Bakry-\'Emery conditions on almost smooth metric measure spaces

In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-\'Emery condition $BE (K, N)$. The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the $BE$ condition is strictly weaker than the $RCD$ condition even in this setting, and that the local dimension is not constant even if the space satisfies the $BE$ condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-\'Emery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be $RCD(K, N)$ spaces.


Introduction
Let (X, d, m) be a compact metric measure space, that is, (X, d) is a compact metric space with supp m = X and m(X) < ∞. There are several de nitions of 'lower Ricci bounds on (X, d, m)', whose studies are very quickly, widely developed now. We refer to, [36] by Lott-Villani, [38] by Sturm,and [2] by Ambrosio-Gigli-Savaré, as their pioneer works.
In this paper, we forcus on two of them. One of them is the Bakry-Émery ( BE ) condition [12] by Bakry-Émery, denoted by BE(K, N), the other is the Riemannian curvature dimension ( RCD ) condition [2] by Ambrosio-Gigli-Savaré (in the case when N = ∞), [26] by Gigli (in the case when N < ∞), denoted by RCD(K, N). Both notions give us meanings that the Ricci curvature of (X, d, m) is bounded below by K, and the dimension of (X, d, m) is bounded above by N in synthetic sense. The BE(K, N) condition is roughly stated by: holds in a weak form for all 'nice' functions f on X (De nition 2.1). It is known that if (X, d) is a n-dimensional smooth Riemannian manifold (M n , g) and m is the Riemannian (or equivalently, the Hausdor ) measure, then, the BE(K, N) condition (1.1) is equivalent to satisfying n ≤ N and Ric g M n ≥ K, and that the BE(K, N) condition is also equivalent to some gradient estimates on the heat ow, so called Bakry-Émery/Bakry-Ledoux gradient estimates.
In general, the implication from RCD(K, N) to BE(K, N) is always satis ed. The converse is true under adding a some property, so-called the 'Sobolev to Lipschitz property', which is introduced in [26] (De nition 2.2). This property (with BE), in a point of view in geometric analysis, plays a role to get the coincidence between the analytic distance d Ch (that is, the induced distance by the Cheeger energy) and the original (ge-ometric) distance d. Moreover, the RCD condition also implies the Sobolev to Lipschitz property. Thus, the following equivalence is known: RCD(K, N) ⇐⇒ BE(K, N) + 'Sobolev to Lipschitz property . (1. 2) The RHS of (1.2) is also called the metric BE(K, N) condition. Thus, to keep the short presentation, we adopt the RHS of (1.2) as the de nition of RCD(K, N) condition in this paper (De nition 2.2). We refer to, [3] by Ambrosio-Gigli-Savaré, [9], [7] by Ambrosio-Mondino-Savaré, and [25] by Erbar-Kuwada-Sturm for the details.
In these observation, more precisely, 'RCD' should be replaced by 'RCD * '. However, since the equivalence between RCD and RCD * spaces is also recently established in [18] by Cavalletti-Milman, we use the notation 'RCD' only for simplicity.
In this paper, we discuss the condition: One of the goals in this paper is to provide an example satisfying (1.3), but it is not an RCD space. More precisely, for any two (not necessary same dimensional) closed pointed Riemannian manifolds (M m i i , g i , p i )(m i ≥ ), the glued metric space M m * M m at their base points with the standard measure is a BE(K, max{m , m }) space, where K := min{inf Ric g M m , inf Ric g M m } (Example 3.18). It is easy to check that this metric measure space does not satisfy the Sobolev to Lipschitz property, thus, it is not a RCD(L, ∞) space for any L ∈ R.
This tells us that (1.3) does not imply the expected Bishop-Gromov inequality (Remark 3.9), and that (1.3) does not imply the constancy of the local dimension. In particular, the glued space gives a rst example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We point out a very recent result in [16] by Bruè-Semola, which states that for any RCD(K, N) space, there exists a unique k such that the k-dimensional regular set R k has positive measure. This generalizes a result of Colding-Naber in [23] for Ricci limit spaces to RCD spaces. Thus, we know that the Sobolev-Lipschitz property is crucial to get the constant dimensional property. Note that in [34], Ketterer-Rajala constructed a metric measure space with the measure contraction property (MCP), which also characterize 'Ricci bounds from below' in a synthetic sense, but the local dimension is not constant. Therefore, in general, MCP and BE spaces are very diferent from RCD spaces.
Moreover, we should pay attention to a similar su cient condition in [7] by Ambrosio-Mondino-Savaré, so-called the local to global property, which states in our compact setting; if (X, d) is a geodesic (or equivalently, length) space and there exists an open covering {U i } i∈I of X such that U i ≠ ∅ and that (U i , d, m U i ) satis es the metric BE(K, N) condition, then, (X, d, m) satis es the metric BE(K, N) condition. In fact, the glued example shows that the openness of U i is essential because although U i := M m i i in M m * M m satis es the assumptions except for their openness properties, but the glued space does not satisfy the metric BE(K, N) condition for all K, N.
In order to justify these, we study almost smooth compact metric measure spaces. See De nition 3.1 for the de nition, which allows us such spaces to have at least the codimension singularities. Thus, compact (Riemannian) orbifolds with the Hausdor measure are typical examples of them. Then, the main result in this paper is roughly stated as follows; if an almost smooth compact metric measure space satis es the Lstrong compactness condition and satis es the gradient estimates on the eigenfunctions, then, a lower bound of the Ricci tensor of the smooth part implies a BE condition (Theorem 3.7). By using this, we can give a necessary and su cient condition for such a space to be a RCD space (Corollary 3.10).
The organization of the paper is as follows.
In section , to keep the short presentation, we give a very quick introduction to calculus on metric measure spaces.
In section , we study our main targets, almost smooth metric measure spaces, and prove the main results.

BE and RCD spaces
We use the notation Br(x) for open balls and Br(x) for {y : d(x, y) ≤ r}. We also use the standard notation LIP(X, d), LIPc(X, d) for the spaces of Lipschitz, compactly supported Lipschitz functions, respectively.
Let us now recall basic facts about Sobolev spaces in metric measure spaces (X, d, m), see [1], [27] and [28] for a more systematic treatment of this topic. We shall always assume that • the metric space (X, d) is compact with supp m = X and m(X) < ∞ for simplicity. The Cheeger energy Ch = Ch d,m : L (X, m) → [ , +∞] is a convex and L (X, m)-lower semicontinuous functional de ned as follows: where Lipf is the so-called slope, or local Lipschitz constant. The Sobolev space H , (X, d, m) then concides with {f : Ch(f ) < +∞}. When endowed with the norm is doubling (see [8]), and separable Hilbert if Ch is a quadratic form (see [2]). According to the terminology introduced in [27], we say that a metric measure space (X, d, m) is in nitesimally Hilbertian if Ch is a quadratic form. By looking at minimal relaxed slopes and by a polarization procedure, one can then de ne a carré du champ playing in this abstract theory the role of the scalar product between gradients (more precisely, the duality between di erentials and gradients, see [27]). In in nitesimally Hilbertian metric measure spaces, the Γ operator satis es all natural symmetry, bilinearity, locality and chain rule properties, and provides integral representation to Ch: We can now de ne a densely de ned operator ∆ : D(∆) → L (X, m) whose domain consists of all functions f ∈ H , (X, d, m) satisfying for some g ∈ L (X, m). The unique g with this property is then denoted by ∆f (see [1]). From the point of view of Riemannian geometry, we will also adopt the following notation instead of Γ; We are now in a position to introduce the BE(K, N) condition (see [9], [7] and [25]): De nition 2.1 (BE spaces). Let (X, d, m) be a compact metric measure space, let K ∈ R and let N ∈ [ , ∞]. We say that ( holds in the weak form, that is, for all φ ∈ D(∆) ∩ L ∞ (X, m) with φ ≥ and ∆φ ∈ L ∞ (X, m).
In order to introduce the class of RCD(K, N) metric measure spaces, we follow the Γ-calculus point of view, based on Bochner's inequality, because this is the point of view more relevant in our proofs. However, the equivalence with the Lagrangian point of view, based on the theory of optimal transport rst proved in [3] (in the case N = ∞) and then in [25], [9] (in the case N < ∞). Moreover, the following de nition should be written as RCD * (K, N) spaces. However, since it is known by [18] that these are equivalent notions, we use the notation RCD(K, N) only for simplicity.
De nition 2.2 (RCD spaces). Let (X, d, m) be a compact metric measure space, let K ∈ R and let N ∈ [ , ∞]. We say that (X, d, m) is a RCD(K, N) space if it is a BE(K, N) space with the Sobolev to Lipschitz property, that is, We end this section by giving the de nition of local Sobolev spaces: In the next section, the local Sobolev spaces will play a role to localize global Sobolev functions to smooth parts via the zero capacity condition.

Almost smooth metric measure space
Let us x a compact metric measure space (X, d, m).

. Constant dimensional case
De nition 3.1 (n-dimensional almost smooth compact metric measure space). Let n ∈ N. We say that (X, d, m) is an n-dimensional almost smooth compact metric measure space associated with an open subset Ω of X if the following three conditions are satis ed;

(Smoothness of Ω)
there exist an n-dimensional (possibly incomplete) Riemannian manifold (M n , g) and a map φ : Ω → M n such that φ is a local isometry between (Ω, d) and (M n , dg), that is, for all p ∈ Ω there exists an open neighborhood U ⊂ Ω of p such that φ| U is an isometry from U to φ(U) as metric spaces; 2. (Hausdor measure condition) The restricition m Ω of m to Ω coincides with the n-dimensional Hausdor measure H n on Ω, that is, m(A) = H n (A) holds for all Borel subset A of Ω; 3. (Zero capacity condition) X \ Ω has zero capacity in the following sense, that is, m(X \ Ω) = is satis ed, there exists a sequence φ i ∈ C ∞ c (Ω) such that the following two conditions hold; (a) for any compact subset A ⊂ Ω, φ i | A ≡ holds for all su ciently large i; The zero capacity condition is a kind of that 'H , -capacity of X \ Ω is zero' whose standard de nition is given by replacing (3.1) by See [35]. In particular, (3.2) is satis ed if ∆φ i L → . Compare with (2) of Proposition 3.3.
Remark 3.2. Whenever we discuss 'analysis/geometry on Ω locally', we can identify (Ω, d) with the smooth Riemannian manifold (M n , g) (thus, sometimes, we will use the notations (Ω, g), Ric g Ω and so on). Note that for all p ∈ M n and all su ciently small r > , B g r (p) is convex and it has a uniform lower bound on Ricci curvature. In particular, the volume doubling condition and the Poincaré inequality hold locally. Thus, Cheeger's theory [19] can be applied locally. In particular, the Lipschitz-Lusin property holds for all f ∈ H , (X, d, m) (this notion is equivalent to that of di erentiability of functions introduced in [31]), that is, for all ϵ > , there exists a Borel subset A of Ω such that m(Ω \ A) < ϵ and that f | A is Lipschitz. Combining this with the locality property of the slope on both theories in [1], in [19], yields where the RHS means the minimal weak upper gradient in [19]. Let us give a quick proof of (3.3) for reader's convenience. By the Lipschitz-Lusin property with the localities of slopes as mentioned above, it su ces to check that under assuming f ∈ LIP(X, d), the LHS of (3.3) is equal to Lipf for m-a.e. x ∈ X. Moreover, since it follows from [1] that |∇f |(x) ≤ Lipf (x) m-a.e. x ∈ X, let us check the converse inequality.
Let x ∈ Ω and x any su ciently small r > as above. Note that by [19] On the other hand, by [2], there exists a sequence Since r is arbitrary, we have the converse inequality, |∇f |(x) ≥ Lipf (x) m-a.e. x ∈ X, which completes the proof. Similarly, the Sobolev space H , (M n , g, H n ), which is de ned by the standard way in Riemannian geometry (that is, the H , -closure of C ∞ c (M n )), coincides with H , (Ω, d, m). We will immediately use these compatibilities below.
From now on, we use the same notation as in De nition 3.1 (e.g. Ω, φ i ) without any attention.
we have (1). Next, let us check (2). It is trivial that the map ι preserves the distances (we identify H , (Ω, d, m) with the image by ι for simplicity). As written in Remark 3.2, it also follows from the smoothness of Ω that H , (Ω, d, m) is a Hilbert space, and that φ i f ∈ H , (Ω, d, m) for all f ∈ LIP(X, d).
Fix f ∈ LIP(X, d). Then, since In particular, m), respectively and that Ch(F i ), Ch(G i ) → Ch(F), Ch(G), respectively. Then, letting i → ∞ in the equality (3.5) for φ = F i , ψ = G i with the lower semicontinuity of the Cheeger energy shows (3.6) Replacing F, G by F + G, F − G, respectively yields the converse inequality, that is, we have the equality in (3.6) for φ = F, ψ = G, which proves that H , (X, d, m) is a Hilbert space. Thus, by [2], LIP(X, d) is dense in H , (X, d, m). Since we already proved that LIP(X, d) ⊂ H , (Ω, d, m), we conclude.
Remark 3.4. Recall that if u i L -weakly converge to u in L (X, m) with sup i u i H , < ∞, then, we see that u ∈ H , (X, d, m) and that ∇u i L -weakly converge to ∇u. Although this statement was already proved in general setting (e.g. [10] and [28]. See also [6] and [32]), for reader's convenience, let us give a proof as follows.
Mazur's lemma yields the rst statement, u ∈ H , (X, d, m). To get the second one, since sup i ∇u i L < ∞, it is enough to check that Then, It is well-known that there are several su cient conditions to satisfy the L -strong compactness condition, for instance, PI-condition (i.e. the volume doubling and the Poincaré inequality are satis ed), which follows from RCD(K, N)-conditions for N < ∞ (see for instance [29] for the proof of the L -strong compactness condition). However, in general, for an n-dimensional almost smooth compact metric measure space, the L -strong compactness condition is not satis ed even if Ω has a uniform lower Ricci bound. To see this, for any two pointed metric spaces (X i , d i , x i )(i = , ), let us denote by (X , d , x ) * (X , d , x ) their glued pointed metric space as x = x , that is, the metric space is with the intrinsic metric, and the base point is the glued point. See [17] for the detail. Sometimes, we denote the metric space by (X * X , d) without any attention on the base points for simplicity.
Then, let us denote by (X, d, x) the pointed Gromov-Hausdor limit space of (X i , d i , x i ). Note that (X, d) is compact, that Ω := X \ {x} satis es the smoothness with Ric g Ω ≥ , and that there exist canonical isometric embeddings T n i → X (we identify T n i with the image). Then, we consider the n-dimensional Hausdor measure H n as the reference measure m on X.
Then, since it is easy to see that for some universal constant C > we see that for all j ≤ i where C is also a universal constant. In particular, which proves the zero capacity condition. Thus, (X, d, H n ) is an n-dimensional almost smooth compact metric measure space. Let us de ne a sequence f i ∈ L (X, H n ) by Then, it is easy to see that f i L -weakly converge to and that f i ∈ H , (X, d, m) with f i L = f i H , = (see also Example 3.8). Since f i does not L -strongly converge to , the L -strong compactness condition does not hold.
It follows from standard arguments in functional analysis that if an in nitesimally Hilbertian compact metric measure space (Y , d, ν) satis es the L -strong compactness condition with dim L (Y , ν) = ∞, then, the spectrum of −∆ is discrete and unbounded (each eigenvalue has nite multiplicities). Thus, we then denote the eigenvalues by counted with multiplicities, and denote the corresponding eigenfunctions by φ Y i with φ Y i L = . We always x an L -orthogonal basis {φ Y i } i consisting of eigenfunctions, immediately. Moreover, it also holds that for all f ∈ L (Y , ν), and that for all f ∈ H , (Y , d, ν), For reader's convenience, we will give proofs of them in the appendix.
We are now in a position to give the main result: Theorem 3.7 (From Ric g Ω ≥ K(n − ) to BE(K(n − ), n)). Let (X, d, m) be an n-dimensional almost smooth compact metric measure space. Assume that (X, d, m) satis es the L -strong compactness condition, that each eigenfunction φ X i satis es |∇φ X i | ∈ L ∞ (X, m) and that Ric g Ω ≥ K(n − ) for some K ∈ R. Then, (X, d, m) satis es the BE(K(n − ), n)-condition.
Proof. Let us use the same notation as above, that is, let f N := N i a i φ X i , where a i := X fφ X i dm. Note that by (3.9) (X, d, m), respectively as N → ∞. In the following, for all h ∈ C ∞ (Ω), the Laplacian tr(Hess h ) de ned in Riemannian geometry is also denoted by the same notation ∆h, without any attention because is satis ed and (3.10) characterizes the function tr(Hess h ) in L loc (Ω, H n ). Fix N ∈ N. Then, let us prove that |∇f N | ∈ H , (X, d, m) as follows. By our assumption on the eigenfunctions, we see that |∇f N | ∈ L ∞ (X, m). Moreover, the elliptic regularity theorem shows that f N | Ω ∈ C ∞ (Ω). Since Ric g Ω ≥ K(n − ), we have Thus, multiplying φ i on both sides and then integrating this over Ω show Since |∇f N | ∈ L ∞ (X, m) and our assumption on the zero capacity, the inequality (3.12) implies In particular, Thus, the monotone convergence theorem yields Ω |Hess f N | dH n < ∞. (3.13) On the other hand, since φ i ∈ C ∞ c (Ω) and f N | Ω ∈ C ∞ (Ω), we have φ i |∇f N | ∈ H , (X, d, m). Moreover, since We are now in a position to nish the proof. Let φ ∈ D(∆) ∩ L ∞ (X, m) with ∆φ ∈ L ∞ (X, m) and φ ≥ . Multiplying φφ i on both sides of (3.11) and integrating this over X show (3.14) Recall that φ i → in L (X, m) and that ∇φ i L -weakly converge to ∇ = with ∇(φφ i ) = φ i ∇φ + φ∇φ i . Thus, we have where we used ∇|∇f N | L ≤ Hess f N L ∇f N L ∞ < ∞ and |∇f N | ∈ H , (X, d, m). Moreover, the dominated convergence theorem yields Thus, combining these with letting i → ∞ in (3.14) shows which completes the proof.
Let us apply Theorem 3.7 to an explicit simple example as follows. S m ( ), respectively, the images of the canonical isometric embeddings S n ( ) → X to the rst sphere and the second one, respectively. Moreover, we denote by p the intersection point of them. It is worth pointing out that (X, d, m) satis es the Ahlfors n-regularity, which is easily checked.
Being an n-dimensional almost smooth compact metric measure space. Let Ω := X \ {p}. Then, it is trivial that Ω satis es the smoothness with Ric g Ω ≥ (n − ) and H n (X \ Ω) = . Let us use ψϵ as in Example 3.6. Then, by an argument similar to that in Example 3.6, it is easy to check that the functions φ i (x) := ψ i − (d(p, x)) satis es the zero capacity condition. Thus, (X, d, m) is an n-dimensional almost smooth compact metric measure space.
Satisfying the L -strong compactness condition. We remark that we see that f | S n j ( ) is an eigenfunction of (S n j ( ), d, H n ). Thus, |∇(f | S n j ( ) )| ∈ L ∞ (S n j ( ), H n ), which implies |∇f | ∈ L ∞ (X, m).
Therefore, we can apply Theorem 3.7 to show that (X, d, m) satis es the BE(n − , n)-condition. Coincidence between the induced distance d Ch by the Cheeger energy and d. Let us prove: Let x ∈ S n ( ) and let y ∈ S n ( ). For any φ as in the RHS of (3.19), where we used the fact that d = d Ch in (S n j ( ), d S n j ( ) , H n ). Thus, taking the supremum in (3.20) with respect to φ shows the inequality '≤' in (3.18).
Then, we see that φ ∈ LIP(X, d), that Lipφ(z) ≤ for all z ∈ X, and that φ(x) − φ(y) = d(x, y), which proves the converse inequality '≥' in (3.18). Similarly, we can prove (3.18) in the remaining case, thus, we have (3.18) for all x, y ∈ X. Poincaré inequality and RCD(K, ∞) condition are not satis ed. Assume that (X, d, m) satis es the ( , )-Poincaré inequality, that is, there exists C > such that for all r > , all x ∈ X and all f ∈ H , (X, d, m), it holds that m(Br(x)) (3.21) Let φ := S n ( ) − S n ( ) . By (3.16), we have φ ∈ H , (X, d, m). Then, by the locality of the slope, we have |∇φ| = m-a.e.. In particular, (3.21) yields that φ must be a constant, which is a contradiction. By the same reason, for all K ∈ R, (X, d, m) does not satisfy RCD(K, ∞)-condition. where the in mum runs over all minimal geodesics γ from x to y. Then, we say that (Y , d, ν) satis es the segment inequality if there exists λ > such that Cheeger-Colding proved in [21] that if (Y , d, ν) satis es the volume doubling condition and the segment inequality, then, the ( , )-Poincaré inequality holds (see also [20] and [30] Proof. Let f ∈ H , (Y , d, ν) with |∇f | ≤ ν-a.e.. As written above, since ( , )-Poincaré inequality is satis ed, f has a representative inf ∈ LIP(Y , d) by the telescope argument (see for instance [19]). Thus, since we have |Lipf | ≤ ν-a.e., which also follows from [19], it su ces to check thatf is -Lipschitz. Let us take a Borel subset A of Y such that ν(Y \ A) = and that Lipf ≤ , ∀x ∈ A.
Applying the segment inequality for Y\A yields that there exists a Borel subset Since ϵ is arbitrary and B is dense in Y × Y, (3.23) yields thatf is -Lipschitz.
Remark 3.15. Let us give remarks on related works. Note that in the following, the spaces are not necessary compact. As we already used, Jiang proved in [33] the gradient estimates on solutions of Poisson's equations (including eigenfunctions) in the setting of metric measure spaces under assuming mild geometric conditions and a heat semigroup curvature condition (or called an weighted Sobolev inequality). Bamler and Chen-Wang proved in [14], in [22], such conditions (including the segment inequality) in their almost smooth settings, independently.
One of interesting questions is; if an n-dimensional almost smooth (compact) metric measure space (X, d, m) satis es that the induced distance dg by g on Ω coincides with d| Ω , then, being Ric g Ω ≥ K(n − ) is equivalent to that (X, d, m) is a RCD(K(n − ), n)-space?

. Nonconstant dimensional case
In this section, let us discuss a variant of n-dimensional almost smooth compact metric measure spaces. Let us recall that we x a compact metric measure space (X, d, m).
De nition 3.16 (Generalized almost smooth compact metric measure space). We say that (X, d, m) is a generalized almost smooth compact metric measure space associated with an open subset Ω of X if the following two conditions are satis ed; (Y , d, ν). Thus, Mazur's lemma shows that f ∈ H , (Y , d, ν) and that f i converge weaky to f in H , (Y , d, ν). Therefore, letting i → ∞ in which shows f ∈ E(λ), where the convergence of the left hand sides comes from the polarization. Thus, (E(λ), · L ) is a Hilbert space.
Then, similar argument with the L -strong compactness condition allows us to prove that S(λ) is a com- Proof. The proof is done by contradiction. Assume that there exists a sequence Then, since f i H , = λ i , by the Lstrong compactness condition, with no loss of generality, we can assume that there exists the L -strong limit function f of f i . Thus, f L = . Moreover, similar argument as in the proof of (2) of Lemma A.1 shows f ∈ E(λ). In particular, λ is an eigenvalue of −∆. Let {g j } j= , ,...,N be an ONB of E(λ).
Assume that E(Y , d, ν) is bounded. Then, Lemma A. On the other hand, it is easy to see that the number λ * := inf f ⊥V Y |∇f | dν Y |f | dν is also an eigenvalue of −∆ and that there exists a minimizer f * of the right hand side with f * ∈ E(λ * ) and f * L = , where we used our assumption, dim L (Y , ν) = ∞, to make sence in the in mum. Thus, since f * ∈ V and f * ⊥ V, we have f * = , which contradicts that f * L = .
Proof. We rst assume that f ∈ H , (Y , d, ν). For all N ∈ N, let f N := N i ( Y fφ Y i dν)φ Y i and let g N := f − f N . With no loss of generality, we can assume that g N ̸ ≡ for all N.
Then, since for all we have g N ⊥ V N , where V N := span {φ Y i } i= , ,...,N . On the other hand, it is easy to see that the number λ N+ := inf h⊥V N Y |∇h| dν Y |h| dν coincides with λ N+ (Y , d, ν) (the inequality λ N+ ≤ λ N+ (Y , d, ν) is trivial. The converse is done by checking that λ N+ is an eigenvalue of −∆, which is similar to the proof of Lemma A.3).
Next, let us check In particular, the Cauchy-Schwartz inequality yields ∇f N L ≤ ∇f L ∇f N L . Thus, since ∇f N L ≤ ∇f L , we have sup i f N H , < ∞. Since f N → f in L (Y , ν) as N → ∞, Mazur's lemma shows (3.9).