Weighted Sobolev Inequalities in CD(0,N) spaces

In this note, we prove global weighted Sobolev inequalities on non-compact CD(0,N) spaces satisfying a suitable growth condition, extending to possibly non-smooth and non-Riemannian structures a previous result by V. Minerbe stated for Riemannian manifolds with non-negative Ricci curvature. We use this result in the context of RCD(0,N) spaces to get a uniform bound of the corresponding weighted heat kernel via a weighted Nash inequality.


Introduction
Riemannian manifolds with non-negative Ricci curvature have strong analytic properties. Indeed, the doubling condition and the local L 2 -Poincaré inequality are satisfied on such spaces, and they imply many important results, like the well-known Li-Yau Gaussian estimates for a class of Green functions including the heat kernel [LY86] or powerful local Sobolev inequalities and parabolic Harnack inequalities (see e.g. [SC02]).
In the recent years, several classes of possibly non-smooth metric measure spaces containing the collection of Riemannian manifolds with non-negative Ricci curvature have been under investigation, both from a geometric and an analytic point of view. For instance, in the context of measure spaces endowed with a suitable Dirichlet form, Sturm proved existence and uniqueness of the fundamental solution of parabolic operators along with Gaussian estimates and parabolic Harnack inequalities [St95,St96], provided the doubling and Poincaré properties hold. Afterwards, general doubling spaces with Poincaré-type inequalities were studied at length by Hajlasz and Koskela [HK00] who proved local Sobolevtype inequalities, a Trudinger inequality, a Rellich-Kondrachov theorem, and many related results.
Approximately a decade ago, Sturm [St06] and Lott and Villani [LV09] independently proposed the curvature-dimension condition CD(0, N ), for N ∈ [1, +∞), as an extension of non-negativity of the Ricci curvature and bound from above by N of the dimension for possibly non-smooth metric measure spaces. Coupled with the infinitesimal Hilbertiannity introduced later on by Ambrosio, Gigli and Savaré [AGS14b] to rule out non-Riemannian structures, the CD(0, N ) condition leads to the stronger RCD(0, N ) condition, where R stands for Riemannian.
The classes of CD(0, N ) and RCD(0, N ) spaces have been extensively studied over the past few years, and it is by now well-known that they both contain the measured Gromov-Hausdorff closure of the class of Riemannian manifolds with non-negative Ricci curvature and dimension lower than N , as well as Alexandrov spaces with non-negative generalized sectional curvature and locally finite and non-zero n-dimensional Hausdorff measure, n being lower that N . Moreover, CD(0, N ) spaces satisfy the doubling and Poincaré properties, and RCD(0, N ) spaces are, in addition, endowed with a regular and strongly local Dirichlet form called Cheeger energy (see Section 2). Therefore, the works of Sturm [St95,St96] imply existence and uniqueness of an heat kernel, which by the way satisfies Gaussian estimates, on RCD(0, N ) spaces.
One of the interest of the CD(0, N ) and RCD(0, N ) conditions, and of the more general CD(K, N ) and RCD(K, N ) conditions for arbitrary K ∈ R, is the possibility of proving classical functional inequalities on spaces with rather loose structure thanks to optimal transport or gradient flow arguments. In this regard, Lott and Villani obtained in [LV07,Th. 5.29] a global Sobolev-type inequality for CD(K, N ) spaces with K > 0 and N ∈ (2, +∞). Later on, in their striking work [CM17, Th. 1.11], Cavaletti and Mondino proved a global Sobolev-type inequality with sharp constant for bounded essentially nonbranching CD * (K, N ) spaces with K ∈ R and N ∈ (1, +∞); in case K > 0 and N > 2, they get the classical Sobolev inequality with sharp constant. This last inequality had been previously justified on RCD * (K, N ) spaces with K > 0 and N > 2 by Profeta [Pr15].
The aim of this note is to provide a new related analytic result, namely a global weighted Sobolev inequality, for certain non-compact CD(0, N ) spaces with N > 2. It is worth underlying that our result does not require the Riemannian synthetic condition RCD(0, N ). Here and throughout the paper, if (X, d, m) is a metric measure space, we write B r (x) for the ball of radius r > 0 centered at x ∈ X, and V (x, r) for m(B r (x)).
Theorem 1 (Weighted Sobolev inequalities). Let (X, d, m) be a CD(0, N ) space with N > 1. Assume that there exists 1 < η ≤ N such that for some o ∈ X. Then for any 1 ≤ p < η, there exists a constant C > 0, depending only on N , η, Θ inf , Θ sup and p, such that for any continuous function u : X → R admitting an upper gradient g ∈ L p (X, m), and µ is the measure absolutely continuous with respect to m with density Theorem 1 extends a result by Minerbe stated for p = 2 on n-dimensional Riemannian manifolds with non-negative Ricci curvature [M09, Th. 0.1]. The motivation there was that the classical L 2 -Sobolev inequality does not hold on those manifolds which satisfy (1.1) with η < N = n, see [M09,Prop. 2.21]. This phenomenon also holds on some metric measure spaces including Finsler manifolds, see the forthcoming [T] for related results.
Our proof is an adaptation of Minerbe's proof to the setting of CD(0, N ) spaces and is based upon ideas of Grigor'yan and Saloff-Coste introduced in the smooth category [GS05] which extend easily to the setting of metric measure spaces. More precisely, we apply an abstract process (Theorem 2) which permits to patch local inequalities into a global one by means of an appropriate discrete Poincaré inequality. In the broader context of metric measure spaces with a global doubling condition, a local Poincaré inequality, and a reverse doubling condition weaker than (1.1), this method provides "adimensional" weighted Sobolev inequalities, as explained in the recent work [T20].
After that, we follow a classical approach (see e.g. [BCLS95]) which was neither considered in [M09] nor in the subsequent related work [He11] to deduce a weighted Nash inequality (Theorem 4) for CD(0, N ) spaces satisfying the growth assumption (1.1), provided η > 2. Let us mention that in the context of non-reversible Finsler manifolds, Ohta put forward an unweighted Nash inequality [Oh17] and that Bakry, Bolley, Gentil and Maheux introduced weighted Nash inequalities in the study of possibly non-ultracontractive Markov semigroups [BBGL12], but these inequalities seem presently unrelated to our.
We conclude this note with a natural consequence in the setting of RCD(0, N ) spaces satisfying a uniform local Ahlfors regularity property, namely a uniform bound for the weighted heat kernel associated with a suitable modification of the Cheeger energy. To the best knowledge of the author, this is the first appearance of this weighted heat kernel whose properties would require a deeper investigation.
The paper is organized as follows. In Section 2, we introduce the tools of non-smooth analysis that we shall use throughout the article. We also define the CD(0, N ) and RCD(0, N ) conditions, and present the aforementioned patching process. Section 3 is devoted to the proof of Theorem 1. Section 4 deals with the weighted Nash inequality and the uniform bound on the weighted heat kernel we mentioned earlier. The final Section 5 provides a non-trivial non-smooth space to which our main theorem applies.

Acknowledgments
I warmly thank T. Coulhon who gave the initial impetus to this work. I am also greatly indebted towards L. Ambrosio for many relevant remarks at different stages of the work. Finally, I would like to thank V. Minerbe for useful comments, G. Carron and N. Gigli for helpful final conversations, and the anonymous referees for precious suggestions.

Preliminaries
Unless otherwise mentioned, in the whole article (X, d, m) denotes a triple where (X, d) is a proper, complete and separable metric space and m is a Borel measure, positive and finite on balls with finite and non-zero radius, such that supp(m) = X. We use the standard notations for function spaces: C(X) for the space of d-continuous functions, Lip(X) for the space of d-Lipschitz functions and L p (X, m) (respectively L p loc (X, m)) for the space of p-integrable (respectively locally p-integrable) functions, for any 1 ≤ p ≤ +∞. If U is an open subset of X, we denote by C c (U ) the space of continuous functions on X compactly supported in U . We also write L 0 (X, m) (respectively L 0 + (X, m)) for the space of m-measurable (respectively non-negative m-measurable) functions. If A is a subset of X, we denote by A its closure. For any x ∈ X and r > 0, we write S r (x) for B r (x)\B r (x). For any λ > 0, if B denotes a ball of radius r > 0, we write λB for the ball with same center as B and of radius λr. If A is a bounded Borel subset of X, then for any locally integrable function u : X → R, we write u A or ffl A u dm for the mean value 1 m(A)´A u dm, and u A for the mean value 1 µ(A)´A u dµ, where µ is as in Theorem 1. Several constants appear in this work. For better readability, if a constant C depends only on parameters a 1 , a 2 , · · · we always write C = C(a 1 , a 2 , · · · ) for its first occurrence, and then write more simply C if there is no ambiguity.

Non-smooth analysis.
Let us recall that a continuous function γ : [0, L] → X is called a rectifiable curve if its length . When γ =γ, we say that γ is parametrized by arc-length.
In the context of metric analysis, a weak notion of norm of the gradient of a function is available and due to Heinonen and Koskela [HeK98]. Building on this, one can introduce the so-called Cheeger energies and the associated Sobolev spaces H 1,p (X, d, m), where p ∈ [1, +∞), in the following way: Definition 2 (Cheeger energies and Sobolev spaces). Let 1 ≤ p < +∞. The p-Cheeger energy of a function u ∈ L p (X, m) is set as where the infimum is taken over all the sequences is then defined as the closure of Lip(X) ∩ L p (X, m) with respect to the norm Remark 1. Following a classical convention, we call Cheeger energy the 2-Cheeger energy and write Ch instead of Ch 2 .
The above relaxation process can be performed with slopes of bounded Lipschitz functions instead of upper gradients, see Lemma 3. Recall that the slope of a Lipschitz function f is defined as and that it satisfies the chain rule, namely |∇(f g)| ≤ f |∇g| + g|∇f | for any f, g ∈ Lip(X). Let us recall that (X, d, m) is called doubling if there exists C D ≥ 1 such that and that it satisfies a uniform weak local L p -Poincaré inequality, where p ∈ [1, +∞), if there exists λ > 1 and C P > 0 such that holds for any ball B of arbitrary radius r > 0, any u ∈ L 1 loc (X, m) and any upper gradient g ∈ L p (X, m) of u. If (2.2) holds with λ = 1, we say that a uniform strong local L p -Poincaré inequality holds.
The next notion serves to turn weak inequalities into strong inequalities, see e.g. [HK00, Sect. 9]. Finally let us introduce a technical property taken from [HK00]. For any v ∈ L 0 (X, m) and 0 < t 1 < t 2 < +∞, we denote by v t 2 t 1 the truncated function min(max(0, v − t 1 ), t 2 − t 1 ) + t 1 . We write χ A for the characteristic function of a set A ⊂ X.
Definition 4 (Truncation property). We say that a pair of m-measurable functions (u, g) such that for some p ∈ [1, +∞), C P > 0 and λ > 1, the inequality (2.2) holds for any ball B of arbitrary radius r > 0, has the truncation property if for any 0 < t 1 < t 2 < +∞, b ∈ R and ε ∈ {−1, 1}, there exists C > 0 such that (2.2) holds for any ball B of arbitrary radius r > 0 with u, g and C P replaced by (ε(u − b)) t 2 t 1 , gχ {t 1 <u<t 2 } and C respectively. The next proposition is a particular case of [HK00, Th. 10.3].
Proposition 1. If (X, d, m) satisfies a uniform weak local L 1 -Poincaré inequality, any pair (u, g) where u ∈ C(X) and g ∈ L 1 loc (X, m) is an upper gradient of u has the truncation property.
The CD(0, N ) and RCD(0, N ) conditions. Let us give the definition of the curvature-dimension conditions CD(0, N ) and RCD(0, N ). For the general condition CD(K, N ) with K ∈ R, we refer to [Vi09,Chap. 29 & 30].
Recall that a curve γ : is called geodesic if for any couple of points (x 0 , x 1 ) ∈ X 2 there exists a geodesic γ such that γ(0) = x 0 and γ(1) = x 1 . We denote by P(X) the set of probability measures on X and by P 2 (X) the set of probability measures µ on X with finite second moment, i.e. such that there exists x o ∈ X for which where the infimum is taken among all the probability measures π on X × X with first marginal equal to µ 0 and second marginal equal to µ 1 . A standard result of optimal transport theory states that if the space (X, d) is geodesic, then the metric space (P 2 , W 2 ) is geodesic too. Let us introduce the Rényi entropies.
We are now in a position to introduce the CD(0, N ) condition, which could be summarized as weak geodesical convexity of all the N -Rényi entropies with N ≥ N .
Then for any function u ∈ C(X) and any upper gradient g ∈ L 1 loc (X, m) of u, for any ball B ⊂ X of arbitrary radius r > 0,ˆB The CD(0, N ) condition does not distinguish between Riemannian-like and non-Riemannian-like structures: for instance, R n equipped with the distance induced by the L ∞ -norm and the Lebesgue measure satisfies the CD(0, N ) condition (see the last theorem in [Vi09]), though it is not a Riemannian structure because the L ∞ -norm is not induced by any scalar product. To focus on Riemannian-like structures, Ambrosio, Gigli and Savaré added to the theory the notion of infinitesimal Hilbertianity, leading to the so-called RCD condition, R standing for Riemannian [AGS14b].
Let us provide some standard facts taken from [AGS14b,G18]. First, note that (X, d, m) is infinitesimally Hilbertian if and only if H 1,2 (X, d, m) is a Hilbert space, whence the terminology. Moreover, for infinitesimally Hilbertian spaces, a suitable diagonal argument justifies for any f ∈ H 1,2 (X, d, m) the existence of a function |∇f | * ∈ L 2 (X, m), called minimal relaxed slope or minimal generalized upper gradient of f , which gives integral representation of Ch, meaning: The minimal relaxed slope is a local object, meaning that |∇f | * = |∇g| * m-a.e. on {f = g} for any f, g ∈ H 1,2 (X, d, m), and it satisfies the chain rule, namely |∇(f g)| * ≤ f |∇g| * + g|∇f | * m-a.e. on X for all f, g ∈ H 1,2 (X, d, m). In addition, the function and defines a strongly local, regular and symmetric Dirichlet form. Finally, the infinitesimally Hilbertian condition allows to apply the general theory of gradient flows on Hilbert spaces, ensuring the existence of the L 2 -gradient flow (h t ) t≥0 of the convex and lower semicontinuous functional Ch, called heat flow of (X, d, m). This heat flow is a linear, continuous, selfadjoint and Markovian contraction semigroup in L 2 (X, m). The terminology 'heat flow' comes from the characterization of (h t ) t≥0 as the only semigroup of operators such that t → h t f is locally absolutely continuous in (0, +∞) with values in L 2 (X, m) and holds for any f ∈ L 2 (X, m), the Laplace operator ∆ being defined in this context by: Patching process Let us present now the patching process [GS05,M09] that we shall apply to get Theorem 1. In the whole paragraph, (X, d) is a metric space equipped with two Borel measures m 1 and m 2 both finite and nonzero on balls with finite and nonzero radius and such that supp(m 1 ) = supp(m 2 ) = X. For any bounded Borel set A ⊂ X and any locally m 2 -integrable function u : X → R, we denote by {u} A the mean value 1 m 2 (A)´A u dm 2 . For any given set S, we denote by Card(S) its cardinality.

Definition 8 (Good covering). Let
For the sake of clarity, we call condition 3. the overlapping condition, condition 4. the embracing condition and condition 5. the measure control condition of the good covering. Note that in [M09] the measure control condition was required also for m 1 though never used in the proofs.
From now on, we consider two numbers p, q ∈ [1, +∞) and two Borel sets A ⊂ A # ⊂ X. We assume that a good covering where V is the set of vertices of the graph, E is the set of edges, and ν is a weight on the graph (i.e. a function ν : V E → R). We define V by associating to each U i a vertex i (informally, we put a point i on each U i ). Then we set E := {(i, j) ∈ V × V : i = j and U i ∩ U j = ∅}. Finally we weight the vertices of the graph by setting ν(i) := m 2 (U i ) for every i ∈ V and the edges by setting ν(i, j) := max(ν(i), ν(j)) for every (i, j) ∈ E.
The patching theorem (Theorem 2) states that if some local inequalities are true on the pieces of the good covering and if a discrete inequality holds on the associated canonical weighted graph, then the local inequalities can be patched into a global one. Let us give the precise definitions.
Definition 9 (Local continuous L q,p -Sobolev-Neumann inequalities). We say that the good covering for all u ∈ L 1 (U * i , m 2 ) and all upper gradients g ∈ L p (U # i , m 1 ).
Definition 10 (Discrete L q -Poincaré inequality). We say that the weighted graph (V, E, ν) satisfies a discrete L q -Poincaré inequality if there exists S d > 0 such that: (2.5) Remark 2. Here we differ a bit from Minerbe's terminology. Indeed, in [M09], the following discrete L q Sobolev-Dirichlet inequalities of order k were introduced for any k ∈ (1, +∞] and any q ∈ [1, k): In the present paper we only need the case k = +∞, in which we recover (2.5): here is why we have chosen the terminology "Poincaré" which seems, in our setting, more appropriate.
We are now in a position to state the patching theorem.
Theorem 2 (Patching theorem). Let (X, d) be a metric space equipped with two Borel measures m 1 and m 2 , both finite and nonzero on balls with finite and nonzero radius, such that supp(m 1 ) = supp(m 2 ) = X. Let A ⊂ A # ⊂ X be two Borel sets, and p, q ∈ [1, +∞) be such that q ≥ p. Assume that (A, A # ) admits a good covering (U i , U * i , U # i ) with respect to (m 1 , m 2 ) which satisfies the local L q,p -Sobolev-Neumann inequalities (2.3) and (2.4) and whose associated weighted graph (V, E, ν) satisfies the discrete L q -Poincaré inequality (2.5). Then there exists a constant C = C(p, q, Q 1 , Q 2 , S c , S d ) > 0 such that for any function u ∈ C c (A # ) and any upper gradient g ∈ L p (A # , m 1 ) of u, Although the proof of Theorem 2 is a straightforward adaptation of [M09, Th. 1.8], we provide it for the reader's convenience.
Proof. Let us consider u ∈ C c (A # ). Then From convexity of the function t → |t| q , we deduce (2.6) From (2.3) and the fact that j x q/p j ≤ ( j x j ) q/p for any finite family of non-negative numbers {x j } (since q ≥ p), we get this last inequality being a direct consequence of the overlapping condition 3. Now the discrete L q -Poincaré inequality (2.5) implies For any (i, j) ∈ E, a double application of Hölder's inequality yields to and as the measure control condition 5. ensures ν(i, j) = max(m 2 (U i ), m 2 (U j )) ≤ Q 2 m 2 (U * k(i,j) ), the embracing condition 4. implies and then where we have used again the convexity of t → |t| q . Summing over (i, j) ∈ E, we get (2.9) Then (2.4) yields to (2.10) Finally, a simple counting argument shows that (2.11) The result follows from combining (2.6), (2.7), (2.8), (2.9), (2.10) and (2.11).
A similar statement holds if we replace the discrete L q -Poincaré inequality by a discrete "L q -Poincaré-Neumann" version: The terminology "Poincaré-Neumann" comes from the mean-value in the left-hand side of (2.12) and the analogy with the local Poincaré inequality used in the study of the Laplacian on bounded Euclidean domains with Neumann boundary conditions, see [SC02, Sect. 1.5.2].
Theorem 3 (Patching theorem -Neumann version). Let (X, d) be a metric space equipped with two Borel measures m 1 and m 2 , both finite and nonzero on balls with finite and nonzero radius, such that supp(m 1 ) = supp(m 2 ) = X. Let A ⊂ A # ⊂ X be two Borel sets such that 0 < m(A) < +∞ and p, q ∈ [1, +∞) such that q ≥ p. Assume that (A, A # ) admits a good covering (U i , U * i , U # i ) with respect to (m 1 , m 2 ) which satisfies the local L q,p -Sobolev-Neumann inequalities (2.3) and (2.4) and whose associated weighted graph (V, E, ν) satisfies the discrete L q -Poincaré-Neumann inequality (2.12). Then there exists a constant C = C(p, q, Q 1 , Q 2 , S c , S d ) > 0 such that for any u ∈ C c (A # ) and any upper gradient The proof of Theorem 3 is similar to the proof of Theorem 2 and writes exactly as [M09, Th. 1.10] with upper gradients instead of norms of gradients, so we skip it.

Proof of the main result
In this section, we prove Theorem 1 after a few preliminary results.
As already pointed out in [M09], the local continuous L 2 * ,2 -Sobolev-Neumann inequalities on Riemannian manifolds (where 2 * = 2n/(n−2) and n is the dimension of the manifold) can be derived from the doubling condition and the uniform strong local L 2 -Poincaré inequality which are both implied by non-negativity of the Ricci curvature. However, the discrete L 2 * -Poincaré inequality requires an additional reverse doubling condition which is an immediate consequence of the growth condition (1.1), as shown in the next lemma.
for some y o ∈ Y and α > 0. Then there exists A > 0 and Proof. The growth condition (3.1) implies the existence of A > 0 such that for any Remark 3. Note that the doubling condition (2.1) easily implies (3.2): see for instance [GHL09,p.9] for a proof giving C RD = (1 + C −4 D ) −1 and α = log 2 (1 + C −4 D ). But in this case, α > 1 if and only if C D < 1 which is impossible. So we emphasize that in our context, in which we want the segment (1, α) to be non-empty, doubling and reverse doubling must be thought as complementary hypotheses.
The next result, a strong local L p -Sobolev inequality for CD(0, N ) spaces, is an important technical tool for our purposes. In the context of Riemannian manifolds, it was proved by Maheux and Saloff-Coste [MS95].
loc (Y, m Y ), and any ball B with arbitrary radius r > 0, Proof. Let u be a continuous function on Y , g ∈ L 1 loc (Y, m Y ) be an upper gradient of u, B be a ball with arbitrary radius r > 0, and p ∈ [1, N ). In this proof u B stands for m Y (B) −1´B u dm Y . Thanks to Hölder's inequality and the doubling property, Proposition 2 implies . Let x 0 , x 1 ∈ Y and r 0 , r 1 > 0 be such that x 1 ∈ B r 0 (x 0 ) and r 1 ≤ r 0 . Then it is applied there to a function f which is Lipschitz on a ball B and extended by 0 outside of B. Note also that Proposition 2 being a L 1 -Poincaré inequality, we can assume α > 1 (a L 2 -Poincaré inequality would have only permit α > 2). CD(0, N ) space satisfying the growth condition (3.1) with α > 1. Then there exists κ 0 = κ 0 (N, α) > 1 such that for any R > 0 such that S R (y o ) is non-empty, for any couple of points (x, x ) ∈ S R (y o ) 2 , there exists a rectifiable curve from x to x that remains inside B R (y o )\B κ −1 0 R (y o ). Let us prove now Theorem 1. Let (X, d, m) be a non-compact CD(0, N ) space with N ≥ 3 satisfying the growth condition (1.1) with parameter η ∈ (1, N ], and p ∈ [1, η). We recall that µ is the measure absolutely continuous with respect to m with density , and that p * = N p/(N − p). Note that Lemma 1 applied to (X, d, m), assuming with no loss of generality that A = 1, implies:

Claim 1. There exists a constant
With no loss of generality, we can assume that min Define then the covering (U i,a , U * i,a , U # i,a ) i∈N,0≤a≤h i where U * i,a is by definition the union of the sets U j,b such that U j,b ∩ U i,a = ∅, and U # i,a is by definition the union of the sets U * j,b such that U * j,b ∩ U * i,a = ∅. Note that (U i,a , U * i,a , U # i,a ) i∈N,0≤a≤h i is not necessarily a good covering, as pieces U i,a might be arbitrary small compared to their neighbors: in this case, the measure control condition 5. would not be true. So whenever U i+1,a ∩ S κ i+1 (o) = ∅ (this condition being satisfied by all "small" pieces), we set U i,a := U i+1,a ∪ U i,a where a is the integer such that U i+1,a ∩ U i,a = ∅; otherwise we set U i+1,a := U i+1,a .
Ui,a Figure 1: for simplicity assume a = a; if U i+1,a ∩ S κ i+1 (o) = ∅, then we glue the small piece U i+1,a to the adjacent piece U i,a to form U i,a We define U * i,a and U # i,a in a similar way from U * i,a and U # i,a respectively. Using the doubling condition, one can easily show that (U i,a , U * i,a , U # i,a ) i∈N,0≤a≤h i is a good covering of (X, d) with respect to (µ, m), with constants Q 1 and Q 2 depending only on N .
Let (V, E, ν) be the weighted graph obtained from (U i,a , U * i,a , U # i,a ) i∈N,0≤a≤h i . Define the degree deg(i, a) of a vertex (i, a) as the number of vertices (j, b) such that U i,a ∩ U j,b = ∅. As a consequence of Claim 1, sup{deg(i, a) : (i, a) ∈ V} ≤ 2h. Moreover: Proof. Take (i, a), (j, b) ∈ E. With no loss of generality we can assume j = i + 1. Take x ∈ U i,a ∩ S (κ i +κ i−1 )/2 (o) and set r = (κ i − κ i−1 )/4, R = 2κ i+1 , so that B r (x) ⊂ U i,a and U i+1,b ⊂ B R (x). Then the doubling condition implies We are now in a position to apply [M09, Prop. 1.12] which ensures that the discrete L 1 -Poincaré inequality implies the L q one for any given q ≥ 1. But the discrete L 1 -Poincaré inequality is equivalent to the isoperimetric inequality ([M09, Prop. 1.14]): there exists a constant I > 0 such that for any Ω ⊂ V with finite measure, in [M09]. Then the discrete L q -Poincaré inequality holds for any q ≥ 1, with a constant S d depending only on q, η, Θ inf , Θ sup and on the doubling and Poincaré constants of (X, d, m), i.e. on N . In case q = p * , we have S d = S d (N, η, p, Θ inf , Θ sup ).
Let us explain how to get the local continuous L p * ,p -Sobolev-Neumann inequalities. We start by deriving from the strong local L p -Sobolev inequality (3.3) a L p -Sobolev-type inequality on connected Borel subsets of annuli.
. Then there exists a constant C = C (N, δ, α, p) > 0 such that for any function u ∈ C(X) and any upper gradient Proof. Define s = δR and choose an s-lattice of A (i.e. a maximal set of points whose distance between two of them is at least s) ( 3s). Using the doubling condition, there is no difficulty in proving that Let us prove that Claim 3 implies the local continuous L p * ,p -Sobolev-Neumann inequalities with a constant S c depending only on N , η and p. Take a piece of the good covering U i,a . Choose δ = (1 − κ −1 )/2 so that [U i,a ] δ ⊂ U * i,a . Take a function u ∈ C(X) and an upper gradient g ∈ L p ([U i,a ] δ , m) of u. Since |u − u U i,a | ≤ |u − c| + |c − u U i,a | for any c ∈ R, convexity of t → |t| p * and Hölder's inequality implŷ As w o is a radial function, we can setw o (r) := w o (x) for any r > 0 and any x ∈ X such that d(o, x) = r. Note that by the Bishop-Gromov theorem,w o is a decreasing function, sô Applying Claim 3 with A = U i,a , R = κ i−1 and α = κ 2 yields tô where we used the same letter C to denote different constants depending only on N , p, and κ. As κ depends only on N , η and p, we get the result. An analogous argument implies the inequalities between levels 2 and 3.
Apply Theorem 2 to get the result.

Weighted Nash inequality and bound of the corresponding heat kernel
In this section, we deduce from Theorem 1 a weighted Nash inequality. We use this result in the context of RCD(0, N ) spaces to get a uniform bound on a corresponding weighted heat kernel.
To prove this theorem, we need a standard lemma which states that the relaxation procedure defining Ch can be performed with slopes of Lipschitz functions with bounded support (we write Lip bs (X) in the sequel for the space of such functions) instead of upper gradients of L 2 -functions. We omit the proof for brevity and refer to the paragraph after Propositon 4.2 in [ACDM15] for a discussion on this result. Note that here and until the end of this section we write L p (m), L p (µ) instead of L p (X, m), L p (X, µ) respectively for any 1 ≤ p ≤ +∞.
In particular, for any u ∈ H 1,2 (X, d, m), there exists a sequence (u n ) n ⊂ Lip bs (X) such that u − u n L 2 (m) → 0 and |∇u n | 2 L 2 (m) → Ch(u) when n → +∞. We are now in a position to prove Theorem 4.
Proof. By the previous lemma it is sufficient to prove the result for u ∈ Lip bs (X). By Hölder's inequality, where 1 2 = θ 1 + 1−θ 2 * i.e. θ = 2 N +2 . Then by Theorem 1 applied in the case p = 2 < η, It follows from the identification between slopes and minimal relaxed gradients established in [Ch99, Theorem 5.1] that Ch(u) = |∇u| 2 L 2 (m) , so the result follows by raising the previous inequality to the power 2(N + 2)/N . Let us consider now a RCD(0, N ) space (X, d, m) satisfying the growth condition (1.1) for some η > 2 and the uniform local N -Ahlfors regularity property : for some C o > 1 and r o > 0. Such spaces are called weakly non-collapsed according to the terminology introduced by Gigli and De Philippis in [DG18]. Note that it follows from [?] that N is an integer which coincides with the essential dimension of (X, d, m). We take the weight which corresponds to the case p = 2 in Theorem 1. Note that (4.1) together with Bishop-Gromov's theorem implies that w o is bounded from above by C ϕf ∈ H 1,2 (X, d, m) ∀ϕ ∈ Lip bs (X)} and note that as an immediate consequence of (4.1) combined with Bishop-Gromov's theorem, w o is bounded from above and below by positive constants on any compact subsets of X, thus f ∈ L 2 loc (m) if and only if f ∈ L 2 loc (µ). Define a Dirichlet form Q on L 2 (µ) as follows. Set and Q is easily seen to be convex. Let us show that it is a L 2 (µ)-lower semicontinuous functional on L 2 (µ). Let {f n } n ⊂ D(Q) and f ∈ L 2 (µ) be such that f n − f L 2 (µ) → 0. Let K ⊂ X be a compact set. For any i ∈ N\{0}, set ϕ i (·) = max(0, 1 − (1/i)d(·, K)) and note that ϕ i ∈ Lip bs (X), 0 ≤ ϕ i ≤ 1, ϕ i ≡ 1 on K and |∇ϕ i | * ≤ (1/i). Then for any i, the sequence {ϕ i f n } n converges to ϕ i f in L 2 (m). The L 2 (m)-lower semicontinuity of the Cheeger energy and the chain rule for the slope implŷ Letting i tend to +∞, then letting K tend to X, yields the result. Then we can apply the general theory of gradient flows to define the semigroup (h µ t ) t>0 associated to Q which is characterized by the property that for any f ∈ L 2 (X, µ), t → h µ t f is locally absolutely continuous on (0, +∞) with values in L 2 (X, µ), and where the self-adjoint operator −A associated to Q is defined on a dense subset D(A) of D(Q) = {Q < +∞} and characterized by: Be aware that although Q is defined by integration with respect to m, it is a Dirichlet form on L 2 (µ), whence the involvement of µ in the above characterization.
Note that by the Markov property, each h µ t can be uniquely extended from L 2 (X, µ) ∩ L 1 (X, µ) to a contraction from L 1 (X, µ) to itself.
We start with a preliminary lemma stating that a weighted Nash inequality also holds on the appropriate functional space when Ch is replaced by Q.
Proof. Let u ∈ L 1 (µ) ∩ D(Q). Then u ∈ L 2 loc (m), ϕu ∈ H 1,2 (X, d, m) for any ϕ ∈ Lip bs (X) and |∇u| * ∈ L 2 (µ). In particular, if we take (χ n ) n as in the proof of Lemma 3, for any n ∈ N we get that χ n u ∈ H 1,2 (X, d, m) and consequently there exists a sequence (u n,k ) k ⊂ Lip bs (X) such that u n,k → χ n u in L 2 (m) and´X |∇u n,k | 2 dm →´X |∇(χ n u)| 2 * dm. Apply Theorem 4 to the functions u n,k to get for any k ∈ N. As the u n,k and χ n u have bounded support, and thanks to (4.1) which ensures boundedness of w o , the L 2 (m) convergence u n,k → χ n u is equivalent to the L 2 loc (m), L 2 loc (µ), L 2 (µ) and L 1 (µ) convergences. Therefore, passing to the limit k → +∞ in (4.2), we get By an argument similar to the proof of Lemma 3, we can show that lim sup n→+∞ˆX |∇(χ n u)| 2 * dm ≤ˆX |∇u| 2 * dm.
Let us apply Lemma 4 to get a bound on the heat kernel of Q.
Theorem 5 (Bound of the weighted heat kernel). Let (X, d, m) be a RCD(0, N ) space with N > 3 satisfying the growth condition (1.1) for some η > 2 and the uniform local N -Ahlfors regular property (4.1) for some C o > 1 and r o > 0. Then there exists C = C(N, η, Θ inf , Θ sup ) > 0 such that Moreover, for any t > 0, h µ t admits a kernel p µ t with respect to µ such that for some C = C(N, η, Θ inf , Θ sup ) > 0, (4.4) To prove this theorem we follow closely the lines of [SC02, Th. 4.1.1]. The constant C may differ from line to line, note however that it will always depend only on η, N , Θ inf and Θ sup .

A non-smooth example
To conclude, let us provide an example beyond the scope of smooth Riemannian manifolds to which Theorem 1 applies. For any positive integer n, let 0 n be the origin of R n .
In [Ha17], Hattori built a complete four dimensional Ricci-flat manifold (M, g) satisfying (1.1) for some η ∈ (3, 4) and whose set of isometry classes of tangent cones at infinity T (M, g) is homeomorphic to S 1 . Of particular interest to us is one specific element of T (M, g), namely (R 3 , d ∞ 0 , 0 3 ), where d ∞ 0 is the completion of the Riemannian metric f g e defined on R 3 \{0 3 } as follows: g e is the Euclidean metric on R 3 and for any x = (x 1 , x 2 , x 3 ) ∈ R 3 \{0 3 }, for some α > 1. Since b x (t) ∼ t −α when t → +∞ and b x (t) ∼ |x| −1 when t → 0 for any x = 0 3 , then f (x) has no singularity on R 3 \{0}; however, b 0 3 (t) = t −α is not integrable on any neighborhood of 0, so f (x) has a singularity at x = 0 3 . In particular, (R 3 , d ∞ 0 , 0 3 ) is a singular space with a unique singularity at 0 3 . Hattori proved that this space is neither a metric cone nor a polar metric space.
Let d g , v g be the Riemannian distance and Riemannian volume measure associated to g, and o ∈ M such that (R 3 , d ∞ 0 , 0 3 ) is a tangent cone at infinity of (M, d g , o). Following a classical method (see e.g. [CC97]), one can equip (R 3 , d ∞