New stability results for sequences of metric measure spaces with uniform Ricci bounds from below

The aim of this paper is to provide new stability results for sequences of metric measure spaces $(X_i,d_i,m_i)$ convergent in the measured Gromov-Hausdorff sense. By adopting the so-called extrinsic approach of embedding all metric spaces into a common one $(X,d)$, we extend the results of Gigli-Mondino-Savar\'e by providing Mosco convergence of Cheeger's energies and compactness theorems in the whole range of Sobolev spaces $H^{1,p}$, including the space $BV$, and even with a variable exponent $p_i\in [1,\infty]$. In addition, building on the results of Ambrosio-Stra-Trevisan, we provide local convergence results for gradient derivations. We use these tools to improve the spectral stability results, previously known for $p>1$ and for Ricci limit spaces, getting continuity of Cheeger's constant. In the dimensional case $N<\infty$, we improve some rigidity and almost rigidity results by Ketterer and Cavaletti-Mondino. On the basis of the second-order calculus by Gigli, in the class of $RCD(K,\infty)$ spaces we provide stability results for Hessians and $W^{2,2}$ functions and we treat the stability of the Bakry-\'Emery condition $BE(K,N)$ and of ${\bf Ric}\geq KI$, with $K$ and $N$ not necessarily constant.

for stability results in the same spirit, obtained from a localization of the Lagrangian definition of curvature/dimension bounds). On the other hand, since our approach is extrinsic, this result becomes of interest from the intrinsic point of view only when ζ's depending on the metric structure, as ϕ • d, are considered. See also Remark 10.7 for an analogous stability property of the BE(K, N ) condition with K and N dependent on x.
We believe that these stability results and the tools developed in this paper could be the basis for the analysis of the stability of the other calculus tools and concepts developed in [G15b], as exterior and covariant derivatives, Hodge laplacian, etc. However, we will not pursue this point of view in this paper.
Organization of the paper. In Section 2 we introduce the main measure-theoretic preliminaries. In Section 3 we discuss convergence of functions f i in different measure spaces relative to m i ; here the main new ingredient is a notion of L p i convergence which allows us also to cover the case when the exponents p i converge to p ∈ [1, ∞). We discuss the case of strong convergence, and of weak convergence when p > 1. Section 4 recalls the main terminology and the main known facts about RCD(K, ∞) spaces and the regularizing properties of the heat flow h t . Less standard facts proved in this section are: the formula provided in Proposition 4.5 for u → X |∇u| dm (somehow reminiscent of the duality tangent/cotangent bundle at the basis of [G15b]), of particular interest for the proof of lower semicontinuity properties, and the weak isoperimetric property of Proposition 4.7.
In Section 5 we enter the core of the paper, somehow "localizing" the Mosco convergence result of Cheeger's energies of [GMS13]. The main result is Theorem 5.7 where we prove, among other things, that the measures |∇f i | 2 i m i weakly converge to |∇f | 2 m whenever f i strongly converge to f in H 1,2 (i.e., f i L 2 -strongly converge to f and the Cheeger energies of f i converge to the Cheeger energy of f ). To prove this, the main difficulty is the localization of the lim inf inequality of [GMS13]; we obtain it using the recent results in [AST16], for families of derivations with convergent L 2 norms (in this case, gradient derivations, see Theorem 5.6 in this paper). Section 6 covers the stability properties of BV functions, the main result is that f ∈ BV (X, d, m i ) whenever f i ∈ BV (X, d, m i ) L 1strongly converge to f , with L = lim inf i |Df i |(X) < ∞. In addition, |Df |(X) ≤ L. The proof of this stability properties strongly relies on the results of Section 5 and, nowithstanding the well-estabilished Eulerian-Lagrangian duality for Sobolev and BV spaces (see [ADM14] for the latter spaces) it seems harder to get from the Lagrangian point of view.
Section 7 covers compactness results for BV and H 1,p , also in the case when p depends on i. In the proof of these facts we use the (local) strong L 2 compactness properties for sequences bounded H 1,2 proved in [GMS13]; passing from the exponent 2 to higher exponents is quite simple, while the treatment of smaller powers and the passage from L p loc to L p convergence (essential for our results in Section 9) requires the existence of uniform isoperimetric profiles. We review the state of the art on this topic in Theorem 7.2. In Section 8 we prove Γ-convergence of the p i -Cheeger energies Ch i p i relative to (X, d, m i ) (set equal to the total variation functional f → |Df |(X) in BV when p = 1), namely whenever f i L p i -strongly converge to f , and the existence of a sequence f i with this property satisfying lim sup i Ch i p i (f i ) ≤ Ch p (f ). The only difference with the case p = 2 considered in [GMS13] is that, in general, we are not able to achieve the lim inf inequality with L p i -weakly convergent sequences, unless a uniform isoperimetric assumption on the spaces grants relative compactness w.r.t. strong L p i convergence. Under this assumption, Mosco and Γ-convergence coincide.
Finally, Section 9, Section 10 and Section 11 cover the above mentioned stability results for p-eigenvalues and eigenfunctions (using Section 7 and Section 8) , for Hessians and Ricci tensors (using Section 5), and the dimensional results relative to the suspension theorems (using Section 9).
Acknowledgement. The first author acknowledges helpful conversations on the subject of this paper with Fabio Cavalletti, Andrea Mondino and Giuseppe Savaré. The second author acknowledges the support of the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, the Grantin-Aid for Young Scientists (B) 16K17585 and the warm hospitality of SNS. The authors warmly thank the referee for the detailed reading of the paper and for the constructive comments.

Notation and basic setting
Metric concepts. In a metric space (X, d), we denote by B r (x) and B r (x) the open and closed balls respectively, by C bs (X) the space of bounded continuous functions with bounded support, by Lip bs (X) ⊂ C bs (X) the subspace of Lipschitz functions. We use the notation C b (X) and Lip b (X) for bounded continuous and bounded Lipschitz functions respectively. which is upper semicontinuous.
The metric algebra A bs . We associate to any separable metric space (X, d) the smallest A ⊂ Lip b (X) containing min{d(·, x), k} with k ∈ Q ∩ [0, ∞], x ∈ D and D ⊂ X countable and dense (2.3) which is a vector space over Q and is stable under products and lattice operations. It is a countable set and it depends only on the choice of the set D (but this dependence will not be emphasized in our notation, since the metric space will mostly be fixed). We shall work with the subalgebra A bs of functions with bounded support.
Measure-theoretic notation. The Borel σ-algebra of a metric space (X, d) is denoted B(X). The Borel signed measures with finite total variation are denoted by M (X), while we use the notation M + (X), M + loc (X), P(X) for nonnegative finite Borel measures, Borel measures which are finite on bounded sets and Borel probability measures. We use the standard notation L p (X, m), L p loc (X, m) for the L p spaces when m is nonnegative (p = 0 is included and denotes the class of m-measurable functions). Notice that, in this context where no local compactness assumption is made, L p loc means p-integrability on bounded subsets.
Given metric spaces (X, d X ) and (Y, d Y ) and a Borel map f : X → Y , we denote by f # the induced push-forward operator, mapping P(X) to P(Y ), M + (X) to M + (Y ) and, if the preimage of bounded sets is bounded, M + loc (X) to M + loc (Y ). Notice that, for all µ ∈ M + (X), f # µ is well defined also if f is µ-measurable.

Convergence of measures.
We say that m n ∈ M loc (X) weakly converge to m ∈ M loc (X) if X v dm n → X v dm as n → ∞ for all v ∈ C bs (X). When all the measures m n as well as m are probability measures, this is equivalent to requiring that X v dm n → X v dm as n → ∞ for all v ∈ C b (X). We shall also use the following well-known proposition.
Proposition 2.1. If m n weakly converge to m in M + loc (X), and if lim sup i→∞ X Θ dm i < ∞ for some Borel Θ : X → (0, ∞], then then (2.4) holds for all v : X → R continuous with |v| ≤ CΘ for some constant C.
Metric measure space. Throughout this paper, a metric measure space is a triple (X, d, m), where (X, d) is a complete and separable metric space and m ∈ M + loc (X). As explained in the introduction, in this paper we always consider metric measure spaces according to the previous definition. When a sequence convergent in the measured-Gromov Hausdorff sense is considered, we shall always assume (up to an isometric embedding in a common space) that the sequence has the structure (X, d, m i ) with m i ∈ M + loc (X) weakly convergent to m ∈ M + loc (X). In particular, this convention forces us to drop the condition supp m = X, used in many papers where individual spaces are considered.

Convergence of functions
In our setting, we are dealing with a sequence (m i ) ⊂ M + loc (X) weakly convergent to m ∈ M + loc (X). Assuming that f i in suitable Lebesgue spaces relative to m i are given, we discuss in this section suitable notions of weak and strong convergence for f i . Motivated by the convergence results of Section 8 and Section 9, we extend the analysis of [GMS13] and [AST16] to the case when also the exponents p i ∈ [1, ∞) are allowed to vary, with p i → p ∈ [1, ∞). For weak convergence we only consider the case p > 1 (we don't need L 1 -weak convergence), while for strong convergence, in connection with the results of Section 6, we also consider the case p = 1.
Weak convergence. Assume that p i ∈ [1, ∞) converge to p ∈ (1, ∞). We say that For R k -valued maps we understand the convergence componentwise. It is obvious that L p i -weak convergence is stable under finite sums. The proof of the following result is very similar to the proof in the case when p and m are fixed, and it omitted.
Moreover, any sequence f i ∈ L p i (X, m i ; R k ) such that (3.1) holds admits a L p i -weakly convergent subsequence. Strong convergence. We discuss the simpler case p i = p first. If p > 1 we say that f i ∈ L p (X, m i ; R k ) L p -strongly converge to f ∈ L p (X, m; R k ) if, in addition to weak L pconvergence, one has lim sup i f i L p (X,m i ;R k ) ≤ f L p (X,m;R k ) . If k = p = 1, we say that In the following remark we see that strong convergence can be written in terms of convergence of the probability measures naturally associated to the graphs of f i ; this holds also for vector valued maps and we will use this fact in the proof of Proposition 3.3.
Remark 3.2 (Convergence of graphs versus L p -strong convergence). If p > 1 one can use the strict convexity of the map z ∈ R k → |z| p to prove that F i : (see for instance [AGS08, Section 5.4], [GMS13]). If p = k = 1, we can use the fact that the signed square root is an homeomorphism of R and the equivalence estabilished in the quadratic case to get the same result.
We recall in the following proposition a few well-known properties of L p -strong convergence, see also [H15], [GMS13] for a more detailed treatment of this topic. Proposition 3.3. For all p ∈ [1, ∞) the following properties hold: -strongly converge to f, g respectively, then f i + g i L p -strongly converge to f + g.
(d) If f i L 2 -strongly converge to f and g i L 2 -weakly converge to g, then If g i are also L 2 -strongly convergent, then f i g i are L 1 -strongly convergent.
(e) If (g i ) is uniformly bounded in L ∞ and L 1 -strongly convergent to g, then Proof. (a) In the case p > 1 this is a simple consequence of Remark 3.2, since in duality with C p (X × R), and then Remark 3.2 applies again to provide the L p -strong convergence of ϕ • f i to ϕ • f . In the case p = 1, since σ(ϕ(f i )) = sign(ϕ•f i ) |ϕ| • f i , from the strong L 2 -convergence of ϕ ± • f i to ϕ ± • f and the additivity of L 2 -strong convergence (proved in (b)) we get the result.
(b) The case p > 1 is dealt with, for instance, in [H15], see Corollary 3.26 and Proposition 3.31 therein. In order to prove additivity for p = 1 we can reduce ourselves, thanks to the stability under left composition proved in (a), to the sum of nonnegative functions u i , v i . Since √ u i and √ v i are L 2 -strongly convergent, using the identity The proof of (c) is a simple consequence of the definitions of L p -strong convergence, splitting ϕ and f i in positive and negative parts to deal also with the case p = 1.
The proof of the first part of statement (d) is a simple consequence of see also Section 8 where a similar argument is used in connection with Mosco convergence. In order to prove L 1 -strong convergence when also g i are L 2 -strongly convergent, we can reduce ourselves to the case when f i and g i are nonnegative. Then, convergence of the L 2 norms of √ f i g i follows by the first part of the statement; weak convergence of For the proof of (e), let N = sup i g i L ∞ (X,m i ) and notice first that (g i ) is uniformly bounded in L p i . Hence, the lim inf inequality follows by the L p i -weak convergence of g i to g. The proof of the lim sup inequality follows by statement (a) with and if for any ǫ > 0 we can find an additive decomposition f i = g i + h i with (i) (g i ) uniformly bounded in L ∞ , and strongly L 1 -convergent; It is obvious from the definition that also L p i -strong convergence is stable under finite sums. In the following proposition we show that stability under composition with Lipschitz maps ϕ holds and that L p i convergence implies convergence of the L p i norms.
Proposition 3.4 (Properties of L p i -strong convergence). The following properties hold: Proof. (a) Possibly splitting ϕ in positive and negative part we can assume ϕ ≥ 0. Since ϕ is a contraction, taking also Proposition 3.3(a) into account, it is immediate to check that decompositions in addition, if ψ is any L p i -weak limit point of (ϕ • f i ), from the lower semicontinuity of L p i convergence we get where g denotes the L p i -strong limit of g i . Since ǫ is arbitrary, we obtain that ψ = ϕ • f , and this proves the L p i -strong convergence of f i to f . (b) The lim inf inequality follows by weak convergence. If f i = g i +h i is a decomposition as in (i), (ii), and if g is the L p i -strong limit of g i , the lim sup inequality is a direct consequence of the inequality f − g L p (X,m) < ǫ and of

Minimal relaxed slopes, Cheeger energy and RCD(K, ∞) spaces
In this section we recall basic facts about minimal relaxed slopes, Sobolev spaces and heat flow in metric measure spaces (X, d, m), see [AGS14a] and [G15a] for a more systematic treatment of this topic. For p ∈ (1, ∞) the p-th Cheeger energy Ch p : is the convex and L p (X, m)-lower semicontinuous functional defined as follows: (4.1) The original definition in [Ch99] involves generalized upper gradients of f n in place of their asymptotic Lipschitz constant, but many other pseudo gradients (upper gradients, or the slope lip(f ) ≤ Lip a (f ), which is a particular upper gradient) can be used and all of them lead to the same definition. Indeed, all these pseudo gradients produce functionals intermediate between the functional in (4.1) and the functional based on the minimal pweak upper gradient of [Sh00], which are shown to be coincident in [ACDM15] (see also the discussion in [AGS14a, Remark 5.12]).
The Sobolev spaces H 1,p (X, d, m) are simply defined as the finiteness domains of Ch p . When endowed with the norm f H 1,p := f p L p (X,m) + pCh p (f ) 1/p these spaces are Banach, and reflexive if (X, d) is doubling (see [ACDM15]). The case p = 2 plays an important role in the construction of the differentiable structure, following [G15b]. For this reason we use the disinguished notation Ch = Ch 2 and it can be proved that H 1,2 (X, d, m) is Hilbert if Ch is quadratic.
In connection with the definition of Ch, for all f ∈ H 1,2 (X, d, m) one can consider the collection RS(f ) all functions in L 2 (X, m) larger than a weak L 2 (X, m) limit of Lip a (f n ), with f n ∈ Lip b (X) and f n → f in L 2 (X, m). This collection describes a convex, closed and nonempty set, whose element with smallest L 2 (X, m) norm is called minimal relaxed slope and denoted by |∇f |. We use the not completely appropriate nabla notation, instead of the notation |Df | of [G15b], since we will be dealing only with quadratic Ch. Notice also that a similar construction can be applied to Ch p , and provides a minimal p-relaxed gradient that can indeed depend on p (see [DmSp15]). However, either under the doubling&Poincaré assumptions [Ch99], or under curvature assumptions [GH14] this dependence disappears and in any case we will only be dealing with the 2-minimal relaxed slope in this paper.
When Ch is quadratic we denote by ∇f, ∇g the canonical symmetric bilinear form from [H 1,2 (X, d, m)] 2 to L 1 (X, m) defined by ∇f, ∇g := lim ǫ→0 |∇(f + ǫg)| 2 − |∇f | 2 2ǫ (4.2) (where the limit is understood in the L 1 (X, m) sense). Notice also that the expression ∇f, ∇g still makes sense m-a.e. for any f, g ∈ Lip b (X) (not necessarily in the H 1,2 space, when m(X) = ∞), since f, g coincide on bounded sets with functions in the Sobolev class, and gradients satisfy the locality property on open and even on Borel sets. Because of the minimality property, |∇f | provides integral representation to Ch, so that X ∇f, ∇g dm = lim ǫ→0 Ch(f + ǫg) − Ch(f ) ǫ and it is not hard to improve weak to strong convergence.
Most standard calculus rules can be proved, when dealing with minimal relaxed slopes. For the purposes of this paper the most relevant ones are: Locality on Borel sets. |∇f | = |∇g| m-a.e. on {f = g} for all f, g ∈ H 1,2 (X, d, m); Pointwise minimality. |∇f | ≤ g m-a.e. for all g ∈ RS(f ); Leibniz rule. If f, g ∈ H 1,2 (X, d, m) and h ∈ Lip b (X), then ∇f, ∇(gh) = h ∇f, ∇g + g ∇f, ∇h m-a.e. in X.
Another object canonically associated to Ch and then to the metric measure structure is the heat flow h t , defined as the L 2 (X, m) gradient flow of Ch, according to the Brezis-Komura theory of gradient flows of lower semicontinuous functionals in Hilbert spaces, see for instance [B70]. This theory provides a continuous contraction semigroup h t in L 2 (X, m) which, under the growth condition extends to a continuous and mass preserving semigroup (still denoted h t ) in all L p (X, m) spaces, 1 ≤ p < ∞. In addition, h t preserves upper and lower bounds with constants, for all t ≥ 0. We shall use h t only in the case when Ch is quadratic, as a regularizing operator. In the sequel we adopt the notation namely D(∆) is the class of functions f ∈ H 1,2 (X, d, m) satisfying − X vg dm = X ∇f, ∇v dm for all v ∈ H 1,2 (X, d, m), for some g ∈ L 2 (X, m) (and then, since g is uniquely determined, ∆f := g). When Ch is quadratis the semigroup h t is also linear (and this property is equivalent to Ch being quadratic) and it is easily seen that lim t↓0 h t f = f strongly in H 1,2 for all f ∈ H 1,2 (X, d, m).
We shall also extensively use the typical regularizing properties (independent of curvature assumptions) as well as the commutation rule Finally, we describe the class of RCD(K, ∞) metric measure spaces of [AGS14b], where thanks to the lower bounds on Ricci curvature even stronger properties of h t can be proved.
(b) Ch is quadratic. This is the axiom added to the Lott-Sturm-Villani theory in [AGS14b].
Remark 4.3 (On the growth condition (4.3)). Notice that (4.3) is needed to give a meaning to the integral in (4.7), as it ensures the integrability of the negative part of ρ log ρ. On the other hand, adopting a suitable convention on the meaning to be given to Ent in these cases of indeterminacy (so that the CD(K, ∞) condition makes anyhow sense), it has been proved in [St06] that (4.3) can be deduced from the CD(K, ∞) condition, and that the constants c i can be estimated in terms of K and of the measure of two concentric balls centered atx ∈ supp m.
It is not hard to prove that the support of any RCD(K, ∞) (or even CD(K, ∞) space) is length, namely the infimum of the length of the absolutely continuous curves connecting any two points x, y ∈ supp m is d(x, y). See [AGS14b] (dealing with finite reference measures), [AGMR15] (for the σ-finite case) and [AGS15] for various characterizations of the class of RCD(K, ∞) spaces. We quote here a few results, which essentially derive from the identification of h t as the gradient flow of Ent w.r.t. the Wasserstein distance and the contractivity properties with respect to that distance.
It is proved in [AGS14b] that the formula whereh t is the dual K-contractive semigroup acting on P 2 (X), provides a pointwise version of the semigroup on L 2 ∩ L ∞ (X, m) with better continuity properties, recalled among other things in the next proposition. Notice also that the formulã provides a canonical extension ofh t to the whole of P(X), used in Proposition 6.3.
In RCD(K, ∞) spaces we have also a useful formula to represent the functional Proposition 4.5. For all f ∈ H 1,2 (X, d, m) one has that |∇f | is the essential supremum of the family ∇f, ∇v as v runs in the family of 1-Lipschitz functions in H 1,2 (X, d, m). Moreover, for all g : X → [0, ∞) lower semicontinuous, one has where the supremum runs among all finite collections of 1-Lipschitz functions v k ∈ H 1,2 (X, d, m) and all w k ∈ C bs (X) with k |w k | ≤ g.
Proof. The proof of the representation of |∇f | as essential supremum has been achieved in [AT14, Lemma 9.2]. We sketch the argument: denoting by M the essential supremum in the statement, one has obviously the inequalities M ≤ |∇f | m-a.e. and | ∇f, ∇v | ≤ M Lip(v) m-a.e. for all v ∈ H 1,2 (X, d, m) Lipschitz and bounded. By localization, this last inequality is improved to | ∇f, ∇v | ≤ M Lip a (v) m-a.e. for all v ∈ H 1,2 (X, d, m) Lipschitz and bounded and then a density argument provides the inequality | ∇f, ∇v | ≤ M |∇v| for all v ∈ H 1,2 (X, d, m) Lipschitz and bounded, which leads to |∇f | ≤ M choosing v = f . In order to prove (4.10) we remark that the representation of |∇f | as essential supremum yields where the supremum runs among all finite Borel partitions B k of X, constants c k ≤ inf B k g and all choices of bounded 1-Lipschitz functions v k ∈ H 1,2 (X, d, m). By inner regularity, the supremum is unchanged if we replace the Borel partitions by finite families of pairwise disjoint compact sets K k . In turn, these families can be approximated by functions w k ∈ C bs (X) with k |w k | ≤ g. Now we recall three useful functional inequalities available in RCD(K, ∞) spaces.
By integration, and then taking the supremum w.r.t. g, we get (4.11).
When the space has finite diameter and K ≤ 0 we will also use, as a replacement of the isoperimetric inequality (presently known in the RCD(K, ∞) setting only when K > 0), the following inequality, which is an easy consequence of Proposition 4.4(c).

Proof. The standard entropy inequality
provides a modulus of continuity ω E , depending only on E ≥ 0, such that g nonnegative Assume first X f dm = 1 and let M > 0. For all t > 0 we apply Proposition 4.6 and Proposition 4.4(c) with r = D to get By a scaling argument, the inequality (4.12) implies Then, given ǫ > 0 we choose first t > 0 sufficiently small such that c(t, K) < ǫ and then M sufficiently large to conclude.
Finally, we close this section by reminding higher order properties, strongly inspired by Bakry's calculus, which played a fundamental role in the recent developments of the theory.

Local convergence of gradients under Mosco convergence
The main goal of this section is to localize the Mosco convergence result of [GMS13], proving convergence results for ∇u i , ∇v i i to ∇u, ∇v when u i are strongly convergent in H 1,2 to u and v i are weakly convergent in H 1,2 to v. When both sequences are strongly convergent, we obtain at least the weak convergence as measures. Besides Theorem 5.4 borrowed from [GMS13], the main tool is the convergence results (in the more general context of derivations) of [AST16], see Theorem 5.6.
Definition 5.1 (Mosco convergence). We say that the Cheeger energies Ch i := Ch m i Mosco converge to Ch if both the following conditions hold: One of the main results of [GMS13] is that Mosco convergence holds if (X, d, for somex ∈ X and c 1 , c 2 > 0. Notice that this result holds even in the larger class of CD(K, ∞) spaces and that the uniform growth condition (5.2), that we prefer to emphasize, is actually a consequence of the local weak convergence of m i to m and of the uniform lower bound on Ricci curvature (see Remark 4.3). Next, we define in a natural way, following [GMS13], weak and strong convergence in the Sobolev space H 1,2 , with a variable reference measure.

Definition 5.2 (Convergence in the Sobolev spaces). We say that
Notice that the sequence f i = h, with h ∈ Lip bs (X) fixed, need not be strongly convergent in H 1,2 , as the following simple example taken from [AST16] shows. The reason is that this sequence should not be considered as a constant one, since the supports of m i can well be pairwise disjoint.
Example 5.3. Take X = R 2 endowed with the Euclidean distance, f (x 1 , x 2 ) = x 2 and let It is immediate to check that weak convergence in H 1,2 is stable under finite sums; it follows from (5.3) below that the same holds for strong convergence in H 1,2 . Also, Theorem 7.4 below (borrowed from [GMS13]) yields that weakly convergent sequences are also L 2 loc -strongly convergent, and provides conditions under which this can be improved to L 2 -strong convergence.
whenever (v i ) strongly converge in H 1,2 to v and (u i ) weakly converge in H 1,2 to u and the heat flows h i relative to (X, d, m i ) converge to the heat flow h relative to (X, d, m) in the following sense: Since sup i Ch i (w i ) is finite, we may let t ↓ 0 to deduce the lim inf inequality; replacing w by −w gives (5.3).
In the following corollary we prove standard consequences of the Mosco convergence of Theorem 5.4, which refine (5.4) (see also [GMS13,Corollary 6.10] for a discrete counterpart of this result, involving the resolvents).
Proof. (a) Using the integration by parts formula we see that f i is weakly convergent in H 1,2 . Let χ ∈ H 1,2 (X, d, m) and let χ i ∈ H 1,2 (X, d, m i ) be strongly convergent to χ in H 1,2 . Let g be a L 2 -weak limit point of ∆ i f i as i → ∞, so that (5.3) gives (along a subsequence, that for simplicity we do not denote explicitly) This proves that f ∈ D(∆) and g = ∆f , so that compactness gives that ∆ i f i L 2weakly converge to ∆f . We can pass to the limit in the integration by parts formula In order to localize the previous results (see in particular (5.3)) we shall use the next theorem, proved in [AST16, Theorem 5.3]. It shows that any sequence (f i ) strongly convergent in H 1,2 to f induces gradient derivations which are strongly converging to the gradient derivation of the limit function, using as class of test functions the family Notice that h Q + A bs depends only on the limit metric measure structure, and it is dense in H 1,2 (X, d, m), see [AST16, Theorem B.1]. Notice also that, since supp m can well be a strict subset of X, the Lip b (X) extension of f ∈ h Q + A bs is not necessarily unique, and therefore ∇v, ∇f i might depend on this extension, when v ∈ H 1,2 (X, d, m i ) (while ∇v, ∇f does not, for v ∈ H 1,2 (X, d, m)). Nevertheless, the following convergence theorem is independent of the extension.
Theorem 5.6 (Strong convergence of gradients). Assume that (X, d, m) is a RCD(K, ∞) metric measure space, that Ch i are quadratic and that Mosco converge to Ch.
Then, for all f ∈ h Q + A bs , ∇v i , ∇f i L 2 -strongly converge to ∇v, ∇f .
Theorem 5.7 (Continuity of the gradient operators). Assume that (X, d, (a) the following tightness on bounded sets holds: Proof. (a) In order to prove (5.6) we choose Using the Leibniz rule once more we get which gives (5.6). Let us now prove (b). Let f ∈ h Q + A bs . Using the Leibniz rule we can write and use (5.3) together with the L 2 -strong convergence of ∇v i , ∇f i to ∇v, ∇f , ensured by Theorem 5.6, to conclude the weak convergence in duality with h Q + A bs of ∇v i , ∇w i i m i . Assuming in addition that ∇v i , ∇w i i satisfy a uniform L p bound for some p > 1, let ξ ∈ L p (X, m) be the L p -weak limit of a subsequence (not relabelled for simplicity of notation). Then, (5.6) gives with ϕ ∈ h Q + A bs , ψ R = 1 − χ R ∈ Lip bs (X), χ R chosen as in the proof of (a), hence we can pass to the limit as i → ∞ to get Since h Q + A bs is dense in L q (X, m), with q dual exponent of p, we can pass to the limit as R → ∞ and use the arbitrariness of ϕ to obtain that ξ = ∇v, ∇w . In order to prove (c), by polarization and the linearity of L 1 -strong convergence it is not restrictive to assume v i = w i . It is then sufficient to apply (5.7) of Lemma 5.8 below (whose proof uses only (a), (b) of this proposition) to obtain the inequality lim inf i A |∇f i | i dm i ≥ A |∇f | dm on any open set A ⊂ X. Assume that ξ ∈ L 2 (X, m) is a L 2 -weak limit point of |∇f i | i ; from the liminf inequality we get A ξ dm ≥ A |∇f | dm for any open set A with m(∂A) = 0. A standard approximation then gives ξ ≥ |∇f | m-a.e. in X. Since the H 1,2 strong convergence gives we obtain the L 2 -strong convergence of |∇f i | i . Combinig the inequality above with lim inf i |∇f i | i L 2 (X,m i ) ≥ ξ L 2 (X,m) we obtain that ξ = |∇f |.
for any lower semicontinuous g : X → [0, ∞] and then for any open set A ⊂ X.
Proof. Since truncation preserves L 2 loc -strong convergence and uniform L 2 bounds, by a truncation argument, in the proof of (5.7) we can assume with no loss of generality that f i are uniformly bounded. Since any lower semicontinuous function is the monotone limit of a sequence of Lipschitz functions with bounded support, we also assume g ∈ Lip bs (X). Also, taking into account the inequality this means that as soon as we have the liminf inequality for Hence, possibly replacing f i by h i t f i we see thanks to (4.8) that we can assume with no loss of generality that f i are uniformly Lipschitz. Under this assumption, we first prove (5.7) in the case when g = χ A is the characteristic function of an open set A ⊂ X, we fix finitely many v k ∈ H 1,2 (X, d, m) with Lip(v k ) ≤ 1, as well as finitely many w k ∈ C bs (X) with supp w k ⊂ A and k |w k | ≤ 1. Let us also fix v k,i strongly convergent in H 1,2 to v k . Now, notice that Indeed, (5.9) follows at once from the weak L 2 convergence of ∇f i , ∇v k,i i to ∇f, ∇v k provided by Theorem 5.7(b). Adding with respect to k, since Lip(v k,i ) ≤ 1 and k |w k | ≤ χ A , from (4.10) with g ≡ χ A we get (5.7). For general g we use the formula gh dµ = It is immediate to check from the definition of total variation that for ϕ•f ∈ BV (X, d, m) for all f ∈ BV (X, d, m) and all ϕ : R → R 1-Lipschitz with ϕ(0) = 0, with |D(ϕ • f )|(X) ≤ |Df |(X). (6.2) In addition, the very definition of |Df |(X) provides the lower semicontinuity property Still using the lower semicontinuity, arguing as in [Mir03], one can prove the coarea formula In the following proposition, whose proof was suggested to the first author by S. Di Marino, we provide a useful equivalent representation of |Df |(X). Proof. By a diagonal argument it is sufficient, for any f ∈ Lip b (X) with lip(f ) ∈ L 1 (X, m), to find f n ∈ Lip bs (X) convergent to f in L 1 (X, m) with Lip a (f n ) → g in L 1 (X, m) and g ≤ lip(f ) m-a.e. in X. By a further diagonal argument, it is sufficient to find f n when f ∈ Lip bs (X). Under this assumption, we know by Theorem 4.1 that there exist m). Since f has bounded support, also f n can be taken with equibounded support, hence both convergences occur in L 1 (X, m). Since |∇f | ≤ lip(f ) m-a.e., we are done.
In the following proposition we list more properties of BV functions in RCD(K, ∞) spaces. Proposition 6.3. Let (X, d, m) be a RCD(K, ∞) space. Then, the following properties hold: Proof. (a) Let f ∈ Lip b (X) ∩ L 1 (X, m) ∩ H 1,2 (X, d, m) and apply (4.9) and the inequality lip(g) ≤ Lip a (g) to get Letting t ↓ 0 provides the inequality ≤ in (a). In order to prove the converse inequality we have to bound from below the number L in (6.1) along all sequences (f n ) ⊂ Lip b (X) convergent to f in L 1 (X, m). It is not restrictive to assume that the lim inf is a finite limit and also, since f is bounded, that f n are uniformly bounded. The finiteness of X |∇f n | dm gives immediately f n ∈ H 1,2 (X, d, m). In addition, for all t > 0 it is easily seen that h t f n weakly converge to h t f in H 1,2 (X, d, m), hence the convexity of and Mazur's lemma give We can use the lower semicontinuity of the total variation to get the inequality ≥ in (a). The proof of (b) in the case of bounded functions uses (4.9) as in the proof of (a) and it is omitted. The general case can be recovered by a truncation argument.
The proof of (c) is an immediate consequence of (4.11) and the definition of BV .
The following theorem provides the stability of the BV property under mGH-convergence. It will be generalized in Theorem 8.1, but we prefer to give a direct proof in the BV case, while the proof of Theorem 8.1 will focus more on the Sobolev case.

Theorem 6.4 (Stability of the BV property under mGH convergence). Let
(6.7) Proof. In the proof it is not restrictive to assume that the functions f i are uniformly bounded. Indeed, since the truncated functions f N i then we could apply (6.2) to f N i and use the lower semicontinuity of the total variation to obtain (6.7).
After this reduction to uniformly bounded sequences, let us fix t > 0 and consider the functions h i t f i , which are uniformly bounded, uniformly Lipschitz (thanks to (4.8)), in H 1,2 (X, d, m i ) and converge to h t f ∈ H 1,2 (X, d, m). If we were able to prove then we could use (6.5) to obtain and we could eventually use once more the lower semicontinuity of the total variation to conclude. Thanks to these preliminary remarks, in the proof of the proposition it is not restrictive to assume that f i are equi-bounded and equi-Lipschitz, with f i ∈ H 1,2 (X, d, m i ), f ∈ H 1,2 (X, d, m). Assuming also with no loss of generality that the lim inf in (6.7) is a finite limit, we have that f i are equi-bounded in H 1,2 , so that they converge weakly to f in H 1,2 . Hence, thanks to the representation (6.4) of the total variation on Lipschitz functions, we need to prove that This is a consequence of Lemma 5.8 with g ≡ 1.

Compactness in H 1,p and in BV
In this section, building upon the basic compactness result in H 1,2 of [GMS13], we provide new compactness results. In order to state them in global form (i.e. passing from L p locstrong to L p -strong convergence) and in order to reach exponents p smaller than 2, suitable uniform isoperimetric estimates along the sequence of spaces will be needed. = o(u) as u ↓ 0, but the formulation (7.1), which involves only the control of sets with sufficiently small measure, is more adapted to our needs. If (X, d, m) has ω as isoperimetric profile, one has the following property: for any ǫ > 0 and any t ∈ R such that m({f > t}) ≤ ω(ǫ), one has In order to prove (7.2) it is sufficent to apply (6.3) to get Eventually, by applying this to g = [(f − t) + ] p , with the Hölder inequality we conclude. By the definition of Ch p we also get {f ≥t} 3) The following theorem provides classes of spaces for which the existence of an isoperimetric profile is known. Notice that RCD(K, N ) spaces with K > 0 and N < ∞ have always finite diameter. Remark 7.3 (Sharp isoperimetric profiles). See also [CM15a] for comparison results and for a description of the sharp isoperimetric profile in the case when N < ∞, in the much more general class of CD(K, N ) spaces (assuming finiteness of the diameter when K ≤ 0).
The following compactness theorem is one of the main results of [GMS13], see Theorem 6.3 therein, we just adapted a bit the statement to our needs, adding also a compactness in L 2 loc independent of the equi-tightness condition (7.5). We say that a sequence (f i ) L 2 loc -strongly converges to f is f i ϕ L 2 -strongly converges to f ϕ for all ϕ ∈ C bs (X).
Theorem 7.4. Assume that (X, d, m i ) are RCD(K, ∞) spaces and f i ∈ H 1,2 (X, d, m i ) satisfy and (for some and thus allx ∈ X) Then (f i ) has a L 2 -strongly convergent subsequence to f ∈ H 1,2 (X, d, m). In general, if only (7.4) holds, (f i ) has a subsequence L 2 loc -strongly convergent to f ∈ H 1,2 (X, d, m). Proof. The first part, as we said, is [GMS13, Theorem 6.3]. For the second part, having fixedx ∈ X, it is sufficient to apply the first part to the sequences f i χ R , where χ R ∈ Lip(X, [0, 1]) with χ R ≡ 1 on B R (x) and χ R ≡ 0 on X \ B R+1 (x), and then to apply a standard diagonal argument.
Under suitable finiteness assumptions, coupled with the existence of a common isoperimetric profile, we can extend this result to L p i compactness, assuming Sobolev or BV bounds, as follows. Assuming the family (f i ) has a L p i(j) -strongly convergent subsequence (f i(j) ). Analogously, if p i = 1 and sup i X then the family (f i ) has a L 1 -strongly convergent subsequence (f i(j) ).
Proof. By L p i -weak compactness we can assume that the weak limit f ∈ L p (X, m) exists. The case p i = 2 for infinitely many i is already covered by Theorem 7.4, indeed the condition (7.5) is automatically satisfied under the isoperimetric assumption, splitting and using (7.3) with p = 2, letting first R → ∞ and then M ↑ ∞.
Hence, in the sequel we need only to consider the cases p i > 2 for i large enough and p i < 2 for i large enough.
In the case when p i > 2 for i large enough the proof is simpler, since for any δ > 0 we can is bounded as well, hence by what we already proved in the case p = 2 we can find a subsequence g i(j) L 2 -strongly convergent and then (since (g i ) are equibounded) L p i -strongly convergent. The decomposition f i = g i + h i can be achieved using (7.3) with p = p i , which gives This is due to the fact that Markov's inequality and the uniform L 1 bound on f i give Hence, we can first choose ǫ > 0 sufficiently small, in such a way that In the case p i < 2 for i large enough the decomposition f i = g i +h i can still be achieved using (7.3) (with ǫ sup i |Df i |(X) < δ in the case p i = 1). Since p i < 2, this time we need one more regularization step to achieve the compactness of g i . More precisely, we write Lipschitz we obtain that sup i Ch 2 (h i t g i ) is uniformly bounded, hence we can extract a L 2 -strongly convergent (and also L p i -strongly convergent) subsequence. It remains to prove that This is an immediate consequence of (6.6) and the uniform boundedness of (g i ).

Mosco convergence of p-Cheeger energies
The definition of Mosco convergence can be immediately adapted to the case when the exponent p is different from 2 and even i-dependent. Adopting the convention Ch 1 (f ) = |Df |(X) to include also the case p = 1, if p i ∈ [1, ∞) converge to p ∈ [1, ∞) we say that the p i -Cheeger energies Ch i p i relative to (X, d, m i ) Mosco converge to Ch p , the p-Cheeger energy relative to (X, d, m), if: (a) (Weak-lim inf). For every f i ∈ L p i (X, m i ) L p i -weakly converging to f ∈ L p (X, m), one has We speak instead of Γ-convergence if the same notions of convergence occurs in (a) and (b), namely the liminf inequality is only required along L p i -strongly convergent sequences. Obviously Mosco convergence implies Γ-convergence and we have provided in Proposition 7.5 a compactness result that allows to improve, under the assumptions on (X, d, m i ) stated in the proposition, Γ to Mosco convergence. Proof. lim inf inequality, p > 1. Possibly replacing f i by their L p i approximations involved in the definition of Ch p i , we need only to prove the weaker inequality Assume first that f i are uniformly bounded in H 1,2 and equi-Lipschitz. Then, Lemma 5.8 and the inequality |∇f for any g lower semicontinuous and nonnegative. This, in combination with the elementary duality identity with q dual exponent of p (applied also to the spaces (X, d, m i ) with p = p i ) provides the inequality In order to remove the additional assumptions on f i we now consider the intermediate case when f i are uniformly bounded in L ∞ and in L 2 . Let us fix t > 0 and consider the functions h i t f i , which are uniformly bounded, uniformly Lipschitz (thanks to (4.8)), in H 1,2 (X, d, m i ) and weakly converge in H 1,2 to h t f ∈ H 1,2 (X, d, m) by Theorem 5.4. Then we can use (4.8), (4.9) and (8. Letting t ↓ 0 then provides (8.2). Eventually we consider the general case f i ; possibly splitting in positive and negative parts, we assume f i ≥ 0. We consider the truncation 1-Lipschitz functions (notice that the quadratic regularization near the origin is necessary in the case p ≥ 2, to get L 2 integrability) By letting N → ∞ we conclude. lim inf inequality, p = 1. The proof is analogous, in the case when the f i are uniformly bounded it is sufficient to prove (8.2) for the regularized functions h i t f i , h t f , without using the duality formula (8.3). Eventually the uniform boundedness assumption on f i can be removed as in the case p > 1, with the simpler truncations ϕ N (z) = min{N, x}. lim sup inequality. For p > 1, let us consider f ∈ H 1,p (X, d, m) and f N ∈ Lip bs (X) with Lip a (f N ) → |∇f | in L p (X, m). For any N one has, by the upper semicontinuity of the asymptotic Lipschitz constant lim sup Since f N L p i converge to f N , by a diagonal argument, we can then define f i = f N (i) with N (i) → ∞ as i → ∞ in such a way that f i L p i converge to f and lim sup i Ch i p i (f i ) ≤ Ch p (f ). For p = 1 the proof is similar and uses Proposition 6.2.

p-spectral gap
Throughout this section we assume that m(X) = 1 when a single space is considered and, when a sequence is considered, also m i (X) = 1. For any p ∈ [1, ∞) and any f ∈ L p (X, m) we put We also recall that for any f ∈ L 1 (X, m) there exists a median of f , i.e. a real number m such that In the following remark we recall a few well-known facts about the minimization problem (9.1) (see also [WWZ10, Lemma 2.2], [C01]).
Remark 9.1. For p ∈ (1, ∞), thanks to the strict convexity of z → |z| p there is a unique minimizer a in (9.1), and it is characterized by It is also well-known that, when p = 1, medians are minimizer in (9.1), the converse seems to be less well-known, so let us provide a simple proof. Assume that a is a minimizer and assume by contradiction that m({f > a}) > 1/2 (if m({f < a}) > 1/2 the argument is similar). We can then find δ > 0 such that m({f > a+δ}) > 1/2 and a simple computation gives contradicting the minimality of a.
In particular, for any p ∈ [1, ∞) there exists a minimizer of (9.1), and it will be denoted by m p (f ); by convention, it will be any median of f when p = 1. Analogously, when we say that m p i (f i ) converge to m p (f ) we understand this convergence in the set-theoretic sense when p = 1 (i.e. limit points of m p i (f i ) are medians).
Lemma 9.2. Let p i converge to p in [1, ∞) and let f i ∈ L p i (X, m i ) be an L p i -strongly convergent sequence to f ∈ L p (X, m). Then taking the infimum with respect to b gives the upper semicontinuity of c p i (f i ).
On the other hand, since it is easily seen that |m p i (f i )| ≤ 2 f i L p i (X,m i ) , the family m p i (f i ) has limit points as i → ∞, and if m p i(k) (f i(k) ) → a as k → ∞ one has If we apply this to limit points of subsequences i(k) on which the lim inf k c p i(k) (f i(k) ) is achieved, this gives that c p i (f i ) → c p (f ). In addition, the inequality (9.2) gives that any limit point of m p i (f i ) is a minimizer.
where the infimum runs among all nonconstant Lipschitz functions f on X. By the very definition of Ch p , the infimum above does not change if we minimize pCh p (f )/c p p (f ) in the class of nonconstant functions f ∈ H 1,p (X, d, m). Furthermore, whenever a minimizer exists, we may normalize it in such a way that c p (f ) = f L p (X,m) = 1 (i.e. the infimum in (9.1) is attained at a = m p (f ) = 0).
For p ∈ (1, ∞), Remark 9.1 and the definition of Ch p gives other characterizations of λ 1,p (X): Then, in [ADG16] it has been proved that where as before the infimum runs among all Borel subsets A of X with 0 < m(A) ≤ m(X)/2 (the same result holds if we use the upper Minkowski content in the definition of h). On the other hand, by applying Lemma 9.2 with m i = m, from Proposition 6.2 we get Eventually, since c 1 (χ A ) = m(A) for m(A) ≤ 1/2, the coarea formula for BV maps shows that the Cheeger constant h coincides also with the quantities in (9.5).
In the following theorem we prove a generalized continuity property (9.6) of the first eigenvalue, allowing also the exponents p i → p ∈ [1, ∞) to depend on i. As the proof shows, this property holds even in the extreme case when diam supp(m) = 0, with the convention (λ 1,p (X, d, m)) 1/p := ∞ if diam supp(m) = 0.
Note that (9.6) in the case when diam supp(m) = 0 will be used in the proof of Corollary 11.6. Proof. For any f ∈ H 1,p (X, d, m) with c p (f ) = f p = 1, by Theorem 8.1, there exists a sequence Taking the infimum with respect to f gives the upper semicontinuity of λ 1,p i (X, d, m i ).
In order to prove the lower semicontinuity, we can assume with no loss of generality that By Proposition 7.5, without loss of generality we can assume that the L p i -strong limit f ∈ L p (X, m) of f i exists. Thus, Theorem 6.4 gives Ch p (f ) ≤ lim inf i Ch i p i (f i ). As a consequence, since Lemma 9.2 gives c p (f ) = f L p (X,m) = 1, we have For p ∈ (1, ∞) and Ω ⊂ X Borel, let us denote : Accordingly, we define λ D 1,p (Ω, d, m) as the infimum of the p-energy with Dirichlet condi- Lemma 9.5. Let p ∈ (1, ∞).
Next we prove (9.9). Let Then, it is easy to check that the implicit function theorem yields that s → m p (f + sg) is differentiable at s = 0. Now, recall that according to [GH14], we can represent pCh p (f ) as X |∇f | p dm, where |∇f | is the 2-minimal relaxed slope (as always, in this paper). Then, the direct calculation of the left hand side of d ds with the differentiability of m p (f + sg) at s = 0 proves (9.9).
In the following stability result we need the extra assumption and f i strongly L p -converge to f for some (and thus all) p ∈ (1, ∞).

(9.12)
This is a kind of extension of Theorem 9.4 to the case p = ∞. We believe that it should be possible to avoid this assumption, possibly making an additional hypothesis on the decay rate of the common isoperimetric profile. Nevertheless, this assumption is harmless for the applications of Theorem 9.6 below in Section 11. Indeed, in the setting of Section 11, as soon as p . (9.13) Proof. Let x 1 , x 2 ∈ supp m; thanks to the weak convergence of m i to m we can find x j,i convergent to x j as i → ∞, j = 1, 2. Let r = d(x 1 , x 2 ), r i = d(x 1,i , x 2,i ) and let us define nonnegative Lipschitz functions δ j,i ∈ Lip(X, d) by Then, since {B r i /2 (x j,i )} j=1,2 are nonempty disjoint subsets of X, and since δ j,i are 1-Lipschitz, for any p ∈ (1, ∞), (9.8) and the Hölder inequality give that for all sufficiently large i. Thus by letting i → ∞ we have By minimizing w.r.t. x 1 and x 2 we get the lim sup inequality in (9.13).
Next we check the lim inf inequality in (9.13). We can assume with no loss of generality that the limit lim i (λ 1,p i (X, d, m i )) 1/p i exists and is finite. For any i such that p i > 2 take a minimizer f i ∈ H 1,p i (X, d, m i ) of the right hand side of (9.4) (whose existence is granted by Proposition 7.5).
, by the compactness property provided by Theorem 8.1 we can also assume thatf + i L pstrongly converge for all p > 1 to a nonnegative g ∈ p>1 H 1,p (X, d, m), thatf − i L pstrongly converge for all p > 1 to a nonnegative h ∈ p>1 H 1,p (X, d, m), so thatf i strongly L p -converge for all p > 1 to f = g − h. For p > 1 fixed, passing to the limit as i → ∞ in the equality we obtain that g = f + and h = f − . We now claim that both f + and f − have unit L ∞ norm. The proof of the upper bound is a simple consequence of the inequalities by letting first i → ∞ and then p → ∞, while the proof of the lower bound is a direct consequence of (9.12).
Recall that derivations, according to [G15b] (the definitions being inspired by [W00]), are linear functionals b : H 1,2 (X, d, m) → L 0 (X, m) satisfying the quantitative locality property |b(u)| ≤ h|∇u| m-a.e. in X, for all u ∈ H 1,2 (X, d, m) for some h ∈ L 0 (X, m). The minimal h, up to m-negligible sets, is denoted |b|. The simplest example of derivation is the gradient derivation b v (u) := ∇v, ∇u induced by v ∈ H 1,2 (X, d, m), which satisfies |b v | = |∇v| m-a.e. in X. By a nice duality argument, it has also been proved in [G15b, Section 2.3.1] that the L ∞ (X, m)-module generated by gradient derivations is dense in the class of L 2 derivations. In the language of [G15b], L 2derivations correspond to L 2 -sections of the tangent bundle T (X, d, m), viewed as dual of the L 2 -sections of cotangent bundle T * (X, d, m) (the latter built starting from differentials of Sobolev functions), see [G15b, Section 2.3] for more details.
Even though higher order tensors will not play a big role in this paper, except for the Hessians, let us describe the basic ingredients of the theory developed for this purpose in [G15b]. In a metric measure space (X, d, m), for p ∈ [1, ∞] let L p (T r s (X, d, m)) denote the space of L p -tensor fields of type (r, s) on (X, d, m), defined as in [G15b]. A tensor field of type (r, s) is a L ∞ (X, m)-multilinear map T : satisfying, for some g ∈ L 0 (X, m) a continuity property with respect to a suitable Hilbert-Schmidt norm on the tensor products. The minimal (up to m-negligible sets) g is denoted |T | and L p tensor fields correspond to tensor fields satisying |T | ∈ L p (X, m).
In particular derivations correspond to (0, 1)-tensor fields. We recall the following facts and definitions: (1) any choice of g 0 , . . . , g r+s ∈ W 1,2 (X, d, m) induces a product tensor field T acting as follows and denoted g 0 r 1 dg k ⊗ r+s r+1 ∇g k . Since derivations correspond to (0, 1)-tensor fields, we recover in particular the concept of gradient derivations.
(2) Denoting, as in [S14], [G15b]  is dense in L p (T r s (X, d, m)) for p ∈ [1, ∞). This is due to the fact that the very definition of tensor product involves a completion procedure of the class of finite sums of elementary products. Notice also that h t maps Lip b (X) into TestF (X, d, m) for all t > 0.
(3) If (X, d, m) is a RCD(K, ∞) space, the space W 2,2 (X, d, m) is defined in [G15b] to be with ρ ∈ C ∞ c ((0, ∞)) convolution kernel and, when necessary, let us define h i ρ in an analogous way. Since (10.6) it is immediately seen that h ρ maps L 2 (X, m) into TestF (X, d, m) and retains many properties of h, namely (with τ = sup supp ρ) if f is bounded and/or Lipschitz, and Then, we define By letting ρ → δ 0 it is immediately seen from (10.7), (10.8), (10.9) that the class Test * F (X, d, m) is dense in TestF (X, d, m), namely for any f ∈ TestF (X, d, m) there exist f n ∈ Test * F (X, d, m) strongly convergent in H 1,2 to f , with sup |f n | ≤ sup |f |, Lip(f n ) ≤ Lip f , and ∆f n → ∆f strongly in H 1,2 .
In the next proposition we show a canonical approximation of test functions in the class TestF (X, d, m) by test functions for the approximating metric measure structures. Notice that we don't know if condition (b) can be improved, getting strong H 1,2 convergence of , such that f i and ∆ i f i strongly converge to f and ∆f in H 1,2 , respectively. Moreover, these properties yield: (a) |∇f i | 2 i L 1 -strongly and L 2 loc -strongly converge to |∇f | 2 ; (b) |∇f i | 2 i weakly converge to |∇f | 2 in H 1,2 . Proof. Let us assume first that f = h ρ g for some g ∈ L 2 ∩ L ∞ (X, m) and some convolution kernel ρ. We define f i as h i ρ g i , with g i L 2 -strongly convergent to g, with g i L ∞ (X,m i ) ≤ g L ∞ (X,m) . It is clear from the construction that f i L ∞ (X,m i ) ≤ f L ∞ (X,m) and that sup i Lip(f i ) < ∞. From (4.5) and (4.6), together with the first formula in (10.6) (applied to h i ρ ), we obtain that both f i and ∆ i f i are bounded in H 1,2 , and their strong convergence is a direct consequence of Corollary 5.5(b) and of (10.6) again.
The weak convergence in H 1,2 of |∇f i | 2 i to |∇f | 2 follows by the apriori estimates (4.13) and (4.14), that ensure the uniform bounds in H 1,2 , and by Theorem 5.7(c) that identifies the L 1 -strong limit (and therefore the weak H 1,2 limit) as |∇f | 2 . Theorem 7.4 provides the relative compactness in L 2 loc of |∇f i | 2 i and then proves L 2 loc -convergence of |∇f i | 2 i to |∇f | 2 as well.
When f ∈ TestF (X, d, m) we apply the previous approximation procedure to h ρ f and then we make a diagonal argument, letting ρ → δ 0 , noticing that the first identity in (10.6) grants the strong convergence in H 1,2 of ∆ i h i ρ f i to ∆h ρ f , while the second identity in (10.6) grants ∆h ρ f L 2 (X,m) ≤ ∆f L 2 (X,m) , ∇∆h ρ f L 2 (X,m) ≤ ∇∆f L 2 (X,m) .
Theorem 10.3 (Stability of W 2,2 regularity and weak convergence of Hessians). Let and assume that f i strongly converge in H 1,2 to f ∈ H 1,2 (X, d, m). Then f ∈ W 2,2 (X, d, m) and Hess i (f i ) L 2 -weakly converge to Hess(f ) in the following sense: whenever g i ∈ H 1,2 (X, d, m i ) are uniformly Lipschitz and strongly converge in H 1,2 to g ∈ H 1,2 (X, d, m), In addition, |Hess(f )| ≤ H m-a.e. for any L 2 -weak limit point H of |Hess i (f i )|, and in particular Proof. Let g ∈ TestF (X, d, m) and let H be a L 2 -weak limit point of |Hess i (f i )|. Let (g i ) be provided by Proposition 10.2. We will first prove convergence of the Hessians under these stronger convergence assumption on g i . In order to identify the L 2 -weak limit of Hess(f i )(g i , g i ) we want to pass to the limit as i → ∞ in the expression Let us analyze the first term. Since div(ϕ∇g i ) = ϕ∆ i g i + ∇g i , ∇ϕ , this term L 2 -strongly converges to div(ϕ∇g) = ϕ∆g + ∇g, ∇ϕ . On the other hand, by Theorem 5.7(b), the term ∇f i , ∇g i i L 2 -weakly converges to ∇f, ∇g . This proves the convergence of the first term. Let us analyze the second term. Since Proposition 10.2(b) shows that |∇g i | 2 i weakly converge in H 1,2 to |∇g| 2 , we can apply Theorem 5.7(b) again to obtain the convergence of X ϕ ∇f i , ∇|∇g i | 2 i i dm i to X ϕ ∇f, ∇|∇g| 2 dm. This completes the proof under the additional assumption on g i . In the general case it is sufficient to apply the already proved convergence result to h i ρ g i , with ρ convolution kernel with support in (0, ∞), noticing the uniform Lipschitz bound on g i yields and that the strong H 1,2 convergence of h i ρ g i to h ρ g yields The inequality |Hess(f )| ≤ H can be proved as follows. We start from the observation that, by bilinearity, for any r, s ∈ [0, π] with r ≤ s. Let us consider eigenfunctions f i ∈ C ∞ (S 2 ) of the first positive eigenvalues of ∆ i with f i L 2 (S 2 ,m i ) = 1, where m i = H 2 /H 2 (S 2 ) with respect to g i . Then, by [CC00] we can assume with no loss of generality that f i strongly converge to f in H 1,2 , with f eigenfunction of the first positive eigenvalue of ∆. It is known that ∆f = 2f and that lim i |Hess i (f i ) + f i g i | L 2 (X,m i ) = 0. Moreover we can prove that f (t) = 3 cos t. Note that these observations correspond to the Bonnet-Mayers theorem and the rigidity on singular spaces. See [CC96,CC00] for the proofs.
In particular lim i |Hess i (f i )| L 2 (S 2 ,m i ) = 2 lim i f i L 2 (S 2 ,m i ) = 2. On the other hand, it was proven in [H15] that g i L 2 -weakly converge to g on [0, π]. Thus Hess(f ) + f g = 0 in L 2 . In particular |Hess(f )| L 2 ([0,π],υ) = f L 2 ([0,π],υ) = 1. Thus these facts give i.e. the Ricci curvatures are strictly increasing even in the case when f i , |∇f i | 2 i , ∆ i f i are uniformly bounded, and strongly converge to f, |∇f | 2 , ∆f in H 1,2 , respectively. In this respect, Theorem 10.5 might be sharp. Moreover this example also tells us that, in general, the condition that ∆ i f i L 2 -strongly converge to ∆f does not imply that |Hess i (f i )| L 2 -strongly converge to |Hess(f )|. Remark 10.7. With a very similar argument one can prove stability of the BE(K, N ) condition with K : X → (−∞, +∞] lower semicontinuous and bounded from below, N : X → (0, ∞] upper semicontinuous. Notice that the strategy of passing to an integral formulation, adopted in [AGS15, Theorem 5.8], seems to work only when K and N are constant.

Dimensional stability results
In this section only we state results that depend on the assumption N < ∞. We recall that the definition of RCD * (K, N ) space has been proposed in [G15a] and deeply investigated and characterized in various ways in [EKS15] (via the so-called Entropy power functional, a dimensional modification of Shannon's logarithmic entropy) and in [AMS15] (via nonlinear diffusion semigroups induced by Rényi's N -entropy), see also [AGS15] in connection with the stability point of view. Starting from RCD(K, ∞), the conditions RCD * (K, N ) amounts to the following reinforcement of Bochner's inequality Hence, Theorem 9.4 and Theorem 9.6 yield that F is continuous. In particular the maximum and the minimum exist. Moreover by the definition these depend only on the parameters N and K. This shows (11.2).
Theorem 11.4. We have the following.
We are now in a position to finish the proof of Corollary 11.6. The proof is done by contradiction via a standard compactness argument. Assume that the assertion is false. Then there exist ǫ > 0, p i ∈ [1, ∞], q i ∈ [1, ∞] and RCD * (N − 1, N )-spaces (X i , d i , m i ) with supp m i = X i and m i (X i ) = 1 such that lim i→∞ (λ 1,p i (X i , d i , m i )) 1/p i − λ N 1,p i 1/p i = 0 and (λ 1,q i (X i , d i , m i )) 1/q i − λ N 1,q i 1/q i ≥ ǫ.
By the sequential compactness of M(N ), without loss of generality we can assume (after embedding isometrically (X i , d i ) into a common metric space (X, d)), that X i = X, d i = d and that the measured Gromov-Hausdorff limit (X, d, m) of the spaces (X, d, m i ) exists, and is an RCD * (N − 1, N )-space. We assume also that the limits p, q ∈ [1, ∞] of p i , q i exist. Then Theorem 9.4 and Theorem 9.6 yield that (λ 1,p (X, d, m)) 1/p = λ N 1,p 1/p and that (λ 1,q (X, d, m)) 1/q = λ N 1,q 1/q .
This contradicts Theorem 11.4 with the argument above.