Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds © 2024 The Author(s). Published by Elsevier Inc. Published version Nobili, Francesco; Violo, Ivan Yuri Nobili, F., & Violo, I. Y. (2024). Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds. Advances in Mathematics, 440, Article 109521. https://doi.org/10.1016/j.aim.2024.109521 2024
Advances in Mathematics 440 (2024) 109521 Contents lists available at ScienceDirect Advances in Mathematics journal homepage: www.elsevier.com/locate/aim Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds Francesco Nobili, Ivan Yuri Violo ∗ University of Jyväskylä, Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014, Finland a r t i c l e i n f o a b s t r a c t Article history: Received 18 October 2022 Received in revised form 9 October 2023 Accepted 27 January 2024 Available online xxxx Communicated by Gang Tian Keywords: Ricci curvature Sobolev inequalities Concentration compactness Stability We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal function of the round sphere. In the setting of non-negative Ricci curvature and Euclidean volume growth, we show an analogous result in comparison with the extremal functions in the Euclidean Sobolev inequality. As an application, we deduce a stability result for minimizing Yamabe metrics. The arguments rely on a generalized Lions’ concentration compactness on varying spaces and on rigidity results of Sobolev inequalities on singular spaces. © 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/). Contents 1. Introduction ....................................................... 2 2. Preliminaries....................................................... 9 2.1. Calculus on metric measure spaces................................... 9 2.2. RCD-spaces................................................... 11 2.3. Sobolev inequalities ............................................. 13 2.4. Convergence and stability under pmGH-convergence....................... 14 *Corresponding author. E-mail addresses: [email protected] (F. Nobili), iv[email protected] (I.Y. Violo). https://doi.org/10.1016/j.aim.2024.109521 0001-8708/© 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).
2F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 3. Pólya-Szegő inequality ................................................ 16 3.1. Non-compact case............................................... 16 3.2. Rigidity...................................................... 19 4. Regularity of extremal functions.......................................... 24 5. Rigidity of extremal functions in the Sobolev inequality ......................... 27 5.1. Compact case ................................................. 27 5.2. Non-compact case............................................... 29 6. Compactness of extremizing sequences ..................................... 30 6.1. Density upper bound ............................................ 30 6.2. Concentration compactness for Sobolev extremals......................... 32 7. Radial functions: technical results......................................... 36 8. Proof of the main results............................................... 40 8.1. Stability in the compact case....................................... 40 8.2. Stability in the non-compact case.................................... 46 Acknowledgments ........................................................ 48 Appendix A. Concentration compactness: non-compact case.......................... 48 A.1. Technical convergence lemmas ...................................... 48 A.2. Concentration compactness principles................................. 51 Appendix B. Technical results .............................................. 53 References ............................................................. 55 1. Introduction The sharp Sobolev inequality on the standard round sphere Sn, n >2, reads as u2 L2∗≤2∗−2 n∇u2 L2+u2 L2,∀u∈W1,2(Sn),(1.1) where 2∗:= 2n/(n −2) and the norms are computed with the renormalized volume measure VolSn VolSn(Sn). This inequality goes back to the work of Aubin [15], who also characterized non-constant extremizers (see also [68, Chapter 5]) having the following expression (denoting by dthe distance induced by the metric): u:= a (1 −bcos(d(·,z 0))n−2 2 ,with a∈R,b∈(0,1),z 0∈Sn.(1.2) We will refer to them as spherical bubbles. A natural question is the one of stability: (Q)Is a function satisfying almost equality in (1.1)close to a spherical bubble? Up to a change of coordinates via the stereographic projection (see e.g. [79,43,45]), this question is equivalent to the stability of the Euclidean Sobolev inequality uL2∗(Rn)≤Eucl(n, 2)∇uL2(Rn),∀u∈˙ W1,2(Rn),(1.3) where ˙ W1,2(Rn) := {u ∈L2∗(Rn): |∇u| ∈L2(Rn)}and Eucl(n, 2) >0is the sharp constant, computed by Aubin [16]and Talenti [97](see (2.10)for its precise value).
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 3 Extremizers, i.e. functions ufor which equality occurs in (1.3), are also in this case completely characterized: u(x):= a (1 + b|x−z0|2)n−2 2 ,a∈R,b>0,z 0∈Rn.(1.4) We shall refer to these functions as Euclidean bubbles (usually called Talenti or AubinTalenti bubbles). The first quantitative stability result was obtained by Bianchi and Egnell [25]who showed that inf ∇(u−w)L2(Rn) ∇uL2(Rn)≤Cn∇uL2(Rn) uL2∗(Rn)−Eucl(n, 2)−11 2,∀u∈˙ W1,2(Rn),(1.5) for a dimensional constant Cn>0and the infimum taken among all was in (1.4). This stability is strong, in the sense that the L2-norm of the difference of gradients is the biggest possible norm that can be controlled, and optimal, as the exponent 1/2is sharp. We mention that quantitative stability for the case of the p-Sobolev inequality in Rnhas also been obtained in sharp form (see [40,47,87,48]). The stability of (1.3)in qualitative form, meaning that if the right-hand side of (1.5)is small then so is the left-hand side (in a non-quantified sense), can be deduced via concentration compactness [80,81]. In this note, we address the analogous stability of (Q)for Sobolev inequalities on more general Riemannian manifolds. Let us consider a closed n-dimensional Riemannian manifold (M, g), n >2, satisfying Ricg≥(n−1)g. Under these assumptions the same Sobolev inequality (1.1)as in the sphere holds [72]: u2 L2∗≤2∗−2 n∇u2 L2+u2 L2,∀u∈W1,2(M),(1.6) where the norms are with the renormalized volume measure. Proofs of this inequality using different methods are also given in [19,21,50,68,20,44]. We can ask an analogous stability: (Q)Is a function satisfying almost equality in (1.6)close to a spherical bubble? Almost equality here means that Q(u):=u2 L2∗−u2 L2 ∇u2 L2∼2∗−2 n. In the previous work [88], we proved that if |Q(u) −2∗−2 n|is small, then Mis qualitatively close in the measure Gromov-Hausdorff sense to a spherical suspension, which
4F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 roughly said is a possibly-singular generalization of the round sphere. In particular, when sup Q(u) =n−1(2∗−2), rigidity occurs, i.e. Mis isometric to Sn. These facts already suggested an affirmative answer to (Q)and in fact here we will confirm that this is indeed the case. More precisely, for Mas above, every a ∈R, b ∈[0, 1) and z∈M, set wa,b,z(·):= a (1 −bcos(d(·,z 0))n−2 2 ,(1.7) with the convention that wa,0,z ≡a. Our main result is then the following (as before, all the norms are with respect to the renormalized volume measure): Theorem 1.1. For every ε >0and n >2there exists δ:= δ(ε, n) >0such that the following holds. Let (M, g)be an n-dimensional Riemannian manifold with Ricg≥(n − 1)gand suppose there exists u ∈W1,2(M)non-constant satisfying Q(u)>2∗−2 n−δ. (1.8) Then, there exist a ∈R, b ∈[0, 1) and z∈Msuch that ∇(u−wa,b,z)L2+u−wa,b,zL2∗ uL2∗≤ε. (1.9) Moreover, if wa,b,z ≡a(i.e. b =0), then a ∈Rcan be chosen so that the reminder R:= u−a satisfies R·R−1 L2−√N+1cos(d(·,p))L2≤Cn(εα+δ)β,(1.10) for some p ∈Mand positive constants α, β, Cndepending only on n. The above theorem is the first stability result for the Sobolev inequality that covers a wide class of Riemannian manifolds; indeed up to our best knowledge only very special symmetric cases had been studied so far: see [24]for the hyperbolic space and [51]for S1(1/√d−2) ×Sn−1(1). Some comments on the above statement are in order. i) The value of δdepends only on nand ε >0, but not on the manifold M. Moreover, up to scaling, an analogous statement holds assuming Ricg≥Kfor some K>0, with δdepending also on K. ii) Even if Theorem 1.1 is stated completely in the smooth-setting, its proof will require the study of the Sobolev inequality also in singular spaces (see below the strategy for more details).
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 5 iii) The result (1.9) actually holds under a slightly weaker assumption than (1.8), namely: u2 L2∗(Volg)≥A∇u2 L2(Volg)+Bu2 L2(Volg),(1.11) with |A −2∗−2 n| +|B−1| ≤δ(see Remark 8.2). iv) The first part of Theorem 1.1 holds also restricting to the class of non constant spherical bubbles, that is wa,b,z with b =0. v) The second part of Theorem 1.1 should be read as follows: if the almost extremal function uis close to a constant, then (up to changing the constant) the remainder is close in L2-sense to a cosine of the distance. Thus, since 1+εcos(d)∼1 (1 + εcos(d))n−2 2 , this means that ustill retains, at a ‘second-order’ approximation, the shape of a spherical bubble. This extra information essentially comes from the fact that the linearization of the Sobolev inequality is the Poincaré inequality, which means that plugging in (1.6) functions of the type 1 +εf and sending ε →0gives the sharp Poincaré inequality for f(see e.g. [88, Lemma 6.7]). Therefore if 1 +εf satisfies almost equality in (1.6), then falmost satisfies equality in the sharp Poincaré inequality and thus should be close to a cosine of the distance (see [38]). vi) When Mis not the round sphere, the existence of an extremizer, that is a function which maximizes Q(u), is unknown in general. This question is contained in [68, Question 4B, Pag. 120] as part of the so-called AB-program around Sobolev inequalities on general Riemannian manifolds. In this direction, we mention the Sobolev-alternative statement proved in [88, Theorem 6.8]. Nevertheless, thanks to the above theorem, we are able to say something about the shape of functions for which this ratio is large, i.e. satisfying (1.8). Remark 1.2. Note that above we deal only with p =2. The reason is that the inequality up Lp∗≤A∇up Lp+up Lp,∀u∈W1,p(M),(1.12) is false for any p >2, A >0and any (M, g)closed manifold (see [68, Prop. 4.1]). As an application of Theorem 1.1, we prove a stability-type result for minimizing Yamabe metrics. Recall that a solution to the Yamabe problem on a Riemannian manifold (M, g)is a smooth positive function usuch that the metric u4 n−2ghas constant scalar curvature (see [99]and also the surveys [78,29]). After the works [98,15,92]it is known that a solution exists on every closed Riemannian manifold and that can be found as a minimizer of
6F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 Y(M,g):= inf u∈W1,2(M) u=0 E(u):= inf u∈W1,2(M) u=0 n(n−1)´2∗−2 n|∇u|2+Scalg n(n−1) u2dVolg ´|u|2∗dVolg2/2∗,(1.13) where Scalgis the scalar curvature of gand Volgis the (non-renormalized) volume measure. Y(M, g)is a called Yamabe constant of (M, g)and it is a conformal invariant. Note that in the case of Sn, the minimizers of E(u)are precisely the spherical bubbles in (1.2). Corollary 1.3. For every n >2and ε >0there exists δ:= δ(ε, n) >0such that the following holds. Let (M, g)be an n-dimensional Riemannian manifold with Ricg≥ (n −1)gand u ∈W1,2(M)non-zero such that dGH(M,Sn)≤δ, |E(u)−Y(M,g)|≤δ. (1.14) Then, there exist a ∈R, b ∈(0, 1) and z0∈Msatisfying u−wa,b,zW1,2 uW1,2≤ε, where wa,b,z is as in (1.7). Here dGH denotes the Gromov-Hausdorff distance. A similar stability for almost minimizers of E(·)has been recently proved in [45]in quantitative form and under no assumptions on the metric. The novelty here is that we have a comparison with an explicit class of functions, while in [45]no information is known about the shape of the minimizers. We discuss now a second stability result on non-compact Riemannian manifolds. Our motivations come from the fact that, to prove Theorem 1.1, non-compact setting will naturally arise in our investigation (see below the main strategy of proof). Let us consider an n-dimensional Riemannian manifolds (M, g), n >2, satisfying Ricg≥0,AV R (M) := lim R→∞ Vol(BR(x)) ωnRn>0,(1.15) for x ∈M. The latter condition is called Euclidean volume growth property and AV R (M) is the asymptotic volume ratio. Notice that the limit exists and is independent of x, by the Bishop-Gromov inequality. In [22], the following sharp Euclidean-type Sobolev inequality was derived under the assumptions (1.15): uL2∗≤AV R (M)−1 nEucl(n, 2)∇uL2,∀u∈˙ W1,2(M).(1.16) Moreover, they proved that equality occurs in (1.16)for some non-zero function u ∈ ˙ W1,2(M), then Mis isometric to Rnand uis in particular an Euclidean bubble. Actually
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 7 in [22]this rigidity requires also u ∈Cn(M)and u ≥0, however these additional assumptions can be removed after the results in [13]and [33](see also Theorem 5.3). The natural stability question is what happens if a function satisfies almost equality in (1.16). Clearly, differently from (1.6), we cannot deduce anything about the geometry of M. Indeed the inequality is sharp on every Mas in (1.15), which means that we can always find functions so that uL2∗ ∇uL2is arbitrary close to AV R (M)−1 nEucl(n, 2). We can prove however that a function for which almost equality occurs in (1.16)is close to a Euclidean bubble. Set va,b,z := a (1 + bd(·,z)2)n−2 2 ,for a∈R,b>0,z ∈M. Theorem 1.4. For every ε >0, V∈(0, 1) and n >2, there exists δ:= δ(ε, n, V) >0 such that the following holds. Let (M, g)be an n-dimensional Riemannian manifold as in (1.15)with AV R (M) ≥Vand assume there exists u ∈˙ W1,2(M)non-zero satisfying uL2∗ ∇uL2 >AV R (M)−1 nEucl(n, 2) −δ. Then, there exist a ∈R, b >0, and z∈Mso that ∇(u−va,b,z)L2 ∇uL2≤ε. Notice that the stability is strong in the sense that we control the gradient norm as in the Euclidean case (1.5). A direct consequence of the above theorem is: Corollary 1.5. Let (M, g)be an n-dimensional Riemannian manifold as in (1.15). Then AV R (M)1 nEucl−1(n, 2) = inf a∈R,b>0,z∈M∇va,b,zL2 va,b,zL2∗ . Remark 1.6. Our main results in Theorem 1.1 and Theorem 1.4, even if stated on smooth Riemannian manifolds, actually hold also in the context of weighted Riemannian manifolds and more generally in the singular setting of metric measure spaces with a synthetic Ricci curvature lower bound. The generalized version of these statements can be found in Theorem 8.1 and Theorem 8.4. Strategy of proof and non-smooth setting. We outline the argument for Theorem 1.1 (Theorem 1.4 is simpler and follows by the same strategy). The underlying idea is classical, that is to argue by contradiction and concentration compactness. However, the novelty is that the space is not homogeneous and also not fixed, since we need to deal with a whole class of Riemannian manifolds. Moreover, singular and non-compact limit
8F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 spaces must also be considered. In particular, the whole analysis will be carried out in the more general setting of RCD spaces, which are metric measure spaces with a synthetic notion of Ricci curvature bounded below (see Section 2for details and references). Suppose that Theorem 1.1 is false. Then, there exist ε >0, a sequence {Mk}k∈N of n-dimensional Riemannian manifolds with Rick≥n −1and non-constant functions uk:Mk→R, ukL2∗=1, which satisfy (1.8)for some δk↓0, but so that for any k∈N inf uk−wL2∗+∇(uk−w)L2>ε, (1.17) where the inf runs among all spherical bubbles w=a(1 −b cos(dk(·, z))2−n 2(dkbeing the distance on Mk). Similarly to the classical concentration compactness [80,81]in Rn, we choose points yk∈Mkand constants σk>0so that, defining (Yk,ρ k,μ k):=(Mk,σ kdk,Volk(Mk)−1σn kVolk),u σk=σ−n/2∗ kuk,(1.18) we have ˆ BYk 1(yk) |uσk|2∗dμk=1 2, (in the actual proof we choose a suitable constant close to 1). The spaces (Yk, ρk, μk)are in particular metric measure spaces which are rescalings of the original manifolds Mk. Note that it can happen that σk↑∞, which corresponds to a concentrating behavior of the sequence uk. In this case, the diameter of Ykgoes to infinity and we are in a sense performing a blow-up along Mk. Thanks to Gromov’s precompactness theorem [64]it is possible to show that, up to a subsequence, (Yk, ρk, μk, yk)converges in the pointed-measure-Gromov-Hausdorff sense to a limit RCD space (Y, ρ, μ, ¯y)(which might be non-smooth). Using a generalized version of Lions’ concentration compactness for a sequence of RCD spaces (see Section 6), we show that up to a further subsequence, uσkconverges L2∗-strongly (on varying spaces, see Definition 2.9 below) to some u ∈L2∗(μ). It also follows that uis extremal for a ‘limit Sobolev inequality’ on Y, that might be both as in (1.6)or of Euclidean-type as in (1.16), depending if there is concentration or not along the original sequence uk. The key point is proving: Concentration ⇒Yis a metric-cone and uis a Euclidean bubble Non-concentration ⇒Yis a spherical suspension and uis a spherical bubble We will show these two facts by proving suitable rigidity theorems for the Sobolev inequalities on RCD spaces (see Section 5). The proof will be then completed by carefully bringing back this information from uto the sequence ukto find a contradiction with (1.17). It is worth noticing that, in case of concentration, the scaled functions uσktend
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 15 (ιn)mn(ι∞)m∞in duality with Cbs(Z) and ιn(xn)→ι∞(x∞)inZ. In the case of a sequence of uniformly locally doubling spaces (as in the case of RCD(K, N)-spaces for fixed K∈R, N<∞) we can also take (Z, d)to be proper. It will be also convenient to adopt the so-called extrinsic approach and identify Xn with their isomorphic copies in (Z, d). This allows writing mnm∞in duality with Cbs(Z). A choice of space (Z, d) together with isomorphic copies of the spaces Xnwill be often called a realization of the convergence. For the scope of this note, it is important to recall the notion of convergence of functions along pmGH-convergence [71,58,8]and their properties. We fix in what follows a pmGH-convergent sequence of pointed metric measure spaces as discussed above. Definition 2.9. Let p ∈(1, ∞)and fix a realization of the convergence in (Z, d). We say: i) fn∈Lp(mn)converges Lp-weak to f∞∈Lp(m∞), provided supn∈NfnLp(mn)<∞ and fnmnf ∞m∞in Cbs(Z); ii) fn∈Lp(mn)converges Lp-strong to f∞∈Lp(m∞), provided it converges Lp-weak and limnfnLp(mn)≤f∞Lp(m∞); iii) fn∈W1,2(Xn)converges W1,2-weak to f∞∈W1,2(X) provided it converges L2-weak and supn∈N∇fnL2(mn)<∞; iv) fn∈W1,2(Xn)converges W1,2-strong to f∞∈W1,2(X) provided it converges L2strong and ∇fnL2(mn)→∇f∞L2(m∞); v) fn∈Lp(mn)converges Lp loc-strong to f∞∈Lp(m∞), provided ηfnconverges Lpstrong to ηf∞for every η∈Cbs(Z). Recall from [71,58,8]the linearity of convergence: if fn, gnconverge Lp-strong to f∞, g∞, respectively, then fn+gnconverges Lp-strong to f∞+g∞.(2.14) We point out the following simple fact: for any p ∈(1, ∞)it holds fnLp-weak converges to f∞⇒f∞L2(m∞)≤lim n→∞fnL2(mn).(2.15) Indeed, if the above liminf above is +∞, then there is nothing to prove. So let us assume it to be finite and also to be a limit, hence fnis L2-bounded. Then there exists an L2-weak convergent subsequence (see [58]) to some h ∈L2(m∞)and in particular hL2(m∞)≤ limnfnL2(mn). By uniqueness of limits we have h =f∞, which shows (2.15). After the works in [95,96,82,52,6,58]and thanks to Gromov’s precompactness theorem [64]we have the following precompactness result. Theorem 2.10. Let (Xn, dn, mn, xn)be a sequence of pointed RCD(Kn, Nn)spaces, n ∈N, with mn(B1(xn)) ∈[v−1, v], for v>1and Kn→K∈R, Nn→N∈[1, ∞).
16 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 Then, there exists a subsequence (Xnk, dnk, mnk, xnk)pmGH-converging to a pointed RCD(K, N)space (X∞, d∞, m∞, x∞). We report from [58]the Mosco-convergence of the Cheeger energies for pmGHconverging RCD-spaces: if fnis L2-weak convergent to f∞, then Ch(f∞)≤lim n→∞ Ch(fn).(2.16) Moreover, for any f∞∈L2(m∞), there exists fn∈L2(mn)converging L2-strong to f∞ and lim n→∞Ch(fn)≤Ch(f∞). In particular, the above is a limit. 3. Pólya-Szegő inequality 3.1. Non-compact case In this part we extend to the non-compact case the Pólya-Szegő inequality of Euclidean-type obtained in [88]. We need first to recall basic notations and facts about monotone decreasing rearrangements for functions in a m.m.s. (X, d, m)(for more details we refer to [86]). Let Ω ⊆X be an open set (possibly unbounded) and u :Ω →[0, +∞)be a Borel function such that m({u >t}) <∞for any t >0. We define μ :[0, +∞) →[0, ∞), the distribution function of uas μ(t) := m({u >t}). For uand μas above, let us consider the generalized inverse u#of μ: u#(s):=ess sup uif s=0, inf {t:μ(t)<s}if s>0. Note that u#is non-increasing. In this note, we will perform rearrangements into the Euclidean model space IN:= ([0, ∞), |.|, mN), equipped with the standard Euclidean distance and weighted measure mN:= σN−1tN−1L1, for N∈(1, ∞). For any open set Ω ⊂Xwe set Ω∗:= [0, r]with mN([0, r]) =m(Ω) (i.e. rN=ω−1 Nm(Ω)), with the convention Ω∗=[0, ∞)if m(Ω) =+∞. The Euclidean monotone rearrangement u∗ N:Ω ∗→R+is then defined by u∗ N(x):=u#(mN([0,x])) = u#(ωNxN),∀x∈Ω∗. Note that u∗ Nis always a non-increasing function, since so is u#. To lighten the notation, we shall often drop the subscript and just write u∗. We collect basic facts about rear-
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 17 rangements, that can be proved by standard arguments as in the Euclidean case (see, e.g. [76]): u≤v⇒u∗≤v∗,(3.1) (ϕ(u))∗=ϕ(u∗),∀ϕ:[0,∞)→[0,∞) non-decreasing. (3.2) uLp(m)=u∗Lp(mN),∀u∈Lp(Ω).(3.3) Lemma 3.1. Let (X, d, m)be a metric measure space and N∈(1, ∞). Let (un): X →R+ be an non-decreasing sequence of Borel functions. Denote u := supnunand suppose that m({u >t}) <+∞for every t >0. Then, u∗ n:IN→R+(which exists by the assumptions) is a monotone non-decreasing sequence and limnu∗ n=u∗a.e. in [0, ∞). Proof. The fact that (u∗ n)is monotone non-decreasing follows by the order preserving property of the rearrangement (3.1). Set g:= supnu∗ n= limnu∗ npointwise on [0, ∞). In particular {u∗ n>t} ↑{g>t}and {un>t} ↑{u >t}for any t >0. Therefore mN({g>t}) = lim n mN({u∗ n>t}) = lim n m({un>t})=m({u>t})=mN({u∗>t}). So g, u∗:[0, ∞) →[0, +∞]are equimeasurable and non-increasing (indeed gis the supremum of non-increasing functions), therefore they coincide a.e. (see e.g. the proof [76, Prop. 1.1.4]). We will need the following approximation result to pass from the bounded to the unbounded case in the Euclidean Pólya-Szegő inequality. It will be needed also in other parts of this note. Lemma 3.2. Let (X, d, m)be a metric measure space and u ∈W1,2 loc (X) such that m({|u| > t}) <+∞for all t >0and |∇u| ∈L2(m). Then there exists a sequence un∈W1,2(X) of functions with bounded support, such that un→um-a.e. and |∇(un−u)| →0in L2(m). Moreover if u ≥0(resp. u ∈Lp(m), p ∈[1, ∞)) we can take (un)non-decreasing (resp. so that un→uin Lp(m)). Proof. We first deal with the case u ≥0and u ∈L∞(m)with m(supp(u)) <+∞. Fix x ∈Xand consider the sequence (ηn) ⊂LIP(X) given by ηn(.) := (2 −d(.,x) n)+∧1. Note that (ηn)is non-decreasing with LIP(ηn) ≤n−1, ηn=1in Bn(x)and supp(ηn) ⊂B2n(x). Take un:= uηn∈W1,2(X) with bounded support. Clearly un↑upointwise and if u ∈Lp(m)also un→uin Lp(m)by dominated convergence. Moreover, by locality ˆ|∇(u−uηn)|2dm≤2ˆ Bc n(x) |∇u|2+|∇(ηnu)|2dm, and by the Leibniz rule
18 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 ∇(ηnu)L2(Bn(x)c)≤2n−1uL∞(m)m(supp(u))1 2+ηn|∇u|L2(Bc n(x)) ≤2n−1uL∞(m)m(supp(u))1 2+∇uL2(Bc n(x)) →0. This proves that |∇u −∇(unηn)| →0in L2(m). If u ≥0, take uk:= ((u −1/k)+) ∧k, k∈N, which is a non-decreasing sequence of functions. Clearly ˆ|∇(u−uk)|2dm≤ˆ {0<u<1/k} |∇u|2dm→0, by dominated convergence. Moreover, since uk∈L∞(m)and m(supp(uk)) <+∞, the conclusion in this case follows from the previous one and a diagonal argument (multiplying by the functions ηn). Monotonicity of the sequence is preserved because ηnf≤η¯ng m-a.e. for every ¯n>nand assuming 0 ≤f≤gm-a.e. The pointwise m-a.e. convergence is also kept, since it remains true on every ball, recalling that ηn=1in Bn(x). Finally for a general uwe approximate first u+and then u−by functions unand vn respectively as we did in the above steps. Clearly if u ∈Lp(m)then un−vn→uin Lp(m). Moreover by construction we have that un−vn=χ{u>0}un−χ{u<0}vn. Therefore |∇(u −(un−vn))| =|∇(u+−un)| +|∇(u−−vn)| →0in L2(m). This concludes the proof also in this case. We can now prove the Pólya-Szegő inequality in the non compact case. Proposition 3.3. Let (X, d, m)be an RCD(0, N)space for some N∈(1, ∞)with AV R (X) >0. Let u ∈W1,2 loc (X) be non-negative and such that m({u >t}) <∞for any t >0. Then, ˆ|∇u|2dm≥AV R (X)2/N ∞ ˆ 0|∇u∗|2dmN,(3.4) meaning that, if the left hand side is finite, then u∗∈W1,2 loc (IN)and (3.4)holds. Proof. First, if ∇uL2(m)=∞, there is nothing to prove. So, suppose |∇u| ∈L2(m). By Lemma 3.2 there exists a non-decreasing sequence un∈W1,2(X) of functions with bounded support, such un→um-a.e. and ∇unL2(m)→∇uL2(m). Applying the Pólya-Szegő inequality for bounded domains in [88, Theorem 3.6], we have u∗ n∈W1,2(IN) and ˆ|∇un|2dm≥AV R (X)2/N ˆ|∇u∗ n|2dmN. Moreover by Lemma 3.1 the sequence u∗ nis non-increasing and supnu∗ n=u∗pointwise. The proof is now concluded since we have that u∗∈W1,2 loc (IN)and limn´|∇u∗ n|2dmN≥ ´|∇u∗|2dmNby semicontinuity (recall (2.1)).
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 19 3.2. Rigidity In this section, we prove the rigidity in the Pólya-Szegő inequality of Proposition 3.3. The idea is that if equality in (3.4)is attained, the superlevel sets are isoperimetric sets, so Theorem 2.6 implies that the space is a cone. This line of thoughts follows classical arguments that date back to the work of [89]in Euclidean contexts and [23]for manifolds with Ricci curvature lower bounds. Moreover, under additional regularity, the function can also be proven to be radial. A similar rigidity result was proved in [86]in the compact case for a different Pólya-Szegő inequality. Theorem 3.4 (Rigidity of the Euclidean Pólya-Szegő inequality). Let (X, d, m)be an RCD(0, N)space for some N∈(1, ∞)with AV R (X) >0. Suppose equality holds in (3.4)(with both sides finite) for u ∈LIPloc(X) non-negative satisfying u(x) →0as d(x, z) →∞, for z∈Xand with (u∗)=0a.e. in {u∗>0}. Then, Xis isomorphic to an N-Euclidean metric measure cone. Moreover, if |∇u| =0m-a.e. on {u >0}, then uis radial, i.e. u(x)=u∗◦AV R (X) 1 Nd(x, x0) for a suitable tip x0of X. Proof. We divide the proof into different steps. Step 1. We establish an improved version of (3.4)for a function uas in the statement. Fix such u. By Theorem 2.8 we know that u ∈L2∗(m). For every n ∈Nset vn:= (u −1/n)+ and notice that they are supported in the open set Ωn:= {u >1/(2n)}, which is bounded. Therefore vn∈LIPc(X). In particular by the Lipschitz-to-Lipschitz property of the rearrangement in the compact case (see [88, Prop. 3.4]) we have v∗ n∈LIPc([0, Rn)) for suitable Rn>0. From (3.2)we also have v∗ n=(u∗−1/n)+, which is non-increasing and (v∗ n)=0a.e. in {v∗ n>0}. In particular u∗∈LIPloc(0, ∞). Define the functions ϕn, ψn, μn:[0, sup vn) →[0, +∞)as ϕn(t):= ˆ {vn>t} |∇vn|2dm,ψ n(t):= ˆ {vn>t} |∇vn|dm,μ n(t):=m({vn>t}) and analogously ϕ, ψ, μ :[0, sup u) →[0, +∞] replacing everywhere vnwith u. Note that, thanks to the locality of the gradient, ϕ(t) =ϕn(t −1/n)for all t ∈(1/n, ∞)and the same holds for ψand μ. We claim that a) μnis absolutely continuous with −μ n(t)= Per({v∗ n>t}) |(v∗ n)|(v∗ n)−1(t),a.e. t∈(0,sup vn).(3.5)
20 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 If moreover |∇u| =0m-a.e. in {u >0}then also −μ n(t)=ˆ|∇vn|−1dPer({vn>t})a.e.t∈(0,sup vn); (3.6) b) ϕn, ψnare absolutely continuous with ϕ n(t)=−ˆ|∇vn|dPer({vn>t}),ψ n(t)=−Per({vn>t}),for a.e. t∈(0,sup vn). (3.7) Claim (3.5)in a) follows from [86, Lemma 3.10-3.11], since μn(t) =mN({v∗ n>t}) and Per({v∗ n>t})is concentrated on the point (v∗ n)−1(t). Claim b) is instead just a direct verification using the coarea formula (see (2.6)), since vn∈LIPc(X). Under the assumption |∇u| =0m-a.e. in {u >0}, by the Hölder inequality (using (3.6)) we have −ϕ n(t)≥−ψ n(t)2(−μ n(t))−1,(3.8) at a.e. t ∈(0, sup vn)which is a differentiability point for μn, ψn, ϕn. If instead we only know that (u∗)=0a.e. in {u∗>0}, we can still deduce (3.8) applying first Hölder inequality and then differentiating (see the argument in [86, Prop. 3.12]). Integrating the above inequality, recalling that Per({v∗ n>t}) =Nω 1 N Nμn(t)N−1 N, we get for every r, s ∈[0, sup vn]with s <r: ˆ {s<vn≤r} |∇vn|2dm≥ r ˆ sPer({vn>t}) Nω 1 N Nμn(t)N−1 N2ˆ|∇v∗ n|dPer({v∗ n>t})dt. (3.9) Hence, the isoperimetric inequality (2.8)gives directly ˆ {s<vn≤r} |∇vn|2dm≥AV R (X)2/N ˆ {s<v∗ n≤r} |∇v∗ n|2dmN,∀0≤s<r≤sup vn,(3.10) having also used coarea formula for the function v∗ nsince it is LIP([0, Rn]) as recalled before. Since vn=(u −1/n)+and v∗ n=(u∗−1/n)+, from the locality of the gradient we can rewrite (3.10)(after a change of variable) as ˆ {s+1/n<u≤r+1/n} |∇u|2dm≥AV R (X)2/N ˆ {s+1/n<u∗≤r+1/n} |∇u∗|2dmN,(3.11) for every s <rwith s, r∈(0, sup u −1/n]. Taking the limit as n →+∞we obtain
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 21 ˆ {s<u≤r} |∇u|2dm≥AV R (X)2/N ˆ {s<u∗≤r} |∇u∗|2dmN,∀0≤s<r≤sup u. (3.12) Step 2. We pass to the proof that Xis a cone. We claim that if equality occurs in (3.10) for some n ∈Nand some r, s ∈[0, sup vn]with r<s, then i) Per({vn>t}) =N(ωNAV R (X)) 1 Nμn(t)N−1 N, for a.e. t ∈(s, r). ii) If |∇u| =0m-a.e. in {u >0}, then |∇vn|is constant Per({vn>t})-a.e. for a.e. t ∈(s, r). Claim i) follows directly from the way we deduced (3.10)from (3.9)using the isoperimetric inequality (2.8). Claim ii) instead follows by the equality case in the Hölder inequality (3.8). We now suppose, as in the hypotheses, that uattains equality in (3.4), which means that equality holds in (3.12)with (s, r) =(0, sup u). We claim that equality must hold in (3.12)also for all s <rwith s, r∈(0, sup u). Suppose it fails for some s <r. Then, calling L(s, r)and R(s, r) respectively the left and right hand sides of (3.12), we have L(0,sup u)=L(0,s)+L(s, r)+L(r, sup u)>R(0,s)+R(s, r)+R(r, sup u)≥R(0,sup u), which contradicts the equality for (0, sup u). This proves the claim. Thus, equality holds in (3.11)for every s <r, with s, r∈(0, sup u −1/n]which is equivalent to equality in (3.10)for every s <rwith r, s ∈[0, sup vn]. Therefore i) holds and, provided |∇u| =0at m-a.e. point in {u >0}, also ii) holds for every s <rwith r, s ∈[0, sup vn]and n ∈N. Putting these together and by arbitrariness of n, implies that Per({u>t})=N(AV R (X)ωN)1/N m(μ(t))N−1 N,a.e. t∈(0,sup(u)),(3.13) and, if |∇u| =0m-a.e. in {u >0}, we get |∇u|≡ctPer({u>t})-a.e. for some constant ct≥0 (3.14) for a.e. t ∈(0, sup u). Therefore, there exists twith μ(t) >0so that equality occurs in (3.13), and recalling the rigidity in Theorem 2.6, we get that Xis isomorphic to an N-Euclidean metric measure cone. Step 3. Here we prove the functional rigidity of u, i.e. we prove that uis radial under the additional assumption: |∇u| =0m-a.e. on {u >0}. We first claim that (3.13) actually holds for every t ∈(0, sup u). Let t ∈(0, sup u) and consider a sequence tn↓tfor which (3.13)holds in every tn. Then, by lowersemicontinuity of the perimeter (see, e.g., [84, Proposition 3.6]) and continuity of μ, we get Per({u>t})≤lim n→∞ Per({u>t n})(3.13) =N(AV R (X)ωN)1/N μ(t)N−1 N.
22 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 Being the converse inequality always true (from (2.8)), the claim follows. Since {u >t} are bounded (recall that utends to zero at infinity), we can apply the rigidity Theorem 2.6 to deduce that for every t ∈(0, sup u) there exists a radius Rt>0and xt∈Xa tip for X (recall that Xis a cone from Step 2) so that m({u >t}BRt(xt)) =0, where denotes the symmetric difference. However {u >t}is open. Thus {u>t}=BRt(xt).(3.15) We stress that the notation xtis chosen because the cone structure may depend a priori on the isoperimetric superlevel set {u >t}. From here, the rest of the proof is devoted to show that xtis in fact independent of tand uis radial. To do so we will follow the lines of the argument used in [86, Theorem 5.1], for the compact case. Using (3.14)and (3.5) (recall that μ(t) =μn(t −1/n)) we get Nc−1 t(AV R (X)ωN)1 Nμ(t)N N−1=ˆ|∇u|−1dPer({u>t})=−μ(t)= Nω 1 N Nμ(t)N N−1 |(u∗)((u∗)−1(t))|, for a.e. t ∈(0, sup u). In particular, |∇u|=AV R (X) 1 N|(u∗)((u∗)−1(t))|Per({u>t})-a.e. and a.e. t∈(0,sup u).(3.16) Let M:= uL∞(m)∈[0, +∞). From the hypotheses u∗is non-negative, strictly decreasing and locally absolutely continuous (in fact locally Lipschitz) in {u∗>0} =[0, A)for some A ∈(0, +∞](in fact A =m({u >0})). Hence it admits a strictly decreasing continuous inverse (u∗)−1:(0, M] →[0, A), locally absolutely continuous in (0, M). Since (u∗)−1(M) =0, we can extend it by zero in [M, ∞)and call H:(0, ∞) →[0, A)this extension. In particular H∈ACloc(0, ∞). Observe that Hmight blow up at zero. Note also that, since u∗is locally Lipschitz in (0, A), it preserves L1-null sets. Hence pre-images of L1-null subsets of (0, M)via H=(u∗)−1are also L1-null. Therefore for a.e. t ∈(0, A) the function u∗is differentiable at (u∗)−1(t), the function His differentiable at tand (u∗)((u∗)−1(t))H(t)=(u∗((u∗)−1(t)))=1.(3.17) To conclude the proof, we need to show that f:= AV R (X)−1 NH◦u :{u >0} →[0, ∞) satisfies f(.)=d(x0,.),(3.18) for some point x0∈{u >0}. Observe that fis continuous. We start proving that: f∈LIPloc({u>0})and|∇f|=1m-a.e. in {u>0}.(3.19)
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 23 To show this we will use the chain rule in Lemma B.3 with u, Ω := {u >0}, ϕ := Hand I:= (0, ∞). To check the hypotheses we observe that by continuity u(Ω) ⊂⊂ (0, ∞)for all Ω⊂⊂ Ω. Moreover by (3.16)and (3.17)we have that for a.e. t ∈(0, M)it holds |H(u)||∇u|=|H(t)||(u∗)((u∗)−1(t))|AV R (X) 1 N=AV R (X) 1 N,Per({u>t})-a.e. Therefore by coarea (recall (2.6)) and the fact that m({|∇u| =0} ∩Ω) =0, we easily deduce that |H(u)||∇u| =AV R (X) 1 Nm-a.e. in Ω. In particular |H(u)||∇u| ∈L2 loc(m) and we can apply Lemma B.3 to deduce that f∈W1,2 loc ({u >0})with |∇f| =1, m-a.e. in {u >0}. Moreover from the local Sobolev-to-Lipschitz property (see [62, Prop. 1.10]) we deduce that f∈LIPloc({u >0})and |f(x)−f(y)|≤d(x, y),∀x, y ∈{u>0},with d(x, y)≤d(x, {u=0}).(3.20) This proves (3.19). Next, we claim that {f<t}=Bt(xt),∀t∈(0,A),(3.21) with xt∈{u >0}. We already know by (3.15)and since His strictly decreasing, that for every t ∈(0, A)the set {f<t}is a ball Brt(xt)for some rt≥0and xttip of X. In particular m({f<t}) =ωNθ(rt)Nand Per({f<t}) =(ωNθ)1 NNθ(rt)N−1, where θ:= AV R (X). Moreover by coarea formula (2.6) applied to −fand using (3.19) ωNθ[(rt)N−(rs)N]=m({f<t})−m({f<s})= ˆ {s≤f<t} |∇f|dm= t ˆ s Per({f<r})dr. Therefore the function (rt)Nis absolutely continuous with d dt(rt)N=(ωNθ)−1Per({f<t})=N(rt)N−1,a.e. t∈(0,A), from which follows that rt=a +t, for all t ∈(0, A), for some constant a ≥0. We claim that a =0. Indeed by continuity and Bishop-Gromov inequality we have aNωNAV R (X) ≤m(∩t>0Ba+t(xt)) = m(∩t>0{f<t})=m({f=0})=m({u=M})=0, where in the last equality we used that |∇u| =0m-a.e. in {u >0}. This proves (3.21). It remains to prove that xt≡x0for all t ∈(0, A). This would show (3.18)and conclude the proof. We argue by contradiction and suppose that xt=x¯ tfor some ¯ t<t <A. Set δ:= d(xt, x¯ t) >0. Recall that xtis a tip of X, hence there is a ray emanating from it and containing ¯xt, i.e. an isometry γ:[0, ∞) →Xwith γ0=xtand γδ=x¯ t. Consider the points x := γt∈∂Bt(xt) ={f=t}and y:= γδ+¯ t∈∂B¯ t(x¯ t) ={f=¯ t}.
24 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 Since γδ+¯ t∈Bt(xt)and γis an isometry, δ+¯ t<t. Therefore applying (3.20), since d(y, {u =0}) ≥d(y, ∂Bt(xt)) =d(x, y), we finally find a contradiction: t−¯ t=f(x)−f(y)≤d(x, y)=t−(¯ t+δ). From Step 1 of the above proof, we deduce the following that has its own interest. Proposition 3.5 (Improved Pólya-Szegő inequality). Let (X, d, m)be an RCD(0, N)space with N∈(1, ∞)and AV R (X) >0. Then for every u ∈LIPloc(X), non-negative, u(x) →0 as d(x, z) →+∞for some z∈X, and with (u∗)=0-a.e. in {u∗>0}, it holds ˆ {s<u<r} |∇u|2dm≥ r ˆ sPer({u>t}) Nω 1 N Nμ(t)N−1 N2ˆ|∇u∗ N|dPer({u∗ N>t})dt, ∀0≤s<r≤sup u. (3.22) Remark 3.6. Even if we shall not need it, we observe that Proposition 3.3, Proposition 3.5 and Theorem 3.4 hold replacing p =2with any p ∈(1, ∞), the proof is the same. We point out that the improved rearrangement inequality (3.22) appeared also in [13, Eq. (3.46)] for non-collapsed spaces and for functions defined on open sets (with finite volume) and with zero-Dirichlet boundary conditions. Remark 3.7 (On the necessity of (u∗)= 0and |∇u| = 0). We point out that, the hypothesis (u∗)=0in Theorem 3.4 is necessary to prove that uis radial. This is wellknown, see e.g. [31, Example 4.6] for an easy counterexample (in Rn) of a Lipschitz function saturating the Pólya-Szegő inequality with (u∗)= 0 occurring on a set of positive measure. In Theorem 3.4 we also assumed |∇u| =0at m-a.e. point of {u >0}. This was needed to carry out key computations by differentiating the distribution functions (see, e.g., (3.6)above), as also done in [86]. It is not clear to us at the moment if this assumption can be removed. 4. Regularity of extremal functions We discuss here the general regularity properties of extremal functions for the Sobolev inequalities (S) considered in this note. Theorem 4.1 (Regularity of extremal functions). Fix N∈(2, ∞)and set 2∗:= 2N/(N− 2). Let (X, d, m)be an RCD(K, N)space, for some K∈R, N∈(2, ∞)supporting a Sobolev inequality (S)with constant A >0, B≥0. Suppose that equality occurs in (S) for some u ∈W1,2 loc (X) satisfying uL2∗(m)=1. Then u ∈D(Δ)and −AΔu=(|u|2∗−2u−Bu).(4.1)
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 31 collapsed RCD-spaces: a reader interested in the case of smooth manifolds can skip this subsection. Lemma 6.1 (Density bound from reverse Sobolev). For every N∈(2, ∞), K∈R, there are constants λN,K ∈(0, 1), rK−,N >0(with r0,N =+∞), CN,K >0such that the following holds. Let (X, d, m)be an RCD(K, N)space and u ∈W1,2 loc (X) ∩L2∗(m), nonconstant satisfying u2 L2∗(m)≥A∇u2 L2(m),(6.1) for some A >0. Assume also that for some η∈(0, λN,K), ρ ∈(0, rK−,N ∧λN,K 8diam(X)) and x ∈Xit holds u2∗ L2∗(Bρ(x)) ≥(1 −η)u2∗ L2∗(m). Then m(Bρ(x)) ρN≤CN,K AN/2.(6.2) Proof. We fix a constant λ =λN,K ∈(0, 1) sufficiently small and to be chosen later. We also fix a constant rK−,N >0, with r0,N =+∞and with rK−,N small and to be chosen later in the case K<0(rK−,N will be chosen after λN,K). Assume ρ ≤rK−,N and η≤λN,K are as in the hypotheses. Observe that B4λ−1ρ(x) X. Up to choosing rK−,N small enough (when K<0) we can assume that 4λ−1ρ ≤˜rK−,N , where ˜rK−,N >0is the one given by Lemma B.2. Set r:= 4λ−1ρ ≥4ρand note that Br(x) X. Fix a cut-off function ϕ ∈LIPc(Br/2(x)) such that ϕ =1in Br/4(x), 0 ≤ϕ ≤1and Lip(ϕ) ≤10/r. Then from (B.2), since r≤˜rK−,N , we have uL2∗(Bρ(x)) ≤uϕL2∗(m)≤CN,Kr m(Br(x))1/N ∇uL2(m)+10CN,K m(Br(x))1/N uL2(Br(x)) ≤CN,Kr m(Br(x))1/N ∇uL2(m)+10CN,K m(Br(x))1/N (uL2(Bρ(x)) +uL2(Br(x)\Bρ(x))) ≤CN,Kr m(Br(x))1/N ∇uL2(m)+10CN,KuL2∗(m) m(Br(x))1/N (m(Bρ(x))1/N +λ1/2∗ m(Br(x))1/N ) Substituting (6.1), applying (2.5)(up to choosing rK−,N small enough so that r≤ RK−,N ), using that 1 −λ <1 −ηand simplifying uL2∗(m), we reach (1 −λ)1/2∗≤CN,Kr √Am(Br(x))1/N +10CN,K((λ/4)γ+λ1/2∗),
32 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 where γ>0is a constant depending only on N. Choosing λsmall enough with respect to Nand Kgives m(Bρ(x)) rN≤m(Br(x)) rN≤CN,K AN/2.(6.3) Recalling that r=4λ−1ρproves (6.2). 6.2. Concentration compactness for Sobolev extremals In the following theorem we show that a sequence of extremizing functions defined on a sequence of RCD(0, N) spaces, after a suitable rescaling of both the function and the space, admits a subsequence converging to a limit extremal function on some limit RCD(0, N) space. The idea is similar to the classical Lions’ concentration-compactness principle ([80,81]). The first step is a characterization of the failure of compactness in the critical Sobolev embedding by specific concentration and splitting of the mass phenomena (see Appendix A.2). The second step is observing that the extra information that the sequence is extremizing for the Sobolev inequality will prevent these pathological phenomena and ensure compactness. A crucial point will be to exploit the strict concavity property of the Sobolev inequality, and in particular of the function t → t2/2∗, to deduce that splitting the mass is not convenient in an extremizing sequence. Theorem 6.2. For every N∈(2, ∞), exists ηN∈(0, 1/2) such that the following holds. Let (Yn, ρn, μn, yn)be a sequence of pointed RCD(0, N)spaces supporting a Sobolev inequality (S)with An→A >0and Bn→B∈[0, ∞)and also satisfying either supnμn(B1(yn)) <+∞or diam(Yn) >η −1 N. Suppose there exist non-constant functions un∈W1,2(Yn)with unL2∗(μn)=1and sup y∈Ynˆ B1(y) |un|2∗dμn=ˆ B1(yn) |un|2∗dμn=1−η, (6.4) un2 L2∗(μn)≥˜ An∇un2 L2(μn)+Bnun2 L2(μn),(6.5) for ˜ An→A, and some η∈(0, ηN). Then, up to a subsequence, it holds: i) YnpmGH-converges to a pointed RCD(0, N)-space (Y, ρ, μ, y)supporting a Sobolev inequality as in (S)with constants A, B; ii) unconverges L2∗-strong to some u ∈W1,2 loc (Y)with |∇u| ∈L2(μ)and ˆ|∇un|2dμn→ˆ|∇u|2dμ, as n↑∞. If B>0, then the convergence is also W1,2-strong.
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 33 iii) It holds u2 L2∗(μ)=A∇u2 L2(μ)+Bu2 L2(μ). Proof. We subdivide the proof into different steps. Step 1. We take ηN:= λ0,N 8∧1 3, with λ0,N as in Lemma 6.1. In light of Theorem 2.10, to extract a subsequence converging pmGH it is sufficient to check that μn(B1(yn)) ∈ (v−1, v)for some v>1. If diam(Yn) >η −1 N≥8λ−1 0,N , thanks to the assumptions (6.4) and (6.5), we can apply Lemma 6.1 to obtain lim nμn(B1(yn)) ≤lim n CN (˜ An)N/2=CN AN/2<+∞, otherwise supnμn(B1(y1)) <+∞is directly true by the assumptions. On the other hand, since by assumption the spaces Ynsatisfy a Sobolev inequality with constants An, Bn, plugging in functions ϕn∈LIP(Yn)such that ϕn=1in B1(yn)with suppϕn⊂B2(yn), 0 ≤ϕn≤1and Lip(ϕn) ≤1, we get μn(B1(yn))2/2∗≤(An+Bn)μn(B2(yn)) ≤2N(An+Bn)μn(B1(yn)), where we used the Bishop-Gromov inequality. Since limn(An+Bn) =A +B>0we also obtain limnμn(B1(yn)) >0. Therefore up to a not relabeled subsequence, the spaces Yn pmGH converge to a pointed RCD(0, N)space (Y, ρ, μ, y). Moreover, the stability of the Sobolev inequalities [88, Lemma 4.1] ensures that Ysupports a Sobolev inequality as in (S)with constants A, B. This settles point i). Step 2. From now on we assume to have fixed a realization of the convergence in a proper metric space (Z, d)(as in Section 2.4). Let νn:= |un|2∗μn∈P(Z). Moreover we will denote by Br(z), z∈Z, and by Bn r(y), y∈Yn, respectively the balls in (Z, d) and in (Yn, ρn), recalling that we are identifying (Yn, ρn)as a subset of (Z, d). From Lemma A.6 we have that, up to a subsequence, (exactly) one of cases i), ii), iii) in the statement of Lemma A.6 holds. We claim i) (i.e. compactness) occurs. First, notice that vanishing as in case ii) cannot occur: lim n→∞ sup y∈Yn νn(BR(y)) ≥lim n→∞νn(B1(yn)) (6.4) =1−η, ∀R≥1. Thus, it remains to exclude the dichotomy case iii). Suppose by contradiction that iii) of Lemma A.6 holds for some λ ∈(0, 1) (with λ ≥limnsupzνn(BR(z)) for all R> 0), sequences Rn↑∞, (zn) ⊂Zand measures ν1 n, ν2 nwith supp(ν1 n) ⊂BRn(zn)and supp(ν2 n) ⊂Z \B10Rn(zn). We claim first that supp(ν1 n) ⊂B3Rn(yn)and supp(ν2 n) ⊂ Z \B4Rn(yn). Indeed λ ≥limnνn(B1(yn)) =1 −ηand lim n νn(BRn(zn)) ≥lim n ν1 n(BRn(zn)) = lim nν1 n(Z) = λ≥1−η.
34 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 Since νn(B1(yn)) =1 −ηand η<1/2, this implies that for nlarge enough BRn(zn) ∩ B1(yn) =0, which implies the claim, provided Rn≥1. Let ϕnbe a Lipschitz cut-off so that 0 ≤ϕn≤1, ϕn≡1on Bn 3Rn(yn), supp(ϕn) ⊂ Bn 4Rn(yn)and Lip(ϕn) ≤R−1 n, for every n ∈N. Since 1≥|ϕn|2+|(1 −ϕn)|2,in Z,(6.6) we can estimate by triangular inequality, the Leibniz rule and Young inequality ∇un2 L2(μn)≥ϕn|∇un|2 L2(μn)+(1 −ϕn)|∇un|2 L2(μn) ≥∇(unϕn)2 L2(μn)+∇(un(1 −ϕn))2 L2(μn) −2(1 + δ−1)un|∇ϕn|2 L2(μn)−2δ∇un2 L2(μn) :=Rn(δ) (6.7) for every δ>0and every n. Setting On:= Bn 4Rn(yn) \Bn 3Rn(yn), we have by the Hölder inequality un|∇ϕn|2 L2(μn)≤Rn−2un2 L2∗(On)μn(On)2/N ≤16v2/N un2 L2∗(On), having used that μn(On) ≤μn(Bn 4Rn(yn)) ≤(4Rn)Nμn(B1(yn)) ≤(4Rn)Nv, by the Bishop-Gromov inequality. Notice that we also have lim n→∞unL2∗(On)≤lim n→∞1−ν1 n(Z) −ν2 n(Z) 1/2∗ =0, from which we get limnun|∇ϕn|2 L2(μn)=0. Therefore, recalling that ∇un2 L2(μn)is uniformly bounded by (6.5), choosing appropriately δn→0, we get Rn(δn)→0.(6.8) Combining (6.7)with (6.8), recalling that limnAn= limn˜ An, we get 1(6.5) ≥lim n→∞An∇(unϕn)2 L2(μn)+An∇(un(1 −ϕn))2 L2(μn)+Bnun2 L2(μn) (S) ≥lim n→∞unϕn2 L2∗(μn)+un(1 −ϕn)2 L2∗(μn) +Bnun2 L2(μn)−unϕn2 L2(μn)−un(1 −ϕn)2 L2(μn) (6.6) ≥lim n→∞ν1 n(Z)2/2∗ +ν2 n(Z)2/2∗ ≥λ2/2∗+(1−λ)2/2∗>1,
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 35 having used the strict concavity of t → t2/2∗and the fact that λ ∈(0, 1). This gives a contradiction, hence dichotomy in iii) cannot happen. Step 3. In the previous step, we proved that case i) in Lemma A.6 occurs, i.e. there exists (zn) ⊂Zsuch that for every ε >0 there exists R:= R(ε)so that ´Bn R(zn)|un|2∗dμn≥ 1 −εfor all n ∈N. As soon as ε <1/2, we have Bn R(zn) ∩Bn 1(yn) =∅and ˆ Bn 2R+1(yn) |un|2∗dμn≥1−ε∀n∈N.(6.9) Moreover yn→yin Z, hence the sequence of probabilities |un|2∗μnis tight (Z is proper) and, along a not relabeled subsequence, converges in duality with Cb(Z) to some ν∈ P(Y). Additionally, up to a further subsequence we have that unis L2∗-weak convergent to some u ∈L2∗(μ)([8]) with supn∇unL2(μn)<∞and also that |∇un|2dμnωin duality with Cbs(Z) for some bounded Borel measure ω. Applying Lemma A.3, up to a further subsequence, we also deduce that unconverges L2 loc-strong to some u ∈L2 loc(μ), together with the facts u ∈W1,2 loc (Y)and |∇u| ∈L2(μ). Note that if B>0then actually u ∈W1,2(Y), by (6.5)and the lower semicontinuity of the L2-norm (2.15). We are in position to invoke Lemma A.7 to infer the existence of countably many points {xj}j∈J⊂Yand positive weights (νj), (ωj) ⊂R+, so that ν=|u|2∗μ +j∈Jνjδxj and ω≥|∇u|2μ +j∈Jωjδxj, with Aωj≥ν2/2∗ jand in particular jν2/2∗ j<∞. Moreover up to passing to a subsequence we can, and will, from now on assume that the limits limn∇un2 L2(μn)and limnBnu2 L2(μn)exist. Finally, by the lower semicontinuity of the L2-norm (see (2.15)) we have Bu2 L2(μ)≤limnBnu2 L2(μn), where Bu2 L2(μ) is taken to be zero when B=0and u2 L2(μ)=+∞. Also limn∇un2 L2(μn)≥ω(Z). Therefore 1 = lim n→∞ˆ|un|2∗dμn≥lim n→∞ ˜ An∇un2 L2(μn)+ lim n→∞Bnun2 L2(μn) ≥Aω(Z) + Bu2 L2(μ) ≥Aˆ|∇u|2dμ+ j∈J ν2/2∗ j+Bu2 L2(μ) (S) ≥ˆ|u|2∗dμ2/2∗ + j∈J ν2/2∗ j ≥ˆ|u|2∗dμ+ j∈J νj2/2∗ =ν(Y)2/2∗=1, having used, in the last inequality, the concavity of the function t2/2∗. In particular, all the inequalities must be equalities and, since t2/2∗is strictly concave, we infer that every term in the sum ´|u|2∗dμ +j∈Jν2/2∗ jmust vanish except one. By the assumption (6.4)and |u|2∗ mnνin Cb(Z), we have νj≤1 −ηfor every j∈J. Hence νj=0
36 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 and uL2∗(μ)=1. This means that unconverges L2∗-strong to u. Moreover, retracing the equalities in the above we have that limn´|∇un|2dμn=´|∇u|2dμand, when B>0, limn´|un|2dμn=´|u|2dμ. This proves point ii). Finally, equality in the fourth inequality is precisely part iii) of the statement. The proof is now concluded. 7. Radial functions: technical results In this section, we prove results about convergence and approximation of radial functions. The first one (Lemma 7.2 below) says that, given a sequence of RCD spaces converging in the pmGH-sense, a radial function on the limit space is the limit of the same radial functions along the sequence. We will need the following simple fact. We omit the proof, which is an easy consequence of Cavalieri’s formula and Bishop-Gromov inequality. In this section, we denote dz(.) := d(z, .)the distance function from a point z. Lemma 7.1. Let (X, d, m)be an RCD(0, N)space for some N∈(2, ∞). Then for every α>N, z∈Xand r>0it holds ˆ Br(z)c dz(·)−αdm≤m(Br(z)) rNCN,αrN−α.(7.1) Lemma 7.2. Let (Yn, ρn, μn, zn)be a sequence of RCD(K, N)spaces, for some K∈ R, N∈(2, ∞), that is pmGH-converging to (Y, ρ, μ, z0). Let p ∈(1, ∞)and f∈C(R) satisfying |f(t)|p≤C|t|−α, for some α>0. Suppose also that lim R→+∞sup nˆ BR(zn)c ρ−α zndμn=0,(7.2) where ρzn(·) := ρn(·, zn). Then, f◦ρznconverges Lp-strong to f◦ρz0. In particular, for any un∈Lp(μn)that converges Lp-strong to f◦ρz0, it holds un−f◦ρznLp(μn)→0.(7.3) Proof. We only need to prove that f◦ρznconverges Lp-strong to f◦ρz0, then (7.3) follows from the linearity of the Lp-convergence (2.14). The assumptions on fimply that fis uniformly continuous and we denote by ω: [0, ∞) →[0, ∞)a global modulus of continuity for f. Observe that fis also bounded. In the sequel, we fix (Z, d)a realization of the convergence and recall that d|Yn×Yn=ρn. We can estimate ˆ|f◦dz0−f◦dzn|pdμn
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 37 ≤ˆ BR(zn) |f◦dz0−f◦dzn|pdμn+2 pˆ Z\BR(zn) |f|p◦dz0+|f|p◦dzndμn ≤μn(BR(zn))ω(d(z0,z n))p+2 pCˆ BR(zn)c d(z0,·)−α+d(zn,·)−αdμn ≤μn(BR(zn))ω(d(z0,z n))p+2 pC·2αsup nˆ BR(zn)c d(zn,.)−αdμn, where in the last step we assume that nis big enough so that d(zn, z0) <R/2, which ensures d−1(z0, ·) ≤2d−1(zn, ·)in BR(zn)c. Since supnμn(BR(z0)) <+∞for every R>0, by the pmGH-convergence, we can send first n ↑∞and then R↑∞to obtain f◦dz0−f◦dznLp(μn)→0. Fix ϕ ∈Cbs(Z) and R>0so that supp(ϕ) ⊂BR(z0), then ˆϕf ◦dz0dμ−ˆϕf ◦dzndμn≤ ≤ϕ∞ˆ suppϕ|f◦dz0−f◦dzn|dμn+ˆϕf ◦dz0dμn−ˆϕf ◦dz0dμ ≤ϕ∞μn(BR(z0))1−1/pf◦dz0−f◦dznLp(μn)+ˆϕf ◦dz0dμn−ˆϕf ◦dz0dμ. Sending n ↑∞we obtain that f◦dzndμnf◦dz0μin duality with Cbs(Z). It remains to prove that f◦ρznLp(μn)→f◦ρz0Lp(μ). Since f◦dz0−f◦dznLp(μn)→0, it is enough to show that f◦dz0Lp(μn)→f◦ρz0Lp(μ). Clearly f◦ρz0Lp(μ)=f◦dz0Lp(μ)≤ limnf◦dz0Lp(μn), hence we only need to show f◦dz0Lp(μ)≥limnf◦dz0Lp(μn). We can assume nis big enough so that d(z0, zn) ≤1. For every R≥4fix a cut-off function ϕR∈Cbs(Z), 0 ≤ϕR≤1, such that ϕR≡1in BR(z0)and with support in B2R(z0). Then ˆ|ϕR(|f|p◦dz0)−|f|p◦dz0|dμn≤ˆ BR(z0)c |f|p◦dz0dμn≤2·2αCsup nˆ BR/2(zn)c d−α zndμn, where we have used that BR(z0)c⊂BR/2(zn)cand d−1 z0≤2d−1 znin BR/2(zn)c. This shows that ˆϕR(|f|p◦dz0)−|f|p◦dz0dμn≤εR→0,as R↑∞, where εRis independent of n. Therefore −εR+lim nˆ|f|p◦dz0dμn≤lim nˆϕR(|f|p◦dz0)dμn=ˆϕR(|f|p◦dz0)dμ≤ˆ|f|p◦ρz0dμ. Sending Rto infinity, we conclude the proof.
38 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 The second result of this section is a technical fact that will play a key role in the proof of our main theorem. It states that a Euclidean bubble which is strongly concentrated around a point is close to a spherical bubble. Lemma 7.3. For every N∈(2, ∞), there are constants CN, α=α(N) >0such that the following holds. Given σ≥1, set 2∗=2N/(N−2) and feu(t):= σN−2 2 1+σ2t2N−2 2 ,f sphere(t):= σN−2 2 1+2σ2(1 −cos(t)N−2 2 ,t∈[0,π]. Let (X, d, m)be RCD(N−1, N), z∈X, dz(.) := d(z, .)and v:= σNm(Bσ−1(z)). Then (feu −fsphere)(dz)L2∗(m)+∇(feu −fsphere)(dz)L2(m)≤CNσ−α(√v+1). Proof. We fix η∈(0, 1) to be chosen later. Denote B:= B1 ησ (z). In what follows CN>0 is a constant depending only on N, its value may vary from line to line without notice and without being relabeled. By Bishop-Gromov and the assumptions, we get m(B)≤v(ησ)−N.(7.4) We divide the proof into two steps, one for the L2∗-norm and one for the L2-norm of the gradient. Step 1. We start estimating (feu −fsphere)(dz)L2∗(m)≤(feu −fsphere)(dz)L2∗(B) +fsphere(dz)L2∗(Bc)+feu(dz)L2∗(Bc)=: I + II + III. We analyze each term separately. We start with I. Recall that |2(1 −cos(t)) −t2|≤ct4,1−cos(t)≤ct2,∀t≥0, for some numerical constant c >0. Using ||x|p−|y|p| ≤Cp|x −y|(|x|p−1+|y|p−1)with p =(N−2)/2and the above estimates we have for all t ∈[0, (ησ)−1)the following: |feu −fsphere|(t) ≤CNσ2(1 −cos(t)) −t21 σ+2σ(1 −cos(t)) N−2 2−1+1 σ+σt2 N−2 2−1 1 σ+σt2N−2 21 σ+2σ(1 −cos(t))N−2 2 ≤CN σ1 (ησ)4·(σ−1+(η2σ)−1)N−2 2−1 σ2−N≤CNη−NσN−2 2−2. This and (7.4) directly implies that
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 39 (I)2∗=ˆ B1 ησ (zn) |feu −fsphere|2∗(dz)dm≤CNvη−N(2∗+1)σ−2·2∗. We pass to II. Note that |fsphere(t)|2∗, |feu(t)|2∗≤CNσ−Nt−2N, having used that 1 − cos(t) ≥ct2in [0, π]for some numerical constant c >0. Hence applying Lemma 7.1 and using (7.4) (II)2∗+ (III)2∗≤CNvσ−N(ση)N≤vCNηN. Step 2. From the chain rule for the gradient and the fact that |∇d(z, .)| =1m-a.e., we have ∇(feu −fsphere)(dz)L2(m)≤(f eu −f sphere)(dz)L2(B) +f sphere(dz)L2(Bc)+f eu(dz)L2(Bc)=: I+II + III. We start with I n. Reasoning similarly to Step 1, we can estimate for all t ∈[0, (ησ)−1) |f eu −f sphere|(t)=(N−2)σ t1 σ+2σ(1 −cos(t))N 2−sin(t)1 σ+σt2N 2 1 σ+σt2N 21 σ+2σ(1 −cos(t))N 2 ≤CNσN+2t2(1 −cos(t)) −t21 σ+2σ(1 −cos(t))N 2−1 +1 σ+σt2N 2−1+CNσN+1|sin(t)−t|1 σ+σt2N 2 ≤CNσN+2t51 σ+1 ση2N 2−1+CNσN+1t31 σ+1 ση2N 2−1 ≤CNσN 2−2η−N−3. Therefore, again using (7.4)we deduce (II n)2≤Cvη−3N−6σ−4. As above we can directly estimate |f eu|,|f sphere|2≤CNσ−N+2t2−2N,t∈[0,π], having used | sin(t)| ≤ct and 1 −cos(t) ≥ct2in [0, π]. Hence by Lemma 7.1 and using (7.4) (II n)2+ (III n)2≤CNvσ−N+2(ση)N−2≤vCNηN−2. Combining all cases and taking η:= σ−βwith β>0small enough depending on Nwe conclude, using also that v1/2∗+v1/2≤2 +2 √v.
40 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 8. Proof of the main results 8.1. Stability in the compact case In this part, we prove the main qualitative stability result of this note. Note that this proves our main Theorem 1.1. We will also provide a proof of Corollary 1.3 at the end. Given N>2the family of spherical bubbles in a metric space (X, d)is denoted by Msphere(X) := {a(1 −bcos d(x, z0))2−N 2:a∈R,b∈(0,1),z 0∈X}∪{u≡a:a∈R}. Theorem 8.1. For every ε >0and N∈(2, ∞)there exists δ:= δ(ε, N) >0such that the following holds. Let (X, d, m)be an RCD(N−1, N)space for some N∈(2, ∞)with m(X) =1, set 2∗=2N/(N−2) and suppose that there exists u ∈W1,2(X) non-constant satisfying u2 L2∗(m)−u2 L2(m) ∇u2 L2(m) >2∗−2 N−δ. (8.1) Then there exists w∈M sphere(X) such that ∇(u−w)L2(m)+u−wL2∗(m) uL2∗(m)≤ε. (8.2) Moreover if w≡a ∈R, then a ∈Rcan be chosen so that the reminder R:= u−a satisfies for some x ∈X R·R−1 L2−√N+1cos(d(·,x))L2≤CN(εα+δ)β,(8.3) for some positive constants α, β, CNdepending only on N. Proof. By scaling invariance, it is not restrictive to assume uL2∗(m)=1. We only need to prove the first part, as the second follows from Proposition 8.3 below. We argue by contradiction and suppose that there exist ε >0, a sequence of RCD(N−1, N) spaces (Xn, dn, mn)and non-constant functions un∈W1,2(Xn)with unL2∗(mm)=1so that un2 L2∗(mn)≥˜ An∇unL2(mn)+un2 L2(mn),(8.4) with ˜ An→2∗−2 Nand satisfying
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 47 1−η=ˆ Btn(yn) |un|2∗dmn=sup y∈Xnˆ Btn(y) |un|2∗dmn,n∈N. Define now σn:= t−1 nand (Yn, ρn, μn, yn) := (Xσn, dσn, mσn, yn), where dσn:= σndn, mσn:= σN nmnand uσn:= σ−N/2∗ nun∈W1,2(Yn). In particular, by scaling, for every n ∈Nwe have 1−η=ˆ B1(yn) |uσn|2∗dμnand uσnL2∗(μn)≥(An−1/n)∇uσnL2(μn). By the assumption, we have the uniform bounds 2V1 NEucl(N, 2) ≤An≤2V−1 NEucl(N, 2). Thus, up to subsequences, we can clearly suppose that An→A, for some A >0 finite. We can now invoke Theorem 6.2 (the assumptions are satisfied as diam(Yn) =+∞) with η:= ηN/2and get that up to a subsequence (Yn, ρn, μn, yn) pmGH-converges to some RCD(0, N)space (Y, ρ, μ, ¯y) supporting a Sobolev inequality (S)with constant A > 0, B=0. Moreover we have L2∗-strong convergence of uσnto a function u ∈W1,2 loc (Y) attaining equality in this said Sobolev inequality and ∇uσnL2(mσn)→∇uL2(μ). From [88, Theorem 4.6] we have AV R (Y) =(Eucl(N, 2)/A)Nand in particular usatisfies the assumptions of Theorem 5.3, which gives that Yis isomorphic to a N-Euclidean metric measure cone with tip z0and u(y)= a (1 + bρ2(y,z0))N−2 2 ,y∈Y, for suitable a ∈R, b >0. Take any zn→z0. Then up to subsequence we can assume that mσn(B1(zn)) ≤ CNAV R (Y)hold for every n. Writing f(t) := a(1 +bt2)2−N 2for every t ∈R+, recalling |f|2∗, |f|2≤Ct−2N+2 and arguing as for (8.12), we see that all the hypotheses of Lemma 7.2 are fulfilled both for f◦ρ(·, z0)and for f◦ρ(·, z0). We therefore apply Lemma 7.2 twice to get that f◦dσn(·, zn)converges L2∗-strong to uand that |f| ◦dσn(·, zn)converges L2-strong to |∇u|. We can thus combine Lemma A.5 with the convergence of the gradient norms to deduce, from the parallelogram identity, that lim n→∞∇uσn−f◦dσn(·,z n)L2(mσn)=0.(8.18) Scaling back, (8.18) becomes lim n→∞∇un−(σN/2∗ nf)◦(σndn(·,z n))L2(mn)=0. This means that the sequence vn:= aσN/2∗ n(1 +bσ2 ndn(·, zn)2)2−N 2∈M eu(Xn), satisfies lim n→∞ ∇(un−vn)L2(mn) ∇unL2(mn) =0,
48 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 having used that ∇unL2(mn)≥CNAV R (Xn)1/N unL2∗≥CNV1/N . This is a contradiction with (8.17)and concludes the proof. From the above stability, the next corollary directly follows (proving also Corollary 1.5). Corollary 8.5. Let (X, d, m)be an RCD(0, N)space with N∈(2, ∞), AV R (X) >0. Then AV R (X) 1 NEucl−1(N,2) = inf v∈Meu(X) ∇vL2(m) vL2∗(m) . Acknowledgments F.N. is supported by the Academy of Finland Grant No. 314789. I.Y.V. is supported by the Academy of Finland projects Incidences on Fractals, Grant No. 321896 and Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups, Grant No. 328846. Appendix A. Concentration compactness: non-compact case Here we extend the concentration compactness tools for a sequence of converging RCD spaces (developed in [88]in compact setting) to the non-compact case. The main difference is that here mass can also escape to infinity and so we need an additional result (see Lemma A.6). Some additional technical convergence results will be also needed and proved in Section A.1. A.1. Technical convergence lemmas Throughout this part we fix a sequence (Xn, dn, mn, xn)of pointed RCD(K, N) spaces, n ∈N∪{∞}, for some K∈R, N∈(1, ∞)with Xn pmGH →X∞. We also fix a proper metric space (Z, d) realizing the convergence via extrinsic approach [58](see Section 2.4). We start with a version of the Brezis-Lieb Lemma [30]. Lemma A.1 (Brezis-Lieb type Lemma). Let q, q∈(1, ∞)and suppose that un∈Lq(mn) satisfy supnunLq(mn)<+∞and that unconverges in Lq-strong to some u∞∈Lq∩ Lq(m∞). Then, for any sequence vn∈Lq(mn)such that vn→u∞strongly both in Lq and Lq, it holds lim n→∞ˆ|un|qdmn−ˆ|un−vn|qdmn=ˆ|u∞|qdm∞.(A.1) Proof. The proof is the same as in [88, Prop. 6.2]. Even if the argument there is done assuming finite reference measure, it is used only at the end when applying the Hölder
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 49 inequality. In that step here is enough to multiply by an arbitrary ϕ ∈Cbs(Z)and argue in the same way. (Note also that the assumptions q∈[2, ∞)and q∈(1, q), even if present in the statement of [88, Prop. 6.2] are actually not used in its proof). We shall need an alternative version of the semicontinuity result (2.16)to deal with locally Sobolev functions; we include a proof since we could not find it in the literature. Lemma A.2. Let p ∈(1, ∞)and suppose (fn) ⊂W1,2 loc (Xn)is Lp-strong converging to f∞. Then ∇f∞2 L2(m∞)≤lim n→∞∇fn2 L2(mn),(A.2) (meaning that, if the right hand side is finite, then f∞∈W1,2 loc (X∞)and (A.2)holds). Proof. Since |fn| →|f∞|Lp-strongly (see [8, a) in Prop. 3.3]) and |∇fn| =|∇|fn|| m-a.e. for every fn, without loss of generality we can suppose fn, f∞nonnegative. If the liminf in (A.2)is infinite, there is nothing to prove. So, let us assume that it is finite. For every k∈N, we consider ϕk∈LIP([0, ∞)with Lip(ϕk) ≤1, ϕk(0) =0, converging point-wise to the identity as k↑∞and such that {ϕk(fn)}nis L2-bounded. For instance we can take ϕk(t) := (t −1/k)+∧k, indeed ϕk(fn)2 L2(mn)≤k2mn({fn>1/k})≤k2+pfnp Lp(mn), for every n ∈N. Again by [8, a) in Prop. 3.3], we have ϕk(fn)is Lp-strong convergent to ϕk(f∞). Moreover is also L2-bounded, thus it is also L2-weak convergent to ϕk(f∞). Then, by (2.16)we have ϕk(f∞) ∈W1,2(X∞)and ∇(ϕk(f∞))2 L2(m∞)≤lim n→∞∇(ϕk(fn))2 L2(mn)≤lim n→∞∇fn2 L2(mn)<∞, having used the fact that ϕkis 1-Lipschitz. By arbitrariness of k>0and since ϕk(f∞) → f∞pointwise, we see by semicontinuity (2.1)that (A.2) follows. The following lemma allows extracting L2 loc-converging subsequences from W1,2boundedness. Lemma A.3. Let p ≥2and suppose un∈W1,2 loc (Xn)converges Lp-weak to u∞∈Lp(m∞) and supn∇unL2(mn)<∞. Then, up to a subsequence unconverges L2 loc-strong to u∞∈W1,2 loc (X∞)with |∇u∞| ∈L2(m∞). Proof. We first prove the L2 loc convergence. Consider ϕ ∈Lipbs(Z) (recall that (Z, d) is a space realizing the convergence). Since supnmn(BR(xn)) <+∞, for every R>0, by Hölder inequality we have supnϕunL2(mn)<+∞. Analogously using the Leibniz
50 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 rule, ϕun∈W1,2(Xn)with supn∇(ϕun)L2(mn)<∞. Thus there exists a subsequence (nk)(see [58, Theorem 6.3]) such that ϕunkconverges L2-strong to some v, which must be equal to ϕu∞by uniqueness of weak limits. Hence the whole sequence ϕunis L2-strongly convergent to ϕu∞. The fact that u∞∈W1,2 loc (X∞) follows by the Mosco convergence of the Cheeger energies (see (2.16)), indeed for every ϕ ∈LIPbs(Z), Ch(ϕu∞) ≤limnCh(ϕun) <∞. It remains to prove that |∇u∞| ∈L2(m∞). Fix a ball B⊂Zand take ϕ ∈Lipbs(Z) equal to 1on B. Using [8, Lemma 5.8], we have ´B|∇u∞|2dm∞=´B|∇(ϕu∞)| dm∞≤limn´B|∇(ϕun)|2dmn≤supn∇unL2(mn)< ∞. Where in the first and last step we used the locality of the gradient. By the arbitrariness of Bthis implies |∇u∞| ∈L2(m∞). Lemma A.4. Let p ≥2and u∞∈W1,2 loc (X∞) ∩Lp(m∞)with |∇u∞| ∈L2(m∞). Then, there exists a sequence un∈W1,2 loc (Xn) ∩Lp(mn)that converges Lpand L2 loc-strong to u∞and so that |∇un|converges L2-strong to |∇u∞|. Proof. By Lemma 3.2 there exists a sequence un∈W1,2(X∞) ∩Lp(m∞)such that un→u∞in Lp(m∞)and |∇un| →|∇u∞|in L2(m∞). From [88, Lemma 6.4] (there written for compact spaces, but the same proof works in the present setting) there exists a sequence uk n∈W1,2(Xn)that converges Lpand W1,2-strong to un. By [8, Theorem 5.7] this implies that |∇uk n|converges L2-strong to |∇(ηkun)|. The conclusion then follows via diagonal argument. Finally the L2 loc-strong convergence follows from Lemma A.3. We prove a convergence result for pairings (the case p = 2 follows from [8, Theorem 5.4]). Lemma A.5. Let p ∈[2, ∞)and un, vn∈Lp(mn) ∩W1,2 loc (Xn)be converging Lpstrong to u∞, v∞respectively. Suppose that u∞∈W1,2 loc (X∞), that ∇unL2(mn)→ ∇u∞L2(m∞)<+∞and limn∇vnL2(mn)<+∞. Then v∞∈W1,2 loc (X∞), |∇v∞| ∈ L2(m∞)and lim n→∞ˆ∇un,∇vndmn=ˆ∇u∞,∇v∞dm∞. Proof. The fact that v∞∈W1,2 loc (X∞)with |∇v∞| ∈L2(m∞) follows from Lemma A.2. In particular by Cauchy-Schwarz ∇u∞, ∇v∞∈L1(m∞). Let t >0and notice that un+tvnconverges Lp-strong to u∞+tv∞by (2.14). Applying again Lemma A.2 we have u∞+tv∞∈W1,2 loc (X∞)and ˆ2t∇u∞,∇v∞+|∇u∞|2+t2|∇v∞|2dm∞=ˆ|∇(u∞+tv∞)|2dm∞ (A.2) ≤lim n→∞ˆ|∇(un+tvn)|2dmn ≤2tlim n→∞ˆ∇un,∇vndmn+2lim nˆ|∇vn|2dmn+ˆ|∇u∞|2dm∞.
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 51 Simplifying ´|∇u∞|2dm∞, dividing by tand sending t ↓0we obtain ´∇u∞, ∇v∞dm∞ ≤limn→∞ ´∇un, ∇vndmn. Arguing analogously for t <0, we conclude. A.2. Concentration compactness principles Here we briefly extend two concentration compactness principles from [80,81](see also [94]) for general sequences of probabilities on metric measure spaces. The first deal with an arbitrary sequence of probability measures on varying ambient space. Compare also with the version [12, Lemma 2.1]. Lemma A.6. Let (Z, d)be a complete and separable metric spaces and let νn∈P(Z), for n ∈N. Then, up to a subsequence, one of the following holds: i) Compactness. There exists (zn) ⊂Zsuch that for all ε >0, there exists R>0 satisfying νn(BR(zn)) ≥1−ε, ∀n∈N. ii) Vanishing. lim n→∞sup z∈Z νn(BR(z)) = 0,∀R>0. iii) Dichotomy. There exists λ ∈(0, 1) with λ ≥limnsupz∈Zνn(BR(z)), for all R>0, so that: there exists Rn↑∞, (zn) ⊂Zand there are ν1 n, ν2 ntwo non-negative Borel measures satisfying 0≤ν1 n+ν2 n≤νn, supp(ν1 n)⊂BRn(zn),supp(ν2 n)⊂Z\B10Rn(zn), lim n→∞ λ−ν1 n(Z)+(1 −λ)−ν2 n(Z)=0. The above can be obtained arguing exactly as in [94, Lemma I in Section 4.3] and therefore its proof is omitted. We briefly comment on the difference in case iii) with respect to [94]: our formulation of case iii) using a sequence Rnfollows from the one used in [94](where Ris fixed depending on a parameter ε >0) with a diagonal argument (this is observed also in the proof of [94, Theorem 4.9]); the condition λ ≥limnsupz∈Zνn(BR(z)) (not present in [94]) instead can be directly checked to hold by the way λis chosen in the proof. The second principle is a concentration compactness result for the Sobolev embedding stating that concentration may occur only at countably-many points. With respect to [88, Lemma 6.6], here we extend the principle to deal with varying pmGH-convergent RCD spaces (hence, the difference arises when considering noncompact limit spaces).
52 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 Lemma A.7. Let (Xn, dn, mn, xn), n ∈N∪{∞}, be pointed RCD(K, N)spaces, K∈R, N∈(1, ∞)with Xn pmGH →X∞and assume that Xnsupports a Sobolev inequality (S) with uniformly bounded constants An>0, Bn≥0. Suppose further that un∈W1,2 loc (Xn) ∩L2∗(mn)with supn∇unL2(mn)<∞is L2 locstrong converging to u∞∈L2∗(m∞)and suppose that |∇un|2mnω, |un|2∗ mnν in duality with Cbs(Z) and Cb(Z), respectively (where (Z, d)is a fixed realization of the convergence). Then, u∞∈W1,2 loc (X∞)with |∇u∞| ∈L2(m∞)and: i) there exists a countable set of indices J, points (xj)j∈J⊂X∞and weights (νj)j∈J⊂ R+so that ν=|u∞|2∗ m∞+ j∈J νjδxj; ii) there exists (ωj)j∈J⊂R+satisfying ν2/2∗ j≤(limnAn)ωjand such that ω≥|∇u∞|2m∞+ j∈J ωjδxj. In particular, we have jν2/2∗ j<∞. Proof. We subdivide the proof into two steps. Step 1. Suppose first that u∞=0. Then, the conclusion follows arguing as in Step 1 of [88, Lemma 6.6] taking here ϕa Lipschitz and boundedly supported (instead of only Lipschitz) cut-off and using the assumed L2 loc-strong convergence. Step 2. For general u∞, the idea is to apply the above to ‘u∞−un’ and then use a Brezis-Lieb lemma to recover the information for u∞. Take ˜una recovery sequence given by Lemma A.4 for u∞. Thus, for every ϕ ∈Lipbs(Z)+, we have ϕunis L2-strong to ϕu∞and L2∗-bounded and ϕ˜unis L2and L2∗-strong convergent to ϕu∞. Therefore Lemma A.1 ensures lim n→∞ˆ|ϕ|2∗|un|2∗dmn−ˆ|ϕ|2∗|un−˜un|2∗dmn=ˆ|ϕ|2∗|u∞|2∗dm∞.(A.3) Now define vn:= un−˜unand notice that all the assumptions ensure that vnis L2 locstrong and L2∗-weak convergent to zero. From the bounds |vn|2∗≤22∗(|un|2∗+|˜un|2∗) and |∇vn|2≤2(|∇un|2+|∇˜un|2)by tightness we can extract a not relabeled subsequence where |vn|2∗ mnconverge in duality with Cb(Z) to ¯νand |∇vn|2mnconverge in duality with Cbs(Z) to a finite Borel measure ¯ω. Then from Step 1, i), ii) hold true for (vn), for suitable weights (νj), (ωj) ⊂R+and points (xj) ⊂X∞. Then passing to the limit in (A.3)
F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 53 ˆϕ2∗dν−ˆϕ2∗d¯ν=ˆϕ2∗|u∞|2∗dm∞,∀ϕ∈Lipbs(Z)+. This in turn implies ν=|u∞|2∗ m∞+¯ν=|u∞|2∗ m∞+jνjδxjthat is point i). We pass to prove ii) and therefore we need to show separately that ω({xj})=¯ω({xj})≥ωj,∀j∈J, ω≥|∇u∞|2m∞. The first can be verified arguing exactly as in Step 2 of [88, Lemma 6.6] replacing the usage of [8, Theorem 5.7] with Lemma A.4 above. For the second, we fix ϕ ∈Cbs(Z), ϕ ≥0, and χ∈LIPbs(Z) be such that χ=1in supp(ϕ). It is easy to check that χun is W1,2-weak converging to χu∞(recall that un→u∞in L2 loc). Then, [8, Lemma 5.8] ensures that ˆϕ|∇u∞|2dm∞=ˆϕ|∇(χu∞)|2dm∞ ≤lim n→∞ˆϕ|∇(χun)|2dmn= lim n→∞ˆϕ|∇un|2dmn By arbitrariness of ϕ, we showed ii) and the proof is now concluded. Appendix B. Technical results In this appendix, we collect basic results about Sobolev inequalities and a version of the chain rule for the weak upper gradient. Lemma B.1. Let (X, d, m)be an RCD(K, N)space, N∈(2, ∞), K∈R, satisfying for A >0 uL2∗(m)≤A∇uL2(m),∀u∈LIPc(X),(B.1) where 2∗:= 2N N−2. Then (B.1)holds also for all u ∈W1,2 loc (X) satisfying m({|u| >t}) < +∞for all t >0. Proof. It is enough to prove (B.1)for non-negative functions. First note that (B.1) holds for every u ∈W1,2(X), by density in energy of Lipschitz functions [5]and by the lower semicontinuity of the L2∗-norm with respect to L2-convergence. For a general u ≥0as in the hypotheses, if ´|∇u|2dm =+∞there is nothing to prove, otherwise take un:= ((u −1/n)+) ∧n ∈W1,2(X) (since un, |∇un| ∈L2(m)) and then send n →+∞). Lemma B.2 (Local Sobolev embedding). Let (X, d, m)be an RCD(K, N)space for some K∈R, N∈(2, ∞)and set 2∗:= 2N/(N−2). Then exists ˜rK−,N >0(with ˜r0,N =+∞) such that for every BR(x) X, R≤˜rK−,N it holds
54 F. Nobili, I.Y. Violo / Advances in Mathematics 440 (2024) 109521 uL2∗(m)≤CN,KR m(BR(x))1/N ∇uL2(m),∀u∈W1,2 0(BR/2(x)).(B.2) Proof. It is enough to prove the statement for u ∈LIPc(BR/2(x)). Thanks to the uniformly locally doubling property of (X, d, m)and the validity of a local (1, 1)-Poincaré inequality ([91]), from the results in [65]the following Sobolev-Poincaré inequality holds BR(x) |f−fBR(x)|2∗dm1 2∗ ≤C(N,K,R0)R B2R(x) |∇f|2dm1 2,∀f∈LIP(X), (B.3) for every R≤R0and where fBR(x):= fflBR(x)fdm(see also [26]). Moreover if K≥0, the constant C(N, K, R0)can be taken independent of R0. Hence applying (B.3)to u ∈LIPc(BR/2(x)) we can write ˆ BR(x) |u|2∗dm1 2∗ ≤CN,KRm(BR(x))1/2∗ m(B2R(x))1/2ˆ B2R(x) |∇u|21 2+m(BR(x))1/2∗−1ˆ BR/2(x) |u|dm ≤CN,KRm(BR(x))−1/N ˆ B2R(x) |∇u|21 2+m(BR/2(x))1−1/2∗ m(BR(x))1−1/2∗ˆ BR/2(x) |u|2∗dm1 2∗ , where we have used that supp(u) ⊂BR/2(x). Thanks to the reverse doubling inequality (recall (2.5)), assuming R≤RK−,N , we can absorb the rightmost term inside the lefthand side of the above to obtain (B.2)as desired. A technical result needed in this note is a chain rule for the composition with an absolutely continuous function ϕ, which we could not find in the literature (see [55]or [59]for the classical one with ϕLipschitz). Lemma B.3 (Chain rule for composition with AC-functions). Let (X, d, m)be a proper metric measure space and u ∈LIPloc(Ω) with Ω ⊂Xopen. Let ϕ ∈ACloc(I)with I open interval such that u(Ω) ⊂⊂ Ifor every Ω⊂⊂ Ω. Suppose also that |ϕ(u)||∇u| ∈ L2 loc(Ω). Then ϕ(u) ∈W1,2 loc (Ω) and |∇ϕ(u)| =|ϕ(u)||∇u|m-a.e. Proof. Up to subtracting a constant, we can assume that 0 ∈Iand ϕ(0) =0. Then with a cut-off argument we can reduce to the case when u ∈LIPc(X) and ϕ ∈AC(R) with compact support and ϕ(0) =0. We argue by approximation and define functions ϕn∈LIP(R)by
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