X f P t ϕ d m = Z X f t ϕ d m for all non-negative Borel ϕ ∈ L ( X, m ) ∩ L ∞ ( X, m ) , where ϕ = ¯ ϕ k ϕ k L ( X, m ) . By (3.58)and Jensen inequality, we can estimate ˜ f t ( x ) log( ˜ f t ( x ) + ε ) = (cid:18)Z X f d H t δ x (cid:19) log (cid:18) ε + Z X f d H t δ x (cid:19) = (cid:18)Z X p t [ x ] d µ (cid:19) log (cid:18) ε + Z X p t [ x ] d µ (cid:19) ≤ Z X p t [ x ]( y ) log( p t [ x ]( y ) + ε ) d µ ( y ) (4.11)for all x ∈ X and ε > . Integrating (4.11), by Lemma 3.24(ii), Tonelli Theorem andCorollary 4.7 we thus get Ent m ( H t µ ) = Z X f t log( f t + ε ) d m ≤ Z X (cid:18)Z X p t [ x ]( y ) log( p t [ x ]( y ) + ε ) d µ ( y ) (cid:19) d m ( x )= Z X (cid:18)Z X p t [ y ]( x ) log( p t [ y ]( x ) + ε ) d m ( x ) (cid:19) d µ ( y ) ≤ Z X log( p t [ y ]( z ) + ε ) + d ( z, y )4 I − ( t ) ! d µ ( y ) (4.12)for m -a.e. z ∈ X . Now let q = m ( B r ( x )) and define ν = q − m B r ( x ) . Integrating (4.12),by Tonelli Theorem and Jensen inequality we get Ent m ( H t µ ) ≤ Z X Z X log( p t [ y ]( z ) + ε ) + d ( z, y )4 I − ( t ) ! d µ ( y ) d ν ( z )= Z X Z X log( p t [ y ]( z ) + ε ) d µ ( y ) d ν ( z ) + 12 I − ( t ) (cid:18) r + Z X d ( y, x ) d µ ( y ) (cid:19) ≤ log (cid:18) ε + Z X Z X p t [ y ]( z ) d ν ( z ) d µ ( y ) (cid:19) + 12 I − ( t ) (cid:18) r + Z X d ( y, x ) d µ ( y ) (cid:19) = log ε + Z X q Z B r ( x ) p t [ y ]( z ) d m ( z ) d µ ( y ) ! + 12 I − ( t ) (cid:18) r + Z X d ( y, x ) d µ ( y ) (cid:19) ≤ log( q − + ε ) + 12 I − ( t ) (cid:18) r + Z X d ( y, x ) d µ ( y ) (cid:19) Passing to the limit as ε → + , we prove (4.4) whenever µ = f m for some non-negative f ∈ L ( X, m ) . Step 2 . Now let µ ∈ P ( X ) . We can find ( µ n ) n ∈ N ⊂ P ac2 ( X ) such that µ n W −→ µ as n → + ∞ . Thanks to the lower semicontinuity property of the entropy and the propertiesof Wasserstein distance, by Step 1 we get that Ent m ( H t µ ) ≤ lim inf n → + ∞ Ent m ( H t µ n ) ≤
12 I − ( t ) (cid:18) r + lim n → + ∞ Z X d ( x, x ) d µ n ( x ) (cid:19) − log m ( B r ( x ))= 12 I − ( t ) (cid:18) r + Z X d ( x, x ) d µ ( x ) (cid:19) − log m ( B r ( x )) and the proof is complete. (cid:3) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 49 Entropic inequalities in groups
We now proceed with the core argument of the proof of the entropic inequalities, adapt-ing the action and the entropy estimates established in [16, Section 4.3] (see also [71, Sec-tion 4.2] for a closely related approach in the finite dimensional case). Since we fre-quently consider curves s µ s = f s m ∈ AC ([0 , P ( X ) , W )) with s f s ∈ C ([0 , p ( X, m )) for some p ∈ [1 , + ∞ ) , we shall denote by s ˙ f s ∈ C([0 , p ( X, m )) the functional derivative in L p ( X, m ) , keeping the notation s
7→ | ˙ µ s | for the metric deriv-ative in ( P ( X ) , W ) .5.1. Strongly regular curves.
Instead of considering regular curves in P ( X ) as in [16,Definition 4.10], we deal with strongly regular curves defined as follows. Definition 5.1 (Strongly regular curve in P ( X ) ) . We say that a curve s µ s ∈ AC ([0 , P ( X )) is strongly regular if µ s = f s m for all s ∈ [0 , with s f s ∈ C ([0 , ( X, m )) . (5.1)Note that the L -integrability property in (5.1) immediately gives that sup s ∈ [0 , Ent m ( µ s ) < + ∞ (5.2)whenever s µ s is a strongly regular curve. Under the standard BE ( K, N ) condi-tion, the uniform upper control of the entropy along regular curves and, most impor-tantly, the absolute continuity property of regular curves at each time with respect tothe reference measure m , are gained from the absolute continuity of heated measuresand the L log L -regularization property of the dual heat flow ( H t ) t ≥ . Having the moregeneral BE w ( c , ∞ ) condition at disposal, the absolute continuity of heated measures andthe L log L -regularization property are still available, recall Corollary 3.22 and Theo-rem 4.4, but the application of the dual heat flow to an arbitrary curve s µ s ∈ AC ([0 , P ( X )) drastically affects its Wasserstein velocity. In more precise terms,by (3.36) we immediately get that the heated curve s µ ts = H t µ s ∈ AC ([0 , P ( X )) satisfies | ˙ µ ts | ≤ c ( t ) | ˙ µ s | for L -a.e. s ∈ [0 , (5.3)for all t ≥ , but, since the general strategy developed in [16, 71] requires the approxi-mation of s µ s with more regular AC -curves in P ( X ) having Wasserstein velocitycloser and closer to the velocity of the original curve (see equation (4.20) in the statementof [16, Proposition 4.11]), the velocity estimate in (5.3) becomes useful only if (at least) c (0 + ) = lim inf t → + ∞ c ( t ) = 1 , a condition which is not available in sub-Riemannian manifolds. For this reason, takinginspiration from the regularization procedure performed in [20, Theorem 4.8], instead ofrelying on the contraction property of the dual heat flow, in our group-modeled frame-work (see Section 5.5 below) we will regularize arbitrary AC -curves in P ( X ) via thegroup (left-)convolution with suitable L -kernels (see also Remark 5.14 below for a strictlyrelated discussion).Since our strategy will be deeply based on the L -regularity property (5.1) of stronglyregular curves, we will frequently take advantage of the following product rule for thefunctional derivative of Lipschitz curves in L ( X, m ) . We leave its elementary proof tothe interested reader. Lemma 5.2 (Product rule for
Lip -curves in L ) . If s a s , b s ∈ Lip([0 , ( X, m )) ,then s R X a s b s d m ∈ Lip([0 , R ) with dd s Z X a s b s d m = Z X ˙ a s b s d m + Z X a s ˙ b s d m for L -a.e. s ∈ [0 , . Action along strongly regular curves.
We begin by fixing two important func-tions. Here and in the rest of the section, we let ϑ ∈ C ([0 , , with ϑ ( i ) = i, i = 0 , , and η ∈ C ([0 , , + ∞ )) with ˙ η ≥ and η ( s ) > for all s > . In the following result, analogous to [16, Lemma 4.13] and [71, Lemma 4.12], we com-pute the derivative of the action along a strongly regular curve. Recall that
Lip ⋆ ( X ) wasdefined in (3.37) as the set of non-negative Lipschitz functions with bounded support. Lemma 5.3 (Derivative of action along strongly regular curves) . Assume ( X, d , m ) sat-isfies BE w ( c , ∞ ) . Let s µ s = f s m ∈ AC ([0 , P ( X )) be a strongly regular curve anddefine s ˜ µ s = H η ( s ) µ ϑ ( s ) = ˜ f s m ∈ C([0 , P ( X )) . (5.4) If ϕ ∈ Lip ⋆ ( X ) and ϕ s = Q s ϕ for all s ∈ [0 , , then s Z X ϕ s d˜ µ s ∈ Lip([0 , R ) (5.5) with dd s Z X ϕ s d˜ µ s = − Z X | D ϕ s | d˜ µ s − ˙ η ( s ) Z X Γ( ˜ f s , ϕ s ) d m + ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ϕ s d m (5.6) for L -a.e. s ∈ (0 , . Note that the function η equals η ( s ) = st for all s ∈ [0 , , t ≥ , in [16, Lemma 4.13],while in [71, Lemma 4.12] η is an increasing C time-change satisfying η (0) = 0 dependingon the dimension N ∈ (0 , + ∞ ) . For similar variations of curves via semigroup operators,we refer the reader to [66, 120].In the proof of Lemma 5.3, having in mind the product rule provided by Lemma 5.2, wewill use the following result about the L -functional derivative of the Hopf–Lax semigroup.For the reader’s convenience, we give a sketch of its proof (see also the discussion in thefirst three paragraphs of the proof of [14, Lemma 6.1]). Lemma 5.4 (Hopf–Lex semigroup is
Lip in L ) . If f ∈ Lip ⋆ ( X ) , then s Q s f ∈ Lip([0 , + ∞ ); L ( X, m )) with d + d s Q s f = − | D Q s f | in L ( X, m ) (5.7) for L -a.e. s > .Proof. If f ∈ Lip ⋆ ( X ) , then also Q s f ∈ Lip ⋆ ( X ) with supp( Q s f ) ⊂ supp f for all s ≥ .Hence by (2.6) we can estimate k Q s + h f − Q s f k L ( X, m ) ≤ q m (supp f ) · f ) h for all < s ≤ s + h , so that the conclusion follows by (2.5) and the Dominated Conver-gence Theorem. (cid:3) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 51
Proof of Lemma 5.3.
By Theorem 3.16(ii), we can estimate W (˜ µ s , ˜ µ s ) = W ( H η ( s ) µ ϑ ( s ) , H η ( s ) µ ϑ ( s ) ) ≤ W ( H η ( s ) µ ϑ ( s ) , H η ( s ) µ ϑ ( s ) ) + W ( H η ( s ) µ ϑ ( s ) , H η ( s ) µ ϑ ( s ) ) ≤ c ( η ( s )) W ( µ ϑ ( s ) , µ ϑ ( s ) ) + W ( H η ( s ) µ ϑ ( s ) , H η ( s ) µ ϑ ( s ) ) for all s , s ∈ [0 , , so that (5.4) readily follows from Lemma 4.3. Since we have s f s ∈ C ([0 , ( X, m )) by (5.1) in Definition 5.1, we also have s ˜ f s = P η ( s ) f ϑ ( s ) ∈ C ((0 , ( X, m )) with dd s ˜ f s = ˙ η ( s ) ∆ d , m P η ( s ) f ϑ ( s ) + ˙ ϑ ( s ) P η ( s ) ˙ f ϑ ( s ) in L ( X, m ) (5.8)for all s ∈ (0 , . Hence, by Lemma 5.2 and Lemma 5.4, we get (5.5) and we can compute dd s Z X ϕ s d˜ µ s = dd s Z X ϕ s ˜ f s d m = Z X ˜ f s dd s ϕ s d m + Z X ϕ s dd s ˜ f s d m = − Z X | D ϕ s | d˜ µ s + Z X (cid:16) ˙ η ( s ) ∆ d , m ˜ f s + ˙ ϑ ( s ) P η ( s ) ˙ f ϑ ( s ) (cid:17) ϕ s d m = − Z X | D ϕ s | d˜ µ s − ˙ η ( s ) Z X Γ( ˜ f s , ϕ s ) d m + ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ϕ s d m for L -a.e. s ∈ (0 , by (5.7) and (2.44), since ϕ s ∈ W , ( X, d , m ) for all s ∈ [0 , ,proving (5.6). This concludes the proof. (cid:3) Entropy along strongly regular curves.
We now introduce some notation. Forevery ε > , we let ℓ ε ( r ) = log( ε + r ) for all r ≥ , and for all µ ∈ P ( X ) we define E ε ( µ ) = Z X ℓ ε ( f ) d µ if µ = f m , + ∞ otherwise . (5.9)Note that E ε ( µ ) ≥ Ent m ( µ ) for all µ ∈ P ( X ) (5.10)and, moreover, that if µ = f m ∈ P ( X ) with f ∈ L ( X, m ) then E ε ( µ ) = log ε + Z X ( ℓ ε ( f ) − log ε ) d µ ≤ log ε + 1 ε Z X f d m < + ∞ . (5.11)For this reason, we set ˆ ℓ ε ( r ) = ℓ ε ( r ) − log ε for all r ≥ . We also let p ε ( r ) = ˆ ℓ ε ( r ) + r ˆ ℓ ′ ε ( r ) for all r ≥ . Note that p ′ ε ( r ) = 2 ℓ ′ ε ( r ) + r ℓ ′′ ε ( r ) = r + 2 ε ( r + ε ) for all r ≥ . (5.12)Finally, note that if ε ∈ (0 , then log( ε + r ) ≤ log(1 + r ) ≤ r for all r ≥ . Thus,if µ = f m ∈ P ( X ) for some f ∈ L ( X, m ) , then [log( ε + f )] − ≤ [log f ] − ∈ L ( X, µ ) by (2.15) and [log( ε + f )] + ≤ f ∈ L ( X, µ ) , so that Ent m ( µ ) ≤ E ε ( µ ) ≤ | E ε ( µ ) | ≤ Z X [log f ] − d µ + Z X f d m and Ent m ( µ ) = lim ε → + E ε ( µ ) , (5.13)by the Dominated Convergence Theorem.In the following result, analogous to [16, Lemma 4.15] and [71, Lemma 4.13], we com-pute the derivative of the truncated entropy E ε defined in (5.9) along a strongly regularcurve. Lemma 5.5 (Derivative of E ε along strongly regular curves) . Assume ( X, d , m ) satisfies BE w ( c , ∞ ) and let ε > . Under the same assumptions of Lemma 5.3 and the notationabove, we have s E ε (˜ µ s ) ∈ C ((0 , R ) (5.14) with dd s E ε (˜ µ s ) ≤ − ˙ η ( s ) Z X Γ( g εs ) d˜ µ s + ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) g εs d m (5.15) for all s ∈ (0 , , where g εs = p ε ( ˜ f s ) for all s ∈ [0 , .Proof. Let ε > be fixed. Since ˆ ℓ ε ∈ C ([0 , + ∞ ); [0 , + ∞ )) ∩ Lip([0 , + ∞ ); [0 , + ∞ )) with ˆ ℓ ε (0) = 0 , by (5.8) and the Mean Value Theorem we get that s ˆ ℓ ε ( ˜ f s ) ∈ C ((0 , ( X, m )) with dd s ˆ ℓ ε ( ˜ f s ) = ˆ ℓ ′ ε ( ˜ f s ) dd s ˜ f s in L ( X, m ) for all s ∈ (0 , . Similarly, since p ε ∈ C ([0 , + ∞ ); R ) ∩ Lip([0 , + ∞ ); R ) with p ε (0) = 0 ,we also have that s g εs = p ε ( ˜ f s ) ∈ C ((0 , ( X, m )) . In addition, recalling that ˜ f s = P η ( s ) f ϑ ( s ) for all s ∈ [0 , , by Lemma 2.1 we also have that s ˜ f s ∈ C((0 , , ( X, d , m )) and thus also s g εs ∈ C((0 , , ( X, d , m )) by the chain rule (2.25). Hence, againby (5.8) and by Lemma 5.2, we get that s E ε (˜ µ s ) = log ε + Z X ˆ ℓ ε ( ˜ f s ) ˜ f s d m ∈ C ((0 , R ) , proving (5.14), and we can compute dd s E ε (˜ µ s ) = dd s Z X ˆ ℓ ε ( ˜ f s ) ˜ f s d m = Z X ˜ f s dd s ˆ ℓ ε ( ˜ f s ) + ˆ ℓ ε ( ˜ f s ) dd s ˜ f s d m = Z X p ε ( ˜ f s ) dd s ˜ f s d m = Z X g εs (cid:16) ˙ η ( s ) ∆ d , m ˜ f s + ˙ ϑ ( s ) P η ( s ) ˙ f ϑ ( s ) (cid:17) d m for all s ∈ (0 , . By the integration-by-part formula (2.44) and the chain rule (2.47), wecan write Z X g εs ∆ d , m ˜ f s d m = − Z X Γ( g εs , ˜ f s ) d m = − Z X Γ( p ε ( ˜ f s ) , ˜ f s ) d m = − Z X p ′ ε ( ˜ f s ) Γ( ˜ f s ) d m Observing that − p ′ ε ( r ) ≤ − r ( p ′ ε ( r )) for all r ≥ , since r p ′ ε ( r ) ≤ for all r ≥ by (5.12),again by the chain rule (2.25) we can estimate Z X g εs ∆ d , m ˜ f s d m = − Z X p ′ ε ( ˜ f s ) Γ( ˜ f s ) d m ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 53 ≤ − Z X ˜ f s ( p ′ ε ( ˜ f s )) Γ( ˜ f s ) d m = − Z X Γ( p ε ( ˜ f s )) d˜ µ s = − Z X Γ( g εs ) d˜ µ s for all s ∈ (0 , , so that dd s E ε (˜ µ s ) = ˙ η ( s ) Z X g εs ∆ d , m ˜ f s d m + ˙ ϑ ( s ) Z X g εs P η ( s ) ˙ f ϑ ( s ) d m ≤ − ˙ η ( s ) Z X Γ( g εs ) d˜ µ s + ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) g εs d m for all s ∈ (0 , , concluding the proof of (5.15). (cid:3) Action and entropy along regular curves.
We now come to the following crucialresult connecting the action estimate obtained in Lemma 5.3 with the entropic inequalityproved in Lemma 5.5. For the same result in the standard BE ( K, N ) framework, we referthe reader to [16, Theorem 4.16] and [71, Proposition 4.16]. Theorem 5.6 (Action and entropy along strongly regular curves) . Assume ( X, d , m ) sati-sfies BE w ( c , ∞ ) and is heat-smoothing as in Definition 3.27. Under the same assumptionsof Lemma 5.3 and Lemma 5.5, if ε > then W (˜ µ , ˜ µ ) − Z ¨ η ( s ) E ε (˜ µ s ) d s + ˙ η (1) E ε (˜ µ ) ≤ ˙ η (0) E ε (˜ µ ) + 12 I − ,η (1) Z | ˙ µ s | d s, where I p,η ( s ) = Z s c p ( η ( r )) d r for all s ∈ [0 , and p ∈ R , and ϑ ( s ) = I − ,η ( s )I − ,η (1) for all s ∈ [0 , . (5.16) Proof.
Let ε > be fixed. On the one hand, recalling (2.29), by Lemma 5.3 we canestimate dd s Z X ϕ s d˜ µ s ≤ − Z X Γ( ϕ s ) d˜ µ s − ˙ η ( s ) Z X Γ( ˜ f s , ϕ s ) d m + ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ϕ s d m for L -a.e. s ∈ (0 , . On the other hand, by Lemma 5.5, we can also estimate ˙ η ( s ) dd s E ε (˜ µ s ) ≤ − ˙ η ( s )2 Z X Γ( g εs ) d˜ µ s + ˙ ϑ ( s ) ˙ η ( s ) Z X ˙ f ϑ ( s ) P η ( s ) g εs d m for all s ∈ (0 , . Hence, we can estimate dd s Z X ϕ s d˜ µ s + ˙ η ( s ) dd s E ε (˜ µ s ) ≤ ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ( ϕ s + ˙ η ( s ) g εs ) d m − Z X (cid:16) Γ( ϕ s ) + ˙ η ( s ) Γ( g εs ) (cid:17) d˜ µ s − ˙ η ( s ) Z X Γ( ˜ f s , ϕ s ) d m for L -a.e. s ∈ (0 , . Recalling that r p ′ ε ( r ) ≤ for all r ≥ by (5.12), by the chainrule (2.47) we can also estimate Z X Γ( g εs , ϕ s ) d˜ µ s = Z X ˜ f s p ′ ε ( ˜ f s )Γ( ˜ f s , ϕ s ) d m ≤ Z X Γ( ˜ f s , ϕ s ) d m for all s ∈ (0 , , so that Z X (cid:16) Γ( ϕ s ) + ˙ η ( s ) Γ( g εs ) (cid:17) d˜ µ s + ˙ η ( s ) Z X Γ( ˜ f s , ϕ s ) d m ≥ Z X (cid:16) Γ( ϕ s ) + ˙ η ( s ) Γ( g εs ) (cid:17) d˜ µ s + ˙ η ( s ) Z X Γ( g εs , ϕ s ) d˜ µ s = 12 Z X Γ( ϕ s + ˙ η ( s ) g εs ) d˜ µ s for all s ∈ (0 , . Thus, by (3.9), we can estimate dd s Z X ϕ s d˜ µ s + ˙ η ( s ) dd s E ε (˜ µ s ) ≤ ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ( ϕ s + ˙ η ( s ) g εs ) d m − Z X Γ( ϕ s + ˙ η ( s ) g εs ) d˜ µ s = ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ( ϕ s + ˙ η ( s ) g εs ) d m − Z X P η ( s ) (cid:16) Γ( ϕ s + ˙ η ( s ) g εs ) (cid:17) d µ ϑ ( s ) ≤ ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ( ϕ s + ˙ η ( s ) g εs ) d m − c − ( η ( s ))2 Z X Γ (cid:16) P η ( s ) ( ϕ s + ˙ η ( s ) g εs ) (cid:17) d µ ϑ ( s ) (5.17)for L -a.e. s ∈ (0 , . Since η ( s ) > for all s ∈ (0 , , we have ϕ s + ˙ η ( s ) g εs ∈ L ∞ ( X, m ) ∩ W , ( X, d , m ) for all s ∈ (0 , so that, thanks to heat-smoothing assumption (3.62), Γ( P η ( s ) ( ϕ s + ˙ η ( s ) g εs )) = | D P η ( s ) ( ϕ s + ˙ η ( s ) g εs ) | m -a.e. in X for all s ∈ (0 , . Thus, by (5.1) in Definition 5.1, by Lemma 3.19 and by Young inequality,we can estimate ˙ ϑ ( s ) Z X ˙ f ϑ ( s ) P η ( s ) ( ϕ s + ˙ η ( s ) g εs ) d m = dd r Z X P η ( s ) ( ϕ s + ˙ η ( s ) g εs ) d µ ϑ ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = s ≤ | ˙ ϑ ( s ) | | ˙ µ ϑ ( s ) | (cid:18)Z X | D P η ( s ) ( ϕ s + ˙ η ( s ) g εs ) | d µ ϑ ( s ) (cid:19) = | ˙ ϑ ( s ) | | ˙ µ ϑ ( s ) | (cid:18)Z X Γ( P η ( s ) ( ϕ s + ˙ η ( s ) g εs )) d µ ϑ ( s ) (cid:19) ≤ c ( η ( s ))2 ˙ ϑ ( s ) | ˙ µ ϑ ( s ) | + c − ( η ( s ))2 Z X Γ( P η ( s ) ( ϕ s + ˙ η ( s ) g εs )) d µ ϑ ( s ) (5.18)for all s ∈ (0 , . In conclusion, by combining (5.17) with (5.18), we get dd s Z X ϕ s d˜ µ s + ˙ η ( s ) dd s E ε (˜ µ s ) ≤ c ( η ( s ))2 ˙ ϑ ( s ) | ˙ µ ϑ ( s ) | (5.19) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 55 for L -a.e. s ∈ (0 , . We now integrate (5.19) in s ∈ [0 , . For the left-hand sideof (5.19), we have Z dd s Z X ϕ s d˜ µ s d s + Z ˙ η ( s ) dd s E ε (˜ µ s ) d s = Z X ϕ d˜ µ − Z X ϕ d˜ µ − Z ¨ η ( s ) E ε (˜ µ s ) d s + (cid:16) ˙ η (1) E ε (˜ µ ) − ˙ η (0) E ε (˜ µ ) (cid:17) . (5.20)For the right-hand side of (5.19), instead, we simply choose ϑ : [0 , → [0 , as in (5.16),so that Z ˙ ϑ ( s ) c ( η ( s )) | ˙ µ ϑ ( s ) | d s = 12 I − ,η (1) Z | ˙ µ ϑ ( s ) | ˙ ϑ ( s ) d s = 12 I − ,η (1) Z | ˙ µ s | d s. (5.21)Combining (5.20) with (5.21), we get Z X ϕ d˜ µ − Z X ϕ d˜ µ − Z ¨ η ( s ) E ε (˜ µ s ) d s + ˙ η (1) E ε (˜ µ ) ≤ ˙ η (0) E ε (˜ µ ) + 12 I − ,η (1) Z | ˙ µ s | d s whenever ε > , and the conclusion follows by taking the supremum on all ϕ ∈ Lip ⋆ ( X ) thanks to (2.13). (cid:3) Remark 5.7 (Errata to the proof of [16, Theorem 4.16]) . We warn the reader that thereis a typo in the last inequality of the long chain of inequalities in the proof of [16, Theo-rem 4.16]: in place of ( ˙ ϑ s ) e − Kst | ˙ ̺ s | , it should be written ( ˙ ϑ s ) e − Kst | ˙ ̺ ϑ ( s ) | . Unfortu-nately, this typo induces the authors to make the wrong choice of ϑ ( s ) at the beginningof [16, p. 393], making the proofs of [16, Theorem 4.16 and Theorem 4.17] not completelycorrected. The reader can easily fix all the computations needed in [16] by adapting theones performed above in the proof of Theorem 5.6.5.5. Admissible groups.
We now focus our attention on some particular admissiblemetric-measure spaces that we call admissible groups . Definition 5.8 (Admissible group) . We say that an admissible metric-measure space ( X, d , m ) is an admissible group if:(i) the metric space ( X, d ) is locally compact;(ii) the set X is a topological group , i.e. the group operations of multiplication ( x, y ) xy and inversion x x − are continuous;(iii) d is left-invariant , i.e. d ( zx, zy ) = d ( x, y ) for all x, y, z ∈ X ;(iv) m is a left-invariant Haar measure , i.e. m is a Radon measure such that m ( xE ) = m ( E ) for all x ∈ X and all Borel set E ⊂ X ;(v) X is unimodular , i.e. m is also right-invariant.For an agile introduction on topological groups and Haar measures, we refer the readerto [75, Section 11.1] and to [76, Chapter 2]. For a more general approach to the subject,see [74, Section 2.7].Note that, since d is left-invariant, we can write B r ( x ) = xB r ( o ) for all x ∈ X and r > , where o ∈ X is the identity element. Since m is right invariant, we thus get that m ( B r ( x )) = m ( B r ( o )) for all x ∈ X and r > , so that if (2.15) is satisfied for one x ∈ X ,then it is satisfied (with the same constants A, B > ) for all x ∈ X . Hence, from nowon, we assume x = o in (2.15) for simplicity. Here and in the rest of the paper, we let L x ( y ) = x − y , x, y ∈ X , be the left-translationmap. The following result is a simple consequence of Definition 5.8 and the definitionsgiven in Section 2. We leave its proof to the interested reader. Proposition 5.9 (Invariance properties of metric-measure objects) . Let ( X, d , m ) be anadmissible group. Let p ∈ [1 , + ∞ ] and x ∈ X be fixed. The following hold.(i) If p < + ∞ , then W p (( L x ) ♯ µ, ( L x ) ♯ ν ) = W p ( µ, ν ) for all µ, ν ∈ P p ( X ) .(ii) If f ∈ Lip( X ) , then also f ◦ L x ∈ Lip( X ) , with | D( f ◦ L x ) | = | D f | ◦ L x and | D ∗ ( f ◦ L x ) | = | D ∗ f | ◦ L x .(iii) If f ∈ W , ( X, d , m ) , then also f ◦ L x ∈ W , ( X, d , m ) , with Ch ( f ◦ L x ) = Ch ( f ) and | D( f ◦ L x ) | w = | D f | w ◦ L x .(iv) If f ∈ Dom(∆ d , m ) , then also f ◦ L x ∈ Dom(∆ d , m ) with ∆ d , m ( f ◦ L x ) = (∆ d , m f ) ◦ L x .(v) If f ∈ L p ( X, m ) , then P t ( f ◦ L x ) = ( P t f ) ◦ L x for all t > . Convolution.
We denote by ( f ⋆ g )( x ) = Z X f ( xy − ) g ( y ) d m ( y ) , x ∈ X, the group convolution of f, g ∈ L ( X, m ) , and we use the notation f ∗ g to denote theusual convolution in R n of f, g ∈ L ( R n , L n ) . Since X is unimodular, by a simple changeof variables we can also write ( f ⋆ g )( x ) = Z X f ( y ) g ( y − x ) d m ( y ) , x ∈ X. Thus, accordingly, for f ∈ L ( X, m ) and µ ∈ P ( X ) we write ( f ⋆ µ )( x ) = Z X f ( xy − ) d µ ( y ) , x ∈ X. For an account on the elementary properties of convolution in locally compact groups,we refer the reader to [76, Section 2.5] (in particular, recall Young inequality in [76,Proposition 2.40]).The following result completes the information provided by Lemma 3.24.
Lemma 5.10 ( (˜ P t ) t> as a right-convolution) . Assume ( X, d , m ) is an admissible groupsatisfying BE w ( c , ∞ ) . If t > , then p t [ x ]( y ) = p t [ o ]( y − x ) for m -a.e. x, y ∈ X. Consequently, if t > , then we can write ˜ P t f ( x ) = Z X f ( y ) p t [ o ]( y − x ) d m ( y ) = ( f ⋆ p t [ o ])( x ) for m -a.e. x ∈ X and for all one-side bounded measurable functions f : X → R .Proof. Let t > be fixed. We start by claiming that H t (( L x ) ♯ µ ) = ( L x ) ♯ ( H t µ ) (5.22)for all µ ∈ P ( X ) and x ∈ X . Indeed, if µ ≪ m , then by (3.21) claim (5.22) is nothingbut Proposition 5.9(v). Since P ac2 ( X ) is W -dense in P ( X ) , claim (5.22) follows fromProposition 5.9(i) and Theorem 3.16(ii) by a simple approximation argument. Thanks toclaim (5.22), we can compute Z X f ( y ) p t [ x ]( y ) d m ( y ) = Z X f ( y ) d H t δ x ( y ) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 57 = Z X f ( y ) d H t (( L x − ) ♯ δ o )( y )= Z X f ( y ) d( L x − ) ♯ ( H t δ o )( y )= Z X f ( xy ) p t [ o ]( y ) d m ( y )= Z X f ( y ) p t [ o ]( x − y ) d m ( y ) for all f ∈ L ∞ ( X, m ) and x ∈ X . Thus p t [ x ]( y ) = p t [ o ]( x − y ) for all x ∈ X and m -a.e. y ∈ X , and the conclusion follows by Lemma 3.24(ii). (cid:3) According to Lemma 5.10, we thus simply write p t [ o ] = p t for all t > and we call ( p t ) t> the ( metric-measure ) heat kernel of the (pointwise version of the) heat flow. Remark 5.11 (Application of (4.10)) . Assume ( X, d , m ) is an admissible group satisfying BE w ( c , ∞ ) with m ∈ P ( X ) . From inequality (4.10) in Corollary 4.7 we immediately have p t ( x ) ≥ exp − d ( x, o )4 I − ( t ) ! (5.23)for all t > and m -a.e. x ∈ X . Inequality (5.23) applies in particular to the (sub-Riemmanian) SU (2) group, see Section 5.9.2 for the precise definition. Up to our knowl-edge, inequality (5.23) provides a new lower bound on the heat kernel in SU (2) .5.7. Approximation by regular curves in admissible groups.
Let ( X, d , m ) be anadmissible group. We say that ̺ ∈ L ( X, m ) is a convolution kernel if it is non-negative,renormalized, symmetric and has bounded support, i.e. ̺ ≥ , Z X ̺ d m = 1 , ̺ ( x − ) = ̺ ( x ) for all x ∈ X, supp ̺ is bounded . (5.24)Since d is left-invariant, d -balls centered at o ∈ X are symmetric, in the sense that x ∈ B r ( o ) ⇐⇒ x − ∈ B r ( o ) whenever x ∈ X and r > . Thus, for all r > , the function ̺ r ∈ L ( X, m ) ∩ L ∞ ( X, m ) defined by ̺ r ( x ) = χ B r ( o ) ( x ) m ( B r ( o )) , x ∈ X, (5.25)is a convolution kernel (and also an approximate identity as r → + , see [76, Proposi-tion 2.44]).The following result provides a simple but useful relation between test plans and con-volution. Lemma 5.12 (Convolution and plans) . Let ( X, d , m ) be an admissible group and let ̺ ∈ L ( X, m ) be as in (5.24) . Let µ , µ ∈ P ( X ) and define ˜ µ , ˜ µ ∈ P ( X ) as ˜ µ = ( ̺ ⋆ µ ) m and ˜ µ = ( ̺ ⋆ µ ) m . If π ∈ Plan ( µ , µ ) , then the measure ˜ π given by Z X × X ϕ ( x , x ) d˜ π ( x , x ) = Z X × X Z X ̺ ( y ) ϕ ( yx , yx ) d m ( y ) d π ( x , x ) (5.26) for all ϕ ∈ C b ( X × X ) is such that ˜ π ∈ Plan (˜ µ , ˜ µ ) . Proof.
Note that ˜ π ∈ P ( X × X ) , since Z X × X d˜ π ( x , x ) = Z X × X Z X ̺ ( y ) d m ( y ) d π ( x , x ) = Z X × X d π ( x , x ) by (5.24) and (5.26). Let us now prove that ( p i ) ♯ ˜ π = µ i , i = 1 , . Let ϕ ∈ C b ( X ) and set ψ i = ϕ ◦ p i ∈ C b ( X × X ) , i = 1 , . By (5.26) and recalling that X is unimodular, we canwrite Z X ϕ d( p i ) ♯ ˜ π = Z X × X ψ i d˜ π = Z X × X Z X ̺ ( y ) ψ i ( yx , yx ) d m ( y ) d π ( x , x )= Z X Z X ̺ ( y ) ϕ ( yx ) d m ( y ) d µ i ( x )= Z X Z X ̺ ( zx − ) ϕ ( z ) d m ( z ) d µ i ( x )= Z X ϕ ( z ) Z X ̺ ( zx − ) d µ i ( x ) d m ( z )= Z X ( ̺ ⋆ µ i )( z ) ϕ ( z ) d m ( z )= Z X ϕ d˜ µ i thanks to Fubini Theorem, concluding the proof. (cid:3) A fundamental consequence of Lemma 5.12 is the following estimate on the Wassersteinvelocity of left-convoluted curves of measures.
Lemma 5.13 (Convolution and W q -velocity) . Let ( X, d , m ) be an admissible group andlet ̺ ∈ L ( X, m ) be as in (5.24) . Let p, q ∈ [1 , + ∞ ) and let I ⊂ R be an interval. If s µ s ∈ AC q ( I ; P p ( X )) , then also s ˜ µ s ∈ AC q ( I ; P p ( X )) , where ˜ µ s = ( ̺ ⋆ µ s ) m forall s ∈ I , with | ˙˜ µ s | ≤ | ˙ µ s | for L -a.e. s ∈ I .Proof. Since X is unimodular and d is left-invariant, by Tonelli Theorem we can estimate Z X d p ( x, o ) d˜ µ s ( x ) = Z X Z X ̺ ( xy − ) d p ( x, o ) d µ s ( y ) d m ( x )= Z X Z X ̺ ( xy − ) d p ( x, o ) d m ( x ) d µ s ( y )= Z X Z X ̺ ( z ) d p ( zy, o ) d m ( z ) d µ s ( y ) ≤ p − Z X Z X ̺ ( z ) (cid:16) d p ( zy, z ) + d p ( z, o ) (cid:17) d m ( z ) d µ s ( y )= 2 p − (cid:18)Z X d p ( y, o ) d µ s ( y ) + Z X d p ( z, o ) ̺ ( z ) d m ( z ) (cid:19) , proving that ˜ µ s ∈ P p ( X ) for all s ∈ I . Now for all k ∈ N let ϕ k ∈ C b ( X × X ) be defined as ϕ k ( x, y ) = d p ( x, y ) ∧ k for all x, y ∈ X . Let s , s ∈ I and let π s ,s ∈ OptPlan( µ s , µ s ) bean optimal plan between µ s and µ s . Let ˜ π s ,s ∈ Plan (˜ µ s , ˜ µ s ) be given by Lemma 5.12accordingly. Since ϕ k ( zx, zy ) = ϕ k ( x, y ) for all x, y, z ∈ X and k ∈ N , we can estimate Z X × X ϕ k ( x, y ) d˜ π s ,s ( x, y ) = Z X × X Z X ̺ ( z ) ϕ k ( zx, zy ) d m ( z ) d π s ,s ( x, y )= Z X × X Z X ̺ ( z ) ϕ k ( x, y ) d m ( z ) d π s ,s ( x, y ) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 59 ≤ Z X × X d p ( x, y ) d π s ,s ( x, y )= W pp ( µ s , µ s ) . By the Monotone Convergence Theorem, we can pass to the limit as k → + ∞ and get W p (˜ µ s , ˜ µ s ) ≤ W p ( µ s , µ s ) whenever s , s ∈ I , concluding the proof. (cid:3) Remark 5.14 (Right-convoluted measures and velocity) . It is not difficult to see that astatement similar to that of Lemma 5.12 holds for the right-convoluted measures ˆ µ =( µ ⋆ ̺ ) m and ˆ µ = ( µ ⋆ ̺ ) m . However, since d is not necessarily right-invariant, it is notclear how to prove a statement similar to that of Lemma 5.13 for the right-convoluted curve s ˆ µ s = ( µ s ⋆̺ ) m . Since in admissible groups the heat semigroup acts on measures as theright-convolution with the heat kernel as seen in Lemma 5.10, the lack of an estimate onthe Wasserstein velocity of right-convoluted curves is a central obstacle for the use of theheat-regularization techniques (which, in the standard BE ( K, N ) framework, inevitablyrely on the crucial fact that c (0 + ) = 1 , recall the discussion made in Section 5.1). Thisalso explains why, in Theorem 5.15 below, we need to assume that the ambient space isan admissible group and rely on left-convolution of measures.We now prove the following crucial approximation result. The line of the proof isclose to that of [20, Theorem 4.8]. For the approximation of curves under the standard BE ( K, N ) condition, we refer the reader to [16, Proposition 4.11] and [71, Lemma 4.11]. Theorem 5.15 (Approximation by strongly regular curves in P ( X ) ) . Let ( X, d , m ) bean admissible group. If s µ s ∈ AC ([0 , P ( X )) , then there exist strongly regularcurves s µ ns ∈ AC ([0 , P ( X )) , n ∈ N , in the sense of Definition 5.1, such that:(i) µ ns W −→ µ s as n → + ∞ for all s ∈ [0 , ;(ii) lim sup n → + ∞ Z | ˙ µ ns | d s ≤ Z | ˙ µ s | d s ;(iii) lim n → + ∞ Ent m ( H t µ ns ) = Ent m ( H t µ s ) for all s ∈ [0 , and t ≥ .Proof. We divide the proof in four steps.
Step 1: time-extension to R . We define R ∋ s ν s ∈ P ( X ) by extending [0 , ∋ s µ s ∈ P ( X ) by continuity with constant values in ( −∞ , ∪ (1 , + ∞ ) . Clearly, we havethat s ν s ∈ AC ( R ; P ( X )) . Step 2: smoothing in the space variable . For all r > , let ̺ r ∈ L ( X, m ) ∩ L ∞ ( X, m ) bedefined as in (5.25). We thus define ν rs = f rs m , where f rs ( x ) = ( ̺ r ⋆ ν s )( x ) = Z X ̺ r ( xy − ) d ν s ( y ) , x ∈ X, for all s ∈ R and r > . By Lemma 5.13, we have s ν rs ∈ AC ( R ; P ( X )) , with | ˙ ν rs | ≤ | ˙ ν s | for L -a.e. s ∈ R , for all r > . Since the family ( ̺ r ) r> is a symmetricapproximation of the identity, we have lim r → + Z X ϕ d ν rs = lim r → + Z X ( ̺ r ⋆ ϕ ) d ν s = Z X ϕ d ν s (5.27)for all ϕ ∈ C b ( X ) by the Dominated Convergence Theorem for all s ∈ R , so that ν rs ⇀ ν s as r → + for all s ∈ R . In addition, we can write Z X d ( x, o ) d ν rs ( x ) = Z X ( ̺ r ⋆ d ( · , o ))( x ) d ν s ( x ) = Z X Z X ̺ r ( y ) d ( x, y ) d m ( y ) d ν s ( x )= Z X − Z B r ( o ) d ( x, y ) d m ( y ) d ν s ( x ) for all s ∈ R and r > , so that (cid:12)(cid:12)(cid:12)(cid:12)Z X d ( x, o ) d ν rs ( x ) − Z X d ( x, o ) d ν s ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z X − Z B r ( o ) (cid:12)(cid:12)(cid:12) d ( x, y ) − d ( x, o ) (cid:12)(cid:12)(cid:12) d m ( y ) d ν s ( x ) ≤ Z X r ( r + 2 d ( x, o )) d ν s ( x ) for all s ∈ R and r > . Hence lim r → + Z X d ( x, o ) d ν rs ( x ) = Z X d ( x, o ) d ν s ( x ) (5.28)for all s ∈ R . Consequently, from (5.27) and (5.28) we infer that ν rs W −→ ν s for all s ∈ R and, in particular, lim inf r → + Ent m ( ν rs ) ≥ Ent m ( ν s ) for all s ∈ R . We can also write ν rs = Z X ( L y ) ♯ ν s ̺ r ( y ) d m ( y ) for all s ∈ R , where L y ( x ) = y − x , x, y ∈ X , denotes the left-translation map. If n = e − c d ( · , o ) m is as in (2.18), then by Jensen inequality we can estimate Ent n ( ν rs ) ≤ Z X Ent n (( L y ) ♯ ν s ) ̺ r ( y ) d m ( y ) for all s ∈ R , so that by (2.17) and (2.18) we can write Ent n (( L y ) ♯ ν s ) = Ent n y − ( ν s ) = Ent m ( ν s ) + c Z X d ( yx, o ) d ν s ( x ) for all y ∈ X and s ∈ R , where n y = ( L y ) ♯ n = e − c d ( · ,y ) m . Since Z X Z X d ( yx, o ) d ν s ( x ) ̺ r ( y ) d m ( y ) = Z X Z X d ( x, y − ) ̺ r ( y ) d m ( y ) d ν s ( x )= Z X Z X d ( x, y ) ̺ r ( y ) d m ( y ) d ν s ( x )= Z X d ( x, o ) d ν rs ( x ) by the symmetry of ̺ r , again by (2.18) we can estimate Ent m ( ν rs ) = Ent ν ( ν rs ) − c Z X d ( x, o ) d ν rs ( x ) ≤ Z X (cid:18) Ent m ( ν s ) + c Z X d ( yx, o ) d ν s ( x ) (cid:19) ̺ r ( y ) d m ( y ) − c Z X d ( x, o ) d ν rs ( x )= Ent m ( ν s ) + c (cid:18)Z X Z X d ( x, y ) ̺ r ( y ) d m ( y ) d ν s ( x ) − Z X d ( x, o ) d ν rs ( x ) (cid:19) = Ent m ( ν s ) for all s ∈ R and r > , so that lim r → + Ent m ( ν rs ) = Ent m ( ν s ) . ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 61
Step 3: smoothing in the time variable . Now let r > be fixed. Let ζ : R → R be asymmetric smooth mollifier in R , i.e. ζ ∈ C ∞ c ( R ) , supp ζ ⊂ [ − , , ≤ ζ ≤ , Z R ζ d τ = 1 . We define ζ j ( τ ) = j ζ ( jτ ) for all τ ∈ R and j ∈ N and consider ν j,rs = f j,rs m with f j,rs = ( ζ j ∗ f r · )( s ) = Z R f rτ ζ j ( τ − s ) d τ for all s ∈ R and all j ∈ N . If s, s ′ ∈ R and π rs,s ′ ∈ Plan ( ν rs , ν rs ′ ) , then the measure π j,rs ∈ P ( X × X ) given by Z X × X ϕ ( x, y ) d π j,rs ( x, y ) = Z R ζ j ( s − τ ) Z X × X ϕ ( x, y ) d π rs,τ ( x, y ) d τ, for any ϕ ∈ C b ( X × X ) , satisfies π j,rs ∈ Plan ( ν rs , ν j,rs ) . Thus, by the convexity propertiesof the squared -Wasserstein distance and Jensen inequality, we get W ( ν j,rs , ν rs ) ≤ Z R ζ j ( s − τ ) W ( ν rs , ν rτ ) d τ for all s ∈ R and j ∈ N , so that ν j,rs W −→ ν rs as j → + ∞ and, in particular, lim inf j → + ∞ Ent m ( ν j,rs ) ≥ Ent m ( ν rs ) for all s ∈ R . In a similar fashion, we can estimate W ( ν j,rs , ν j,rs ′ ) = W (cid:18)Z R ν rs − τ ζ j ( τ ) d τ, Z R ν rs ′ − τ ζ j ( τ ) d τ (cid:19) ≤ Z R W (cid:16) ν rs − τ , ν rs ′ − τ (cid:17) ζ j ( τ ) d τ for all s, s ′ ∈ R and j ∈ N , so that s ν j,rs ∈ AC ( R ; P ( X )) with | ˙ ν j,rs | ≤ ( ζ j ∗ | ˙ ν r · | )( s ) for L -a.e. s ∈ R and all j ∈ N . As in Step 2, let n = e − c d ( · , o ) m be as in (2.18). Sincethe function H ( u ) = u log u + (1 − u ) , defined for all u ≥ , is convex and non-negative,by Jensen inequality we can estimate Ent n ( ν j,rs ) = Z X H (cid:16) f j,rs d m d n (cid:17) d n = Z X H (cid:16) ( ζ j ∗ f r · )( s ) d m d n (cid:17) d n = Z X H (cid:16) ( ζ j ∗ ( f r · d m d n ))( s ) (cid:17) d n ≤ Z X (cid:16) ζ j ∗ H (cid:16) f r · d m d n (cid:17)(cid:17) ( s ) d n = ( ζ j ∗ Ent n ( ν r · ))( s ) for all s ∈ R and j ∈ N . Arguing as in Step 2, we immediately deduce that Ent m ( ν j,rs ) ≤ ( ζ j ∗ Ent m ( ν r · ))( s ) for all s ∈ R and j ∈ N , so that lim j → + ∞ Ent m ( ν j,rs ) = Ent m ( ν rs ) for all s ∈ R . Step 4: time-restriction to [0 , and conclusion . Define the curve s µ j,ks as therestriction of the curve s ν j,ks to the interval [0 , . Note that the regularization map R j,r sending the original curve s µ s to the regularized curve s µ j,ks is linear (with respect to convex combinations) and thus commutes with the dual heat flow map, i.e. R j,r ◦ H t = H t ◦ R j,r for all t ≥ . Since by construction lim j,r Ent m ( µ j,rs ) = Ent m ( µ s ) for all s ∈ [0 , , we thus have lim sup j,r Ent m ( H t µ j,rs ) = lim sup j,r Ent m ( R j,r ( H t µ s )) = Ent m ( H t µ s ) ≤ Ent m ( µ s ) for all t ≥ and s ∈ [0 , by Lemma 4.1(i). The conclusion thus follows by a standarddiagonalization argument. (cid:3) Entropic inequalities in admissible groups.
We can now state and prove themain result of this paper. We refer the reader to [16, Theorem 4.17] and to [71, Theo-rem 4.19] for the analogous results in the standard BE ( K, N ) framework. Theorem 5.16 (Entropic inequalities) . Let ( X, d , m ) be an admissible group satisfyingthe heat-smoothing property as in Definition 3.27. The following are equivalent.(i) ( X, d , m ) satisfies BE w ( c , ∞ ) .(ii) The dual heat semigroup ( H t ) t ≥ in (3.21) satisfies W ( H t µ , H t µ ) −
12 R( t , t ) W ( µ , µ ) ≤ ( t − t ) (cid:18) Ent m ( H t µ ) − Ent m ( H t µ ) (cid:19) (5.29) for all µ ∈ Dom(
Ent ) , µ ∈ P ( X ) and ≤ t ≤ t , with also µ ∈ Dom(
Ent m ) in the particular case t = t = 0 , where R( t , t ) = Z c − ((1 − s ) t + st ) d s. (5.30) (iii) The dual heat semigroup ( H t ) t ≥ in (3.21) uniquely extends to a map on P ( X ) satisfying (3.36) and such that Ent m ( H t + h µ s ) ≤ (1 − s ) Ent m ( H t µ ) + s Ent m ( H t µ )+ s (1 − s )2 h t, t + h ) W ( µ , µ ) − W ( H t µ , H t µ ) ! (5.31) for all t ≥ and h > , whenever s µ s ∈ Geo([0 , P ( X )) is a W -geodesicjoining µ , µ ∈ Dom(
Ent m ) and R is as in (5.30) . Mimicking the standard framework, thanks to Theorem 5.16 we can introducing thefollowing notation.
Definition 5.17 ( EVI w ( c ) and RCD w ( c , ∞ ) conditions) . An admissible metric-measurespace ( X, d , m ) is said to satisfy the weak Evolution Variation Inequality with respectto the function c : [0 , + ∞ ) → (0 , + ∞ ) in (3.8), EVI w ( c ) for short, if inequality (5.29) inTheorem 5.16(ii) holds. Analogously, ( X, d , m ) is said to satisfy the ( dimension-free ) weakRiemannian Curvature-Dimension Condition with respect to the function c , RCD w ( c , ∞ ) for short, if inequality (5.31) in Theorem 5.16(iii) holds.With this terminology, one can rephrase Theorem 5.16 simply writing that, for anadmissible heat-smoothing group ( X, d , m ) , it holds BE w ( c , ∞ ) ⇐⇒ EVI w ( c ) ⇐⇒ RCD w ( c , ∞ ) . ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 63
Proof of Theorem 5.16.
We prove each implication separately.
Proof of (i) ⇒ (ii) . Let ≤ t ≤ t be fixed. Let s µ s ∈ AC ([0 , P ( X )) be a curvejoining µ ∈ Dom(
Ent m ) and µ ∈ P ( X ) (with also µ ∈ Dom(
Ent m ) in the particularcas t = t = 0 ). We can find strongly regular curves s µ ns ∈ AC ([0 , P ( X )) , n ∈ N , as in Definition 5.1, approximating the curve s µ s as stated in Theorem 5.15.By Theorem 5.6 applied to each s µ ns with η ( s ) = (1 − s ) t + st for all s ∈ [0 , , weget W ( H t µ n , H t µ n ) + ( t − t ) E ε ( H t µ n ) ≤ ( t − t ) E ε ( H t µ n ) + 12 I − ,η (1) Z | ˙ µ ns | d s (5.32)for all n ∈ N and ε > . On the one hand, recalling (5.10), we have E ε ( H t µ n ) ≥ Ent m ( H t µ n ) for all ε > and n ∈ N . On the other hand, by (5.13) we have that lim ε → + E ε ( H t µ n ) = Ent m ( H t µ n ) for all n ∈ N , since µ n = f n m with f n ∈ L ( X, m ) . Thus we can pass to the limitas ε → + in (5.32) and get W ( H t µ n , H t µ n ) + ( t − t ) Ent m ( H t µ n ) ≤ ( t − t ) Ent m ( H t µ n ) + 12 I − ,η (1) Z | ˙ µ ns | d s (5.33)for all n ∈ N . By (3.36) in Theorem 3.16 and Theorem 5.15(i), we have H t i µ ni W −→ H t i µ i as n → + ∞ for i = 0 , , so that lim n → + ∞ W ( H t µ n , H t µ n ) = W ( H t µ , H t µ ) . (5.34)Also, by the lower semicontinuity of the entropy, we have Ent m ( H t µ ) ≤ lim inf n → + ∞ Ent m ( H t µ n ) . (5.35)Finally, by Theorem 5.15(iii), we can estimate lim sup n → + ∞ Ent m ( H t µ n ) ≤ Ent m ( H t µ ) . (5.36)By (5.34), (5.35) and (5.36), we can thus pass to the limit as n → + ∞ in (5.33) getting W ( H t µ , H t µ ) + ( t − t ) Ent m ( H t µ ) ≤ ( t − t ) Ent m ( H t µ ) + 12 R( t , t ) Z | ˙ µ s | d s, (5.37)so that (ii) follows by minimizing (5.37) with respect to the curves µ ∈ AC ([0 , P ( X )) joining µ and µ . Proof of (ii) ⇒ (i) . Choosing t = t = t > in (5.29), we get W ( H t µ , H t µ ) ≤ c ( t ) W ( µ , µ ) for all µ , µ ∈ Dom(
Ent m ) . This proves the validity of (3.35) in Theorem 3.16(ii) for D = Dom( Ent m ) . Since Dom(
Ent m ) is a W -dense subset of P ac2 ( X ) , (i) immediatelyfollows by Theorem 3.16. Proof of (i) ⇒ (iii) . Since we already know that (i) ⇔ (ii), by Theorem 3.16 we havethat the dual heat semigroup uniquely extends to a map on P ( X ) satisfying (3.36) andwe can thus argue as in the proof of [66, Theorem 3.2]. So let t ≥ and h > befixed and let s µ s ∈ Geo([0 , P ( X )) be a W -geodesic joining µ , µ ∈ Dom(
Ent m ) .By (5.29) applied respectively to the couple µ ∈ Dom(
Ent m ) , µ s ∈ P ( X ) , and to thecouple µ ∈ Dom(
Ent m ) , µ s ∈ P ( X ) , both with the choice t = t and t = t + h for all s ∈ [0 , (recall that H t µ ∈ Dom(
Ent m ) for all µ ∈ P ( X ) and t > by Theorem 4.4), weget − s W ( H t + h µ s , H t µ ) + s W ( H t + h µ s , H t µ ) −
12 R( t, t + h ) (cid:16) (1 − s ) W ( µ s , µ ) + s W ( µ s , µ ) (cid:17) ≤ h (cid:18) (1 − s ) Ent m ( H t µ ) + s Ent m ( H t µ ) − Ent m ( H t + h µ s ) (cid:19) (5.38)for all s ∈ [0 , . Since s µ s is a W -geodesic, we can estimate (1 − s ) W ( µ s , µ ) + s W ( µ s , µ ) = s (1 − s ) W ( µ , µ ) (5.39)for all s ∈ [0 , . Thanks to the elementary inequality (1 − s ) a + sb ≥ s (1 − s )( a + b ) for all a, b ∈ R , s ∈ [0 , , by the triangular inequality we can also estimate (1 − s ) W ( H t + h µ s , H t µ ) + s W ( H t + h µ s , H t µ ) ≥ s (1 − s ) (cid:18) W ( H t + h µ s , H t µ ) + W ( H t + h µ s , H t µ ) (cid:19) ≥ s (1 − s ) W ( H t µ , H t µ ) . (5.40)By combining (5.39) and (5.40) with (5.38), we immediately deduce (iii). Proof of (iii) ⇒ (i) . Since ( H t ) t ≥ satisfies (3.36), (i) trivially follows by Theorem 3.16. (cid:3) Application to Carnot groups and the SU (2) group. We conclude this sectionwith the application of Theorem 5.16 to Carnot groups and to the SU (2) group.5.9.1. Carnot groups.
We recall that a Carnot group G is a connected, simply connectedand nilpotent Lie group whose Lie algebra g of left-invariant vector fields has dimen-sion n ∈ N and admits a stratification of step κ ∈ N , g = V ⊕ V ⊕ · · · ⊕ V κ with V i = [ V , V i − ] for i = 1 , . . . , κ, [ V , V κ ] = { } . We set m i = dim( V i ) and h i = m + · · · + m i for i = 1 , . . . , κ , with h = 0 and h κ = n .We fix an adapted basis of g , i.e. a basis X , . . . , X n such that X h i − +1 , . . . , X h i is a basis of V i , i = 1 , . . . , κ. ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 65
Using the exponential coordinates, it is possible to identify G with R n endowed withthe group law determined by the Campbell–Hausdorff formula (in particular, the identity o ∈ G corresponds to ∈ R n and x − = − x for x ∈ G ), and it is not restrictive to assumethat X i (0) = e i for any i = 1 , . . . , n . In particular, by left-invariance, for any x ∈ G weget X i ( x ) = dL x e i , i = 1 , . . . , n, where L x : G → G is the left-translation by x ∈ G . We endow g with the left-invariantRiemannian metric h· , ·i G that makes the basis X , . . . , X n orthonormal. We let H G ⊂ T G be the horizontal tangent bundle of the group G , i.e. the left-invariant sub-bundle of thetangent bundle T G such that H G = { X (0) : X ∈ V } , and we let ∇ G f = m X j =1 ( X j f ) X j ∈ V be the horizontal gradient of f .For any i = 1 , . . . , n , we define the degree d ( i ) ∈ { , . . . , κ } of the basis vector field X i as d ( i ) = j if and only if X i ∈ V j . Using this notation, the one-parameter family of groupdilations ( δ λ ) λ ≥ : G → G is given by δ λ ( x ) = δ λ ( x , . . . , x n ) = ( λx , . . . , λ d ( i ) x i , . . . , λ κ x n ) , for all x ∈ G . (5.41)The bi-invariant Haar measure of the group G coincides (up to a multiplicative con-stant) with the n -dimensional Lebesgue measure L n and has the homogeneity property L n ( δ λ ( E )) = λ Q L n ( E ) , where the integer Q = P κi =1 i dim( V i ) is called the homogeneousdimension of the group.We endow the group G with the canonical Carnot–Carathéodory metric structure in-duced by H G . More precisely, the Carnot–Carathéodory distance between x, y ∈ G isthen defined as d cc ( x, y ) = inf (cid:26)Z k ˙ γ ( t ) k G dt : γ is horizontal , γ (0) = x, γ (1) = y (cid:27) . (5.42)Here and in the following, we say that a Lipschitz curve γ : [0 , → G is a horizontalcurve if ˙ γ ( t ) ∈ H γ ( t ) G for a.e. t ∈ [0 , . By Chow–Rashevskii’s Theorem, the function d cc is in fact a distance, which is also left-invariant and homogeneous with respect to thedilations defined in (5.41), in the sense that d cc ( zx, zy ) = d cc ( x, y ) , d cc ( δ λ ( x ) , δ λ ( y )) = λ d cc ( x, y ) , for all x, y, z ∈ G and λ ≥ . The resulting metric space ( G , d cc ) is a complete, separable,locally compact and geodesic space. Note that L n ( B r ( x )) = c n r Q for all x ∈ G and r ≥ , where c n = L n ( B (0)) .The standard sub-Laplacian operator is ∆ G = P m i =1 X i . Since the horizontal vectorfields X , . . . , X h satisfy the Hörmander condition , by Hörmander Theorem the sub-elliptic heat operator ∂ t − ∆ G is hypoelliptic , meaning that its fundamental solution, theheat kernel p : (0 , + ∞ ) × G → (0 , + ∞ ) , is a smooth function. For the main properties ofthe heat kernel, we refer to [20, Theorem 2.3] and to the references therein. Here we onlyrecall that, given a function f ∈ L ( G , L n ) , the function P t f ( x ) = f t ( x ) = ( f ⋆ p t )( x ) = Z G f ( y ) p t ( y − x ) d y, ( t, x ) ∈ (0 , + ∞ ) × G , is smooth and is a solution of the heat diffusion problem ∂ t f t = ∆ G f t in (0 , + ∞ ) × G ,f = f, on { } × G , where the initial datum is assumed in the L -sense, i.e. lim t → f t = f in L ( G , L n ) . Accord-ingly, we can define P t µ ( x ) = ( µ ⋆ p t )( x ) = Z G p t ( y − x ) d µ ( y ) , ( t, x ) ∈ (0 , + ∞ ) × G , whenever µ ∈ P ( X ) , so that we can identify H t = P t for all t ≥ .It is not difficult to recognize that the space W , ( G , d cc , L n ) induced by the metric-measure structure ( G , d cc , L n ) actually coincides with the well-known horizontal Sobolevspace W , G ( G ) = n f ∈ L ( G , L n ) : X i f ∈ L ( G , L n ) , i = 1 , . . . , m o , where X i f stands for the derivative of the function f in the direction X i defined in theusual weak sense via integration by parts against test functions. In particular, the Sobolevspace W , ( G , d cc , L n ) is Hilbertian and the Cheeger energy coincides with the horizontalDirichlet energy Ch ( f ) = Z G k∇ G f ( x ) k G d x for all f ∈ W , G ( G ) , so that | D f | w = k∇ G f k G . We refer the reader to [92, Theorem 1.3], [102, Theorem 1.2]and [108, Theorem 6.3] for more general results in this direction (note that strictly relatedobservations are made in [9, Section 3.2] for the BV space in the sub-Riemannian context).By a standard regularization argument via group convolution, we thus immediatelydeduce that if f ∈ W , G ( G ) with k∇ G f k G ≤ L , then f agrees L n -a.e. with a d cc -Lipschitzfunction with Lipschitz constant not larger than L .In conclusion, ( G , d cc , L n ) is an admissible metric-measure space in the sense of Sec-tion 2.11 which is also a heat-smoothing admissible group as in Definitions 3.27 and 5.8.Therefore, combining [109, Theorem 1.8] with Theorem 3.16 and Theorem 5.16, we getthe following result. Theorem 5.18 (Equivalence in Carnot groups) . Let ( G , d cc , L n ) be a Carnot group.There exists an optimal constant C G ≥ (depending only on the group structure andsuch that C G = 1 if and only if G is commutative) satisfying the following four equivalentproperties.(i) [ BE w ( C G , ∞ ) ] If f ∈ C ∞ ( G ) , then Γ G ( P t f ) ≤ C G P t Γ G ( f ) for all t ≥ .(ii) [Kuwada] If µ, ν ∈ P ( G ) , then W ( P t µ, P t ν ) ≤ C G W ( µ, ν ) for all t ≥ .(iii) [ EVI w ( C G ) ] If µ , µ ∈ Dom(
Ent L n ) , then W ( P t µ , P t µ ) − C G W ( µ , µ ) ≤ t − t ) (cid:16) Ent L n ( P t µ ) − Ent L n ( P t µ ) (cid:17) for all ≤ t ≤ t .(iv) [ RCD w ( C G , ∞ ) ] If s µ s ∈ Geo([0 , P ( G )) connects µ , µ ∈ Dom(
Ent L n ) ,then Ent L n ( P t + h µ s ) ≤ (1 − s ) Ent L n ( P t µ ) + s Ent L n ( P t µ )+ s (1 − s )2 h (cid:16) C G W ( µ , µ ) − W ( P t µ , P t µ ) (cid:17) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 67 for all s ∈ [0 , , t ≥ and h > . Thanks to the entropy dissipation along the heat flow proved in [20, Proposition 4.2]and to the integrability property of the Fisher information along the heat flow given byLemma 4.1(ii), from Theorem 5.18(iv) we deduce the following weak convexity propertyof the entropy along W -geodesics in Carnot groups. Here and in the following, we let F G ( f ) = Z G ∩{ f> } k∇ G f k G f d x be the Fisher information in the Carnot group G . Corollary 5.19 (Weak convexity of
Ent L n in Carnot groups) . Let ( G , d cc , L n ) be aCarnot group and let C G ≥ be as in Theorem 5.18. Let s µ s ∈ Geo([0 , P ( G )) bea W -geodesic connecting µ , µ ∈ Dom(
Ent L n ) . If µ s ∈ Dom(
Ent L n ) for some s ∈ (0 , ,then Ent L n ( µ s ) ≤ (1 − s ) Ent L n ( P t µ ) + s Ent L n ( P t µ )+ s (1 − s )2 h (cid:16) C G W ( µ , µ ) − W ( P t µ , P t µ ) (cid:17) + Z t + h F G ( P r µ s ) d r (5.43) for all t ≥ and h > . It is interesting to compare inequality (5.43) when t = 0 with the entropic inequalityobtained in [29, Corollary 3.4] when G = H n , the n -dimensional Heisenberg group. Notethat similar comparisons can be made for several others Carnot groups thanks to theresults obtained in [30, 35].The Heisenberg group H n , n ∈ N , is the non-commutative Carnot group of step whoseLie algebra satisfies g = V ⊕ V with m = 2 n , m = 1 and X i = ∂ x i − x n + i ∂ x n +1 , X n + i = ∂ x n + i + x i ∂ x n +1 , X n +1 = ∂ x n +1 , for all i = 1 , . . . , n . Thanks to [29, Corollary 3.4], the unique W -geodesic s µ s ∈ Geo([0 , P ( X )) joining two compactly supported measures µ , µ ∈ Dom(
Ent L n +1 ) satisfies Ent L n +1 ( µ s ) ≤ (1 − s ) Ent L n +1 ( µ ) + s Ent L n +1 ( µ ) + w ( s ) (5.44)for all s ∈ (0 , , where w ( s ) = − (cid:16) (1 − s ) (1 − s ) s s (cid:17) (5.45)for all s ∈ (0 , . Note that the function in (5.45) is concave and such that lim s → + w ( s ) = lim s → − w ( s ) = 0 and it satisfies < w ( s ) ≤ w ( ) = log 4 for all s ∈ (0 , . Therefore, as a consequence of (5.44), we get that µ s ∈ Dom(
Ent L n +1 ) and hence, by Corollary 5.19, we can also estimate Ent L n +1 ( µ s ) ≤ (1 − s ) Ent L n +1 ( µ ) + s Ent L n +1 ( µ ) + σ ( s ) (5.46)where σ ( s ) = inf ( h > s (1 − s ) C H n − h W ( µ , µ ) + Z h F H n ( P r µ s ) d r ) (5.47)for all s ∈ [0 , . Although we are not able to give a more explicit formula for the function in (5.47),it appears that, at least in some cases when µ and µ are very close to each other,inequality (5.46) is more precise than inequality (5.44) for intermediate times, that is, σ ( s ) < w ( s ) for some s ∈ (0 , . As a trivial example, if µ = µ , then µ s = µ for all s ∈ (0 , and thus inequality (5.44) reduces to ≤ w ( s ) for all s ∈ (0 , , while σ ( s ) = 0 for all s ∈ [0 , . As a less trivial example, exploiting the fact that right translations areoptimal transport maps in H n , see [18, Example 5.7] and [72, Section 2.1], we can provethe following result. Here we let ˜ F H n ( f ) = Z H n ∩{ f> } k ˜ ∇ H n f k H n f d x be the Fisher information in the Heisenberg group H n relative to the right-invarianthorizontal gradient ˜ ∇ H n . Proposition 5.20 (An estimate of σ for right translations) . Let µ = f L n +1 ∈ Dom(
Ent L n +1 ) be such that f ∈ C c ( R n +1 ) with ˜ F H n ( f ) < + ∞ . Let u ∈ H n be ahorizontal point in H n , i.e. u n +1 = 0 , and define T s ( x ) = xu s for all x ∈ H n and s ∈ [0 , , where u s = su . Then s µ s = ( T s ) ♯ µ is the unique W -geodesic joining µ with µ = ( T ) ♯ µ and σ ( s ) ≤ d cc ( u, o ) q s (1 − s ) ( C H n −
1) ˜ F H n ( f ) for all s ∈ (0 , . In particular, for any ε ∈ (0 , there exists δ > such that d cc ( u, o ) q ˜ F H n ( f ) < δ = ⇒ σ ( s ) < w ( s ) for all s ∈ ( ε, − ε ) . Proof.
The fact that s µ s = ( T s ) ♯ µ is the unique W -geodesic joining µ with µ =( T ) ♯ µ can be proved arguing as in [72, Section 2.1] since, up to a rotation fixing thevertical axis { x ∈ R n +1 : x i = 0 for all i = 1 , . . . , n } , one can also assume u i = 0 for all i = 2 , . . . , n . We thus omit the details. By the optimality of right translations, we have W ( µ , µ ) = Z H n d ( x, T ( x )) d µ ( x ) = Z H n d ( x, xu ) d µ ( x ) = d ( u, o ) . Moreover, we can write µ s = f s L n +1 , where f s = f ◦ T s ∈ C c ( R n +1 ) , so that ∇ H n ( P r f s ) = ∇ H n ( f s ⋆ p r ) = ( ˜ ∇ H n f s ) ⋆ p r for all r > and s ∈ (0 , , and thus F H n ( P r f s ) = Z H n | (( ˜ ∇ H n f s ) ⋆ p r )( x ) | H n ( f s ⋆ p r )( x ) d x ≤ Z H n | ˜ ∇ H n f s ( x ) | H n f s ( x ) d x = Z H n | ˜ ∇ H n f ( x ) | H n f ( x ) d x for all r > and s ∈ (0 , by Jensen inequality, arguing as in the proof of [20, Lemma 4.5](for right convolutions instead of left ones). Hence, from (5.47), we get σ ( s ) ≤ s (1 − s ) C H n − h W ( µ , µ ) + Z h F H n ( P r µ s ) d r ≤ s (1 − s ) C H n − h d ( u, o ) + h ˜ F H n ( f ) for all h > and s ∈ (0 , . The conclusion thus follows by optimizing with respectto h > and by recalling (5.45). (cid:3) ENERALIZED BAKRY–ÉMERY CURVATURE CONDITION 69
The SU (2) group. The group SU (2) is the Lie group of × complex unitarymatrices with determinant . Its Lie algebra su (2) consists of all × complex unitaryskew-Hermitian matrices with trace . Following the notation of [38], a basis of su (2) isgiven by the Pauli matrices X = − ! , Y = ii ! , Z = i − i ! , satisfying the relations [ X, Y ] = 2 Z, [ Y, Z ] = 2 X, [ Z, X ] = 2 Y. We keep the same notation X , Y and Z for the left invariant vector fields on SU (2) corresponding to the Pauli matrices. Similarly as before, we let ∇ SU (2) f = ( Xf ) X + ( Y f ) Y be the horizontal gradient of f . Using the cylindric coordinates ( r, ϑ, z ) exp( r cos ϑ X + r sin ϑ Y ) exp( ζ Z ) = e iζ cos r e i ( ϑ − ζ ) sin r − e − i ( ϑ − ζ ) sin r e − iζ cos r ! valid for r ∈ [0 , π ) , ϑ ∈ [0 , π ] and ζ ∈ [ − π, π ] (originally introduced in [65]), thenormalized bi-invariant Haar measure m ∈ P ( SU (2)) can be written as d m = 14 π sin(2 r ) d r d ϑ d ζ . Once the left-invariant Riemannian metric h· , ·i SU (2) making the basis X , Y , Z orthonormalis introduced, we can endow the group SU (2) with the Carnot–Carathéodory distance d cc defined analogously as in (5.42). The resulting metric space ( SU (2) , d cc ) is a complete,separable, locally compact and geodesic space.The standard sub-Laplacian operator is ∆ SU (2) = X + Y and, again by HörmanderTheorem, the heat operator ∂ t − ∆ SU (2) has a smooth fundamental solution p : (0 , + ∞ ) × SU (2) → (0 , + ∞ ) which induces the associated heat flow ( P t ) t ≥ by right convolution (sothat we can still identify H t = P t for all t ≥ ).Arguing similarly in the case of Carnot groups (recall the previously cited [92, 102, 108]),it is possible to identify the space W , ( SU (2) , d cc , m ) with the horizontal Sobolev space W , SU (2) ( SU (2)) = n f ∈ L ( SU (2) , m ) : Xf, Y f ∈ L ( SU (2) , m ) o defined using integration by parts against test functions, so that W , ( SU (2) , d cc , µ ) isHilbertian, the Cheeger energy coincides with the horizontal Dirichlet energy Ch ( f ) = Z SU (2) k∇ SU (2) f k SU (2) d m , for all f ∈ W , SU (2) ( SU (2)) , and | D f | w = k∇ SU (2) f k SU (2) . We again refer the reader to [102, Theorem 1.2] for a proofof these identifications.Exploiting the group structure of SU (2) similarly to the case of Carnot groups, weget that if f ∈ W , SU (2) ( SU (2)) with k∇ SU (2) f k SU (2) ≤ L , then f agrees µ -a.e. with a d cc -Lipschitz function with Lipschitz constant not larger than L .Therefore, ( SU (2) , d cc , m ) is a heat-smoothing admissible group and, combining [38,Theorem 4.10] with Theorem 3.16 and Theorem 5.16, we get the following result. Theorem 5.21 (Equivalence in SU (2) ) . Let ( SU (2) , d cc , m ) be as above. There exists aconstant C SU (2) ≥ √ satisfying the following four equivalent properties. (i) [ BE w ( C SU (2) e − t , ∞ ) ] If f ∈ C ∞ ( SU (2)) , then Γ SU (2) ( P t f ) ≤ C SU (2) e − t P t Γ SU (2) ( f ) for all t ≥ .(ii) [Kuwada] If µ, ν ∈ P ( SU (2)) , then W ( P t µ, P t ν ) ≤ C SU (2) e − t W ( µ, ν ) for all t ≥ .(iii) [ EVI w ( C SU (2) e − t ) ] If µ , µ ∈ Dom(
Ent m ) , then W ( P t µ , P t µ ) − C SU (2) t − t ) e t − e t W ( µ , µ ) ≤ t − t ) (cid:16) Ent m ( P t µ ) − Ent m ( P t µ ) (cid:17) for all ≤ t ≤ t .(iv) [ RCD w ( C SU (2) e − t , ∞ ) ] If s µ s ∈ Geo([0 , P ( SU (2))) is a geodesic connecting µ , µ ∈ Dom(
Ent m ) , then Ent m ( P t + h µ s ) ≤ (1 − s ) Ent m ( P t µ ) + s Ent m ( P t µ )+ s (1 − s )2 h C SU (2) he t + h ) − e t W ( µ , µ ) − W ( P t µ , P t µ ) ! for all s ∈ [0 , , t ≥ and h > . Since ( SU (2) , d cc , m ) is a Sasakian manifold , the resulting sub-Riemannian structureon SU (2) is ideal , see [36, Definition 13 and Section 7.4] for the precise definitions. Thus,according to [36, Theorem 39], if µ , µ ∈ Dom(
Ent m ) are two compactly supported prob-ability measures, then the unique Wassestein geodesic s µ s ∈ Geo([0 , P ( SU (2))) joining them satisfies µ s ≪ m for all s ∈ [0 , . Thanks to [36, Theorem 9 and Corol-lary 67] (see also [5, 103]), it actually holds that µ s ∈ Dom(
Ent m ) for all s ∈ (0 , and thefunction s Ent m ( µ s ) satisfies an inequality similar to (5.44).Up to our knowledge, there is no analogue of the entropy dissipation for L -densitiesproved in [20, Proposition 4.2] for the SU (2) group and we can only rely on the generalresult for L ∩ L -densities obtained in [14, Proposition 4.22]. Thus, at the present mo-ment, an inequality for the function s Ent m ( µ s ) similar to (5.43) holds in the SU (2) group under the additional assumption that d µ s d m ∈ L ( X, m ) for some s ∈ (0 , . Also,up to our knowledge, it is not known if right translations in the SU (2) group are optimaltransport maps. For this reason, a comparison of the entropic inequalities in the SU (2) group analogous to the one done above for Carnot groups is not easily reachable at thepresent moment. We will hopefully come back to this topic in a future work. References [1] A. Agrachev, D. Barilari, and U. Boscain,
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