Sharp Cheeger-Buser type inequalities in $ \mathsf{RCD}(K,\infty)$ spaces

The goal of the paper is to sharpen and generalise bounds involving the Cheeger's isoperimetric constant $h$ and the first eigenvalue $\lambda_{1}$ of the Laplacian. \\A celebrated lower bound of $\lambda_{1}$ in terms of $h$, $\lambda_{1}\geq h^{2}/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on $\lambda_{1}$ in terms of $h$ was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below. \\ The goal of the paper is two fold. First: we sharpen the inequalities obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality for spaces with (Bakry-\'Emery weighted) Ricci curvature bounded below by $K\in {\mathbb R}$ (the inequality is sharp for $K>0$ as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called $ \mathsf{RCD}(K,\infty)$ spaces.


Introduction
Throughout the paper (X, d) will be a complete metric space and m will be a non-negative Borel measure on X, finite on bounded subsets. The triple (X, d, m) is called metric measure space, m.m.s. for short. We denote by Lip(X) the space of real-valued Lipschitz functions over X and we write f ∈ Lip b (X) if f ∈ Lip(X) and f is bounded with bounded support. Given f ∈ Lip(X) its slope |∇f |(x) at x ∈ X is defined by • If m(X) < ∞, the non-zero constant functions are in L 2 (X, m) and are eigenfunctions of the Laplacian with eigenvalue 0. In this case, the first non-trivial eigenvalue is given by • When m(X) = ∞, 0 may not be an eigenvalue of the Laplacian and the first eigenvalue is characterized by Note that λ 0 may be zero (for instance if m(X) < ∞ or (X, d, m) is the Euclidean space R d with the Lebesgue measure) but there are examples when λ 0 > 0: for instance in the Hyperbolic plane λ 0 = 1/4 and more generally on an n-dimensional simply-connected Riemannian manifold with sectional curvatures bounded above by k < 0 it holds λ 0 ≥ (n − 1) 2 k/4 (see [27]).
Given a Borel subset A ⊂ X with m(A) < ∞, the perimeter Per(A) is defined as follows (see for instance [24]): Per(A) := inf lim inf n→∞ X |∇f n |dm : f n ∈ Lip b (X), f n → χ A in L 1 (X) .
In 1970, Cheeger [15] introduced an isoperimetric constant, now known as Cheeger constant, to bound from below the first eigenvalue of the Laplacian. The The lower bound obtained in [15] for compact Riemannian manifolds, now know as Cheeger inequality, reads as As proved by Buser [9], the constant 1/4 in (5) is optimal in the following sense: for any h > 0 and ε > 0, there exists a closed (i.e. compact without boundary) two-dimensional Riemannian manifold (M, g) with h(M ) = h and such that λ 1 ≤ 1 4 h(M ) 2 + ε. The paper [15] is in the framework of smooth Riemannian manifolds; however, the stream of arguments (with some care) extend to general metric measure spaces. For the reader's convenience, we give a self-contained proof of (5) for m.m.s. in the Appendix (see Theorem 4.2). Cheeger's inequality (5) revealed to be extremely useful in proving lower bounds on the first eigenvalue of the Laplacian in terms of the isoperimetric constant h. It was thus an important discovery by Buser [10] that λ 1 and h are actually equivalent, up to a constant depending on the lower bound on the Ricci curvature of the smooth Riemannian manifold. More precisely, Buser [10] proved that for any compact Riemannian manifold of dimension n and Ric ≥ K, K ≤ 0 it holds Note that the constant here is dimension-dependent. For a complete connected Riemannian manifold with Ric ≥ K, K ≤ 0, Ledoux [23] remarkably showed that the constant can be chosen to be independent of the dimension: The goal of the present work is two fold: (1) The main results of the paper (Theorem 1.1 and Corollary 1.2) improve the constants in both the Buser type inequalities (6)-(7) in a way that now the inequality is sharp for K > 0 (as equality is attained on the Gaussian space). (2) The inequalities are established in the higher generality of (possibly non-smooth) metric measure spaces satisfying Ricci curvature lower bounds in synthetic sense, the so called RCD(K, ∞) spaces.
For the precise definition of RCD(K, ∞) space, we refer the reader to Section 2. Here let us just recall that the RCD(K, ∞) condition was introduced by Ambrosio-Gigli-Savaré [5] (see also [3]) as a refinement of the CD(K, ∞) condition of Lott-Villani [25] and Sturm [32]. Roughly, a CD(K, ∞) space is a (possibly infinite dimensional, possibly non smooth) metric measure space with Ricci curvature bounded from below by K, in a synthetic sense. While the CD(K, ∞) condition allows Finsler structures, the main point of RCD is to reinforce the axiomatization (by asking linearity of the heat flow) in order to rule out Finsler structures and thus isolate the "possibly non-smooth Riemannian structures with Ricci curvature bounded below". It is out of the scopes of this introduction to survey the long list of achievements and results proved for CD and RCD spaces (to this aim, see the Bourbaki seminar [33] and the recent ICM-Proceeding [1]). Let us just mention that a key property of both CD and RCD is the stability under measured Gromov-Hausdorff convergence (or more generally D-convergence of Sturm [32,5], or even more generally pointed measured Gromov convergence [19]) of metric measure spaces. In particular pmGH limits of Riemannian manifolds with Ricci bounded below, the so-called Ricci limits, are examples of (possibly non-smooth) RCD spaces. Let us also recall that weighted Riemannian manifolds with Bakry-Émery Ricci tensor bounded below are also examples of RCD spaces; for instance the Gaussian space It is also worth recalling that if (X, d, m) is an RCD(K, ∞) space for some K > 0, then m(X) < ∞; since scaling the measure by a constant does not affect the synthetic Ricci curvature lower bounds, when K > 0, without loss of generality one can then assume m(X) = 1.
In order to state our main result, it is convenient to set The aim of the paper is to prove the following theorem. Let (X, d, m) be an RCD(K, ∞) space, for some K ∈ R.
• In case m(X) = 1, then The inequality is sharp for K > 0, as equality is achieved for the Gaussian space Using the expression (8) of J K , in the next corollary we obtain more explicit bounds.
Comparison with previous results in the literature. Theorem 1.1 and Corollary 1.2 improve the known results about Buser-type inequalities in several aspect. First of all the best results obtained before this paper are the aforementioned estimates (6)-(7) due to Buser [10] and Ledoux [23] for smooth complete Riemannian manifolds satisfying Ric ≥ K, K ≤ 0. Let us stress that the constants in Corollary 1.2 improve the ones in both (6)- (7) and are dimension free as well. In addition, the improvements of the present paper are: • In case K > 0, the inequalities (9) and (11) are sharp (as equality is attained on the Gaussian space). • The results hold in the higher generality of (possibly non-smooth) RCD(K, ∞) spaces. The proof of Theorem 1.1 is inspired by the semi-group approach of Ledoux [22,23], but it improves upon by using Proposition 3.1 in place of: • A dimension dependent Li-Yau inequality, in [22].  [26], under the RCD * (K, N ) condition by Erbar-Kuwada-Sturm [17], and generalized by Cavalletti and Mondino [12] to a sharp spectral gap for the p-Laplacian for essentially non-branching CD * (K, N ) spaces involving also an upper bound on the diameter (together with rigidity and almost rigidity statements). Jiang-Zhang [20] independently showed, for p = 2, that the improved version under an upper diameter bound holds for RCD * (K, N ). The rigidity of the Lichnerowitz spectral gap for RCD * (K, N ) spaces, K > 0, N ∈ (1, ∞), known as Obata's Theorem was first proved by Ketterer [21]. The rigidity in the Lichnerowitz spectral gap for RCD(K, ∞) space, K > 0, was recently proved by Gigli-Ketterer-Kuwada-Ohta [18]. Local Poincaré inequalities in the framework of CD(K, N ) and CD(K, ∞) spaces were proved by Rajala [29]. Finally various lower bounds, together with rigidity and almost rigidity statements for the Dirichlet first eigenvalue of the Laplacian, have been proved by Mondino-Semola [28] in the framework of CD and RCD spaces. Lower bounds on the Cheeger's isoperimetric constant have been obtained for (essentially non-branching) CD * (K, N ) spaces by Cavalletti-Mondino [11,12,13] and for RCD(K, ∞) spaces (K > 0) by Ambrosio-Mondino [2].

Acknowledgements
The work has been developed when the N. DP. was visiting the Mathematics Institute at the University of Warwick during fall term 2018. He wishes to thank the Institute for the excellent working conditions and the stimulating atmosphere. A.M. is supported by the EPSRC First Grant EP/R004730/1 "Optimal transport and Geometric Analysis" and by the ERC Starting Grant 802689 "CURVATURE".

Preliminaries
Throughout the paper, (X, d) is a complete and separable metric space and B is the completion of the Borel σ-algebra of (X, d). We endow (X, d) with a reference σ-finite nonnegative measure m over (M, B), with supp(m) = X and satisfying an exponential growth condition: namely that there exist x 0 ∈ X, M > 0 and c ≥ 0 such that We denote by P 2 (X) the space of probability measures on X with finite second moment and we endow this space with the Kantorovich-Wasserstein distance W 2 defined as follows: for µ 0 , µ 1 ∈ P 2 (X) we set where the infimum is taken over all π ∈ P(X × X) with µ 0 and µ 1 as the first and the second marginal.
The relative entropy functional Ent m : In the sequel we use the notation: We now define the CD(K, ∞) condition, coming from the seminal works of Lott-Villani [25] and Sturm [32].
We denote by Lip(X) the space of real-valued Lipschitz functions over X and we write with the convention |∇f |(x) = 0 if x is an isolated point. The Cheeger energy (introduced in [14] and further studied in [4]) is defined as the L 2lower semicontinuous envelope of the functional f → 1 2 X |∇f | 2 dm, i.e.: Ch m (f ) := inf lim inf If Ch m (f ) < ∞, it was proved in [14,4] that the set is closed and convex, therefore it admits a unique element of minimal norm called minimal weak upper gradient and denoted by |Df | w . The Cheeger energy can be then represented by integration as Ch is a dense linear subspace of L 2 (X, m). It then admits an L 2 gradient flow which is a continuous semi-group of contractions (H t ) t≥0 in L 2 (X, m), whose continuous trajectories t → H t f , for f ∈ L 2 (X, m), are locally Lipschitz curves from (0, ∞) with values into L 2 (X, m). We now define the RCD(K, ∞) condition, introduced and throughly analyzed in [5] (see also [3] for the present simplified axiomatization and the extension to the σ-finite case).
We say that the metric measure space (X, d, m) is RCD(K, ∞) if it satisfies the CD(K, ∞) condition and moreover the Cheeger energy Ch m is quadratic, i.e. it satisfies the parallelogram identity If (X, d, m) is an RCD(K, ∞) space, then the Cheeger energy induces the Dirichlet form E(f ) := 2Ch m (f ) which is strongly local, symmetric and admits the Carré du Champ Hilbert. Moreover, the sub-differential ∂Ch m is single-valued and coincides with the linear generator ∆ of the heat flow semi-group (H t ) t≥0 defined above. In other terms, the semigroup can be equivalently characterized by the fact that for any where the limit in the strong L 2 (X, m)-topology.
The semigroup H t extends uniquely to a strongly continuous semigroup of linear contractions in L p (X, m), p ∈ [1, ∞), for which we retain the same notation. Regarding the case p = ∞, it was proved in [5, Theorem 6.1] that there exists a version of the semigroup such that H t f (x) belongs to C ∩ L ∞ ((0, ∞) × X) whenever f ∈ L ∞ (X, m). We will implicitly refer to this version of H t f when f is essentially bounded. Moreover, for any f ∈ L 2 ∩ L ∞ (X, m) and for every t > 0 we have H t f ∈ V ∩ Lip(X) with the explicit bound (see [5,Theorem 6.5] for a proof) Two crucial properties of the heat flow are the preservation of mass and the maximum principle (see [4]): 0 ≤ H t f ≤ C, for any 0 ≤ f ≤ C m-a.e., C > 0.
A result of Savaré [30,Corollary 3.5] ensures that, in the RCD(K, ∞) setting, for every f ∈ V and α ∈ [ In particular, We will suppose that (X, d, m) admits a compact embedding of V in L 2 (X, m). By spectral theory (see [16] for a general reference and [19] for some results in the RCD setting), this is equivalent to ask that −∆ has discrete spectrum consisting of an increasing sequence of non-negative eigenvalues {λ n } ∞ n=0 such that lim n→∞ λ n → +∞. In particular, the first positive eigenvalue of −∆ is well defined and non-negative.
Given K ∈ R, we define the function j K : (0, ∞) → (0, ∞) as Notice that j K is increasing as a function of K. The next proposition was proved in the smooth setting by Bakry, Gentil and Ledoux (see [8], [6] and [7, Proposition 8.6.1]).
We notice that φ ε (0) = 0 and φ ε (x) ≥ 0 for every x ∈ [0, 1−η]. Moreover, using the property (24), φ ε is Lipschitz in the range of H t−s f . Since t → H t f is a locally Lipschitz map with values in L p (X, m) for 1 < p < ∞ ([31, Theorem 1, Section III], we have that Ψ ε is a locally Lipschitz map with values in L 1 (X, m). Let ψ ∈ L 1 ∩ L ∞ (X, m) be a non-negative function. By the chain rule for locally Lipschitz maps, the fundamental theorem of calculus for the Bochner integral and the properties of the semigroup H t we have that for any ε > 0 it holds Applying the Cauchy-Schwarz inequality and the identity I(x)I ′′ (x) = −1, for all x ∈ (0, 1), we get that the right hand side of (33) is bounded below by The expression (34) is non-negative and dominated by which is uniformly integrable for any fixed δ > 0, thanks to the bound (22). Since I is continuous, I(0) = 0 and I(x) > 0 for every x ∈ (0, 1), using the locality property (see [5,

equation 2.18])
|DH t−s f | w = 0 m-a.e. on the set {H t−s f = 0}, the Dominated Convergence Theorem yields for every δ ∈ (0, t/2). Now, we can bound the right hand side of (35) using the inequality (26) in order to obtain Applying the continuity of the function I and the continuity of the semigroup we can now pass to the limit as δ ↓ 0 and obtain for every η > 0 sufficiently small, every f ∈ L 2 (X, m), f : , consider the truncation f η := min(f, 1−η). Applying (36) to f η , we have From we can then pass to the limit as η ↓ 0 in (37) and obtain Since ψ ∈ L 1 ∩ L ∞ (X, m), ψ ≥ 0 is arbitrary, the desired estimate (29) follows. Recalling that 0 ≤ I ≤ 1 √ 2π , the inequality (29) yields , m-a.e., for every t > 0, for any f ∈ L 2 (X, m), f : We next recall the definition of the first non-trivial eigenvalue of the laplacian −∆. First of all, if m(X) < ∞, the non-zero constant functions are in L 2 (X, m) and are eigenfunctions of −∆ with eigenvalue 0. In this case, the first non-trivial eigenvalue is given by λ 1 When m(X) = ∞, 0 may not be an eigenvalue of −∆ and the first eigenvalue is characterized by Note that λ 0 may be zero (for instance if m(X) < ∞ or (X, d, m) is the Euclidean space R d with the Lebesgue measure) but there are examples when λ 0 > 0: for instance in the Hyperbolic plane λ 0 = 1/4 and more generally on an n-dimensional simply-connected Riemannian manifold with sectional curvatures bounded above by k < 0 it holds λ 0 ≥ (n − 1) 2 k/4 (see [27]). Observe that, by the very definition of Cheeger energy (19), the definition (2) of λ 1 (resp. (3) of λ 0 ) given in the Introduction in terms of slope of Lipschitz functions, is equivalent to (39) (resp. (40)). It is also convenient to set where j K was defined in (28).
The inequality is sharp for K > 0, as equality is achieved for the Gaussian space. Proof.
First of all, we claim that for any f ∈ L 2 (X, m) with zero mean it holds To prove (44) let 0 ≡ f ∈ L 2 (X, m) such that 0 = X f dm = X H t f dm. Then and the Gronwall's inequality yields (44). Next we claim that, by duality, the bound (30) implies where J K (t) was defined in (41).
To prove (46) we take a function g, g ∞ ≤ 1, and observe that Since g is arbitrary, the claimed (46) follows from the last estimate combined with (30). We now combine the above claims in order to conclude the proof. Let A ⊂ X be a Borel subset and let f n ∈ Lip b (X) be a recovery sequence for the perimeter of the set A, i.e.: Inequality (46) passes to the limit since H t is continuous in L 1 (X, m) [4, Theorem 4.16] and we can write , (47) where we used properties (23), (24), together with the semigroup property and the selfadjointness of the semigroup. Observing that X H t/2 (χ A − m(A)) dm = 0 thanks to (23) and applying (44), we can bound H t/2 (χ A ) 2 2 in the following way A direct computation gives χ A − m(A) 2 2 = m(A)(1 − m(A)), so that the combination of (47) and (48) yields Recalling that in the definition of the Cheeger constant h(X) one considers only Borel subsets A ⊂ X with m(A) ≤ 1/2, the last inequality (49) gives (42).
Step 2: Proof of (43), the case m(X) = ∞. Arguing as in (45) using Gronwall Lemma, for any f ∈ L 2 (X, m) it holds Note that in order to establish (47), the finiteness of m(X) played no role. Now we can directly use (50) to bound the right hand side of the equation (47) in order to achieve for any Borel subset A ⊂ X with m(A) < ∞. The estimate (43) follows.
Remark 3.3. It was proved in [19] that an RCD(K, ∞) space, with K > 0 (or with finite diameter) has discrete spectrum (as the Sobolev imbedding V into L 2 is compact). Even in case of infinite measure the embedding of V in L 2 may be compact. An example is given by R with the Euclidean distance d(x, y) = |x − y| and the measure m := 1 √ 2π e x 2 /2 dL 1 . It is a RCD(−1, ∞) space and a result of Wang [34] ensures that the spectrum is discrete.
3.1. From the implicit to explicit bounds and sharpness in case K > 0. Proof of Corollary 1.2. In this section we show how to derive explicit bounds of λ 1 (resp. λ 0 ) in term of the Cheeger constant h, starting from (42) (resp. (43)). We also show that (42) is sharp, since equality is achieved on the Gaussian space. First of all, the expression of the function J K defined in (41) can be explicitly computed as: Case K = 0. When K = 0, the estimate (42) combined with (51) gives where we set T = λ 1 t in the last identity.
A good lower estimate of M is given by 2/π. Using this bound, we obtain Case K > 0. We start with the following where T > 0 is a fixed number. Then f 1 is an increasing function and We put y := √ e T x − 1 so that we have to prove y arctan(y) − log(y 2 + 1) ≥ 0, y > 0.
Called g 1 (y) the function g 1 (y) := y arctan(y) − log(y 2 + 1), we have that g 1 (0) = 0 and g ′ 1 (y) = arctan(y) − y 1 + y 2 ≥ 0, so that the inequality (55) is proved and f 1 is increasing for any T > 0. The proof is finished since Rewriting the estimate (42) using (51) in case K > 0, we obtain Thanks to the Lemma 3.4 it is clear that we can always obtain the same lower bound of the case K = 0 (as expected), but this can be improved as soon as we have a positive lower bound of the quotient K/λ 1 . Indeed, let us suppose K/λ 1 ≥ c > 0. Then, observing that from (56), we obtain When X = R d endowed with the Euclidean distance d(x, y) = |x − y| and the Gaussian we have that h = 2 π , K = 1 and λ 1 = 1 (see [7,Section 4.1]). Thus, we can take c = 1 and the equality in (57) is achieved, making sharp the lower bound.
Case K < 0. We begin by noticing that The following lemma holds: where T > 0 is a fixed number. Then f 2 is a decreasing function.
The combination of (42), (51) and (58) implies that if (X, d, m) is an RCD(K, ∞) space with K < 0 and m(X) = 1 then In case (X, d, m) is an RCD(K, ∞) space with K < 0 and m(X) = ∞ then, using (43) instead of (42), the estimate (62) holds with λ 1 replaced by λ 0 and h(X) replaced by h(X)/2. We make two different choices: where we used the inequality Applying now Lemma 3.5, we obtain The combination of (63) and (64) gives that, if (X, d, m) is an RCD(K, ∞) space with K < 0 and m(X) = 1 h(X), By the same arguments, using (43) instead of (42), one gets that for (X, d, m) an RCD(K, ∞) space with K < 0 and m(X) = ∞ it holds Remark 3.6. Another bound, like in the case K > 0, can be obtained in the presence of a lower bound for K/λ 1 , if m(X) = 1 (resp. a lower bound for K/λ 0 , if m(X) = ∞). To see this, let us suppose K/λ 1 ≥ −c, c > 0 (resp. K/λ 0 ≥ −c). Then, using (42) (resp. (43)), (51) and Lemma 3.5, we have that (resp. the left hand side can be improved to h/ √ 2π) 4. Appendix A: Cheeger's inequality in general metric measure spaces The Buser-type inequalities of Theorem 1.1 and Corollary 1.2 give an upper bound on λ 1 (resp. on λ 0 , in case m(X) = ∞) in terms of the Cheeger constant h(X). It is natural to ask if also a reverse inequality holds, namely if it possible to give a lower bound on λ 1 (resp. on λ 0 , in case m(X) = ∞) in terms of h(X). The answer is affirmative in the higher generality of metric measure spaces with a non-negative locally bounded measure without curvature conditions, see Theorem 4.2 below. This generalizes to the metric measure setting a celebrated result by Cheeger [15], known as Cheeger's inequality.
A key tool in the proof of the Cheeger's inequality is the co-area formula; more precisely, in the arguments it is enough to have an inequality in the co-area formula. For the reader's convenience, we give below the statement and a self-contained proof. (68) Proof. The proof is quite standard, but since we did not find it in the literature stated at this level of generality (tipically one assumes some extra condition like measure doubling and gets a stronger statement, namely equality in the co-area formula; see for instance [24]) we add it for the reader's convenience. Let E t := {u > t} and set V (t) := Et |∇u| dm. The function t → V (t) is non-increasing and bounded, thus differentiable for L 1 -a.e. t > 0. Since X u dm < ∞, we also have that m({u = t}) = 0 for L 1 -a.e. t > 0.
For h > 0 define u h (x) = ψ(h, u(x)) and observe that the sequence (u h ) h ⊂ Lip b (X).
We first claim that u h → χ E(t) in L 1 (X, m) as h ↓ 0.
by Dominated Convergence Theorem, since by assumption u has bounded support, m is finite on bounded sets and χ {t−h<u≤t} → χ {u=t} point-wise as h ↓ 0.
In order to prove that E t is a set of finite perimeter it is then sufficient to show that lim sup h↓0 X |∇u h | dm < ∞. To this aim observe that Since by assumption t > 0 is a differentiability point for V , we obtain that E t is a finite perimeter set satisfying Using that (71) holds for L 1 -a.e. t > 0 and that V is non-increasing, we get Theorem 4.2 (Cheeger's Inequality in metric measure spaces). Let (X, d) be a complete metric space and let m be a non-negative Borel measure finite on bounded subsets.
(1) If m(X) < ∞ then (2) If m(X) = ∞ then As proved by Buser [9], the constant 1/4 in (73) is optimal in the following sense: for any h > 0 and ε > 0, there exists a closed (i.e. compact without boundary) two-dimensional Riemannian manifold (M, g) with h(M ) = h and such that λ 1 ≤ 1 4 h(M ) 2 + ε. Proof. We give a proof of (73), the arguments for showing (74) being analogous: the only difference is that, in case m(X) = ∞, f in the arguments below is taken 0 ≡ f ∈ Lip b (X), without the extra condition X f dm = 0. Indeed the condition X f dm = 0 in case m(X) < ∞ is necessary in order to rule out constant functions. By the very definition of λ 1 as in (2), for every ε > 0 there exists f ∈ Lip b (X) with Since |∇f 2 | ≤ 2|f | |∇f |, by Cauchy-Schwartz inequality it holds X |∇f 2 | dm ≤ 2 Plugging (76) into (75), gives Applying the co-area inequality (68) to u = f 2 and recalling the definition of Cheeger's constant h(X) as in (4), we obtain Plugging (78) into (77) yields Since ε > 0 is arbitrary, the claim (73) follows.