Some functional inequalities on non-reversible Finsler manifolds

Transcription

1 Some functional inequalities on non-reversible Finsler manifolds Shin-ichi Ohta Abstract We continue our study of geometric analysis on possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the Γ-calculus à la Bakry et al, we show the dimensional versions of the Poincaré Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti ondino in the framework of curvature-dimension condition by means of the localization method. We show that the same sharp) estimates hold also for non-reversible metrics. 1 Introduction The aim of this article is to put forward geometric analysis on possibly non-reversible Finsler manifolds in the sense of F v) F v)) of Ricci curvature bounded below. Geometric analysis on spaces of Ricci curvature bounded below is a classical as well as active area. The classical Riemannian theory has been generalized to weighted Riemannian manifolds, linear arkov diffusion semigroups the Γ-calculus à la Bakry, Émery, and others, see the recent comprehensive book [BGL]), and to metric measure spaces satisfying Lott, Sturm and Villani s curvature-dimension condition [St1, St2, LV, Vi]). In fact, a reinforced version of the curvature-dimension condition, called the Riemannian curvature-dimension condition, is equivalent to the Bochner inequality which is the starting point of the Γ-calculus [AGS, EKS]). The class of Finsler manifolds takes an interesting place in the above picture. First of all, the distance function on a Finsler manifold can be asymmetric dy, x) dx, y) is allowed), thus it is not precisely a metric space in the usual sense. A non-riemannian Finsler manifold of weighted Ricci curvature bounded below satisfies the curvature-dimension condition in the naturally extended form to asymmetric distances), but the Riemannian curvature-dimension condition never holds. Finally, the natural Finsler Laplacian turns out nonlinear, and hence the Γ-calculus does not directly apply. onetheless, we Department of athematics, Kyoto University, Kyoto, , Japan sohta@math.kyotou.ac.jp), Current address: Department of athematics, Osaka University, Osaka, , Japan s.ohta@math.sci.osaka-u.ac.jp); Supported in part by JSPS Grant-in-Aid for Scientific Research KAK- EHI) 15K

2 investigated the nonlinear heat flow associated with the nonlinear Laplacian in [OS1], and established the Bochner inequality in [OS3]. This Bochner inequality has many applications including gradient estimates [OS3, Oh7]) and various eigenvalue estimates [WX, YH1, YH2, Xi]). We have developed the nonlinear Γ-calculus approach in [OS3, Oh7], the current article could be regarded as a continuation. It is not always possible to generalize a linear argument to the Finsler setting the most important example would be the noncontraction of heat flow in [OS2]), however, there are also many positive results. For example, the aforementioned gradient estimates were shown indeed in this line, and we proved Bakry Ledoux s Gaussian isoperimetric inequality under Ric K > 0 in the sharp form [Oh7], see [BL] for the original Riemannian result). The latter is a particular case of the Lévy Gromov type isoperimetric inequality, which was studied in [Oh6] by means of the localization method. The localization is another powerful technique reducing an inequality on a space to those on geodesics see [Kl, C1, C2]), however, it gives only non-sharp estimates in the non-reversible situation see [Oh6] for details). We will give sharp functional inequalities even in the non-reversible case, these generalize results in [C2] which cover reversible Finsler manifolds see also [Pr] for a related work on metric measure spaces enjoying the Riemannian curvature-dimension condition). Let us also remark that Ric for < 0 is not treated in [C2]. Our arguments essentially follow the known lines of Γ-calculus. There arise, however, some technical difficulties due to the nonlinearity, while the smoothness of the space sometimes gives additional help. The organization of the article is as follows. We briefly review the basics of Finsler geometry in Section 2. In Section 3, we prove the Poincaré Lichnerowicz inequality ) 2 f 2 dm f dm 1 F 2 f) dm 1.1) K under Ric K > 0 for, 0) [n, ), where n is the dimension of the manifold. Section 4 is devoted to the logarithmic Sobolev inequality f log f dm 1 F 2 f) dm 1.2) 2K f under Ric K > 0 with [n, ). In Section 5, we show the Sobolev inequality f 2 L p f 2 L 2 1 F 2 f) dm p 2 K for 1 p 2 + 1)/ under Ric K > 0 with [n, ] see also Remark 5.7 for a slight improvement of the admissible range of p). We finally discuss further related problems in Section 6. We remark that 1.1) with [n, ] and 1.2) with = were obtained in [Oh2] as applications of the curvature-dimension condition. 2 Preliminaries for Finsler geometry We briefly review the basics of Finsler geometry we refer to [BCS, Sh, SS] for further reading), and some facts from [Oh2, OS1, OS3]. Interested readers can consult [Oh7] for more elaborate preliminaries, as well as related surveys [Oh3, Oh4, Oh8]. 2

3 Throughout the article, let be a connected, n-dimensional C -manifold without boundary such that n 2. We also fix an arbitrary positive C -measure m on. 2.1 Finsler manifolds Given local coordinates x i ) n i=1 on an open set U, we will always use the fiber-wise linear coordinates x i, v j ) n i,j=1 of T U such that v = n j=1 v j x T x j x, x U. Definition 2.1 Finsler structures) We say that a nonnegative function F : T [0, ) is a C -Finsler structure of if the following three conditions hold: 1) Regularity) F is C on T \ 0, where 0 stands for the zero section; 2) Positive 1-homogeneity) It holds F cv) = cf v) for all v T and c > 0; 3) Strong convexity) The n n matrix gij v) ) n 1 := 2 F 2 ) i,j=1 2 v i v v) j ) n i,j=1 2.1) is positive-definite for all v T \ 0. We call such a pair, F ) a C -Finsler manifold. If F v) = F v) holds for all v T, then we say that F is reversible or absolutely homogeneous. Define the dual inkowski norm F : T [0, ) to F by F α) := sup αv) = sup αv) v T x, F v) 1 v T x, F v)=1 for α T x. It is clear by definition that αv) F α)f v). We remark that, however, αv) F α)f v) does not hold in general. Let us denote by L : T T the Legendre transform. amely, L is sending α T x to the unique element v T x such that F v) = F α) and αv) = F α) 2. We can write down in coordinates L α) = 1 2 n j=1 [F ) 2 ] α) x, where α = α j x j n α j dx j. j=1 The map L T x is linear if and only if F Tx comes from an inner product. For x, y, we define the asymmetric) distance from x to y by dx, y) := inf η 1 0 F ηt) ) dt, where the infimum is taken over all piecewise C 1 -curves η : [0, 1] such that η0) = x and η1) = y. ote that dy, x) dx, y) can happen since F is only positively 3

4 homogeneous. A C -curve η on is called a geodesic if it is locally minimizing and has a constant speed with respect to d. Given each v T x \ {0}, the positive-definite matrix g ij v)) n i,j=1 in 2.1) induces the Riemannian structure g v of T x as n x n ) x n g v a i, b x i j := g x j ij v)a i b j. 2.2) i=1 j=1 i,j=1 otice that this definition is coordinate-free, and we have g v v, v) = F 2 v). One can similarly define gα : Tx Tx R for α Tx \ {0}. The main tools in this article are the nonlinear Laplacian, the associated heat flow, the integration by parts and the Bochner inequality. Thus we will not use coordinate calculations of covariant derivative, etc., so that we do not recall them. For later use we also introduce the following quantity associated with, F ): g v w, w) S F := sup sup x v,w T x\0 F 2 w) = sup x sup α,β T x \0 F β) 2 g αβ, β) [1, ]. 2.3) Since g v w, w) S F F 2 w) and g v is the Hessian of F 2 /2 at v, the constant S F measures the fiber-wise) concavity of F 2 and is called the 2-)uniform smoothness constant see [Oh1]). We remark that S F = 1 holds if and only if, F ) is Riemannian. 2.2 Weighted Ricci curvature We begin with a useful interpretation of the Ricci curvature of, F ) found in [Sh, 6.2]. Given a unit vector v T x F 1 1), we extend it to a non-vanishing C -vector field V on a neighborhood U of x such that every integral curve of V is geodesic, and consider the Riemannian structure g V of U induced from 2.2). Then the Finsler Ricci curvature Ricv) of v with respect to F coincides with the Riemannian Ricci curvature of v with respect to g V in particular, it is independent of the choice of V ). Inspired by the above interpretation and the theory of weighted Ricci curvature also called the Bakry Émery Ricci curvature) of Riemannian manifolds, the weighted Ricci curvature for, F, m) was introduced in [Oh2] as follows. Definition 2.2 Weighted Ricci curvature) Given a unit vector v T x, let V be a C -vector field on a neighborhood U of x as above. We decompose m as m = e Ψ vol gv on U, where Ψ C U) and vol gv is the volume form of g V. Denote by η : ε, ε) the geodesic such that η0) = v. Then, for, 0) n, ), define Ric v) := Ricv) + Ψ η) 0) Ψ η) 0) 2. n We also define Ric v) and Ric n v) as the limits and set Ric cv) := c 2 Ric v) for c 0. We will denote by Ric K, K R, the condition Ric v) KF 2 v) for all v T. otice that multiplying a positive constant with m does not change Ric, thereby, when m) <, we will normalize m so as to satisfy m) = 1 without loss of generality. In 4

5 the Riemannian case, Ric with [n, ] has been well studied, see [Li, Ba, Qi] among many others. The study of the negative range, 0) and even, 1]) is more recent, see [K, i1, i2, Oh5, Wy, WY]. It is established in [Oh2, Oh5, Oh6] for [n, ], < 0, = 0, respectively) that, for K R, the bound Ric K is equivalent to Lott, Sturm and Villani s curvaturedimension condition CDK, ). This characterization extends the corresponding result on weighted Riemannian manifolds and has many geometric and analytic applications. 2.3 onlinear Laplacian and heat flow For a differentiable function f : R, the gradient vector at x is defined as the Legendre transform of the derivative of f: fx) := L Dfx)) T x. Define the divergence, with respect to the measure m, of a measurable vector field V with F V ) L 1 loc ) in the weak form as ϕ div m V dm := DϕV ) dm for all ϕ Cc ). Then we define the distributional Laplacian of f Hloc 1 ) by f := div m f), that is, ϕ f dm := Dϕ f) dm for all ϕ Cc ). Since the Legendre transform is nonlinear, our Laplacian is a nonlinear operator unless F is Riemannian. In [OS1, OS3], we have studied the associated nonlinear heat equation t u = u. In order to recall some results in [OS1], we define the energy of u Hloc 1 ) by Eu) := 1 F 2 u) dm = 1 F Du) 2 dm. 2 2 Define H0) 1 as the closure of Cc ) with respect to the absolutely homogeneous) norm u H 1 := u L 2 + {Eu) + E u)} 1/2. We can construct global solutions to the heat equation as gradient curves of the energy functional E in the Hilbert space L 2 ). We summarize the existence and regularity properties established in [OS1, 3, 4] in the next theorem see also [Oh7, 2]). Theorem 2.3 i) For each initial datum f H0) 1 and T > 0, there exists a unique global solution u = u t ) t [0,T ] to the heat equation with u 0 = f. oreover, the distributional Laplacian u t is absolutely continuous with respect to m for all t > 0. ii) The continuous version of a global solution u enjoys the Hloc 2 -regularity in x as well as the C 1,α -regularity in both t and x. oreover, t u t lies in Hloc 1 ) C) for all t > 0, and further in H0) 1 if S F <. 5

6 We set as usual u t := ut, ). otice that the standard elliptic regularity yields that u C {t, x) 0, ) Du t x) 0} ). ote also that, by the construction of heat flow as the gradient flow of E: If c u 0 C almost everywhere, then c u t C almost everywhere for all t > 0. lim t Eu t ) = 0. We will use the following equation found in [OS3, 4.2)] see also [OS1, Theorem 3.4]). Lemma 2.4 Let u t ) t 0 be a global solution to the heat equation. Then holds almost everywhere for all t > Bochner inequality [ F 2 u t ) ] = 2D[ u t ] u t ) t We finally recall the origin of our estimates, the Bochner inequality, established in [OS3] and slightly generalized in [Oh5, Oh7]. Theorem 2.5 Bochner inequality) Assume Ric K for some K R and, 0) [n, ]. Given f H0) 1 Hloc 2 ) C1 ) such that f H0), 1 we have [ ]) F Dϕ f 2 f) dm 2 } ϕ {D[ f] f) + KF 2 f) + f)2 dm 2.4) for all bounded nonnegative functions ϕ H 1 loc ) L ). Recall that f H 1 0) is achieved by solutions to the heat equation if S F <. The operator f in the LHS is a linearization of the gradient given by f u := n i,j=1 g ij f) u x j x, i where g ij f)) is the inverse matrix of g ij f)). Let us similarly define f u := div m f u). ote that f f = f and f f = f hold. We will sometimes use the following common notation for brevity: Γ 2 f) := f [ F 2 f) 2 ] D[ f] f). 2.5) Then 2.4) is written in a shorthand way as Γ 2 f) KF 2 f) + f) 2 /. 6

7 3 Poincaré Lichnerowicz inequality We start with the Poincaré Lichnerowicz inequality under the curvature bound Ric K > 0. The [n, ] case was shown in [Oh2] as a consequence of the curvaturedimension condition. In the Riemannian setting, the case of, 0) was shown independently in [K] and [Oh5]. For simplicity we will assume that is compact it automatically holds when [n, ) and is complete), and normalize m as m) = 1. ote that the ergodicity: u 0 dm in L 2 ) 3.1) u t holds for all global solutions u t ) t 0 to the heat equation since lim t Eu t ) = 0). Proposition 3.1 Assume that is compact and satisfies Ric K for some K R and, 0) [n, ]. Then we have, for any f H 2 ) C 1 ) such that f H 1 ), } {D[ f] f) + KF 2 f) + f)2 dm 0. In particular, if K > 0, then we have F 2 f) dm 1 K If =, then the coefficient in the RHS of 3.2) is read as 1/K. f) 2 dm. 3.2) Proof. The first assertion is a direct consequence of Theorem 2.5 with ϕ 1. We further observe, by the integration by parts, K F 2 f) dm + f 2 L 2 D[ f] f) dm = f 2 L 2. Rearranging this inequality yields 3.2) when K > 0. ow the Poincaré Lichnerowicz inequality is obtained by a technique similar to [BGL, Proposition 4.8.3]. We define the variance of f L 2 ) under m) = 1) as Var m f) := f 2 dm f dm) 2. Theorem 3.2 Poincaré Lichnerowicz inequality) Let be compact, m) = 1, and suppose Ric K > 0 for some, 0) [n, ]. Then we have, for any f H 1 ), ) 2 f 2 dm f dm 1 F 2 f) dm. K 7

8 Proof. Let u t ) t 0 be the solution to the heat equation with u 0 = f, and put Φt) := u t 2 L 2 for t 0. Observe that Φ t) = 2 u t u t dm = 2 F 2 u t ) dm = 4Eu t ), and by Lemma 2.4, Φ t + δ) Φ t) lim δ 0 δ for all t > 0. Then 3.2) implies 0 2Φ t) 1 K = 4 lim δ 0 Eu t+δ ) Eu t ) δ lim δ 0 = 4 u t 2 L 2 Φ t + δ) Φ t). 3.3) δ otice now that the ergodicity 3.1) implies lim t Φt) = f dm)2. Thus the differential inequality 3.3) yields Var m f) = Φ t) dt 1 ) lim 2K t Φ t) Φ 2 1) 0) = K Ef). This completes the proof. 4 Logarithmic Sobolev inequality We next study the logarithmic Sobolev inequality. From here on we consider only K > 0 and [n, ). Then Ric K implies the compactness of, thus we normalize m as m) = 1 without loss of generality. We first consider a sufficient condition for the logarithmic Sobolev inequality as in [BGL, Proposition 5.7.3]. Proposition 4.1 Assume that is compact, Ric K > 0, and F 2 { [ ]) } u) F dm C Du u 2 u) + ud[ log u)] [log u]) dm 4.1) u 2u 2 holds for some constant C > 0 and all functions u H 2 ) C 1 ) such that u H 1 ) and inf u > 0. Then the logarithmic Sobolev inequality {f>0} f log f dm C 2 {f>0} F 2 f) f holds for all nonnegative functions f H 1 ) with f dm = 1. dm 4.2) The assumed inequality 4.1) is written in shorthand as recall 2.5)) uf 2 [log u]) dm C uγ 2 log u) dm. 4.3) 8

9 ote that both sides are well-defined thanks to inf u > 0 and F u) L ). The LHS of the logarithmic Sobolev inequality is the relative entropy of the probability measure fm with respect to m: Ent m fm) := f log f dm. 4.4) Since m) = 1, we have Ent m fm) 0 and Ent m fm) = 0 holds if and only if f = 1 almost everywhere. Proof. Since m) = 1, by the truncation argument, one can assume f ε for some ε > 0. Then the solution u t ) t 0 to the heat equation with u 0 = f satisfies u t ε for all t > 0, thereby 4.1) admits u t. Then the claim follows from a similar argument to Theorem 3.2 applied to Ψt) := Ent m u t m). We see that Ψ F 2 u t ) t) = log u t + 1) t u t dm = D[log u t ] u t ) dm = dm. u t We further deduce from Lemma 2.4 and the integration by parts that { Ψ ut t) = F 2 u u 2 t ) 2 } D[ u t ] u t ) dm t u t [ ]) F 2 u = Du t ut t ) dm 2 D[ u u 2 t ] [log u t ]) dm. t Since D[ u t ] [log u t ]) dm = log u t ) u t dm = D[ log u t )] u t ) dm, the supposed inequality 4.1) means 2Ψ t) CΨ t). We observe from the logarithmic Sobolev inequality under Ric K > 0 that u t log u t dm 1 F 2 u t ) dm 1 F 2 u t ) dm. 2K u t 2Kε Together with lim t Eu t ) = 0, we have lim t Ψt) = lim t Ψ t) = 0 and thus f log f dm = Ψ t) dt C Ψ t) dt = C 2 2 Ψ 0). We complete the proof. 0 ow we can show the sharp, dimensional logarithmic Sobolev inequality along the lines of [BGL, Theorem 5.7.4]. 0 Theorem 4.2 Logarithmic Sobolev inequality) Assume that Ric some [n, ) and m) = 1. Then we have f log f dm 1 F 2 f) dm 2K f {f>0} {f>0} for all nonnegative functions f H 1 ) with f dm = 1. K > 0 for 9

10 Proof. Fix h C ) and consider the function e ah for a > 0. We begin with some preliminary and useful equations. ote first that, since a > 0, e ah ) = ae ah h, e ah ) = ae ah { h + af 2 h)}. 4.5) Thus, on the one hand, [ ] a Γ 2 e ah ) = h 2 e 2ah F 2 h) a 2 e ah D [ e ah { h + af 2 h)} ] h) 2 [ [ ] ] F = a 2 div m e 2ah h 2 h) + ae 2ah F 2 h) h 2 } a 2 e {a{ h 2ah + af 2 h)}f 2 h) + D[ h] h) + ad[f 2 h)] h) { [ ] } F = a 2 e 2ah h 2 h) + ad[f 2 h)] h) + a 2 F 4 h) D[ h] h) 2 = a 2 e 2ah{ Γ 2 h) + ad[f 2 h)] h) + a 2 F 4 h) }. 4.6) On the other hand, it follows from the integration by parts that Γ 2 e ah ) dm = e ah ) 2 L = 2 a2 e 2ah { h) 2 + 2aF 2 h) h + a 2 F 4 h)} dm = a 2 e 2ah { h) 2 2aD[F 2 h)] h) 3a 2 F 4 h)} dm. Comparing this with 4.6), we have e 2ah h) 2 dm = e 2ah{ Γ 2 h) + 3aD[F 2 h)] h) + 4a 2 F 4 h) } dm. 4.7) ext we apply the Bochner inequality 2.4) to e ah and find, by 4.5) and 4.6), Γ 2 h) + ad[f 2 h)] h) + a 2 F 4 h) KF 2 h) + 1 { h)2 + 2aF 2 h) h + a 2 F 4 h)}. 4.8) To be precise, 4.8) holds in the weak sense as in Theorem 2.5, we will employ e h as a test function. Then the RHS is being, by 4.7) with a = 1/2 and the integration by parts, { e h KF 2 h) + 1 { h) 2 + 2aF 2 h) h + a 2 F 4 h) }} dm { } = e h KF 2 h) + a2 F 4 h) dm + 1 e {Γ h 2 h) + 3 } 2 D[F 2 h)] h) + F 4 h) dm 2a e h{ F 4 h) + D[F 2 h)] h) } dm = K e h F 2 h) dm + 1 ) } 3 e {Γ h 2 h) + 2 2a D[F 2 h)] h) + a 1) 2 F 4 h) dm. 10

11 Hence we obtain from 4.8) that 1 1 ) e h Γ 2 h) dm K e h F 2 h) dm )a + e h D[F 2 h)] h) dm 2 + a 1)2 a 2 e h F 4 h) dm. 4.9) Choosing a = 3/{2 + 2)} > 0 gives that 1 e h Γ 2 h) dm K e h F 2 4 1) 1) h) dm + e h F 4 h) dm 4 + 2) 2 K e h F 2 h) dm. 4.10) This is the desired inequality 4.1) recall 4.3)) for u = e h with C = 1)/K. Then u C ) and inf u > 0. By approximation this implies 4.1) for all u in the required class, therefore we complete the proof by Proposition 4.1. Remark 4.3 Different from the Poincaré Lichnerowicz inequality in the previous section, the calculation in the above proof does not admit being negative. Precisely, a < 0 is acceptable in the reversible case, whereas the last inequality 4.10) fails for < 0. In fact, the logarithmic Sobolev inequality of the form 4.2) does not hold under Ric K > 0 with < 0. This is because the model space given in [i1] satisfies only the exponential concentration, while 4.2) or the Talagrand inequality below) implies the normal concentration see [Le] for the theory of concentration of measures). As a corollary, we have the dimensional Talagrand inequality on the relation between the Wasserstein distance W 2 and the relative entropy 4.4). See [Oh2, Vi] for the definition of W 2 as well as the Talagrand inequality under Ric K > 0. We will denote by P) the set of all Borel probability measures on. Corollary 4.4 Talagrand inequality) Assume that Ric K > 0 for [n, ) and m) = 1. Then we have, for all µ P), W 2 2 µ, m) 2 1) K Ent mµ). Proof. This is the well known implication going back to [OV]. Since our distance is asymmetric, we give an outline along [GL] for completeness. ote that it is enough to show the claim for µ = fm with f H 1 ). Let u t ) t 0 be the global solution to the heat equation with u 0 = f, and put µ t := u t m P) and Ψt) := Ent m µ t ). Then, as we saw in the proof of Proposition 4.1, Ψ t) = F 2 u t ) u t 11 dm 2K 1 Ψt)

12 by Theorem 4.2. We can rewrite this inequality as 2 1) ) t). Ψ t) Ψ K Arguing as in [GL] following the lines of [GKO]), we obtain ) 2 W 2 µ t, µ t+ε ) lim ε 0 ε F 2 u t ) u t dm = Ψ t) for almost every t > 0. Thus we have, since lim t W 2 µ t, m) = lim t Ψt) = 0, W 2 µ, m) = lim W 2 µ 0, µ t ) t 0 2 1) = Ψ0). K This completes the proof. Ψ t) dt 2 1) K 0 Ψ ) t) dt 5 Sobolev inequality This section is devoted to the Sobolev inequality. We first derive a non-sharp Sobolev inequality followed by qualitative consequences. Then, with the help of these qualitative properties, we proceed to the sharp estimate. 5.1 on-sharp Sobolev inequality and applications We start with a useful inequality as in [BGL, Theorem 6.8.1]. Proposition 5.1 Logarithmic entropy-energy inequality) Assume Ric K > 0 for some [n, ) and m) = 1. Then we have Ent m f 2 m) 2 log ) F 2 f) dm K for all f H 1 ) with f 2 dm = 1. Proof. Let us first consider nonnegative f. By truncation we can assume f L ) and inf f > 0, in particular f 2 H 1 ). Consider the solution u t ) t 0 to the heat equation with u 0 = f 2, and put Ψt) := Ent m u t m). Then we have, by Proposition 4.1, Ψ F 2 u t ) t) = dm = u t F 2 [log u t ] ) dm 0. u t oreover, recall also from the proof of Proposition 4.1 that Ψ t) = Du t [ u t F 2 [log u t ]) ]) dm 2 u t D [log u t ]) [log u t ]) dm. 12

13 Then the Bochner inequality 2.4) shows that Ψ t) 2KΨ t) + 2 u t [log u t ]) 2 dm. By the Cauchy Schwarz inequality and u t dm = 1, we find Ψ t) 2KΨ t) + 2 Therefore the function is nondecreasing, and hence ) 2 u t [log u t ] dm = 2KΨ t) + 2 Ψ t) ) t e 2Kt 1 K ) Ψ t) Ψ t) K {e 2Kt 1 K ) 1 1}. Ψ 0) Integrating this inequality gives Ψ0) Ψt) ) 2 log 1 1 e 2Kt ) Ψ 0). K ote finally that Ψ 0) = F 2 f 2 )) f 2 dm = 4 F 2 f) dm since f 0. Thus letting t completes the proof for f 0. In general, we divide f into f + := max{f, 0} and f := max{ f, 0}, and apply the above inequality with respect to F and F v) := F v), respectively see [Oh2, Corollary 8.4] for details). Then the concavity of the function log1 + s) for s 0 shows the claim. Remark 5.2 The inequality in 5.1) is invalid for < 0 since the Cauchy Schwarz inequality cannot be reversed. ext we follow the strategy in [BGL, Proposition 6.2.3] to show the ash inequality and then a non-sharp Sobolev inequality. Lemma 5.3 ash inequality) Assume that Ric K > 0 for some [n, ) and m) = 1. Then we have, for all f H 1 ), f +2 L 2 f 2 L + 4 ) /2 2 K Ef) f 2 L 1. 13

14 Proof. ormalize f so as to satisfy f 2 dm = 1. Put ψθ) := log f L 1/θ) for θ 0, 1], and notice that ψ is a convex function due to the Hölder inequality: ψ 1 λ)θ + λθ ) = 1 λ)θ + λθ ) ) log f 1 λ)/1 λ)θ+λθ ) f λ/1 λ)θ+λθ ) dm 1 λ)θ + λθ ) ) 1 λ)θ/1 λ)θ+λθ ) ) ) λθ log f 1/θ /1 λ)θ+λθ ) dm f 1/θ dm = log f 1 λ L 1/θ f λ L 1/θ ) = 1 λ)ψθ) + λψθ ) for all θ, θ 0, 1] and λ 0, 1). Therefore ) 1 ψ1) ψ + 1 ) ψ = 1 ) ψ 2 since we supposed f L 2 = 1. Combining this with ) 1 ψ = 1 4 log f f 2 ) dm = Ent m f 2 m) 2 2 and Proposition 5.1, we obtain f L 1 exp 1 ) 2 Ent mf 2 m) ) /4 K Ef) ) /2 K Ef). This completes the proof. Proposition 5.4 on-sharp Sobolev inequality) Assume that Ric some [n, ) 2, ) and m) = 1. Then we have K > 0 for f 2 L p C 1 f 2 L 2 + C 2Ef) for all f H 1 ), where p = 2/ 2), C 1 = C 1 ) > 1 and C 2 = C 2 K, ) > 0. Proof. By the same reasoning as Proposition 5.1 and f + 2 L p + f 2 L p f + p L p + f p L p)2/p, 5.2) we can assume that f L ) and inf f > 0. For k Z consider the decreasing sequence A k := {f > 2 k } and set f k := min { max{f 2 k, 0}, 2 k} 2 k on A k+1, = f 2 k on A k \ A k+1, 0 on \ A k. otice that 2 k χ Ak+1 f k 2 k χ Ak χ A denotes the characteristic function of A) and hence 2 2k ma k+1 ) f k 2 L 2 22k ma k ), f k L 1 2 k ma k ). 14

15 Together with the ash inequality Lemma 5.3) applied to f k, we have 2 2k ma k+1 ) ) +2)/2 fk +2 L 2 f k 2 L k ma k ) + 4 K Ef k) Let us rewrite this by using p = 2/ 2) as ) /2 K Ef k) f k 2 ) /2 2 2k ma k ) 2. 2 pk+1) ma k+1 ) 2 p 2 2k ma k ) + 4 ) /+2) K Ef k) 2 pk ma k ) ) 4/+2). Combining this with the Hölder inequality implies 2 pk+1) ma k+1 ) 2 p 2 p 2 p Hence we have 2 pk ma k ) that { 2 2k ma k ) + 4 { 2 2k ma k ) + 4 K Ef k) { 2 2k ma k ) + 4 K Ef k) ) 2)/+2) 2 p ) /+2) K Ef k) 2 2pk ma k ) 2) 2/+2)} }) /+2) L 1 2 2pk ma k ) 2 ) 2/+2) }) /+2) 4/+2) 2 pk ma k )). On the one hand, we deduce from Ef k) = Ef) and { 2 2k ma k ) + 4 }) /+2) K Ef k). 5.3) 2 2k ma k ) = 4 2 2k {ma k ) ma k+1 )} f 2 L 2 On the other hand, we similarly find { 2 2k ma k ) + 4 } K Ef k) 4 3 f 2 L K Ef). 2 pk ma k ) = 1 2 p 1 2 pk+1) {ma k ) ma k+1 )} 2 p f p L p. Substituting these into 5.3) yields 4 2 p f p L p) 2)/ 2 p+2)/ 3 f 2 L + 4 ) 2 K Ef). 15

16 Recalling p = 2/ 2), we finally obtain 4 f 2 L p 2p 2)/ 2 p+2)/ 3 f 2 L + 4 ) 2 K Ef) 4 = 2 4/ 2) 3 f 2 L + 4 ) 2 K Ef). We have several qualitative consequences from the non-sharp Sobolev inequality in Proposition 5.4. In fact, one can reduce these qualitative arguments to the Riemannian case by virtue of the uniform smoothness 2.3). Corollary 5.5 Assume that Ric K > 0 for some [n, ) 2, ) and m) = 1. Then there exists a C -Riemannian metric g for which f 2 L p C 1 f 2 L 2 + C 2S F E g f) holds for all f H 1 ), with p = 2/ 2), C 1 > 1 and C 2 > 0 as in Proposition 5.4. Proof. Let {U i } i be an open cover of, V i a non-vanishing C -vector field on U i, and {ρ i } i a partition of unity subordinate to {U i } i. Consider the Riemannian metric g := i ρ ig Vi. For any f H 1 ), we have 2E F f) = ρ i F Df) 2 dm ρ i S F gv i Df, Df) dm = 2S F E g f). i U i i U i Combining this with Proposition 5.4, we complete the proof. 5.2 Sharp Sobolev inequality We finally show the sharp Sobolev inequality along the lines of [BGL, Theorem 6.8.3]. Theorem 5.6 Sobolev inequality) Assume that Ric K > 0 for some [n, ) and m) = 1. Then we have f 2 L p f 2 L 2 p 2 1 K for all 1 p 2 + 1)/ and f H 1 ). F 2 f) dm The case of p = 2 is understood as the limit, giving the logarithmic Sobolev inequality Theorem 4.2). The p = 1 case amounts to the Poincaré Lichnerowicz inequality Theorem 3.2), whereas only for n. Proof. Let us assume > 2 for simplicity. This certainly covers the case of = n = 2 just by taking the limit as 2. otice also that, due to 5.2) for p > 2 and its converse for p < 2, it suffices to show the claim for nonnegative f by a similar argument to Proposition

17 Take the smallest possible constant C > 0 satisfying f 2 L p f 2 L 2 p 2 2CEf) 5.4) for all nonnegative functions f H 1 ). Our goal is to show C 1)/K. Let us suppose that we find a extremal nonconstant) function f 0 enjoying equality in 5.4) as well as f L ) and inf f > 0, and normalize it as f L p = 1. For any ϕ C ) and ε > 0, by the choice of f, f + εϕ 2 L p f 2 L p f + εϕ 2 L 2 + f 2 L 2 p 2 2C{Ef + εϕ) Ef)}. Dividing both sides by ε and letting ε 0, we have ) 1 2 p 2 p pf p 1 ϕ 2fϕ dm 2C Dϕ f) dm = 2C Since ϕ was arbitrary, it follows that f is well-defined and) Put f = e u. Then 5.5) is rewritten as therefore ϕ f dm. f p 1 f = Cp 2) f. 5.5) e p 1)u e u = Cp 2) [e u ] = Cp 2)e u { u + F 2 u)}, e p 2)u = 1 Cp 2){ u + F 2 u)}. 5.6) ultiply both sides by e bu u with b 0 and use the integration by parts to find p 2 + b) e p 2+b)u F 2 u) dm = b e bu F 2 u) dm + Cp 2) e bu{ u) 2 bf 4 u) D[F 2 u)] u) } dm. Recall 4.7) with a = b/2 0: { e bu u) 2 dm = e bu Γ 2 u) + 3b } 2 D[F 2 u)] u) + b 2 F 4 u) dm. 5.7) Thus p 2 + b) e p 2+b)u F 2 u) dm = b e bu F 2 u) dm + Cp 2) ) 3b + Cp 2) Cp 2)bb 1) 17 e bu Γ 2 u) dm e bu D[F 2 u)] u) dm e bu F 4 u) dm.

18 Substituting 5.6) to e p 2)u in the LHS, we have on the one hand Cp 2) e bu Γ 2 u) dm = p 2) e bu F 2 u) dm Cp 2)p 2 + b) e bu F 2 u){ u + F 2 u)} dm ) 3b Cp 2) 2 1 e bu D[F 2 u)] u) dm Cp 2)bb 1) e bu F 4 u) dm = p 2) e bu F 2 u) dm + Cp 2) p 1 b ) e bu D[F 2 u)] u) dm 2 + Cp 2) 2 b 1) e bu F 4 u) dm. 5.8) On the other hand, multiplying the RHS of 4.8) by e bu and integrating it gives, together with 5.7), { } e bu KF 2 u) + a2 F 4 u) dm + 1 { e bu Γ 2 u) + 3b } 2 D[F 2 u)] u) + b 2 F 4 u) dm 2a e bu{ bf 4 u) + D[F 2 u)] u) } dm = K e bu F 2 u) dm + 1 ) } 3b e {Γ bu 2 u) + 2 2a D[F 2 u)] u) + a b) 2 F 4 u) dm. Thus we obtain from 4.8) the following variant of 4.9): 1 1 ) e bu Γ 2 u) dm K e bu F 2 u) dm ) 3b 4a + 2 a e bu D[F 2 u)] u) dm ) a b) 2 + a 2 e bu F 4 u) dm. 5.9) Comparing the coefficients in 5.8) and 5.9), we would like to choose a and b enjoying p 1 b 2 = 3b 2 + 2)a 2 1), p 2)b 1) = a b)2 a ) 1 18

19 At this point we need an additional care, because of the non-reversibility, on the ranges of a and b. As we mentioned, they necessarily satisfy a 0 and b 0. We first deduce from the first equation in 5.10) that a = b 2 p 1) Substituting this into the latter inequality in 5.10) yields ) ) b 2 2 p b p 2) + p 1) 2 = ) + 2 Let us denote the LHS by hb). The discriminant of h is given by ) p 1) 2 p 1) + 1, which vanishes at p = 1, 2/ 2) and is nonnegative for p [1, 2/ 2)]. ote also that the axis of hb) is positive since p 2 + 1)/ 2/ 2)) implies 2 1 p 1 ) ) > Thus we have the positive solution b p 1 ) + 2 of 5.11), and put a 0 := b 0 1 p 1) Then we observe from p 2 + 1)/ that a 0 1 p 1 1 p 1) ) Plugging above a 0 and b 0 into 5.8) and 5.9), we obtain C 1)/K as desired. Finally, since there may not be a good extremal function, one needs an extra discussion on the approximation procedure. This step, needing only the non-sharp Sobolev inequality, can be reduced to the Riemannian case by virtue of Corollary 5.5. See the latter half of the proof of [BGL, Theorem 6.8.3] for details. Remark 5.7 In 5.12) we used p 2 + 1)/ which is slightly more restrictive than p 2/ 2) in [BGL, C2]. A more precise estimate gives hb 0 2a 0 ) 0 for [ 2 + 1) p, ) ] 2 + 8, 4 1) which means a 0 0 and slightly improves the acceptable range of p. This is, however, still more restrictive than p 2/ 2). Indeed, in the extremal case of p = 2/ 2), one can explicitly calculate see [BGL, Theorem 6.8.3]) b 0 = 2 1 p ) = 2 3) 2, a 0 = 2 2 < 0. 19

20 Thanks to the smoothness of and m, we have the following corollary. Corollary 5.8 Sobolev inequality for = ) Assume that is compact and satisfies Ric K > 0 and m) = 1. Then we have f 2 L p f 2 L 2 1 F 2 f) dm p 2 K for all 1 p 2 and f H 1 ). Proof. By the smoothness and the compactness, for any ε > 0, we have Ric ε K ε for sufficiently large ε <. Then the claim is derived from Theorem 5.6 as the limit of ε 0. 6 Further problems A) Though we did not pursue that direction in this article, the p-spectral gap is also treated in [C2]. It is worthwhile to study such a problem on non-reversible Finsler manifolds. See [YH1] for a related work. B) There remain many open problems in the case of < 0. We saw that the Poincaré Lichnerowicz inequality admits < 0 and the logarithmic Sobolev inequality does not recall Remark 4.3). onetheless, we had in [Oh5] certain variants of the logarithmic Sobolev and Talagrand inequalities under the entropic curvature-dimension condition CD e K, ). The condition CD e K, ) is seemingly stronger than Ric K when < 0, whereas the precise relation is still unclear. One of the widely open problems for < 0 is a gradient estimate for the heat semigroup. The model space given in [i1] would give a clue to the further study. References [AGS] [Ba] L. Ambrosio,. Gigli and G. Savaré, Bakry Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab ), D. Bakry, L hypercontractivité et son utilisation en théorie des semigroupes. French) Lectures on probability theory Saint-Flour, 1992), 1 114, Lecture otes in ath., 1581, Springer, Berlin, [BGL] D. Bakry, I. Gentil and. Ledoux, Analysis and geometry of arkov diffusion operators. Springer, Cham, [BL] [BCS] D. Bakry and. Ledoux, Lévy Gromov s isoperimetric inequality for an infinitedimensional diffusion generator. Invent. ath ), D. Bao, S.-S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry. Springer-Verlag, ew York, [C1] F. Cavalletti and A. ondino, Sharp and rigid isoperimetric inequalities in metricmeasure spaces with lower Ricci curvature bounds. Invent. ath ),

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