Limits of manifolds with a Kato bound on the Ricci curvature
aa r X i v : . [ m a t h . DG ] F e b LIMITS OF MANIFOLDS WITH A KATO BOUND ON THERICCI CURVATURE
GILLES CARRON, ILARIA MONDELLO, AND DAVID TEWODROSE
Abstract.
We study the structure of Gromov-Hausdorff limits of sequencesof Riemannian manifolds { ( M nα , g α ) } α ∈ A whose Ricci curvature satisfies a uni-form Kato bound. We first obtain Mosco convergence of the Dirichlet energiesto the Cheeger energy and show that tangent cones of such limits satisfy the RCD(0 , n ) condition. When assuming a non-collapsing assumption, we in-troduce a new family of monotone quantities, which allows us to prove thattangent cones are also metric cones. We then show the existence of a well-defined stratification in terms of splittings of tangent cones. We finally provevolume convergence to the Hausdorff n -measure. Introduction
We study the structure of Gromov-Hausdorff or measured Gromov-Hausdorfflimits of manifolds whose Ricci curvature satisfy a Kato type bound. Our resultsextend previous results proven by J. Cheeger and T. Colding for limits of manifoldscarrying a uniform lower bound on the Ricci curvature.
Gromov-Hausdorff convergence of manifolds.
In the 1980s, Mr Gromovshowed a compactness result for Riemannian manifolds satisfying a uniform lowerbound on the Ricci curvature. It states that, if { ( M nα , g α , o α ) } α is a sequence ofpointed complete Riemannian manifolds satisfying uniformly Ric ≥ Kg (1)for some K ∈ R and o α ∈ M α , then up to extracting a sub-sequence, the sequence { ( M nα , d g α , o α ) } α converges in the pointed Gromov-Hausdorff topology to a com-plete proper metric space ( X, d , o ) . A natural question was then to describe thestructure of such metric spaces arising as limits of smooth manifolds. In the 1990s, aseries of results by J. Cheeger and T. Colding [Col97, CC96, CC97, CC00a, CC00b]made it possible to better understand this problem and their work launched a vastresearch program. Many recent results have led to a significant understanding ofthe so-called “Ricci limit spaces” [Che01, CN15, JN21, CJN18].In the study of limit spaces, there are two different scenarios, depending onwhether the sequence of manifolds is collapsing or non-collapsing . In the first case,the volume of unit balls B ( o α ) goes to as α tends to infinity, while in the non-collapsed case there is a uniform lower bound on this volume. Since the work ofK. Fukaya [Fuk87] it is known that, in the collapsed case, the Gromov-Hausdorfftopology is not sufficient to recover good geometric information on the limit space,such as, for instance, information about the spectrum of its Laplacian. K. Fukayathen introduced the measured Gromov-Hausdorff topology. For that, one re-scalesthe Riemannian measure d ν g α and considers sequences of manifolds as sequences ofmetric measure spaces ( M nα , d g α , d µ α , o α ) , where d µ α = ν − g α ( B ( o α )) d ν g α . Thenagain, up to extracting a sub-sequence, there is convergence to a metric measurespace ( X, d , µ, o ) in the pointed measured Gromov-Hausdorff topology. This allowsfor finer results on the structure of the limit space. In the 2000s the works of J. Lott, K.-T. Sturm and C. Villani showed that it ispossible to define a generalization of a Ricci lower bound in the setting of metricmeasure spaces. This lead to the notion of
CD(
K, n ) metric measure spaces, thatare known to include Ricci limit spaces [LV09, Stu06a, Stu06b, Vil09]. Later on,L. Ambrosio, N. Gigli and G. Savaré introduced a refinement of the CD(
K, n ) con-dition, the so-called Riemannian curvature dimension condition RCD(
K, n ) , whichis also satisfied by manifolds carrying the lower bound (1) and is preserved un-der measured Gromov-Hausdorff convergence [AGS14]. Under this more restrictivecondition, it is possible, for instance, to define a Laplacian operator and thus re-formulate classical inequalities of Riemannian geometry in the setting of metricmeasure spaces. Nowadays, the structure of RCD(
K, n ) spaces is fairly well under-stood [DPG18, MN19, GP16, DPMR17, KM18, BS20], and such spaces provide agood conceptual framework comprising Ricci limit spaces. Ricci limit spaces and beyond.
The work of J. Cheeger and T. Coldingrelies on several crucial tools and results. The first one is the Bishop-Gromovvolume comparison theorem, which provides a monotone quantity given by the volume ratio r ν g ( B r ( x )) V n,K ( r ) , Monotone quantities play a crucial role when investigating blow-up phenomena ingeometric analysis. In the case of Ricci limit spaces, the monotonicity of the vol-ume ratio is the keystone for understanding the local geometry of limit spaces. Twoother very important results are the almost splitting theorem [CC97] and the the-orem now known as “almost volume cones implies almost metric cone” (see [CC96,Theorem 3.6]). Together with the monotonicity of the volume ratio, they imply inparticular that tangent cones of non-collapsed Ricci limit spaces are genuine metriccones. Many additional technical results are involved in the study of Ricci limitspaces. For example, the existence of good cut-off functions (with bounded gradientand Laplacian) plays an important role in many proofs for exploiting the Bochnerformula. The construction of cut-off functions relies on strong analytic propertiesof manifolds with Ricci curvature bounded from below, such as Laplacian compar-isons, Gaussian heat kernel bounds, the Cheng-Yau gradient estimate.However, there are some very interesting contexts in which a good understandingof the convergence of smooth manifolds is needed, but a uniform lower bound on theRicci curvature is not satisfied and the previous tools are not available: for example,in the study of the Ricci flow [Bam17, Bam18, BZ17, Sim20a, Sim20b] and of crit-ical metrics [TV05a, TV05b, TV08]. It is then important to investigate situationsin which weaker assumptions on the curvature are made. The case in which oneassumes some L p bound on the Ricci curvature, for p > n/ , has been well studiedsince the end of the 1990s and one gets a number of results about the structure ofthe limit space under an additional smallness assumption [PW97, PW01, TZ16].In particular, the two above almost rigidity results hold. Very recently, C. Kettereropened the way to a new interesting perspective in the study of limits of manifoldswith the appropriate L p bound on the Ricci curvature, by showing, among others,that tangent cones of the limit space are RCD(0 , n ) -spaces [Ket20]. One questionhe raises in this work is whether the same result holds when assuming a Kato con-dition on the Ricci curvature: we give a positive answer to this question in one ofour results (see Theorem B). Where V n,K ( r ) is the volume of a geodesic balls with radius r in a simply connected space ofconstant curvature equals to K/ ( n − . TRUCTURE OF KATO LIMITS 3
Kato type bound.
It has been recently remarked [Car19, Ros19] that a Katotype bound on the Ricci curvature makes it possible to use ideas of Q.S. Zhangand M. Zhu [ZZ18] and, as a consequence, to obtain a Li-Yau type bound forsolutions of the heat kernel. Many geometric estimates, that are known when theRicci curvature is bounded from below, follow under such less restrictive Kato typebound.Let ( M n , g ) be a closed Riemannian manifold and introduce the function Ric - : M → R + to be the best function such that Ric ≥ −
Ric - g. Define for all τ > k τ ( M n , g ) = sup x ∈ M ˆ [0 ,τ ] × M H ( t, x, y ) Ric - ( y ) d ν g ( y ) d t, where H ( t, · , · ) is the Schwartz kernel (the heat kernel) of the heat operator e − t ∆ .If for some T > the following Kato type boundk T ( M n , g ) ≤ n , (2)holds, then one gets geometric and analytic results similar to the ones implied bythe condition (1) with K = − /T. Moreover, as noticed in [Car19], the set of closedmanifolds satisfying (2) is pre-compact in the Gromov-Hausdorff topology. As aconsequence, it is natural to ask under which extent results on the structure ofRicci limit spaces can be obtained under this weaker assumption.The Kato condition was introduced with the aim of studying Schrödinger oper-ators in the Euclidean space L = ∆ − V, where V ≥ and our convention is that ∆ = - P i ∂ /∂x i on R n . A non negativepotential V : R n → R + is said to be in the Kato class, or to satisfy the Katocondition, if lim T → sup x ∈ R n ¨ [0 ,T ] × R n e − k x − y k t (4 πt ) n V ( y ) d y d t = 0 . At the regularity level, this condition only requires that V is the Laplacian of acontinuous function. Moreover, if V is in the Kato class, when t tends to 0 one cancompare the semi-groups e − t ∆ and e − t (∆ − V ) and thus recover good properties of e − t (∆ − V ) for t small enough. We refer to the beautiful survey of B. Simon [Sim82]for a extensive overview on the Kato condition in the Euclidean setting, and to thebook of B. Guneysu [G¨17] for an account of the Kato condition in the Riemanniansetting. In our context, the potential V is chosen to be Ric - .Now, an assumption in the spirit of the Kato condition in R n would requirenot only that k T is uniformly bounded along the sequence of manifolds for a fixed T > , but also some uniform control on the way that k τ goes to when τ goes to . This kind of control is actually required in our analysis in order to be able tocompare infinitesimal geometry of limit spaces with Euclidean geometry. In partic-ular, this plays an important role to get the appropriate monotone quantities thatwe rely on for studying the geometry of tangent cones. Main results.
We begin by illustrating our main results in the non-collapsedcase.
Theorem A.
Let ( X, d , o ) be the pointed Gromov-Hausdorff limit of a sequence ofclosed Riemannian manifolds { ( M nα , d g α , o α ) } α satisfying the uniform Kato bound ∀ τ ∈ (0 ,
1] : k τ ( M nα , g α ) ≤ f ( τ ) , (3) GILLES CARRON, ILARIA MONDELLO, AND DAVID TEWODROSE where f : [0 , → R + is a non-decreasing function such that ˆ p f ( τ ) dττ < ∞ , (4) and the non-collapsing condition ν g α ( B ( o α )) ≥ v > . (5) Then the following holds. (i)
Volume convergence:
For any r > and x α ∈ M α such that x α → x ∈ X we have lim α →∞ ν g α ( B r ( x α )) = H n ( B r ( x )) . (ii) Structure of tangent cones:
For any x ∈ X , tangent cones of X at x are RCD(0 , n ) metric cones. (iii) Almost everywhere regularity :
For H n -a.e. x ∈ X , ( R n , d eucl ) is theunique tangent cone of X at x . (iv) Stratification:
Let S k be the set consisting of the points x ∈ X such that X does not carry any tangent cone at x that splits isometrically a factor R k +1 . Then S k satisfies dim H S k ≤ k. The first point is a generalization of the volume continuity showed in [Col97].The fact that tangent cones are metric cones in the analog of [CC97, Theorem5.2] and the two last points correspond to [CC97, Theorem 4.7] (see also [Che01,Theorem 10.20]). We also conjecture that, under the same assumptions, we have S n − = S n − , that is the singular set has codimension at least two. As for the caseof Ricci limit spaces, we expect the existence of an open subset that is n -rectifiableand bi-Hölder homeomorphic to a manifold. We plan to address these questions inour subsequent work.Observe that the uniform Kato bound (3) with a function satisfying (4) is guar-anteed as soon as one has an appropriate estimate on the L p norm of the Riccicurvature. This is due to C. Rose and P. Stollmann [RS17]: thanks to their work,it is possible to show that if p > n/ and ε ( p, n ) is small enough, then the followingestimate diam ( M α , g α ) (cid:18) M (cid:12)(cid:12) Ric - − κ (cid:12)(cid:12) p + d ν g α (cid:19) p ≤ ε ( p, n ) implies (3) and (4). Similarly, our non-collapsing assumption and the uniform Katobound is ensured under the assumptions considered by G. Tian and Z. Zhang inthe study of Kähler-Ricci flow g ( t ) [TZ16], that are an a-priori bound on the L p norm of Ricci curvature for p > n/ ∀ t ≥ ˆ M (cid:12)(cid:12) Ric g ( t ) (cid:12)(cid:12) p d ν g ( t ) ≤ Λ and a non-collapsing condition ∀ t ≥ , ∀ x ∈ M, ∀ r ∈ (0 , , ν g ( t ) ( B r ( x )) ≥ vr n . In the collapsed case, our results give less information about the structure of thelimit space, but apply with a weaker hypothesis.
Theorem B.
Let ( X, d , µ, o ) be the pointed measured Gromov-Hausdorff limit ofa sequence { ( M nα , d g α , µ α , o α ) } α , satisfying the uniform Kato bound (3) for somenon-decreasing positive function f : [0 , → R + such that lim τ → f ( τ ) = 0 , (6) with x + = max { x, } . TRUCTURE OF KATO LIMITS 5 with the re-scaled measure d µ α = d ν g α ν g α ( B ( o α )) . (7) Then we have: (i) the Cheeger energy is quadratic and ( X, d , µ ) is an infinitesimally Hilbertianspace in the sense of [Gig15] ; (ii) for any x ∈ X , metric measure tangent cones of X at x are RCD(0 , n ) ; (iii) if X is compact, then the spectrum of the Laplace operators of ( M nα , g α ) converges to the spectrum of the Laplacian associated to ( X, d , µ ) . The last point extends [Fuk87, Theorem 0.4]. The second point generalizesC. Ketterer’s result [Ket20, Corollary 1.7]: under the same assumptions of [TZ16]that we recalled above, he proved that tangent cones are
RCD(0 , n ) spaces. Partof his proof relies on an almost splitting theorem of [TZ16]. In our case, we donot use an almost splitting theorem. Nonetheless, we point out that our proofshows that whenever the sequence of manifolds ( M α , g α ) is such that, for some τ > , k τ ( M α , g α ) tends to zero as α goes to infinity, then the limit ( X, d , µ ) isan RCD(0 , n ) space. As a consequence, Gigli’s splitting theorem for RCD(0 , n ) spaces applies [Gig13, Gig14]. Moreover, a contradiction argument based on pre-compactness leads to an almost splitting theorem for manifolds with k τ smallenough. Then we do have an almost splitting theorem in our setting, but in con-trast to what happens in the study of Ricci limit and RCD spaces, where suchtheorem represents a key tool, we obtain it as a consequence of our results ratherthan relying on it on our proofs.
Outline of proofs.
We now describe some of the ideas playing a role in ourproofs and their organization, starting from Theorem B. The Kato type bound (2)provides very good heat kernel estimates (see for example Proposition 2.3) whichimply in particular that a sequence of manifolds satisfying (2), when considered asa sequence of
Dirichlet spaces, is uniformly doubling and carries a uniform Poincaréinequality. This, together with the results of A. Kasue [Kas05] and K. Kuwae andT. Shioya [KS03], ensures that the measured Gromov-Hausdorff convergence can bestrengthened, in the sense that one additionally obtains Mosco convergence of theDirichlet energies. More precisely, assume that ( X, d , µ, o ) is a pointed measuredGromov-Hausdorff limit of a sequence of closed manifolds { ( M nα , d g α , µ α , o α ) } α ,where d µ α is either the Riemannian volume in the non-collapsing case, or its re-scaled version (7) in the collapsing one. Up to extraction of a sub-sequence, it ispossible to define a closed, densely defined quadratic form E on L ( X, µ ) which isobtained as the Mosco limit of the Dirichlet energies: u ˆ M α | du | g α d µ α . A priori, different sub-sequences could lead to different quadratic forms. Moreover,the space ( X, d , µ, o ) carries both the Dirichlet energy E and the Cheeger energycanonically associated to d and µ . In general, these two energies do not need tocoincide, see for instance [ACT18, Theorem 7.1]: it gives an example of a limitspace such that the distance is a Finsler metric and thus the Cheeger energy, notbeing quadratic, cannot coincide with any Dirichlet form.However, under the Kato bound (3) together with (6), we can use the Li-Yautype inequality in order to get estimates for the solutions of the heat equation onthe manifolds ( M α , g α ) . We show that such estimates pass to the limit and hold onthe Dirichlet limit space ( X, d , µ, E ) . As a consequence, we can apply a result due toL. Ambrosio, N. Gigli, G. Savaré [AGS15] and to P. Koskela, N. Shanmugalingam, GILLES CARRON, ILARIA MONDELLO, AND DAVID TEWODROSE
Y. Zhou [KSZ14] and we obtain that the limit Dirichlet energy E coincides infact with the Cheeger energy of the metric measure space ( X, d , µ ) . Hence, underconditions (3) and (6), measured Gromov-Hausdorff convergence implies Moscoconvergence of the Dirichlet energies to the Cheeger energy.Our proof also applies when for some τ > α →∞ k τ ( M nα , g α ) = 0 . (8)Under this condition, we additionally show that the Bakry-Ledoux gradient esti-mate holds on the limit space and thus ( X, d , µ ) is an RCD(0 , n ) space. Thanksto the re-scaling properties of the heat kernel, if ( X, d , µ, o ) is a limit of manifoldssatisfying (3) and (6), then any tangent cone of X is a limit of re-scaled manifoldsfor which (8) holds for all τ > . Therefore, this implies Theorem B(ii).As for the non-collapsed case, we prove that the limit measure µ coincides withthe n -dimensional Hausdorff measure, so that Gromov-Hausdorff convergence un-der conditions (4) and (5) not only implies Mosco converge of the energies, butalso measured Gromov-Hausdorff convergence. To prove this, we introduce a newfamily of monotone quantities that, when the Ricci curvature is non-negative, in-terpolates between the Li-Yau’s inequality and Bishop-Gromov volume comparisontheorem. Our quantities are modeled on Huisken’s entropy for the mean curvatureflow [Hui90]. In order to define them, for a closed manifold ( M n , g ) with heat kernel H , we define the function U by setting H ( t, x, y ) = exp (cid:16) − U ( t,x,y )4 t (cid:17) (4 πt ) n . We then introduce for any s, t > the Gaussian’s type entropy θ x ( s, t ) = ˆ M exp (cid:16) − U ( t,x,y )4 s (cid:17) (4 πs ) n d ν g ( y ) . When the Ricci curvature is non-negative, we show that for all x ∈ M the function λ θ x ( λs, λt ) , is monotone non-increasing for s ≥ t , non-decreasing for s ≤ t . This interpolatesbetween the Bishop-Gromov and Li-Yau’s inequalities in the following sense. When t = 0 , we can write θ x ( s,
0) = ˆ M exp (cid:16) − d ( x,y ))4 s (cid:17) (4 πs ) n d ν g ( y ) = 12 ˆ ∞ e − ρ ρ ν g (cid:0) B ρ √ s ( x ) (cid:1) V n, ( ρ √ s ) dρ. Then, Bishop-Gromov volume comparison implies that for any s ≥ the function λ θ x ( λs, is monotone non-increasing. Moreover, one of the consequences ofLi-Yau’s inequality is that for all x ∈ M the map t (4 πt ) n H ( t, x, x ) is monotone non-decreasing. When noticing that the semi-group law allows one towrite H (2 t, x, x ) = ˆ M H ( t, x, y ) d ν g ( y ) , a simple computation shows that for any t > and s = t/ the function λ θ x ( λt/ , λt ) is monotone non-decreasing.Observe that by Varadhan’s formula (22) we have d ( x, y ) = lim t → U ( t, x, y ) , TRUCTURE OF KATO LIMITS 7 so that, when t tends to zero, our quantities θ x tend to Θ x ( s ) = (4 πs ) n ˆ M e − d ( x,y )24 s d ν g ( y ) . This corresponds to Huisken’s entropy and to the H s volume considered by W. Jiangand A. Naber in [JN21], where it is shown to be monotone non-increasing if the Riccicurvature is non-negative. Moreover, in the case of a Ricci limit space ( X, d , µ ) , thelimit of Θ x as s tends to 0 coincides with the volume density at x , that is ϑ X ( x ) = lim r → µ ( B ( x, r )) ω n r n , where ω n is the volume of the unit ball in R n . Bishop-Gromov inequality guaranteesthat such limit does exist.In our setting, with the uniform Kato bounds (6) and (4), the Li-Yau typeinequality allows us to show that our quantities θ x are almost monotone , in thesense that there exists a function F of λ , tending to 1 as λ tends to 0, such thatthe map λ θ x ( λs, λt ) F ( λ ) has the same monotonicity as θ x when the Ricci curvature is non-negative. Thereis a limitation on the range of parameter where our monotonicity holds, when t ≤ s we also need s (cid:22) t/ p f ( t ) . As a consequence, the quantity Θ x is not monotone andwe do not get a monotone quantity based on the volume ratio.Observe that the only bound (3), with a function tending to 0 as t goes tozero, is not enough to obtain the above family of monotone quantities: due to thedependence of the Li-Yau type inequality on k τ , some kind of integral bound onk τ is needed. Moreover, for a sequence of smooth manifolds ( M nα , g α ) , the uniformbound (4) implies that function F is the same for all ( M α , g α ) , so that we get acorresponding family of monotone quantities on the limit space ( X, d , µ ) .Thanks to this almost monotonicity, we are able to show that on a tangent coneat x ∈ X the quantity Θ x is constant. Then for all r > the measure of ballscentered at x is equal to Θ x ω n r n . This, together with the fact that tangent conesare RCD(0 , n ) spaces and with the main result of [DPG16], allows us to obtain thattangent cones are metric cones.We also prove that the almost monotonicity of θ x implies that the volume den-sity ϑ X ( x ) is well defined on the limit space, despite the lack of monotonicity of thevolume ratio. We then show that the volume density is lower semi-continuous undermeasured Gromov-Hausdorff convergence. As a consequence, we obtain the strati-fication result from arguments inspired by B. White [Whi97] and G. De Philippis,N. Gigli [DPG18]. In the same proof, we get that µ -almost everywhere tangentcones are Euclidean, with a measure given by ϑ X ( x ) H n . In order to prove volumeconvergence, we then show that µ − a.e. x ∈ X : ϑ X ( x ) = 1 . For this purpose, we prove the existence at almost every point x ∈ X of har-monic ε -splitting maps H : B r ( x ) → R n . Splitting maps are “almost coordinates”,in the sense that they are (1 + ε ) -Lipschitz, ∇ H is close to the identity and theHessian is close to zero in L . They have been extensively used in the study ofRicci limit spaces and were recently proven to exist on RCD spaces too [BPS20].In our case, we obtain a very good control of ∇ H thanks to the Mosco convergenceof Dirichlet energy. This is still not enough to prove, as in [Che01] or [Gal98], that ϑ X ( x ) = 1 . But we are also able to obtain the Hessian bound thanks to one of ourmain technical tools, that is the existence of good cut-off functions when just the GILLES CARRON, ILARIA MONDELLO, AND DAVID TEWODROSE
Kato type bound (2) is satisfied.
Outline of the paper.
Section 2 includes the main preliminary tools that werely on throughout the paper. After introducing the convergence notions that weneed, we focus on Dirichlet spaces. We state a compactness result for PI Dirichletspaces, originally observed in [Kas05], for which we give a proof in the Appendix,and we collect the assumptions under which a Dirichlet space satisfies the
RCD condition.In Section 3, we introduce the different Kato type conditions that we considerin the rest of the paper, we state pre-compactness results and show that underassumptions (3) and (4) the intrinsic distance associated to the Dirichlet energycoincides with the limit distance. In the non-collapsing case, we recall a usefulAhlfors regularity result due to the first author that also holds in the limit.Section 4 is devoted to proving some technical tools obtained under assumption(2), in particular the existence of good cut-off functions and the resulting Hessianbound.In Section 5 we prove Theorem B, first by showing that under assumptions (3)and (6) the Dirichlet energies converge to the Cheeger energy. This immediatelyimplies convergence of the spectrum when X is compact. We then prove that ifk τ ( M α , g α ) tends to zero for some fixed τ > , the limit space is an RCD(0 , n ) space.In Section 6, we introduce and study the quantity θ x ( t, s ) . We show its almostmonotonicty and then obtain that, in the non-collapsing case, under assumptions(3) and (4), tangent cones are metric cones and the volume density is well-defined.Section 7 is devoted to proving Theorem A (iv). In particular, we obtain that µ -a.e. tangent cones are unique and coincides with ( R n , d e , ϑ X ( x ) H n , . In the lastsection we show that ϑ X ( x ) is equal to one almost everywhere: we prove existenceof harmonic splitting maps and as a consequence we get volume convergence.In the Appendix we show the convergence results that are needed in Section 5,for passing to the limit the appropriate estimates on manifolds, and in Section 4, toget the existence of ε -splitting harmonic maps with a good bound on the gradient.We also give an explicit proof of the compactness theorem for PI Dirichlet spaces. Acknowledgement.
The first author thanks the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment; he wasalso partially supported by the ANR grants:
ANR-17-CE40-0034 : CCEM and
ANR-18-CE40-0012 : RAGE . The second author was partially funded by theANR grant
ANR-17-CE40-0034 : CCEM . Contents
Introduction 11. Preliminaries 91.1. The doubling condition 101.2. Dirichlet spaces 101.3. The Poincaré inequality and PI Dirichlet spaces 131.4. Notions of convergence 141.5. A compactness result for Dirichlet spaces 191.6. Dirichlet spaces satisfying an
RCD condition 192. Kato limits 212.1. Dynkin limits 232.2. Kato limits 252.3. Ahlfors regularity 27
TRUCTURE OF KATO LIMITS 9 L p -Kato condition 283. Analytic properties of manifolds with a Dynkin bound 293.1. Good cut-off functions 293.2. Hessian estimates 313.3. Gradient estimates for harmonic functions 334. Curvature-dimension condition for Kato limits 354.1. Differential inequalities 354.2. Convergence of the Energy 394.3. The RCD condition for a certain class of Kato limit spaces 415. Tangent cones are metric cones 435.1. On a doubling space 435.2. On a Dirichlet space 465.3. A differential inequality 475.4. Consequences on non-collapsed strong Kato limits 506. Stratification 557. Volume continuity 607.1. Existence of splitting maps 617.2. Proof of Theorem 7.2 66Appendix 68A. Approximation of functions 68B. Convergence of integrals 70C. Heat kernel characterization of PI -Dirichlet spaces. 73D. Proof of Theorem 1.17. 78E. Further convergence results 80References 881. Preliminaries
Throughout this paper, n is a positive integer, and A is a countable, infinite,directed set like N , for instance. We choose to denote sequences with countableinfinite sets: this means that if { u α } α ∈ A is a sequence in a topological space ( X, T ) ,then { u α } converges to u if and anly if for any neighborhood U of u there existsa finite subset C ⊂ A such that α / ∈ C implies u α ∈ U . Similarly, a sequence { u α } α ∈ A admits a convergent sub-sequence if and only if there exists an infinitesubset B ⊂ A such that the sequence { u β } β ∈ B converges.All the manifolds we deal with in this paper are smooth and connected, andthe Riemannian metrics we consider on these manifolds are all smooth. We oftenuse the notation M n to specify that a manifold M is n -dimensional. We callclosed any Riemannian manifold which is compact without boundary. Whenever ( M, g ) is a Riemannian manifold, we write d g for its Riemannian distance, ν g forits Riemannian volume measure, ∆ g for its Laplacian operator which we choose todefine as a non-negative operator, i.e. ˆ M g ( ∇ u, ∇ v ) d ν g = ˆ M (∆ g u ) v d ν g for any compactly supported smooth functions u, v : M → R .We recall that a metric space ( X, d ) is called proper if all closed balls are compactand that it is called geodesic if for any x, y ∈ X there exists a rectifiable curve γ joining x to y whose length is equal to d ( x, y ) , in which case γ is called a geodesicfrom x to y . We also recall that the diameter of a metric space ( X, d ) is set as diam( X ) := sup { d ( x, y ) : x, y ∈ X } . If f : X → R is a locally d -Lipschitz function,we define its local Lipschitz constant Lip d f by setting Lip d f ( x ) := lim sup y → x | f ( x ) − f ( y ) | d ( x,y ) if x ∈ X is not isolated , otherwise . We call metric measure space any triple ( X, d , µ ) where ( X, d ) is a metric spaceand µ is a Radon measure finite and non-zero on balls with positive radius, andwe write B r ( x ) for the open metric ball centered at x ∈ X with radius r > , and ¯ B r ( x ) for the closed metric ball. We may often implicitly consider a Riemannianmanifold ( M, g ) as the metric measure space ( M, d g , ν g ) in which case metric ballsare geodesic balls.We use standard notations to denote several classical function spaces: L p ( X, µ ) , L ploc ( X, µ ) , C ( X ) , Lip( X ) or Lip( X, d ) , C ∞ ( M ) and so on. We use the subscript c to denote the subspace of compactly supported functions of a given function space,like C c ( X ) for compactly supported functions in C ( X ) , for instance. We write C ( X ) for the space of continuous functions converging to at infinity, which is the L ∞ ( X, µ ) -closure of C c ( X ) .We write A for the characteristic function of some set A ⊂ X . By supp f (resp. supp µ ) we denote the support of a function f (resp. a measure µ ). If f isa measurable map from a measured space ( X, µ ) to a measurable one Y , we write f µ for the push-forward measure of µ by f .For any s > we write ω s for the constant π s/ / Γ( s/ , where Γ is the usualgamma function; as well-known, in case s is an integer k , then ω k coincides withthe Hausdorff measure of the unit Euclidean ball in R k .1.1. The doubling condition.
Let us begin with recalling the definition of adoubling metric measure space.
Definition 1.1.
Given R ∈ (0 , + ∞ ] and κ ≥ a metric measure space ( X, d , µ ) iscalled κ -doubling at scale R if for any ball B r ( x ) ⊂ X with r ≤ R we have µ ( B r ( x )) ≤ κ µ ( B r ( x )) . When R = + ∞ , we simply say that ( X, d , µ ) is doubling. Doubling metric measure spaces have the following useful properties.
Proposition 1.2.
Assume that ( X, d , µ ) is κ -doubling at scale R for some κ ≥ and R ∈ (0 , + ∞ ] and that ( X, d ) is geodesic. Then there exists c, λ, δ > dependingonly on κ such that:i) µ ( B r ( x )) ≤ ce λ d ( x,y ) r µ ( B r ( y )) for any x, y ∈ X and < r ≤ R ,ii) µ ( B r ( x )) ≤ c (cid:0) rs (cid:1) λ µ ( B s ( x )) for any x ∈ X and < s ≤ r ≤ R ,iii) µ ( B S ( x )) ≤ e λ Ss µ ( B s ( x )) for any x ∈ X and < s ≤ R ≤ S ,iv) µ ( B s ( x )) e − λ (cid:0) rs (cid:1) δ ≤ µ ( B r ( x )) for any x ∈ X and < s < r < min { R, D/ } ,where D = diam( X, d ) ,v) µ ( B r ( x ) \ B r − τ ( x )) ≤ c (cid:0) τr (cid:1) δ µ ( B r ( x )) for any x ∈ X , r > and < τ < min { r, R } . We refer to [HSC01, Subsection 2.3] for the first four properties and to [CM98,Lemma 3.3] or [Tes07] for the last one.1.2.
Dirichlet spaces.
Let us recall now some classical notions from the theoryof Dirichlet forms; we refer to [FOT11] for details.
TRUCTURE OF KATO LIMITS 11
Let H be a Hilbert space of norm | · | H . We recall that a quadratic form Q : H → [0 , + ∞ ] is called closed if its domain D ( Q ) equipped with the norm | · | Q :=( | · | H + Q ( · )) / is a Hilbert space.Let ( X, T ) be a locally compact separable topological space equipped with a σ -finite Radon measure µ fully supported in X . A Dirichlet form on L ( X, µ ) with dense domain D ( E ) ⊂ L ( X, µ ) is a non-negative definite bilinear map E : D ( E ) × D ( E ) → R such that E ( f ) := E ( f, f ) is a closed quadratic form satisfyingthe Markov property, that is for any f ∈ D ( E ) , the function f = min(max( f, , belongs to D ( E ) and E ( f ) ≤ E ( f ) ; we denote by h· , ·i E the scalar product associatedwith | · | E . We call such a quadruple ( X, T , µ, E ) a Dirichlet space. When T isinduced by a given distance d on X , we write ( X, d , µ, E ) instead of ( X, T , µ, E ) and call ( X, d , µ, E ) a metric Dirichlet space.Any Dirichlet form E is naturally associated with a non-negative definite self-adjoint operator L with dense domain D ( L ) ⊂ L ( X, µ ) defined by D ( L ) := (cid:26) f ∈ D ( E ) : ∃ h =: Lf ∈ L ( X, µ ) s.t. E ( f, g ) = ˆ X hg d µ ∀ g ∈ D ( E ) (cid:27) . The spectral theorem implies that L generates an analytic sub-Markovian semi-group ( P t := e − tL ) t> acting on L ( X, µ ) where for any f ∈ L ( X, µ ) , the map t P t f is characterized as the unique C map (0 , + ∞ ) → L ( X, µ ) , with valuesin D ( L ) , such that ( dd t P t f = − L ( P t f ) ∀ t > , lim t → k P t f − f k L ( X,µ ) = 0 . Moreover, we get the property that when ≤ f ≤ then ≤ P t f ≤ . Standardfunctional analytic theory shows that ( P t ) t> extends uniquely for any p ∈ [1 , + ∞ ) to a strongly continuous semi-group of linear contractions in L p ( X, µ ) . Moreover,the spectral theorem yields a functional calculus which justifies the following esti-mate: for any t > and f ∈ D ( E ) , k f − P t f k L ≤ √ t E ( f ) . (9) Heat kernel.
We call heat kernel of E any function H : (0 , + ∞ ) × X × X → R such that for any t > the function H ( t, · , · ) is ( µ ⊗ µ ) -measurable and P t f ( x ) = ˆ X H ( t, x, y ) f ( y ) d µ ( y ) for µ -a.e. x ∈ X, (10)for all f ∈ L ( X, µ ) . If E admits a heat kernel H , then it is non negative andsymmetric with respect to its second and third variable, and for any t > thefunction H ( t, · , · ) is uniquely determined up to a ( µ ⊗ µ ) -negligible subset of X × X .Moreover, the semi-group property of ( P t ) t> results into the so-called Chapman-Kolmogorov property for H : ˆ X H ( t, x, z ) H ( s, z, y ) d µ ( z ) = H ( t + s, x, y ) , ∀ x, y ∈ X, ∀ s, t > . (11)The space ( X, T , µ, E ) – or the heat kernel H – is called stochastically completewhenever for any x ∈ X and t > it holds ˆ X H ( t, x, y ) d µ ( y ) = 1 . (12) Strongly local, regular Dirichlet spaces.
Let us recall now an important definition.
Definition 1.3.
A Dirichlet form E on L ( X, µ ) is called strongly local if E ( f, g ) =0 for any f, g ∈ D ( E ) such that f is constant on a neighborhood of supp g , andregular if C c ( X ) ∩ D ( E ) contains a core, that is a subset which is both dense in C c ( X ) for k · k ∞ and in D ( E ) for | · | E . If ( X, T , µ, E ) is a Dirichlet space where E is strongly local and regular, we say that ( X, T , µ, E ) is a strongly local, regularDirichlet space. By a celebrated theorem from A. Beurling and J. Deny [BD59], any stronglylocal, regular Dirichlet form E on L ( X, µ ) admits a carré du champ , that is a non-negative definite symmetric bilinear map Γ : D ( E ) × D ( E ) → Rad , where
Rad isthe set of signed Radon measures on ( X, T ) , such that E ( f, g ) = ˆ X dΓ( f, g ) ∀ f, g ∈ D ( E ) , where ´ X dΓ( f, g ) denotes the total mass of the measure Γ( f, g ) . Moreover, Γ islocal, meaning that ˆ A dΓ( u, w ) = ˆ A dΓ( v, w ) holds for any open set A ⊂ X and any u, v, w ∈ D ( E ) such that u = v on A . Thanksto this latter property, Γ extends to any µ -measurable function f such that for anycompact set K ⊂ X there exists g ∈ D ( E ) such that f = g µ -a.e. on K ; we denoteby D loc ( E ) the set of such functions. Then Γ satisfies the Leibniz rule and the chainrule. If we set Γ( f ) := Γ( f, f ) , this implies Γ( f g ) ≤ f ) + Γ( g )) (13)for any f, g ∈ D loc ( E ) ∩ L ∞ loc ( X, µ ) and Γ( η ◦ h ) = ( η ′ ◦ h ) Γ( h ) (14)for any h ∈ D loc ( E ) and η ∈ C ( R ) bounded with bounded derivative.A final consequence of strong locality and regularity is that the operator L canon-ically associated to E satisfies the classical chain rule: L ( φ ◦ f ) = ( φ ′ ◦ f ) Lf − ( φ ′′ ◦ f )Γ( f ) ∀ f ∈ G , ∀ φ ∈ C ∞ ([0 , + ∞ ) , R ) , (15)where G is the set of functions f ∈ D ( L ) such that Γ( f ) is absolutely continuouswith respect to µ with density also denoted by Γ( f ) . In particular: Lf = 2 f Lf − f ) ∀ f ∈ G . (16) Intrinsic distance.
The carré du champ operator of a strongly local, regularDirichlet form E provides an extended pseudo-metric structure on X given by thenext definition. Definition 1.4.
The intrinsic extended pseudo-distance d E associated with E isdefined by d E ( x, y ) := sup {| f ( x ) − f ( y ) | : f ∈ C ( X ) ∩ D loc ( E ) s.t. Γ( f ) ≤ µ } (17) for any x, y ∈ X , where Γ( f ) ≤ µ means that Γ( f ) is absolutely continuous withrespect to µ with density lower than µ -a.e. on X . A priori, d E ( x, y ) may be infinite, hence we use the word “extended”. Of coursethe case where d E does provide a metric structure on X is of special interest. Inthis regard, if ( X, T , µ, E ) is a strongly local, regular Dirichlet space where d E is adistance inducing T , we denote it by ( X, d E , µ, E ) . TRUCTURE OF KATO LIMITS 13
The Poincaré inequality and PI Dirichlet spaces.
Given R ∈ (0 , + ∞ ] ,we say that a strongly local, regular Dirichlet space ( X, d E , µ, E ) satisfies a R -scale-invariant Poincaré inequality if there exists γ > such that k u − u B k L ( B ) ≤ γ r ˆ B dΓ( u ) (18)for any u ∈ D ( E ) and any ball B with radius r ∈ (0 , R ] . When R = + ∞ , we simplysay that ( X, d E , µ, E ) satisfies a Poincaré inequality. The next definition is centralin our work. Definition 1.5.
Given R ∈ (0 , + ∞ ] , κ ≥ and γ > , we say that a stronglylocal, regular Dirichlet space ( X, d E , µ, E ) is PI κ , γ ( R ) if it satisfies the followingconditions: • ( X, d E , µ ) is κ -doubling at scale R , • ( X, d E , µ, E ) satisfies a R -scale- invariant Poincaré inequality (18) with con-stant γ . We may use the terminology
PI( R ) if no reference to the doubling or Poincaréconstant is required, or even PI if we do not need to mention the scale R . Geometry and analysis of PI Dirichlet spaces.
Assume that ( X, d E , µ, E ) is a PI κ , γ ( R ) Dirichlet space for some given R ∈ (0 , + ∞ ] , κ ≥ and γ > . Accordingto [Stu96], the strong locality and regularity assumptions on E imply that the metricspace ( X, d E ) is geodesic and that it satisfies the Hopf-Rinow theorem: it is properif and only if it is complete. Moreover, there is a relationship between the localLipschitz constant and the carré du champ of d E -Lipschitz functions, see [KZ12,Theorem 2.2] and [KSZ14, Lemma 2.4]: when u ∈ Lip( X, d E ) , then u ∈ D loc ( E ) and the Radon measure Γ( u ) is absolutely continuous with respect to µ ; moreover,there exists a constant η ∈ (0 , depending only on κ , γ such that η (Lip d E u ) ≤ dΓ( u )d µ ≤ (Lip d E u ) µ -a.e. on X. (19)In addition, it follows from [KZ12, Theorem 2.2] that Lip c ( X, d E ) is dense in D ( E ) and that for any u ∈ D ( E ) the Radon measure Γ( u ) is absolutely continuous withrespect to µ with density ρ u ∈ L loc ( X, µ ) comparable to the approximate Lipschitzconstant of u .For a strongly local, regular Dirichlet space ( X, d E , µ, E ) , to be PI κ , γ ( R ) impliesto have a Hölder continuous heat kernel H satisfying Gaussian upper and lowerbounds: there exists C , C > depending only on κ , γ such that C − µ ( B √ t ( x ) e − C d E ( x,y ) t ≤ H ( t, x, y ) ≤ C µ ( B √ t ( x )) e − d E ( x,y )5 t (20)for all t ∈ (0 , R ) and x, y ∈ X . This implication is actually an equivalence: seeTheorem C.1 in the Appendix where we provide references. In fact such a Dirichletspace satisfies the Feller property: the heat semi-group extends to a continuoussemi-group on C ( X ) .Moreover, a PI( R ) Dirichlet space is necessary stochastically complete: thiswas proved on Riemannian manifolds by A. Grigor’yan [Gri99, Theorem 9.1] andextended to Dirichlet spaces by K-T. Sturm [Stu94, Theorem 4].The above Gaussian upper bound can be improved to get the optimal Gaussianrate decay.
Proposition 1.6.
Let ( X, d E , µ, E ) be a PI κ , γ ( R ) Dirichlet space. Then there exist
C, ν > depending only on κ , γ such that for any x, y ∈ X and t ∈ (0 , R ) , H ( t, x, y ) ≤ Cµ ( B R ( x )) R ν t ν (cid:18) d E ( x, y ) t (cid:19) ν +1 e − d E ( x,y )4 t (21) Moreover, Varadhan’s formula holds: for any x, y ∈ X , d E ( x, y ) = − t → t log H ( t, x, y ) . (22)The Gaussian upper bound can be found in [Gri94, Theorem 5.2 ] (see also[Sik96, Cou93] for optimal versions) and Varadhan’s formula is due to ter Elst,D. Robinson, and A. Sikora [tERS07] (see also [Ram01] for an earlier result).1.4. Notions of convergence.
We provide now our working definitions of con-vergence of spaces and of points, functions, bounded operators and Dirichlet formsdefined on varying spaces.1.4.1.
Convergence of spaces.
Let us start with some classical definitions.
Pointed Gromov-Hausdorff convergence.
For any ε > , an ε -isometry betweentwo metric spaces ( X, d ) and ( X ′ , d ′ ) is a map Φ : X → X ′ such that | d ( x , x ) − d ′ (Φ( x ) , Φ( x )) | < ε for any x , x ∈ X and X ′ = S x ∈ X B ε (Φ( x )) . A sequence ofpointed metric spaces { ( X α , d α , o α ) } α converges in the pointed Gromov-Hausdorfftopology (pGH for short) to another pointed metric space ( X, d , o ) if there existtwo sequences { R α } α , { ε α } α ⊂ (0 , + ∞ ) such that R α ↑ + ∞ , ε α ↓ , and, for any α , an ε α -isometry Φ α : B R α ( o α ) → B R α ( o ) such that Φ α ( o α ) = o . We denote thisby ( X α , d α , o α ) pGH −→ ( X, d , o ) . Pointed measured Gromov-Hausdorff convergence.
Let us assume that the spaces { ( X α , d α , o α ) } α , ( X, d , o ) are equipped with Radon measures { µ α } α , µ respectively.Then the sequence of pointed metric measure spaces { ( X α , d α , µ α , o α ) } α convergesto ( X, d , µ, o ) in the pointed measured Gromov-Hausdorff topology (pmGH forshort) if there exist two sequences { R α } α , { ε α } α ⊂ (0 , + ∞ ) such that R α ↑ + ∞ , ε α ↓ , and, for any α , an ε α -isometry Φ α : B R α ( o α ) → B R α ( o ) such that Φ α ( o α ) = o and (Φ α ) µ α ⇀ µ, where we recall that (Φ α ) µ α ⇀ µ means that for any ϕ ∈ C c ( X ) , lim α ˆ X α ϕ ◦ Φ α d µ α = ˆ X ϕ d µ. We denote this by ( X α , d α , µ α , o α ) pmGH −→ ( X, d , µ, o ) . Precompactness results.
Gromov’s well-known precompactness theorem yieldsthe following.
Proposition 1.7.
For any
R > and κ , η ≥ , the space of pointed proper geodesicmetric measure spaces ( X, d , µ, o ) satisfying ( X, d , µ ) is κ -doubling at scale R, (23) η − ≤ µ ( B R ( o )) ≤ η (24) is compact in the pointed measured Gromov-Hausdorff topology, i.e. for every se-quence of pointed proper geodesic metric measure spaces { ( X α , d α , µ α , o α ) } α satis-fying (23) and (24) , there is a subsequence B ⊂ A and a pointed proper geodesicmetric measure space ( X, d , µ, o ) satisfying (23) and (24) such that ( X β , d β , µ β , o β ) pmGH −→ ( X, d , µ, o ) . TRUCTURE OF KATO LIMITS 15
We point out that Gromov’s precompactness theorem is usually stated for com-plete, locally compact, length metric spaces [Gro99, Proposition 5.2], but the Hopf-Rinow theorem ensures that these assumptions are equivalent to being proper andgeodesic.Remark that the condition (24) is stated for the radius R but the doublingcondition implies that when R > then there is some η ≥ such that (24) holds ifand only if there is some η ≥ , depending only on η , R, R and κ such that : η − ≤ µ (cid:0) B R ( o ) (cid:1) ≤ η . Note that the doubling condition is stable with respect to multiplication of themeasure by a constant factor. Therefore, if { ( X α , d α , µ α , o α ) } α is a sequence ofpointed proper geodesic metric measure spaces satisfying (23) but not (24), we mayrescale each measure µ α into m α µ α for some m α > in such a way that the sequence (cid:8) m − α µ α ( B R ( o α )) + m α µ − α ( B R ( o α )) (cid:9) α is bounded; then (cid:8) ( X α , d α , m − α µ α , o α ) (cid:9) α admits a pmGH convergent subsequence. We can choose m α = µ α ( B R ( o α )) , forinstance. Tangent cones of doubling spaces.
We recall the classical definition of a tangentcone.
Definition 1.8.
Let ( X, d , µ ) be a metric measure space and x ∈ X . The pointedmetric space ( Y, d Y , x ) is a tangent cone of X at x if there exists a sequence { ε α } α ∈ A ⊂ (0 , + ∞ ) such that ε α ↓ and ( X, ε − α d , x ) pGH −→ ( Y, d Y , x ); it can always be equipped with a limit meaure µ Y such that, up to a subsequence, ( X, ε − α d , µ ( B d ε α ( y )) − µ, x ) pmGH −→ ( Y, d Y , µ Y , x ) . (25) The pointed metric measure space ( Y, d Y , µ Y , x ) is then called a measured tangentcone of X at x . If ( Y, d Y , µ Y , y ) is a measured tangent cone of X at x and y ∈ Y , we refer to a measured tangent cone ( Z, d Z , µ Z , y ) of Y as y as an iteratedmeasured tangent cone of X . Remark 1.9.
We often use ( X x , d x , µ x , x ) to denote a measured tangent cone of ( X, d , µ ) at x .As well-known, on a geodesic proper metric measure space ( X, d , µ ) that is κ -doubling at scale R for some κ ≥ and R ∈ (0 , + ∞ ) , the existence of measuredtangent cones at any point x is guaranteed and any of these measured tangentcones is κ -doubling. Indeed, for any ε > , the rescaled space ( X, ε − d , µ ) is κ -doubling at scale R/ε . Hence when ε ≤ R , the the rescaled space ( X, ε − d , µ ) is κ -doubling at scale . Hence Proposition 1.7 applies to the rescaled spaces { ( X, ε − d , µ ( B ε ( x )) − µ, x ) } ε> and yields the existence of measured tangent coneswhich are κ -doubling at any scale S ≥ .When for some m > the space ( X, d , µ ) additionally satisfies a (local) m -Ahlforsregularity condition, i.e. for each ρ > there exists c ρ > such that for any x ∈ X ,any r ∈ (0 , and y ∈ B ρ ( x ) , c ρ r m ≤ µ ( B r ( y )) ≤ r m /c ρ , then it is convenient to rescale the measure by ε − m to study measured tangentcones. In this case, the tangent measures are only changed by a multiplicativeconstant positive factor. Convergence of points and functions.
A natural way to formalize the notionsof convergence of points and functions defined on varying spaces is the following.We let { ( X α , d α , µ α , o α ) } α , ( X, d , µ, o ) be proper pointed metric measure spacessuch that ( X α , d α , µ α , o α ) pmGH −→ ( X, d , µ, o ) . (26)As the ε α -isometries between X α and X are not unique (they can be composedfor instance with isometries of X α or X ), we make a specific choice by using thefollowing characterization: Characterization 1.
The pmGH convergence (26) holds if and only if there exist { R α } α , { ε α } α ⊂ (0 , + ∞ ) with R α ↑ + ∞ , ε α ↓ and ε α -isometries Φ α : B R α ( o α ) → B R α ( o ) such that:(1) Φ α ( o α ) = o ,(2) (Φ α ) µ α ⇀ µ . From now on and until the end of this section, we work with the notations pro-vided by this characterization.
Convergence of points.
Let x α ∈ X α for any α and x ∈ X be given. We say thatthe sequence of points { x α } α converges to x if d (Φ α ( x α ) , x ) → . We denote thisby x α → x . Uniform convergence.
Let u α ∈ C ( X α ) for any α and u ∈ C ( X, µ ) be given. Wesay that the sequence of functions { u α } α converges uniformly on compact sets to u if k u α − u ◦ Φ α k L ∞ ( B ( o α ,R )) → for any R > . It is easy to show the followinguseful criterion for uniform convergence on compact sets. Proposition 1.10.
Let u α ∈ C ( X α ) for any α and u ∈ C ( X ) be given. Then { u α } α converges uniformly on compact sets to u if and only if u α ( x α ) → u ( x ) whenever x α → x . In case ϕ α ∈ C c ( X α ) for any α and ϕ ∈ C c ( X ) , we write ϕ α C c −→ ϕ if there is R > such that supp ϕ α ⊂ B R ( o α ) for any α large enough and if { ϕ α } α converges uniformly to ϕ . When the spaces { ( X α , d α , µ α ) } α , ( X, d , µ ) are all κ -doubling at scale R , then for every ϕ ∈ C c ( X ) we can build functions ϕ α ∈ C c ( X α ) such that ϕ α C c −→ ϕ : see Proposition A.1 in the Appendix. Weak L p convergence. Let p ∈ (1 , + ∞ ) . Let f α ∈ L p ( X α , µ α ) for any α and f ∈ L p ( X, µ ) be given. We say that the sequence of functions { f α } α convergesweakly in L p to f , and we note f α L p ⇀ f, if sup α k f α k L p < + ∞ and ϕ α C c −→ ϕ = ⇒ ˆ X α ϕ α f α d µ α = ˆ X ϕf d µ. We have the following compactness result:
Proposition 1.11. If sup α k f α k L p < + ∞ , then there exists a subsequence B ⊂ A and f ∈ L p ( X, µ ) such f β L p ⇀ f .Strong L p convergence and duality. Let p ∈ (1 , + ∞ ) . Let f α ∈ L p ( X α , µ α ) forany α and f ∈ L p ( X, µ ) be given. We say that the sequence of functions { f α } α converges strongly in L p to f , and we note f α L p → f, TRUCTURE OF KATO LIMITS 17 if f α L p ⇀ f and k f α k L p → k f k L p . For every f ∈ L p ( X, µ ) , we can build functions f α ∈ L p ( X α , µ α ) converging to f strongly in L p : this follows from approximating f with functions { f i } ⊂ C c ( X ) , approximating each f i with functions f i,α ⊂ C c ( X α ) as mentioned before, and using a diagonal argument.Moreover there is a duality between weak convergence in L p and strong conver-gence in L q when p and q are conjugate exponent, as detailed in the next proposi-tion. Proposition 1.12.
Let p, q ∈ (1 , + ∞ ) be satisfying /p + 1 /q = 1 . Consider f α ∈ L p ( X α , µ α ) for any α and f ∈ L p ( X, µ ) . Then • f α L p → f if and only if ϕ α L q ⇀ ϕ = ⇒ ´ X α ϕ α f α d µ α = ´ X ϕf d µ, • f α L p ⇀ f if and only if ϕ α L q → ϕ = ⇒ ´ X α ϕ α f α d µ α = ´ X ϕf d µ. Convergence of bounded operators.
When B α : L ( X α , µ α ) → L ( X α , µ α ) for any α and B : L ( X, µ ) → L ( X, µ ) are bounded linear operators, we say that { B α } α converges weakly to B if f α L ⇀ f = ⇒ B α f α L ⇀ Bf and that { B α } α converges strongly to B if f α L → f = ⇒ B α f α L → Bf.
By duality, { B α } α converges weakly to B if and only if the sequence of the adjointoperators { B ∗ α } α converges strongly to the adjoint operator B ∗ . In particular, ifthe operators B α and B are all self-adjoint, weak and strong convergences areequivalent. Convergence in energy.
When each metric measure space is endowed with aDirichlet form so that { ( X α , d α , µ α , E α ) } α , ( X, d , µ, E ) are Dirichlet spaces, we cansimilarly define convergence in energy of functions. Let f α ∈ D ( E α ) for any α and f ∈ D ( E ) be given. We say that the sequence { f α } α converges weakly in energy to f , and we note f α E ⇀ f, if f α L ⇀ f and sup α E α ( f α ) < + ∞ . We say that { f α } α converges strongly in energyto f , and we note f α E → f if it converges weakly in energy to f and additionally f α L → f and lim α E α ( f α ) = E ( f ) . Using the non negative selfadjoint operator L α (resp. L ) associated to E α (resp. to E ), we have f α E ⇀ f ⇐⇒ (1 + L α ) f α L ⇀ (1 + L ) f and f α E → f ⇐⇒ (1 + L α ) f α L → (1 + L ) f. Remark 1.13.
All the above definitions have also a localized version where eachfunction f α is defined only on a ball centered at o α with a fixed radius. For instancefor a given ρ > , if f α ∈ L p ( B ρ ( o α )) for any α and f ∈ L p ( B ρ ( o )) , we say that thesequence { f α } converges weakly to f in L p ( B ρ ) , and we note f α L p ( B ρ ) ⇀ f, provided • sup α ´ B ρ ( o α ) | f α | p d µ α < ∞ , • for any sequence { ϕ α } α where ϕ α ∈ C c ( B ρ ( o α )) for any α and any ϕ ∈C c ( B ρ ( o )) , ϕ α C c → ϕ ⇒ lim α ˆ X α ϕ α f α d µ α = ˆ X ϕf d µ. Similarly, we define L ploc convergence of functions through pmGH convergence ofspaces in the following way: if f α ∈ L ploc ( X α , µ α ) for any α and f ∈ L ploc ( X, µ ) , wesay that the sequence { f α } α converges weakly to f in L ploc , and we note f α L ploc ⇀ f, if for any ρ > , f α | B ρ ( o α ) L p ( B ρ ) ⇀ f | B ρ ( o ) . Mosco convergence.
We recall the following notion of convergence that wasintroduced by U. Mosco in [Mos94] for quadratic forms. We formulate it if forDirichlet forms as this is sufficient for our purposes.
Definition 1.14.
Let { ( X α , T α , µ α , E α ) } α , ( X, T , µ, E ) be Dirichlet spaces. We saythat the sequence of Dirichlet forms {E α } α converges to E in the Mosco sense if thetwo following properties hold:(1) for any sequence { u α } α where u α ∈ D ( E α ) for any α and any u ∈ D ( E ) , u α L ⇀ u = ⇒ E ( u ) ≤ lim inf α E ( u α ) , (2) for any u ∈ D ( E ) there exists u α ∈ D ( E α ) for any α such that u α E → u. Mosco convergence of Dirichlet forms is equivalent to the convergence of manyrelated objects: this follows from [KS03, Theorem 2.4]. Recall that for a sequenceof self-adjoint operators { B α } α weak and strong convergence are equivalent. Proposition 1.15.
Let { ( X α , d α , µ α , E α ) } α , ( X, d , µ, E ) be metric Dirichlet spaces.For any α let L α (resp. L ) be the non-negative self-adjoint operator associated with E α (resp. E ) and let ( P αt ) t> (resp. ( P t ) t> ) be the generated semi-group.Then thefollowing statements are equivalent:(1) E α → E in the Mosco sense,(2) there exists t > such that the sequence of bounded operators { P αt } α strongly/ weakly converges to P t ,(3) for all t > the sequence of bounded operators { P αt } α strongly/ weaklyconverges to P t ,(4) the sequence of operators { ξ ( L α ) } strongly converges to ξ ( L ) for any smoothfunction ξ : [0 , + ∞ ) → R with supp ξ ⊂ [0 , R ] for some R > ,(5) the sequence of operators { ξ α ( L α ) } strongly converges to ξ ( L ) for any se-quence { ξ α : [0 , + ∞ ) → R } of continuous functions vanishing at infinitywhich converges to a continuous function ξ : [0 , + ∞ ) → R vanishing atinfinity. Definition 1.16.
Let { ( X α , d α , µ α , E α , o α ) } α , ( X, d , µ, E , o ) be pointed metric Dirich-let spaces. We say that the sequence { ( X α , d α , µ α , E α , o α ) } α converges to ( X, d , µ, E , o ) in the pointed Mosco-Gromov-Hausdorff sense if ( X α , d α , µ α , o α ) pmGH → ( X, d , µ, o ) and E α → E in the Mosco sense . We note ( X α , d α , µ α , E α , o α ) pMGH → ( X, d , µ, E , o ) . TRUCTURE OF KATO LIMITS 19
A compactness result for Dirichlet spaces.
The next theorem is a keytool in our analysis. It was already observed by A. Kasue [Kas05, Theorem 3.4]and extended [KS03, Theorem 5.2] in showing that the limit space is regular andstrongly local. We refer to Section D in the Appendix for the proof.
Theorem 1.17.
Let κ , η ≥ , γ > and R ∈ (0 , + ∞ ] be given. Assume that { ( X α , d E α , µ α , o α , E α ) } α ∈ A is a sequence of complete PI κ , γ ( R ) pointed Dirichletspaces such that for all α , η − ≤ µ α ( B R ( o α )) ≤ η. Then there exist a complete pointed metric Dirichlet space ( X, d , µ, o, E ) and a subse-quence B ⊂ A such that { ( X β , d E β , µ β , o β , E β ) } β ∈ B Mosco-Gromov-Hausdorff con-verges to ( X, d , µ, o, E ) ; moreover, E is regular, strongly local, the intrinsic pseudo-distance d E is a distance, and there is a constant c ∈ (0 , depending only on κ and γ such that: c d E ≤ d ≤ d E . Moreover, the space ( X, d E , µ, E ) is PI κ , γ ′ ( R ) for some constant γ ′ ≥ . Further-more, if H β (resp. H ) is the heat kernel of ( X β , d β , µ β , E β ) (resp. ( X, d , µ, o, E ) ) forany β , then for any t > H β ( t, · , · ) → H ( t, · , · ) uniformly on compact sets, (27) where we make an implicit use of the obvious convergence ( X β × X β , d β ⊗ d β , ( o β , o β )) pGH −→ ( X × X, d × d , ( o, o )) . If ( X, d E , µ, E ) is a strongly local and regular Dirichlet space satisfying a R -scale-invariant Poincaré inequality for some R > , it is not difficult to check that forany x ∈ X and any ρ > , the rescaled quadratic form E ρ := ρ µ ( B ρ ( x )) − E isa strongly local and regular Dirichlet form on ( X, d E , µ ρ := µ ( B ρ ( x )) − µ ) suchthat d E ρ = ρ − d E , and the space ( X, d E ρ , µ ρ , E ρ ) satisfies a ( R/ρ ) -scale-invariantPoincaré inequality. This observation coupled with the previous theorem leads tothe next result. Corollary 1.18.
Let ( X, d E , µ, E ) be a complete PI κ , γ ( R ) Dirichlet space and x ∈ X . If ( X x , d x , µ x , x ) is a measured tangent cone of ( X, d E , µ ) at x . Then it canendowed with a strongly local and regular Dirichlet form E x such that d E x is adistance bi-Lispchitz equivalent to d x and the space ( X x , d E x , µ x , E x ) is PI κ , γ ′ ( ∞ ) for some γ ′ . Moreover, there exists a sequence { ρ α } ⊂ (0 , + ∞ ) such that ρ α → and ( X, d E ρα , µ ρ α , x, E ρ α ) pMGH −→ ( X x , d E x , µ x , x, E x ) where µ ρ α := µ ( B ρ α ( x )) − µ and E ρ α := ρ α µ ( B ρ α ( x )) − E for any α . Note that different sequences could lead to different Dirichlet form on the samemeasured tangent cone.1.6.
Dirichlet spaces satisfying an
RCD condition.
Let us conclude these pre-liminaries with some facts concerning the Riemannian Curvature Dimension con-dition
RCD ∗ ( K, n ) , where K ∈ R , in the setting of Dirichlet spaces.The Cheeger energy [Che99] of a metric measure space ( X, d , µ ) is the convexand L ( X, µ ) -lower semicontinuous functional Ch : L ( X, µ ) → [0 , + ∞ ] defined by Ch ( f ) = inf f n → f (cid:26) lim inf n → + ∞ ˆ X |∇ f n | d µ (cid:27) (28)for any f ∈ L ( X, µ ) , where the infimum is taken over the set of sequences { f n } n ⊂ L ( X, µ ) ∩ Lip( X ) such that k f n − f k L ( X,µ ) → . We set H , ( X, d , µ ) := D ( Ch ) = { Ch < + ∞} and call H , ( X, d , µ ) the Sobolev space of ( X, d , µ ) . A suitable diagonal argumentshows that for any f ∈ H , ( X, d , µ ) there exists a unique L -function | df | called minimal relaxed slope of f such that Ch ( f ) = ˆ X | df | d µ and | df | = | dg | µ -a.e. on { f = g } for any g ∈ H , ( X, d , µ ) . Moreover, this func-tion | df | coincides µ -a.e. with the local Lipschitz function of f in case f is locallyLipschitz.There is no reason a priori for the Cheeger energy to be a Dirichlet form or evena quadratic form. In this respect, we provide the next definition and the subsequentproposition which are taken from [AGS14]. Definition 1.19.
A Polish metric measure space ( X, d , µ ) is called infinitesimallyHilbertian if Ch is a quadratic form. Proposition 1.20.
Let ( X, d , µ ) be an infinitesimally Hilbertian space. Then H , ( X, d , µ ) endowed with the norm k · k H , = p k · k L + Ch ( · ) is a Hilbert space.Moreover, the Cheeger energy Ch is a strongly local and regular Dirichlet form; itscarré du champ operator takes values in the set of absolutely continuous Radonmeasures, and for any f , f ∈ H , ( X, d , µ ) , h df , df i := dΓ( f , f )d µ = lim ǫ → | d ( f + ǫf ) | − | df | ǫ in L ( X, µ ) . In particular, dΓ( f ) = | df | d µ for any f ∈ H , ( X, d , µ ) . When ( X, d , µ ) is infinitesimally Hilbertian, we call Laplacian of ( X, d , µ ) thenon-negative, self-adjoint operator associated to Ch , and we denote it by ∆ . Wealso write ( e − t ∆ ) t ≥ for the semi-group generated by ∆ .For the scope of our work, we must know under which conditions does the Dirich-let form E of a Dirichlet space ( X, d , µ, E ) coincide with the Cheeger energy of ( X, d , µ ) . The next result brings us such a condition in the context of PI Dirichletspaces; it follows from [KSZ14, Th. 4.1]. Proposition 1.21.
Let ( X, d E , µ, E ) be a PI Dirichlet space. Assume that forsome
T > there exists a locally bounded function κ : [0 , T ] → [0 , + ∞ ) such that lim inf t → κ ( t ) = 1 and ˆ X ϕ dΓ( P t u ) ≤ κ ( t ) ˆ X P t ϕ dΓ( u ) (29) for all u ∈ D ( E ) , nonnegative ϕ ∈ D ( E ) ∩ C c ( X ) and t ∈ [0 , T ] . Then Ch = E . To perform our analysis in Sections 5 and 6, we need some results from the theoryof spaces satisfying a Riemannian Curvature Dimension condition
RCD ∗ ( K, n ) ,where K ∈ R is fixed from now on. In our setting, the original formulation of the RCD(
K, n ) and RCD ∗ ( K, n ) conditions based on optimal transport [Stu06b, LV09,BS10, AGS14] is less relevant than the one provided by a suitable combination of[AGS15] and [EKS15, Section 5]. In a general framework which covers our needs,these two articles discuss on how to recover a distance d from a measure space ( X, µ ) equipped with a suitable Dirichlet energy E in such a way that ( X, d , µ ) isan RCD ∗ ( K, n ) space. When particularized to our context, these works provide uswith the following definition. Definition 1.22. A PI Dirichlet space ( X, d E , µ, E ) is called an RCD ∗ ( K, n ) spaceif and only if there exists T > such that for any t ∈ (0 , T ) and f ∈ D ( E ) ,
12 ( P t f − ( P t f ) ) ≥ I K ( t ) |∇ P t f | + J K ( t ) (∆ P t f ) n (BL(K,n)) TRUCTURE OF KATO LIMITS 21 holds in a weak sense, namely against any non-negative test function ϕ ∈ C c ( X ) ∩D ( E ) , where I K ( t ) := ( e K t − / K and J K ( t ) := ( e Kt − Kt − / K for any K = 0 and I ( t ) = t , J ( t ) = t / . Remark 1.23.
Inequality BL(K,n) is one of many equivalent forms of an estimatedue to Bakry and Ledoux [BL06] which is equivalent to the well-known Bakry-Émery condition [AGS15, Cor. 2.3].To conclude this section, let us consider an
RCD(0 , n ) space ( X, d , µ ) – we pointout that in case K = 0 , the RCD(
K, n ) and RCD ∗ ( K, n ) conditions are equivalent.The Bishop-Gromov theorem for RCD(0 , n ) spaces (known even for the broaderclass of CD(0 , n ) spaces, see [Vil09, Th. 30.11]) ensures that for any x ∈ X thevolume ratio r µ ( B ( x, r )) /r n is non-increasing, hence we can define the volumedensity at a point x as follows. Definition 1.24.
Let ( X, d , µ ) be an RCD(0 , n ) space. Then the volume density at x ∈ X is defined as ϑ X ( x ) := lim r → µ ( B ( x, r )) ω n r n ∈ (0 , + ∞ ] · Note that without any particular assumption ϑ X ( x ) may be infinite. On thismatter, N. Gigli and G. De Philippis introduced in [DPG18] an important defini-tion. Definition 1.25.
We say that an
RCD(0 , n ) space ( X, d , µ ) is weakly non-collapsedif the volume density ϑ X ( x ) is finite for all x ∈ X . Weakly non-collapsed
RCD(0 , n ) spaces are important to us because they enjoythe so-called volume-cone-implies-metric-cone property. To state this latter, wemust recall a couple of definitions.If ( Z, d Z ) is a metric space, then the metric cone over Z is the metric space ( C ( Z ) , d C ( Z ) ) defined in the usual way, see e.g. [BBI01, Section 3.6.2.]. Definition 1.26.
We say that a metric measure space ( X, d , µ ) is a α -metricmeasure cone with vertex x ∈ X for some α ≥ if there exists a metric measurespace ( Z, d Z , µ Z ) and an isometry ϕ : X → C ( Z ) sending x to the vertex of thecone C ( Z ) and such that d( ϕ µ )( r, z ) = d r ⊗ r α − d µ Z . Here is the property of weakly non-collapsed
RCD(0 , n ) spaces [DPG18, Theorem1.1] we use in a crucial way in our analysis. Proposition 1.27.
Let ( X, d , µ ) be a weakly non-collapsed RCD(0 , n ) space suchthat the function (0 , + ∞ ) ∋ r µ ( B r ( x )) /r n is constant for some x ∈ X . Then ( X, d , µ ) is a n -metric measure cone with vertex x . Kato limits
Let ( M n , g ) be a closed Riemannian manifold. The k · k , -closure H , ( M ) of the space of smooth functions and the distributional Sobolev space W , ( M ) coincide and do not depend on the metric g , see e.g. [Heb96, Chapter 2], hence weindifferently use both notations in the rest of the article. Moreover, since the spaceof smooth functions is k · k , -dense in the one of Lipschitz functions, we also have H , ( M, d g , ν g ) = H , ( M ) , and the Cheeger energy Ch g of ( M, d g , ν g ) coincidewith the usual Dirichlet energy defined by E ( u, v ) = ˆ M g ( ∇ u, ∇ v ) d ν g , E ( u ) = ˆ M |∇ u | d ν g , for any u, v ∈ W , ( M ) . As well-known, Ch g is a strongly local and regular Dirichletform with core C ∞ ( M ) and associated operator the Laplacian ∆ g . Moreover, d Ch g is a distance that coincides with d g . We denote by H : R + × M × M → R the heatkernel of Ch g which we call heat kernel of ( M, g ) , and by ( P t ) t> the associatedsemi-group. For any x ∈ M , we define ρ ( x ) = inf v ∈ T x M,g x ( v,v )=1 Ric x ( v, v ) and Ric - ( x ) = max {− ρ ( x ) , } . For all t > we introduce the following quantity:k t ( M n , g ) = sup x ∈ M ˆ t ˆ M H ( s, x, y ) Ric - ( y ) d ν g ( y ) d s. (30)This quantity is defined more generally as k t ( V ) for a Borel function V , whereRic - is replaced by V . Then V is said to be in the contractive Dynkin class when k t ( V ) < and in the Kato class if k t ( V ) tends to 0 as t goes to zero (see for example[G¨17, Chapter VI]). In our case, since the manifold is compact, Ric - always belongsto the Kato class.We point out that k t ( M n , g ) has a useful scaling property given by ∀ ε, t > , k t ( M n , ε − g ) = k ε t ( M n , g ) , (31)It is an easy consequence of the scaling property of the heat kernel: if H ε is the heatkernel of ( M n , ε − g ) , then H ε ( s, x, y ) = ε n H ( ε s, x, y ) for all s > and x, y ∈ M .In the following, we consider the next uniform bounds for sequences of closedsmooth manifolds. Definition 2.1.
Let { ( M nα , g α ) } α ∈ A be a sequence of closed manifolds. We say that { ( M nα , g α ) } α ∈ A satisfies • a uniform Dynkin bound if there exists T > such that sup α k T ( M α , g α ) ≤ n ; (UD)• a uniform Kato bound if there exists a non-decreasing function f :(0 , T ] → R + such that f ( t ) → when t → and for all t ∈ (0 , T ]sup α k t ( M α , g α ) ≤ f ( t ); (UK)• a strong uniform Kato bound if there exist a non-decreasing function f : (0 , T ] → R + , T, Λ > such that for all t ∈ (0 , T ]sup α k t ( M α , g α ) ≤ f ( t ) , ˆ T p f ( s ) s d s ≤ Λ . (SUK)We observe that obviously a Kato bound implies a Dynkin bound, and a strongKato bound implies a Kato bound. Indeed, if f is as in (SUK), then for any t ∈ (0 , T ] we have p f ( t ) ≤ Λ log (cid:18) Tt (cid:19) − . Therefore f tends to zero when t goes to zero. Without loss of generality, we canalways assume that the function f is bounded by (16 n ) − . Remark 2.2.
The scaling property of k t ( M, g ) ensures that the previous boundsare preserved when rescaling the metrics by a factor ε − for ε ∈ (0 , . Indeed, forany t > and ε ∈ (0 , k t ( M n , ε − g ) = k ε t ( M n , g ) < k t ( M n , g ) . TRUCTURE OF KATO LIMITS 23
Dynkin limits.
In this section, we prove a pre-compactness result for se-quences of manifolds satisfying a uniform Dynkin bound. We start by provingthat a uniform Dynkin bound leads to a uniform volume estimate and a uniformPoincaré inequality.
Proposition 2.3.
Let ( M n , g ) be a closed Riemannian manifold, and T > . As-sume k T ( M n , g ) ≤ n and set ν := e n . Then there exists θ ≥ and γ > depending only on n such that for any x ∈ M and < s < r ≤ √ T , ν g ( B r ( x )) ν g ( B s ( x )) ≤ θ (cid:16) rs (cid:17) ν , for any ball B ⊂ M with radius r ≤ √ T and any ϕ ∈ C ( B ) , ˆ B ( ϕ − ϕ B ) d ν g ≤ γ r ˆ B | dϕ | d ν g . In particular, ( M n , d g , ν g , Ch g ) is a PI κ , γ ( √ T ) Dirichlet space for κ = 2 ν θ . Remark 2.4.
The previous proposition is a minor variation of [Car19, Propositions3.8 and 3.11] where similar estimates were shown for balls with radii lower than diam( M ) / but with constants that additionally depended on diam( M ) . Proof.
Step 1.
Observe that ν > . We begin with proving the following Sobolevinequality: there exists λ > depending only on n such that for any ball B ⊂ M with radius r ≤ √ T and any ϕ ∈ C c ( B ) , (cid:18) ˆ B | ϕ | νν − d ν g (cid:19) − ν ≤ λ r ( ν g ( B )) ν (cid:20) ˆ B | dϕ | d ν g + 1 r ˆ B | ϕ | dν g (cid:21) . (32)To this aim, take r ∈ (0 , √ T ) , x ∈ M and y ∈ B r ( x ) . From [Car19, Theorem3.5], we know that there exists c n > depending only on n such that for any s ∈ (0 , r / , e − s/r H ( s, y, y ) ≤ H ( s, y, y ) ≤ c n ν g ( B r ( x )) (cid:18) r s (cid:19) ν/ · Moreover, since the function s H ( s, y, y ) is non-increasing, for any s > r / , e − s/r H ( s, y, y ) ≤ e − s/r H ( r / , y, y ) ≤ e − s/r c n ν g ( B r ( x )) 2 ν/ . As the function ξ e − ξ ξ − ν/ is bounded from above on [1 / , + ∞ ) by some con-stant c ′ n > depending only on n , we get e − s/r H ( s, y, y ) ≤ c ′ n (cid:18) r s (cid:19) ν/ c n ν g ( B r ( x )) 2 ν/ . Setting c ′′ n := max( c n , c ′ n ) , we obtain for any s > e − s/r H ( s, y, y ) ≤ c ′′ n r η ν g ( B r ( x )) 1 s η/ · In particular the heat kernel of the operator ∆+1 /r acting on L ( B, µ ) with Dirich-let boundary condition satisfies the same estimate and one deduces the Sobolevinequality (32) from a famous result of Varopoulos [Var85, Section 7]. Step 2.
Let us prove 1. To this aim, we follow the argument of [Aku94, Car96]:for given < s < r ≤ √ T , apply the Sobolev inequality (32) in the case B = B r ( x ) and ϕ = dist( · , M \ B s ( x )) to obtain (cid:16) s (cid:17) (cid:0) ν g ( B s ( x )) (cid:1) − ν ≤ λ r ( ν g ( B r ( x )) ν ν g ( B s ( x )) . (33)Set Θ( τ ) := ν g ( B τ ( x )) /τ η for any τ > and use elementary manipulations to turn(33) into Θ( s/ − /ν ≤ ΛΘ( s ) with Λ = 2 ν +1 λ Θ( r ) /ν · (34)Iterating, we get for any positive integer ℓ Θ( s/ ℓ ) (1 − /ν ) ℓ ≤ Λ P ℓ − k =0 (1 − /ν ) k Θ( s ) . As lim ℓ →∞ Θ( s/ ℓ ) (1 − ν ) ℓ = 1 and P + ∞ k =0 (1 − /ν ) k = ν/ we obtain ≤ Λ ν/ Θ( s ) = [2 ν +1 λ ] ν/ Θ( r ) Θ( s ) which is . with θ = [2 ν +1 λ ] ν/ .Let us now prove 2. We recall a classical result (see e.g. [HSC01, Th. 2.7]): ametric measure space ( X, d , µ ) doubling at scale √ T equipped with a strongly localand regular Dirichlet form E with heat kernel H satisfies a Poincaré inequality atscale √ T if and only if there exists c, C, ε , ε > such that for all x ∈ M and t ∈ (0 , ε √ T ) ,(i) H ( t, x, x ) ≤ Cµ ( B √ t ( x )) − ,(ii) cµ ( B √ t ( x )) − ≤ inf { H ( t, x, y ) : y ∈ B ε √ t ( x ) } .In our context, (i) and (ii) hold with ε = ε = 1 and C, c depending only on n :indeed, (i) is a direct consequence of [Car19, Theorem 3.5] while (ii) follows fromthe same argument as in the proof of [Car19, Proposition 3.11] based on a methodfrom [CG97]. (cid:3) The previous proposition ensures that we can apply Theorem 1.17 to a sequenceof manifolds satisfying a uniform Dynkin bound (UD), and obtain the followingpre-compactness result.
Corollary 2.5.
Let
T > and { ( M α , g α ) } α ∈ A be a sequence of closed manifoldssatisfying the uniform Dynkin bound (UD) . For all α ∈ A let o α ∈ M α and set µ g α = ν g α ν g ( B √ T ( x )) , E g α = ˆ M α | du | d µ α , for all u ∈ C ( M α ) . Then there exist a PI κ , γ ( √ T ) Dirichlet space ( X, d , µ, o, E ) and a subsequence B ⊂ A such that { ( M β , d g β , µ β , o β , E β ) } β ∈ B Mosco-Gromov-Hausdorff converges to ( X, d , µ, o, E ) . Definition 2.6.
A metric Dirichlet space ( X, d , µ, o, E ) is called a Dynkin limit space if it is obtained as the Mosco-Gromov-Hausdorff limit of a sequence of closedRiemannian manifolds satisfying a uniform Dynkin bound.
Remark 2.7.
For any
T > , the class of Dynkin limit spaces obtained as lim-its of manifolds satisfying (UD) is closed under pointed Mosco-Gromov-Hausdorffconvergence: this follows from a direct diagonal argument.As a consequence, tangent cones of Dynkin limit spaces equipped with their in-trinsic distance are Dynkin limit spaces too. Indeed, if ( X, d , µ, o, E ) is the pointedMosco-Gromov-Hausdorff limit of pointed manifolds { ( M α , g α , o α ) } α ∈ A satisfying(UD) and ( X x , d x , µ x , x, E x ) is a tangent cone at x ∈ X provided by Corollary TRUCTURE OF KATO LIMITS 25 { ( M β , ε − β d g β , ν g β ( B ε β ( x β )) − ν β , x β , ε β ν g β ( B ε β ( x β )) − E β ) } β ∈ B where B ⊂ A , x β ∈ M β and ε β > for any β ∈ B , and x β → x , ε β → .Another consequence is that if { z α } α ∈ A belongs to a compact set of X and { ε α } ⊂ (0 , + ∞ ) satisfies ε α → , then there exists a subsequence B ⊂ A such that the se-quence { ( X, ε − β d , µ ( B ε β ( z β )) − µ, z β , ε β µ ( B ε β ( z β )) − E ) } β converges to a Dynkinlimit space ( Z, d Z , µ Z , z, E Z ) which may also be written as the limit of a sequenceof rescaled manifolds { ( M β , ε − β d g β , ν g β ( B ε β ( x β )) − ν g β , x β , ν g β ( B ε β ( x β )) − E β ) } β ,with x β ∈ M β for any β ∈ B and d ( z β , Φ β ( x β )) → , where Φ β is as in Char-acterization 1.2.2. Kato limits.
In this section we consider manifolds with a uniform Katobound. In this case, some better properties can be proved for the distance in thelimit and for tangent cones (see the next remark and Proposition 2.12). Thanks tothe previous pre-compactness result we can give the following definition.
Definition 2.8.
A Dirichlet space ( X, d , µ, o, E ) is called a Kato limit space if itis obtained as a Mosco-Gromov-Hausdorff limit of manifolds with a uniform Katobound (UK) . Remark 2.9.
A Kato limit space is obviously a PI κ , γ ( √ T ) Dirichlet space for any
T > such that f ( T ) ≤ / (16 n ) . Remark 2.10.
As in the case of Dynkin limits, tangent cones of Kato limits areKato limits as well. Not only, if ( X, d , µ, o, E ) is a Kato limit and ( X x , d x , µ x , x, E x ) is a tangent cone at x ∈ X provided by Corollary 1.18, then this latter is a limitof rescaled manifolds { ( M α , ε − α d g α , ν g α ( B ε α ( x α )) − ν α , x α , ε α ν g α ( B ε α ( x α )) − E α )) } such that for all t > k t ( M α , ε − α g α ) → as α → ∞ . Indeed, we have k t ( M α , ε − α g α ) = k ε α t ( M α , g α ) ≤ f ( ε α t ) → as α → ∞ . Thisobservation also applies to spaces ( Z, d Z , µ Z , z, E Z ) obtained as limits of rescalingsof X centered at varying but convergent points, as considered in Remark 2.7.As a consequence of Theorem 1.17, any Dynkin limit space ( X, d , µ, o, E ) satisfies d ≤ d E . But for Kato limit spaces, this inequality turns out to be an equality. Toprove this fact, we need the following Li-Yau inequality which was proved in [Car19,Proposition 3.3]. Proposition 2.11.
Let ( M n , g ) be a closed Riemannian manifold, and T > .Assume k T ( M n , g ) ≤ n · If u is a positive solution of the heat equation on [0 , T ] × M , then for any ( t, x ) ∈ [0 , T ] × M , e − √ n k t ( M,g ) | du | u − u ∂u∂t ≤ n t e √ n k t ( M,g ) . (35)We are now in a position to prove the following. Proposition 2.12.
Let ( X, d , µ, o, E ) be a Kato limit space. Then d = d E .Proof. We only need to prove d ≥ d E . Let { ( M nβ , g β , o β ) } β ∈ B be a sequence ofclosed Riemannian manifolds satisfying a uniform Kato bound and such that thesequence { ( M β , d β , µ β , o β , E β ) } converges in the Mosco-Gromov-Hausdorff sense to ( X, d , µ, o, E ) . In particular, there exists T > such that { ( M nβ , g β ) } β ∈ B satisfies the uniform Dynkin bound (UD). We claim that for any x, y ∈ X , t ∈ (0 , T ) and θ ∈ (0 , , log (cid:18) H ( θt, x, x ) H ( t, x, y ) (cid:19) ≤ n /θ ) e + d ( x, y )4(1 − θ ) t e √ nf ( t ) · (36)Let us explain how to conclude from there. Multiply by − t and apply Varadhan’sformula (22) as t → to get d E ( x, y ) ≤ d ( x, y )1 − θ · The desired inequality follows from θ ↓ .In the following we thus prove (36). Take β ∈ B . Let u be a positive solutionof the heat equation on [0 , T ] × M β . Take x, y ∈ M β , t ∈ (0 , T ] and s ∈ (0 , t ) . Let γ : [0 , t − s ] → M β be a minimizing geodesic from y to x . For any τ ∈ [0 , t − s ] set φ ( τ ) := log u ( t − τ, γ ( τ )) and note that φ (0) = log u ( t, y ) and φ ( t − s ) = log u ( s, x ) . Differentiate φ at τ and apply the Li-Yau inequality (35) and the simple fact k t − τ ≤ k t to get the firstinequality in the following calculation, where u and its derivatives are implicitlyevaluated at ( t − τ, γ ( τ )) and where we omit ( M β , g β ) in the notation k t ( M β , g β ) for the sake of simplicity: ˙ φ ( τ ) = − u ∂u∂t + h ˙ γ ( τ ) , d log u i≤ ne √ n k t − τ t − τ ) − e − √ n k t | du | u + h ˙ γ ( τ ) , d log u i = ne √ n k t − τ t − τ ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − √ n k t d log u − e √ n k t γ ( τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + e √ n k t | ˙ γ ( τ ) | ≤ ne √ n k t − τ t − τ ) + e √ n k t | ˙ γ ( τ ) | ≤ ne √ n k t − τ t − τ ) + e √ n k t d β ( x, y )4( t − s ) · Hence, when integrating between and t − s and changing variables in the firstterm, we obtain: log (cid:18) u ( s, x ) u ( t, y ) (cid:19) ≤ n ˆ ts e √ n k τ dττ + e √ n k t d β ( x, y )4( t − s ) · Write s = θt for some θ ∈ (0 , . The uniform Dynkin bound (UD) allows us tobound e √ n k τ in the first term of the right-hand side by e , while the uniform Katobound lets us bound e √ n k t by e √ nf ( t ) in the second term. Thus log (cid:18) u ( s, x ) u ( t, y ) (cid:19) ≤ n e log( t/s ) + d β ( x, y )4(1 − θ ) t e √ nf ( t ) . Choose s = θt and u ( τ, z ) = H β ( τ, x, z ) for any ( τ, z ) ∈ (0 , T ) × M to get log (cid:18) H β ( θt, x, x ) H β ( t, x, y ) (cid:19) ≤ n /θ ) e + d β ( x, y )4(1 − θ ) t e √ nf ( t ) · The Mosco-Gromov-Hausdorff convergence ( M β , d β , µ β , o β , E β ) → ( X, d , µ, o, E ) even-tually yields (36). (cid:3) TRUCTURE OF KATO LIMITS 27
Remark 2.13.
Observe that the previous proof also applies more generally inthe case of a sequence of manifolds { ( M α , g α ) } α ∈ A such that there exists a non-decreasing function f : (0 , T ) → R + , tending to as t goes to 0 and for which lim sup α →∞ k t ( M α , g α ) ≤ f ( t ) for all t ∈ (0 , T ] . In particular, we have d = d E whenever ( X, d , µ, o, E ) is the Mosco-Gromov-Hausdorfflimit of a sequence { ( M α , g α ) } α ∈ A such that for some T > α →∞ k T ( M α , g α ) = 0; in this case f is constantly equal to . As a consequence of Remark 2.10, Proposition2.12 applies to tangent cones (and rescalings centered at convergent points) of Katolimits.2.3. Ahlfors regularity.
We now discuss volume estimates for closed Riemannianmanifolds ( M n , g ) satisfying the strong Kato condition integral bound k T ( M n , g ) ≤ n and ˆ T p k s ( M n , g ) s d s ≤ Λ , (37)for some T, Λ > . This condition was also considered in [Car19].The proof of [Car19, Proposition 3.13] gives the following volume estimate whichimproves the one given in Proposition 2.3. Proposition 2.14.
Let ( M n , g ) be a closed Riemannian manifold satisfying (37) for some T, Λ > . Then there exists a constant C n > depending only on n suchthat for all x ∈ X and ≤ r ≤ s ≤ √ T then ν g ( B r ( x )) ≤ C Λ+1 n r n and ν g ( B s ( x )) ν g ( B r ( x )) ≤ C Λ+1 n (cid:16) sr (cid:17) n . (38) Proof.
The upper bound is proven in [Car19, Page 3144]. The other estimate is aconsequence of the proof of Proposition 2.3 and of the estimate (see again [Car19,Page 3144]): < s < t ≤ √ T : s n H ( s, x, x ) ≤ C Λ+1 n t n H ( t, x, x ) , that holds for any x ∈ M and < s < t ≤ √ T . (cid:3)
Then using the doubling properties [Proposition 1.2-ii)], we get the followinguniform local Ahlfors regularity result:
Corollary 2.15.
Let ( M n , g ) be a closed Riemannian manifold satisfying (37) forsome T, Λ > . Then there exists a constant C n > depending only on n such thatfor all o, x ∈ X and ≤ r ≤ √ T then ν g (cid:0) B √ T ( o ) (cid:1) T n ≤ C (Λ+1) d ( x,o ) √ T n ν g ( B r ( x )) r n (39) Remark 2.16.
A metric measure space ( X, d , µ ) for which there exists C , C > such that C ≤ µ ( B r ( x )) /r n ≤ C for any x ∈ X and r > is usually calledAhlfors n -regular. Thus (38) and (39) tell that for any R > B R ( o ) , d g , ν g ) isAhlfors n -regular with constants depending on n, Λ , R, T and ν g ( B √ T ( o )) /T n .2.4. Non-collapsed strong Kato limits.
We introduce a last class of limit spacesthat we are going to deal with, that is strong Kato limits with a non-collapsingassumption. In this case, the limit measure carries the local Ahlfors regularitydescribed above. This will be important in proving that tangent cones are metriccones and for our stratification result.
Definition 2.17.
A Dirichlet space ( X, d , µ, o, E ) is called a strong Kato limit ifit is obtained as a Mosco-Gromov-Hausdorff limit of pointed manifolds ( M α , g α , o α ) with a strong uniform Kato bound. It is called non-collapsed strong Kato limit if moreover there exists v > such that for all αv g α ( B √ T ( o α )) ≥ vT n , (NC) where T is given in Definition 2.1. Remark 2.18.
The convergence of the measure ensures that if the manifolds ( M α , g α ) satisfy a strong uniform Kato bound, then inequalities (38) and (39)passes to the limit. In particular, if ( X, d , µ, o, E ) is a non-collapsed strong Katolimit, then there exists constants C, λ such that we have for all < r ≤ s ≤ √ T and x ∈ X µ ( B r ( x )) ≤ Cr n , µ ( B s ( x )) µ ( B r ( x )) ≤ C (cid:16) sr (cid:17) n , and the lower bound µ ( B r ( x )) ≥ ve − λ d ( x,o ) √ T r n . As a consequence, for any
R > , ( B R ( o ) , d , µ ) is Ahlfors n -regular, with constantsdepending on n, Λ , R, T and v . Remark 2.19.
As in the previous cases, tangent cones of strong Kato limits arestrong Kato limits. Under the non-collapsing assumption (NC), the previous remarkensures a local Ahlfors n -regularity. Then as observed in Section 2, we can considertangent cones as limits of re-scaled manifolds { ( M α , ε − α d g α , ε − nα ν g α , x α } , that is tosay that we can replace the re-scaling factor ν g α ( B √ T ( x α )) of the measures by ε − nα .This sequence of re-scaled manifolds also satisfies the non-collapsing condition.This also applies to limits ( Z, d Z , µ Z , z, E Z ) of rescalings of non-collapsed strongKato manifolds centered at varying but convergent points.2.5. L p -Kato condition. The strong Kato condition is implied by a uniform onthe L p -Kato constant for p > . Introduce k p,T ( M, g ) := sup x ∈ M T p − ˆ T ˆ M H ( s, x, y ) Ric - ( y ) p d ν g ( y ) ds ! p . (40)When p > , and using Hölder inequality, we obtain k t ( M, g ) ≤ (cid:18) tT (cid:19) − p k p,T ( M, g ) . Hence a sequence { ( M nα , g α ) } α ∈ A of closed manifolds satisfying sup α ∈ A k p,T ( M, g ) < ∞ satisfies a strong uniform Kato bound.As noticed in [Car19, Proposition 3.15], we can estimate the L Kato constantin terms of the Q − curvature.Recall that if ( M, g ) is Riemannian manifold of dimension n ≥ , its Q − curvatureis defined by: Q g = 12( n −
1) ∆Scal g − n − | Ric | + c n Scal g , where c n = n − n +16 n − n ( n − ( n − . TRUCTURE OF KATO LIMITS 29
Proposition 2.20.
Let ( M n , g ) be a closed Riemannian manifold of dimension n ≥ such that : − κ ≤ Q g and | Scal g | ≤ κ , where κ > . Then k ,T ( M, g ) ≤ C ( n ) κ √ T (cid:16) κ √ T (cid:17) . Analytic properties of manifolds with a Dynkin bound
In this section we develop some analytic tools in the setting of manifolds withDynkin bound on k T ( M n , g ) , that is for which there exists T > such that k T ( M n , g ) ≤ n . (D)3.1. Good cut-off functions.
The existence of cut-off functions with suitablybounded gradient and Laplacian is a key technical tool in the theory of Ricci limitspaces ([CC97]) and
RCD ∗ ( K, N ) spaces ([MN19]). Our next proposition tells thatsuch functions also exist in the context of manifolds with a Dynkin bound; it alsoprovides an alternative proof to the one given by Cheeger and Colding on manifoldswith Ricci curvature bounded from below. Proposition 3.1.
Let ( M n , g ) be a closed Riemannian manifold satisfying (D) forsome T > . Then for any ball B r + s ( x ) ⊂ M there exists a function χ ∈ C ∞ ( M ) such that ≤ χ ≤ and(1) χ = 1 on B r ( x ) ,(2) χ = 0 on M \ B r + s ( x ) ,(3) there exists a constant C = C ( n ) > such that |∇ χ | ≤ C ( n )min( s, √ T ) and | ∆ χ | ≤ C ( n )min( s , T ) · We start with the following useful consequence of a Kato bound:
Lemma 3.2. If ( M n , g ) be a closed Riemannian manifold satisfying (D) for some T > . Assume that u : M → R is a Λ -Lipschitz function then for any t ∈ (0 , T ] and x ∈ M : (cid:12)(cid:12) ∇ e − t ∆ u (cid:12)(cid:12) ( x ) ≤ . Proof.
It is well known (see [CR20, Remark 1.3.2], or [Voi77, Proof of Theorem 1,step i)], the Kato bound (D) implies that for any t ∈ (0 , T ] (cid:13)(cid:13)(cid:13) e − t (∆ − Ric - ) (cid:13)(cid:13)(cid:13) L ∞ → L ∞ ≤ n n − ≤ . Then the result follows by the domination properties for the Hodge-Laplacian ~ ∆ = d ∗ d + dd ∗ = ∇ ∗ ∇ + Ric on forms: (cid:12)(cid:12) ∇ e − t ∆ u (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e − t~ ∆ ∇ u (cid:12)(cid:12)(cid:12) ≤ e − t (∆ − Ric - ) |∇ u | . (cid:3) The inequalities of Li-Yau have the following consequence which will be veryuseful to us
Lemma 3.3. If ( M n , g ) be a closed Riemannian manifold satisfying (D) for some T > . The heat kernel of ( M n , g ) satisfies for any x ∈ M and t ∈ (0 , T ] : ˆ M |∇ z H ( t, x, z ) d ν g ( z ) ≤ r e n t and ˆ M |∇ z H ( t, x, z ) | H ( t, y, z ) d ν g ( z ) ≤ e n t . Proof.
Using Hölder’s inequality and the stochastic completeness of H , we get ˆ M |∇ z H ( t, x, z ) d ν g ( z ) ≤ ˆ M |∇ z H ( t, y, z ) | H ( t, y, z ) d ν g ( z ) ! / (cid:18) ˆ M H ( t, y, z ) d ν g ( z ) (cid:19) = ˆ M |∇ z H ( t, y, z ) | H ( t, y, z ) d ν g ( z ) ! / . Hence the first estimate follows from the second one’s. By the Li-Yau estimate (35), |∇ z H ( t, y, z ) | H ( t, y, z ) ≤ e n t H ( t, y, z ) + e ∂H ( t, y, z ) ∂t · (41)Since ˆ M ∂H ( t, y, z ) ∂t d ν g ( z ) = ∂∂t ˆ M H ( t, y, z ) d ν g ( z ) | {z } =1 = 0 , the second estimate follows. (cid:3) Proof of Proposition 3.1.
Take x ∈ M and r, s > . Let ρ be the distance functionto x (i.e. ρ ( · ) = d g ( x, · ) ) and set ρ t := e − t ∆ ρ for any t ∈ (0 , T ] . Note that from the previous lemma, ρ t is -Lipschitz. Then forany y ∈ M , (cid:12)(cid:12)(cid:12)(cid:12) ∂ρ t ∂t ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = | ∆ ρ t ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ M h∇ z H ( t, y, z ) , ∇ ρ ( z ) i d z (cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ M |∇ z H ( t, y, z ) | |∇ ρ ( z ) | | {z } =1 ν g -a.e. d z ≤ e r n t Where we used Lemma 3.3. This estimate implies | ρ t ( y ) − ρ ( y ) | ≤ e √ n √ t. Therefore, if y ∈ B r ( x ) then ρ t ( y ) ≤ r + e √ n √ t while if y ∈ M \ B r + s ( x ) then ρ t ( y ) ≥ r + s − e √ n √ t. Hence defining t = (cid:16) s e √ n (cid:17) , we choose t = t o if t o ≤ T and t = T if t o > T . Sothat ρ t ( y ) ≤ r + s if y ∈ B r ( x ) and ρ t ( y ) ≥ r + 3 s if y ∈ M \ B r + s ( x ); TRUCTURE OF KATO LIMITS 31
Let u : R + → R + be a smooth function such that u = ( on [0 , / , on [3 / , + ∞ ) . Set χ ( y ) := u (cid:18) ρ t ( y ) − rs (cid:19) for any y ∈ M . Then χ = ( on B r ( x ) , on M \ B r + s ( x ) . Since dχ ( y ) = 1 r u ′ (cid:18) ρ t ( y ) − rs (cid:19) dρ t ( y ) and ∆ χ ( y ) = 1 s u ′ (cid:18) ρ t ( y ) − rs (cid:19) ∆ ρ t ( y ) − s u ′′ (cid:18) ρ t ( y ) − rs (cid:19) | dρ t | ( y ) for any y ∈ M , setting L := sup R | u ′ | + | u ′′ | ), we get k dχ k ∞ ≤ Ls and k ∆ dχ k ∞ ≤ L (cid:18) e √ ns √ t + 4 s (cid:19) . (cid:3) Remark 3.4 (Complete Kato manifolds) . Note that the above proof makes useof an integrated version of the Li-Yau inequality only. In this regard, it wouldbe interesting to study whether the assumption (D) on a complete Riemannianmanifold implies ˆ M |∇ z H ( t, y, z ) | H ( t, y, z ) d ν g ( z ) ! ≤ C ( n ) √ t · This would ensure the existence of good cut-off functions which would in turnprovide the Li-Yau inequality, and then make possible the study of limits of completeRiemannian manifolds satisfying the uniform bound (UD).3.2.
Hessian estimates.
Good cut-off functions are particularly relevant to de-duce the following powerful Hessian estimates, that we will use in Section 8.
Proposition 3.5.
Let ( M n , g ) be a closed Riemannian manifold satisfying (D) for some T > . Then there exists a constant C = C ( n ) > such that for any u ∈ C ∞ ( M ) and any ball B r ( x ) ⊂ M , ˆ B r ( x ) |∇ du | d ν g ≤ C ( n ) ˆ B r ( x ) (cid:20) (∆ u ) + 1min( r , T ) | du | (cid:21) d ν g . (42) If u is additionally harmonic, then ˆ B r ( x ) |∇ du | d ν g ≤ C ( n )min( r , T ) ˆ B r ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | du | − B r ( x ) | du | d ν g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ν g . (43) Proof.
Step 1.
Using again [CR20, Remark 1.3.2], or [Voi77, Proof of Theorem 1,step i)], we obtain the following estimate: (cid:13)(cid:13)(cid:13) e − T (∆ − Ric - ) (cid:13)(cid:13)(cid:13) L → L ≤ n n − ≤ . (44) Step 2.
All the integrals in these steps are taken with respect to ν g , hence weskip the notation d ν g for the sake of brevity. Take u ∈ C c ( B r ( x )) . The estimate(44) implies that the bottom of the spectrum of ∆ − Ric - is bounded from belowby − log 2 T . Thus for any v ∈ C ∞ ( M ) , ˆ M Ric - v ≤ ˆ M (cid:20) | dv | − log(2) T v (cid:21) . (45)Let χ be a cut-off function as built in Proposition 104 such that χ = 1 on B r ( x ) and χ = 0 on M \ B r ( x ) . Apply Bochner’s formula to u : |∇ du | + 12 ∆ | du | + Ric( du, du ) = h d ∆ u, du i , multiply by χ and integrate over M . This gives ˆ M χ |∇ du | + 12 ˆ M χ ∆ | du | ≤ ˆ M Ric - | du | χ + ˆ M χ h d ∆ u, du i . (46)We control the second term in the right-hand side of (46) as follows, using succes-sively integration by parts, the Cauchy-Schwarz inequality, and the elementary fact ab ≤ a + b : ˆ M h d ∆ u, χ du i = ˆ M χ (∆ u ) − ˆ M ∆ u h dχ , du i≤ ˆ M χ (∆ u ) + ˆ M χ | ∆ u | | dχ || du |≤ ˆ M χ (∆ u ) + ˆ M | dχ | | du | . (47)Now we control the first term in the right-hand side of (46) as follows. Thanks to(45) we have ˆ M Ric - | du | χ ≤ (cid:18) ˆ M |∇ ( χ | du | ) | + log 2 T ˆ M ( χ | du | ) (cid:19) . Since ˆ M |∇ ( χ | du | ) | = ˆ M h χ ∇| du | , ∇ ( χ | du | ) i + h| du |∇ χ, ∇ ( χ | du | ) i = ˆ M χ |∇| du || + χ | du |h∇ χ, ∇| du |i + | du |h∇ χ, ∇ ( χ | du | ) i = ˆ M χ |∇| du || + ˆ M h∇ χ, χ | du |∇| du | + | du |∇ ( χ | du | ) | {z } = ∇ ( χ | du | ) i and |∇| du || ≤ |∇ du | , we get ˆ M Ric - | du | χ ≤ (cid:18) ˆ M χ |∇ du | + ˆ M (∆ χ ) χ | du | + log 2 T ˆ M ( χ | du | ) (cid:19) . (48)Combining (46) with (47) and (48), we get ˆ M χ |∇ du | ≤ ˆ M χ (∆ u ) + ˆ M (cid:20) | dχ | + log 22 T χ + 12 | ∆ χ | χ −
12 (∆ χ ) (cid:21) | du | which eventually leads to (42) thanks to the properties of χ . TRUCTURE OF KATO LIMITS 33
The second estimate (43) is obtained in a similar way by replacing ∆ | du | with ∆( | du | − c ) in (46), where c = ffl B r ( x ) | du | d ν g . (cid:3) Gradient estimates for harmonic functions.
We conclude with the fol-lowing gradient estimates, that will also be useful in Section 8.
Lemma 3.6.
Let ( M n , g ) be a closed Riemannian manifold satisfying (D) for some T > , and let h : B r ( x ) → R be a harmonic function. Then for some constant c n > depending only on n :(1) sup B r ( x ) |∇ h | ≤ c r/ √ Tn B r ( x ) |∇ h | d ν g ! / ,(2) sup B r ( x ) |∇ h | ≤ c r/ √ Tn r sup B r ( x ) | h | .Proof. We first proof the result when r ≤ √ T . Consider the operator A = (cid:18) ∆ g + 1 T (cid:19) − Ric - . Assumption (D) ensures that k A k L ∞ → L ∞ ≤ n e − ≤ n . See for example [Car19, Lemma 3.18].The same is true when replacing the Laplacian on M by the Laplacian ∆ B onthe ball B = B r ( x ) with the Dirichlet boundary conditions : that is introducing A B = (cid:18) ∆ B + 1 T (cid:19) − Ric - , we get k A B k L ∞ → L ∞ ≤ / (8 n ) . As a consequence, if f = A B (1) then we find aunique ϕ ∈ L ∞ ( B ) solving ϕ = A B ϕ + f. Note that A B preserves the positivity: v ≥ ⇒ A B v ≥ . Then it is not difficultto show that ϕ satisfies the inequality ≤ ϕ ≤ n − n ≤ . Moreover by construction ϕ ∈ W , ( B ) and is zero along ∂B .Consider J = 1 + ϕ . By definition, J solves the equation (cid:18) ∆ B + 1 T − Ric - (cid:19) J = 1 T , and ≤ J ≤ . Now consider the Laplacian ∆ J associated to the quadratic form E J (Ψ) = ˆ B | d Ψ | J d ν g , on the space L ( B, J d ν g ) . Then for any Ψ ∈ L ( B, J d ν g ) we have J − (cid:18) ∆ + 1 T − Ric - (cid:19) ( J Ψ) = ∆Ψ − J − h dJ, d Ψ i + Ψ JT = ∆ J Ψ + Ψ
JT .
When choosing
Ψ = | dh | J we obtain ∆ J (cid:18) | dh | J (cid:19) + | dh | J T = J − (cid:18) ∆ | dh | + | dh | T − Ric - | dh | (cid:19) ≤ | dh | JT , where we used that ∆ | dh | ≤ Ric - | dh | because of Bochner inequality. We thenconclude that ∆ J Ψ ≤ T Ψ . (49)Now for ν = e n , the following Sobolev inequality holds for all ϕ ∈ C ∞ ( B ) (cid:18) ˆ B | ϕ | νν − d ν g (cid:19) − ν ≤ C ( n ) r ν g ( B ) ν (cid:20) ˆ B | d ϕ | d ν g + 1 r ˆ B | ϕ | d ν g (cid:21) , this implying the analog Sobolev inequality for the measure J d ν g : (cid:18) ˆ B | ϕ | νν − J d ν g (cid:19) − ν ≤ C ( n ) r ν g ( B ) ν (cid:20) ˆ B | d ϕ | J d ν g + 1 r ˆ B | ϕ | J d ν g (cid:21) . Together with inequality (49) and De Giorgi-Nash-Moser iteration, this leads to sup B r ( x ) Ψ ≤ C ( n ) s ν g ( B ) ˆ B r/ ( x ) Ψ J d ν g . Since J is bounded between 1 and 2, we then obtain sup B r ( x ) | dh | ≤ C ( n ) s ν g ( B ) ˆ B r/ ( x ) | dh | d ν g , (50)thus the first inequality.As for the second inequality, take ξ ∈ C ∞ c ( M ) such that ξ ≡ on B r/ ( x ) and | dξ | ≤ C/r for some C > . The integration by parts formula applied to ξh , thatis ˆ B | d ( ξh ) | d ν g = ˆ B | dξ | h d ν g + ˆ B ξ h ∆ h d ν g , and the fact that h is harmonic imply ˆ B | d ( ξh ) | d ν g ≤ Cr ˆ B | h | d ν g ≤ Cr ν g ( B ) sup B | h | . The left-hand side is bounded from below by ´ B r/ ( x ) | dh | d ν g , hence the right-hand side of (50) is bounded from above by the previous right-hand side.The proof when r > √ T follows. Indeed the doubling property implies thatwhen y ∈ B r ( x ) we have ν g ( B r ( x ) ≤ C r √ T n ν g ( B √ T ( y )) . As a consequence we have: B √ T ( y ) | dh | d ν g ≤ (cid:0) ν g ( B √ T ( y ) (cid:1) − ˆ B r ( x ) | dh | d ν g ≤ C r √ T n B r ( x ) | dh | d ν g . But we have already proved that | dh | ( y ) ≤ C B √ T ( y ) | dh | d ν g . (cid:3) Remark 3.7.
Notice that whenever r ≤ √ T we do not need to consider theexponent r/ √ T in the previous estimates. TRUCTURE OF KATO LIMITS 35 Curvature-dimension condition for Kato limits
In this section we prove that the Cheeger energy built from the metric measurestructure of a Kato limit space ( X, d , µ, o, E ) always coincides with the limit Dirich-let energy E . Moreover, we show that tangent cones of a Kato limit space are all RCD(0 , n ) spaces, and that they are additionally weakly non-collapsed in case thespace is a non-collapsed strong Kato limit. We obtain these two latter statementsby establishing the Bakry-Ledoux gradient estimate BL(0 , n ) on a specific class ofKato limit spaces, namely those obtained from a sequence of closed Riemannianmanifolds { ( M nα , g α ) } for which there exists T > such that k T ( M α , g α ) tends tozero as α → ∞ .For these purposes and following [AGS15, Subsection 2.2], we define the followingquantity A on a strongly local, regular Dirichlet space ( X, d , µ, E ) . For all t > and u ∈ D ( E ) , ϕ ∈ L ( X ) ∩ L ∞ ( X ) with ϕ ≥ , we set A t ( u, ϕ )( s ) := 12 ˆ X ( P t − s u ) P s ϕ d µ (51)for any s ∈ [0 , t ] . As it is shown in [AGS15, Lemma 2.1], the function s A t ( u, ϕ )( s ) is continuous on [0 , t ] and continuously differentiable on (0 , t ] with de-rivative given by dd s A t ( u, ϕ )( s ) = ˆ X P s ϕ dΓ( P t − s u ) . (52)for any s ∈ (0 , t ] . Whenever ϕ additionally belongs to D ( E ) , the map s A t ( u, ϕ )( s ) is in C ([0 , t ]) and the previous formula is valid for s = 0 .4.1. Differential inequalities.
We first need to prove some differential inequali-ties on closed manifolds ( M n , g ) with a smallness condition on k t ( M n , g ) . Theorem 4.1.
Let ( M n , g ) be a closed Riemannian manifold satisfying k t ( M n , g ) < (SK) for some t > . Then for all v ∈ W , ( M ) with ∆ g v ∈ L ( M ) ∩ L ∞ ( M ) , for anynon-negative ϕ ∈ W , ( M ) ∩ L ∞ ( M ) , we have the inequality ˆ M ( P t ϕv − ϕ ( P t v ) ) d ν g ≥ e − k t ( M,g ) (cid:18) t ˆ M ϕ | dP t v | d ν g + t n ˆ M ϕ (∆ g P t v ) d ν g (cid:19) . (53) In the next subsection, under the assumption k T ( M α , g α ) → , we aim to passinequality (53) to a limit space in order to get the Bakry-Ledoux gradient estimate BL(0 , n ) . To this aim, it is useful to rewrite this inequality in terms of A andits derivative with respect to s ∈ (0 , t ) . Since on a closed manifold ( M n , g ) , thederivative of A is simply dd s A t ( u, ϕ )( s ) = ˆ M | dP t − s u | P s ϕ d ν g , we can rephrase Theorem 4.1 as follows. Theorem 4.2.
Let ( M n , g ) be a closed Riemannian manifold satisfying (SK) forsome t > . Then for all v ∈ W , ( M ) with ∆ g v ∈ L ( X ) ∩ L ∞ ( X ) , for anynon-negative ϕ ∈ W , ( M ) ∩ L ∞ ( M ) and any s ∈ (0 , t ) , A t ( v, ϕ )( t ) − A t ( v, ϕ )( s ) ≥ e − k t ( M,g ) (cid:18) ( t − s ) dd s A t ( v, ϕ )( s ) + 2 n ( t − s ) A t (∆ v, ϕ )( s ) (cid:19) . (54) Indeed, with A and its derivative, inequality (53) writes as A t ( v, ϕ )( t ) − A t ( v, ϕ )(0) ≥ e − k t ( M,g ) (cid:18) t ˆ M ϕ | dP t v | d ν g + 2 n t A t (∆ v, ϕ )( t ) (cid:19) . Now for any s ∈ [0 , t ] we also have k t − s ( M, g ) ≤ k t ( M, g ) < / so the previousholds with t − s instead of t : A t − s ( v, P s ϕ )( t − s ) − A t − s ( v, P s ϕ )(0) ≥ e − k t ( M,g ) (cid:20) ( t − s ) ˆ M P s ϕ | dP t − s v | d ν g + 2 n ( t − s ) A t − s (∆ v, P s ϕ )( t − s ) (cid:21) and this rewrites easily as inequality (54).The proof of Theorem 4.1 relies on a modified version of the function s dd s A t ( u, ϕ )( s ) . For any closed Riemannian manifold ( M n , g ) , any t > and anypositive “gauging” function J : [0 , t ] × M → R , we define B J ( u, ϕ )( s ) := ˆ M | dP t − s u | ( P s ϕ ) J t − s d ν g for any s, u, ϕ as above, where J t − s ( · ) = J ( t − s, · ) . For the sake of simplicity, fromnow on in this section we denote P τ u, P τ ϕ by u τ , ϕ τ for any τ > . Lemma 4.3.
Let ( M n , g ) be a closed Riemannian manifold, t > and J : [0 , t ] × M → (0 , + ∞ ) be a smooth function. Then for any ε > , dd s B J ( u, ϕ )( s ) ≥ ˆ M ϕ s (cid:18)(cid:18) − ∆ g J − ˙ J − ε | dJ | J − J Ric - (cid:19) | du t − s | +2(1 − ε ) J (∆ g u t − s ) n (cid:19) d ν g (55) for any s ∈ [0 , t ] , where ˙ J is a shorthand for dd s J .Proof. When deriving B J ( u, ϕ ) with respect to s we obtain: dd s B J ( u, ϕ )( s ) = ˆ M ϕ s (cid:16) − ∆ g ( J | du t − s | ) − ˙ J | du t − s | + 2 J h d ∆ g u t − s , du t − s i (cid:17) d ν g = ˆ M ϕ s (cid:0) − (∆ g J ) | du t − s | − J ∆ g | du t − s | + 2 h dJ, ∇| du t − s | i− ˙ J | du t − s | + 2 J h d ∆ g u t − s , du t − s i (cid:17) d ν g , where we have used the Leibniz formula ∆ g ( J | du t − s | ) = (∆ g J ) | du t − s | + J ∆ g ( | du t − s | ) − h∇| du t − s | , dJ i = (∆ g J ) | du t − s | + J ∆ g ( | du t − s | ) − ∇ du t − s ( du t − s , dJ ) . Then by using Bochner’s formula − ∆ g | df | + 2 h d ∆ g f, df i = 2( |∇ df | + Ric( df, df )) with f = u t − s we obtain dd s B J ( u, ϕ )( s ) = ˆ M ϕ s (cid:16) ( − ∆ g J − ˙ J ) | du t − s | + 2 J ( |∇ du t − s | + Ric( du t − s , du t − s ))+4 ∇ du t − s ( dJ, du t − s )) d ν g . TRUCTURE OF KATO LIMITS 37
By using the fact that, for all x ∈ M , Ric x ≥ − Ric - ( x ) we get the lower bound dd s B J ( u, ϕ )( s ) ≥ ˆ M ϕ s h ( − ∆ g J − ˙ J − J Ric - ) | du t − s | + 2 J |∇ du t − s | +4 √ J ∇ du t − s (cid:18) dJ √ J , du t − s (cid:19)(cid:21) Now, for any ε > we have: √ J ∇ du t − s (cid:18) dJ √ J , du t − s (cid:19) ≥ − (cid:18) εJ |∇ du t − s | + 1 ε | dJ | J | du t − s | (cid:19) , Therefore we get dd s B J ( u, ϕ )( s ) ≥ ˆ M ϕ s (cid:20)(cid:18) − ∆ g J − ˙ J − ε | dJ | J − J Ric - (cid:19) | du t − s | +2(1 − ε ) J |∇ du t − s | (cid:3) d ν g . We conclude by using |∇ du t − s | ≥ (∆ g u t − s ) /n . (cid:3) Lemma 4.4.
Let ( M n , g ) be a closed Riemannian manifold satisfying (SK) forsome t > . Set ε := 4 k t ( M, g ) . Then there exists a unique solution J : [0 , t ] × M → (0 , + ∞ ) to the problem ∆ g J + ˙ J + 2 ε | dJ | J + 2 J Ric - = 0 J (0 , x ) = 1 (EJ) which satisfies e − t ( M,g ) ≤ J ≤ . Proof.
Consider δ = ε − ≥ , we have ∆ g ( J − δ ) = − δJ − δ − (cid:18) ∆ g J + ( δ + 1) | dJ | J (cid:19) = − δJ − δ − (cid:18) ∆ g J + 2 ε | dJ | J (cid:19) . Define I := J − δ . Then J solves (EJ) if and only if I is a solution of ( ∆ g I + ˙ I − δI Ric - = 0 I (0 , x ) = 1 This latter equation is equivalent to the following integral equation: I ( t, x ) = 1 + 2 δ ˆ t ˆ M H ( t − s, x, y ) Ric - ( y ) I ( s, y ) d ν g ( y ) d s. (56)Consider the map f ∈ L ∞ ([0 , t ] × M ) T f ∈ L ∞ ([0 , t ] × M ) defined by T f ( s, x ) = 2 δ ˆ t ˆ M H ( t − s, x, y ) Ric - ( y ) f ( s, y ) d ν g ( y ) d s. By definition of k t ( M, g ) and of δ , the operator norm of T satisfies: k T k L ∞ → L ∞ ≤ δ k t ( M, g ) = 2 (cid:18) ε − (cid:19) k t ( M, g ) . Recall that ε = 4 k t ( M, g ) , so k T k L ∞ → L ∞ ≤ − t ( M, g ) < . As a consequence, the operator Id − T is invertible and I = (Id − T ) − is theunique solution of the equation (56) that satisfies k I k L ∞ ≤ . Then the integral equation (56) implies that ≤ I ≤ δ k t ( M, g ) k I k L ∞ ≤ δ k t ( M, g ) ≤ e δ k t ( M,g ) . Therefore we get e − k t ( M,g ) ≤ J ≤ for J = I − /δ , as we wished. (cid:3) Corollary 4.5.
Let ( M n , g ) be a closed Riemannian manifold satisfying (SK) forsome t > . Then for all u ∈ C ( M ) , ϕ ∈ C ( M ) with ϕ ≥ and τ ∈ (0 , t ] : , ˆ M ϕ | dP τ u | d ν g ≤ e k t ( M,g ) ˆ M P τ ϕ | du | d ν g . (57) Proof.
We only need to prove (57) for τ = t as condition (SK) and our proof remainstrue if t is replaced by any τ ∈ (0 , t ] . First observe that inequality (55) and thelower bound for J given by Lemma 4.4 imply dd s B J ( u, ϕ )( s ) ≥ e − k t ( M,g ) − t ( M, g )) n ˆ M ϕ s (∆ g u t − s ) d ν g ≥ e − k t ( M,g ) n ˆ M ϕ s (∆ g u t − s ) d ν g , (58)where we used that − x ≥ e − x on (cid:2) , (cid:3) , with x = 4 k t ( M, g ) . In particular thelatter inequality yields that dd s B J ( u, ϕ )( s ) ≥ . Therefore, when integrating between and t , we get ˆ M ϕ ( P t | du | − J | dP t u | ) d ν g ≥ , which leads to ˆ M P t ϕ | du | d ν g ≥ ˆ M J | dP t u | d ν g . Inequality (57) then immediately follows by using the lower bound J ≥ e − k t ( M,g ) . (cid:3) Remark 4.6.
Corollary 4.5 can be rephrased in the following way: if ( M n , g ) isa closed Riemannian manifold satisfying (SK), then for any u ∈ W , ( M ) and any t > , | dP t u | ≤ e k t ( M,g ) P t ( | du | ) holds in the weak sense. Of course, the right-hand side can be bounded from aboveby e / n P t ( | du | ) . Thus, as a direct consequence of P t being non-negative andsub-Markovian, if u is κ -Lipschitz, then P t u is e / n κ -Lipschitz.We are now in position to prove Theorem 4.1. Proof of Theorem 4.1.
We consider again inequality (58). By definition of A andsince P t − s ∆ g u = ∆ g P t − s u , we can write it as dd s B J ( u, ϕ )( s ) ≥ e − k t ( M,g ) n A (∆ g u, ϕ )( s ) . Since A is monotone non decreasing in s , A (∆ g u, ϕ )( s ) is bounded from below byits value in s = 0 . Then we get dd s B J ( u, ϕ )( s ) ≥ e − k t ( M,g ) n ˆ M ϕ (∆ g u t ) d ν g . We integrate this latter inequality between and t , so that we get ˆ M ( ϕ t | du | − ϕJ t | du t | ) d ν g ≥ e − k t ( M,g ) n t ˆ M ϕ (∆ g u t ) d ν g . TRUCTURE OF KATO LIMITS 39
Using the lower bound of Lemma 4.4 for J , we get ˆ M ϕ t | du | d ν g ≥ e − k t ( M,g ) ˆ M ϕ | du t | d ν g + e − k t ( M,g ) n t ˆ M ϕ (∆ g u t ) d ν g ≥ e − k t ( M,g ) (cid:18) ˆ M ϕ | du t | d ν g + 2 n t ˆ M ϕ (∆ g u t ) d ν g (cid:19) · We also have k s ( M, g ) ≤ k t ( M, g ) < / for any s ∈ (0 , t ] . Hence if s ∈ (0 , t ] then ˆ M ϕ s | du | d ν g ≥ e − k t ( M,g ) (cid:18) ˆ M ϕ | du s | d ν g + 2 n s ˆ M ϕ (∆ g u s ) d ν g (cid:19) · Apply this with t = s and u = P t − s v = v t − s to get ˆ M ϕ s | dv t − s | d ν g ≥ e − k t ( M,g ) (cid:18) ˆ M ϕ | dv t | d ν g + 2 n s ˆ M ϕ (∆ g v t ) d ν g (cid:19) . (59)Observe that the left-hand side of the previous inequality can be rewritten as thefollowing derivative with respect to s : ˆ M ϕ s | dv t − s | d ν g = dd s ˆ M ϕ s v t − s ν g . Taking this into account while integrating (59) between and t yields (53). (cid:3) Convergence of the Energy.
Let us prove now that the Cheeger energy ofa Kato limit space ( X, d , µ, o, E ) coincides with the limit Dirichlet energy E . Theorem 4.7.
Let ( X, d , µ, E , o ) be a Kato limit space. Then E = Ch d . Remark 4.8.
Theorem 4.7 implies that for Kato limit spaces ( X, d , µ, o ) , thepmGH convergence of the approximating sequence of manifolds implies the Moscoconvergence of the associated energies. As a consequence, if X is compact then wehave convergence of the spectrum of the rescaled Laplacians of the approximatingmanifolds to the spectrum of the Laplacian associated with the Cheeger energy.This generalizes results of J. Cheeger and T. Colding [CC00b, Section 7], where auniform lower bound on the Ricci curvature is assumed, and of K. Fukaya [Fuk87],under a uniform bound on the curvature. Proof of Theorem 4.7.
Let { ( M α , g α , o α ) } α be a sequence of pointed Riemannianmanifolds satisfying a uniform Kato bound and such that { ( M α , d g α , µ α , o α , E α ) } α ∈ A converges in the Mosco-Gromov-Hausdorff sense to ( X, d , µ, E , o ) . Let T > and f : (0 , T ] → [0 , + ∞ ) be the non-decreasing function in Definition 2.1. We know fromProposition 2.12 that d = d E . Moreover, by Remark 2.9, we get that ( X, d E , µ, E ) is a PI ( R ) Dirichlet space. Therefore, thanks to Proposition 1.21, we are left withshowing that for any u ∈ D ( E ) , any non-negative ϕ ∈ D ( E ) ∩ C c ( X ) and any t ∈ [0 , T ] , ˆ X ϕ dΓ( P t u ) ≤ e f ( t ) ˆ X P t ϕ dΓ( u ) . (60)Let u, ϕ be as above. Set L := k ϕ k L ∞ and let R > be such that supp ϕ ⊂ B R ( o ) . Let { u α } α , { ϕ α } be two sequences, where u α ∈ D ( E α ) and ϕ α ∈ C c ( X α ) ∩D ( E α ) for any α , such that • u α E −→ u , • the sequence { ϕ α } α converges uniformly to ϕ , • ≤ ϕ α ≤ L and supp ϕ α ⊂ B R +1 ( o α ) for any α . The Mosco convergence E α → E guarantees the existence of { u α } while PropositionA.1 ensures the existence of { ϕ α } . Let ( P αt ) t ≥ (resp. ( P t ) t ≥ ) be the heat semi-group of the Dirichlet space ( M α , d α , µ α , E α ) (resp. ( X, d , µ, E ) ) for any α . Fix t ∈ (0 , T ] and s ∈ [0 , t ] . Set a ( s ) := A t ( u, ϕ )( s ) and a α ( s ) := A αt ( u α , ϕ α )( s ) for any α . We claim that lim α a α ( s ) = a ( s ) . (61)Indeed, by (3) in Proposition 1.15; the sequence { P αt − s u α } α converges strongly in L to P t − s u . Moreover, let us prove that { P αs ϕ α } α converges uniformly on compactsets to P s ϕ . Since P s ϕ is continuous, this follows from showing that for any x ∈ X and any given sequence { x α } such that M α ∋ x α → x ∈ X , lim α P αs ϕ α ( x α ) = P s ϕ ( x ) . (62)Let H α (resp. H ) be the heat kernel of ( M α , d α , µ α , E α ) (resp. ( X, d , µ, E ) ) for any α . We have P αs ϕ α ( x α ) = ˆ M α H α ( s, x α , y ) ϕ α ( y ) d µ α ( y ) for any α , and similarly P s ϕ ( x ) = ˆ X H ( s, x, y ) ϕ ( y ) d µ ( y ) . The heat kernel estimate (20) ensures that we can apply Proposition B.3 to thecontinuous functions { H α ( s, x α , · ) ϕ α ( · ) } α , H ( s, x, y ) ϕ ( y ) : this directly establishes(62). We are then in a position to apply Proposition B.1 to make the integrals a α ( s ) = ˆ X ( P αt − s u α ) P αs ϕ α converge to a ( s ) , as claimed in (61).From (52), we have a ′ α ( s ) = ˆ M α P αs ϕ α (cid:12)(cid:12) dP αt − s u α (cid:12)(cid:12) d µ α . Assume from now on that s ∈ (0 , t/ . According to Corollary 4.5, we know that a ′ α ( s ) ≤ e f ( t ) a ′ α ( t − s ) . Integrating this inequality between and s and dividing by s yields a α ( s ) − a α (0) s ≤ e f ( t ) a α ( t ) − a α ( t − s ) s . Letting α → ∞ , then s → , and using expression (52) for a ′ ( s ) = A ′ t ( u, ϕ )( s ) imply inequality (60). (cid:3) Remarks 4.9.
Similarly to what we pointed out in Remark 2.13, it is clear thatTheorem 4.7 also holds when we assume the following more general assumption:there exists a non-decreasing function f : (0 , T ] → R + such that lim f = 0 and lim sup α →∞ k t ( M α , g α ) ≤ f ( t ) for all t ∈ (0 , T ] . In particular, when lim α k T ( M α , g α ) = 0 we have both d = d E and E = Ch .According to Remark 2.10, tangent cones (and rescalings ( Z, d Z , µ Z , z, E Z ) cen-tered at convergent points) of a Kato limit are obtained as limits of manifolds suchthat for all t > , k t ( M α , g α ) → as α → ∞ . Thus Theorem 4.7 then applies tosuch spaces. TRUCTURE OF KATO LIMITS 41
The
RCD condition for a certain class of Kato limit spaces.
In the nextkey result we prove that any Kato limit space obtained from manifolds { ( M α , g α ) } with k T ( M nα , g α ) → for some T > satisfies the Riemannian Curvature Dimen-sion condition RCD(0 , n ) . Theorem 4.10.
Let ( X, d , µ, o, E ) be a Dirichlet space obtained as pointed Mosco-Gromov-Hausdorff limit of spaces { ( M α , d g α , µ α , o α , E α ) } , where { ( M nα , g α , o α ) } isa sequence of closed pointed Riemannian manifolds such that k T ( M nα , g α ) → (H) for some T > . Then ( X, d , µ ) is an RCD(0 , n ) space.Proof. Our goal is to establish the Bakry-Ledoux
BL(0 , n ) estimate: for t ∈ (0 , T ) and u ∈ D ( E ) and ϕ ∈ D ( E ) ∩ C c ( X ) with ϕ ≥ , we aim at showing ˆ X ϕ ( P t u − ( P t u ) ) d µ ≥ ˆ X ϕ (cid:18) t dΓ ( P t u )d µ + t n (∆ P t u ) (cid:19) d µ, (63)starting from inequality (54) in Theorem 4.2. As in the proof of Theorem 4.7, welet { u α } α , { ϕ α } be two sequences where u α ∈ D ( E α ) , ϕ α ∈ C c ( X α ) ∩ D ( E α ) for any α such that • u α E −→ u , • the sequence { ϕ α } α converges uniformly to ϕ , • ≤ ϕ α ≤ L and supp ϕ α ⊂ B R +1 ( o α ) for any α , for some L, R > .Take t ∈ (0 , T ] and ≤ s ≤ t . Like in the proof of the previous Theorem 4.7, weset a α ( s ) := A αt ( u α , ϕ α )( s ) for any α and a ( s ) := A t ( u, ϕ )( s ) . We know from therethat lim α a α ( s ) = a ( s ) . For any α , we let ( P αt ) t ≥ be the heat semi-group associated with E α and we set c α ( s ) = ˆ M α P αs ϕ α (cid:0) ∆ α P αt − s u α (cid:1) d µ α and c ( s ) = ˆ X P s ϕ (∆ P t − s u ) d µ. Thanks to (5) in Proposition 1.15, we know that ∆ α P αt strongly converges to ∆ P t in the sense of bounded operators. In particular for any s ∈ [0 , t ) , we have thestrong convergence ∆ α P αt − s u α L −→ ∆ P t − s u, hence the same argument to get the convergence a α ( s ) → a ( s ) gives lim α c α ( s ) = c ( s ) . Now observe that dividing inequality (54) by ν g α ( B √ T ( o α ) implies, for any < s Corollary 4.11. Let ( X, d , µ, o, E ) be a (resp. non-collapsed strong) Kato limitspace. Then for any x ∈ X , any measured tangent cone ( X x , d x , µ x , x ) is a (resp.weakly non-collapsed) RCD(0 , n ) metric measure space. The fact that tangent cones of non-collapsed strong Kato limits are weakly non-collapsed follows from the local Ahlfors regularity given in Remark 2.18. Remark 4.12. The previous Corollary applies in particular to iterated tangentcones and to convergent rescalings (not necessarily centered at a same point) ofa Kato limit. Indeed, let ( X, d , µ, o, E ) be a Kato limit and assume that for somesequence { ε α } ⊂ (0 , + ∞ ) such that ε α ↓ and some points { x α } ⊂ X , the sequenceof pointed rescalings { ( X, ε − α d , ε − nα µ, x α ) } converges to some pointed metric mea-sure space ( Z, d Z , µ Z , z ) . Thanks to Remark 2.19, we know that ( Z, d Z , µ Z ) hasthe same properties as a tangent cone. As a consequence, ( Z, d Z , µ Z ) is a weaklynon-collapsed RCD(0 , n ) space. TRUCTURE OF KATO LIMITS 43 Tangent cones are metric cones In this section, we prove that the tangent cones of a non-collapsed strong Katolimit space are all metric cones. To this aim, we introduce two crucial quantities.Let ( X, d , µ, E ) be a metric Dirichlet space admitting a heat kernel H . We recallthat n is a given positive integer. For any x ∈ X and s, t > , we define the Θ -volume by Θ x ( s ) := (4 πs ) − n ˆ X e − d x,y )4 s d µ ( y ) and similarly the θ -volume by θ x ( s, t ) := (4 πs ) − n ˆ X e − U ( t,x,y )4 s dµ ( y ) where U is defined by H ( t, x, y ) = (4 πt ) − n e − U ( t,x,y )4 t (64)for any y ∈ X . Remark 5.1. We point out the following properties of θ x and Θ x .(i) θ x ( t, t ) = ´ X H ( t, x, y ) d µ ( y ) is identically if the Dirichlet space is stochas-tically complete, for instance for PI Dirichlet space.(ii) The Chapman-Kolmogorov property yields θ x (cid:18) t , t (cid:19) = (4 πt ) n H ( t, x, x ) . (65)(iii) We always have the lower bound: Θ x ( s ) ≥ (4 πs ) − n e − r s µ ( B r ( x )) , (66)hence if µ is non trivial then Θ x is always positive (but perhaps infinite).When it is necessary, we specify the space X to which the previous quantitiesare associated by writing Θ Xx ( s ) , θ Xx ( s, t ) .We start by recalling an elementary property of the Θ -volume that relates it tothe density of volume at a given point.5.1. On a doubling space.Lemma 5.2. Let ( X, d , µ ) be κ -doubling at scale R .i) The function ( s, x ) ∈ R ∗ + × X Θ x ( s ) is continuous.ii) For all x ∈ X and s > we have that Θ x ( s ) is finite. More precisely if t ∈ (0 , R ] , there is a constant A depending only κ such that for any x ∈ X and s ∈ (0 , R ] : Θ x ( s ) ≤ A µ ( B √ s ( x )) s n . (67) iii) If there exists ϑ ∈ (0 , + ∞ ) such that lim r → + µ ( B r ( x )) ω n r n = ϑ, then lim s → + Θ x ( s ) = ϑ .Proof. For any x ∈ X , by Cavalieri’s formula we can write Θ x ( s ) = ˆ + ∞ e − r s r s µ ( B r ( x )) d r (4 πs ) n A simple computation using Proposition 1.2-iii) ensures that this integral converges,hence Θ x ( s ) is well defined for any s > . Moreover, Lebesgue’s dominated convergence theorem implies that Θ x ( s ) de-pends continuously on ( s, x ) ∈ R ∗ + × X . In addition, when < s ≤ R , theestimate (67) follows from Θ x ( s ) ≤ µ ( B √ s ( x ))(4 πs ) n + ˆ X \ B √ s ( x ) e − d x,y )4 s (4 πs ) n d µ ( y ) ≤ µ ( B √ s ( x ))(4 πs ) n + ˆ + ∞√ s e − ρ s ρ s µ ( B ρ ( x )) d ρ (4 πs ) n ≤ µ ( B √ s ( x ))(4 πs ) n (cid:18) ˆ + ∞√ s e − ρ s + λ ρ √ s ρ s d ρ (cid:19) ≤ µ ( B √ s ( x ))(4 πs ) n (cid:18) ˆ + ∞ e − ρ + λρ ρ ρ (cid:19) . Now assume that lim r → + µ ( B r ( x )) ω n r n = ϑ. We have similarly Θ x ( s ) = ˆ B R ( x ) e − d x,y )4 s (4 πs ) n d µ ( y ) + ˆ M \ B R ( x ) e − d x,y )4 s (4 πs ) n d µ ( y ) . The same estimate yields ˆ M \ B R ( x ) e − d x,y )4 s (4 πs ) n d µ ( y ) ≤ µ ( B √ R ( x ))(4 πs ) n ˆ ∞ R e − ρ s + λ ρR ρ s d ρ = µ ( B √ R ( x ))(4 πs ) n ˆ ∞ R √ s e − ρ + λ ρ √ sR ρ ρ. We clearly have lim s → + ˆ M \ B R ( x ) e − d x,y )4 s (4 πs ) n d µ ( y ) = 0 . Moreover we have ˆ B R ( x ) e − d x,y )4 s (4 πs ) n d µ ( y ) = ˆ R √ s e − ρ ω n ρ n +1 π ) n µ ( B ρ √ s ( x )) ω n ( ρ √ s ) n d ρ. Note that there is some constant C (depending on x ) such that for r ≤ R : µ ( B r ( x )) ≤ Cr n . Hence the dominated convergence theorem and the fact that ˆ + ∞ e − ρ ω n ρ n +1 π ) n dρ = Θ R n (1) = 1 imply that lim s → + ˆ B R ( x ) e − d x,y )4 s (4 πs ) n d µ ( y )) = ϑ. (cid:3) We also have the following assertion about the comparison between the measureof balls and the Θ -volume: Lemma 5.3. Let ( X, d , µ ) be a metric measure space, and n > .i) let x ∈ X and c > , then Θ x ( s ) = c for all s > if and only if µ ( B r ( x )) = cω n r n for all r > . TRUCTURE OF KATO LIMITS 45 ii) If ( X, d , µ ) is κ -doubling at scale R , then for any s ∈ (0 , R ] : a µ ( B √ s ( x ))( √ s ) n ≤ Θ x ( s ) ≤ A µ ( B √ s ( x ))( √ s ) n . (68) where the constant a > depend only on n and the constant A depends onlyon κ .iii) Assume that for for some R > and x ∈ X : ∀ s ∈ (0 , R ] : c ≤ Θ x ( s ) ≤ C then for any r ∈ (0 , R ] : vr n ≤ µ ( B r ( x )) ≤ V r n . where the constant v, V depends only on n, c, C .Proof. The first equivalence follows from Cavalieri’s principle and some propertiesof the Laplace transform, see e.g. the proof of [CT19, Lem. 3.2]. The second assertion is a consequence of the two inequalities (66) and (67). For the third assertion : The upper bound is a consequence of (66). The upperbound follows from an argument used in the proof of [CT19, Lem. 5.1]. The estimate(66) also implies that for any ρ ≥ √ t : µ ( B ρ ( x )) ≤ (4 πs ) n e ρ s C. Hence for any κ ∈ (0 , , Θ x ( κs ) ≤ µ ( B √ s ( x ))(4 πκs ) n + ˆ X \ B √ s ( x ) e − d x,y )4 κs (4 πκs ) n d µ ( y ) ≤ µ ( B √ s ( x ))(4 πκs ) n + ˆ + ∞√ s e − ρ κs ρ κs µ ( B ρ ( x )) d ρ (4 πκs ) n ≤ µ ( B √ s ( x ))(4 πκs ) n + Cκ n ˆ + ∞√ s e − ( κ − ) ρ s ρ κs d ρ ≤ µ ( B √ s ( x ))(4 πκs ) n − Cκ n e − ( κ − ) ρ s − κ + ∞√ s ≤ µ ( B √ s ( x ))(4 πκs ) n + Ce − ( κ − ) (1 − κ ) κ n . Choosing κ small enough so that e − ( κ − ) (1 − κ ) κ n ≤ cC yields that c πκs ) n ≤ µ ( B √ s ( x )) . (cid:3) The Θ -volume is continuous with respect to pointed measured Gromov-Hausdorffconvergence: Proposition 5.4. Let { ( X α , d α , µ α , o α ) } α , ( X, d , µ, o ) be proper geodesic pointedmetric measure spaces κ -doubling at scale R such that ( X α , d α , µ α , o α ) → ( X, d , µ, o ) in the pmGH sense. Let x α ∈ X α → x ∈ X . Then for any s > : lim α Θ X α x α ( s ) = Θ Xx ( s ) . Proof. This is a direct consequence of the Proposition B.3. (cid:3) On a Dirichlet space. The following gives a relationship between the Θ -and θ -volume on a PI Dirichlet space. Proposition 5.5. If ( X, d E , µ, E ) is a PI( R ) -Dirichlet space then when the Θ -volume is defined with the intrinsic distance then for any s > : Θ x ( s ) = lim t → θ x ( s, t ) . Proof. According to the Varadhan’s formula (22), we get that for any x, y ∈ X d E ( x, y ) = lim t → U ( t, x, y ) . The Fatou’s lemma implies that Θ x ( s ) ≤ lim inf t → θ x ( s, t ) . (69)To get the limsup inequality, we will use the heat kernel upper bound (1.6) (see[Gri94, Theorem 5.2]): H ( t, x, y ) ≤ Cµ ( B R ( x )) R ν t ν (cid:18) d E ( x, y ) t (cid:19) ν +1 e − d E ( x,y )4 t for any x, y ∈ X and t ∈ (0 , R ) .Since there exists C > such that ξ ξ ν +1 e − ξ ≤ C for any ξ > , then choosing ξ = ε (1 + d E ( x, y ) /t ) where ε ∈ (0 , leads to H ( t, x, y ) ≤ Cµ ( B R ( x )) R ν t ν ε − ν − e ε +( ε − d E ( x,y )4 t , which implies that there is a constant C not depending on t ∈ (0 , R ] and y ∈ X such that − U ( t, x, y )4 ≤ t ( C + log(1 /t )) − (1 − ε ) d E ( x, y )4 (70)and then, for any s > , θ x ( s, t ) ≤ (cid:18) e C t (cid:19) ts Θ x ( s/ (1 − ε )) . (71)So that for any ε ∈ (0 , 1) :lim sup t → θ x ( s, t ) ≤ Θ x (cid:18) s − ε (cid:19) . Then the assertion follows from the continuity of Θ with respect to s . (cid:3) Remark 5.6. It is worth pointing out that if ( X, d E , µ, E ) is a PI κ , γ ( R ) -Dirichletspace then there is a constant α such that for any t ∈ (0 , R ) and any x, y ∈ X , weget the heat kernel bound t n µ ( B √ t ( x )) 1 α t n e − α d x,y ) t ≤ H ( t, x, y ) ≤ t n µ ( B √ t ( x )) α t n e − d x,y ) α t . This easily implies that (cid:18) (4 π ) n α (cid:19) ts Θ x (cid:16) s α (cid:17) ≤ (cid:18) t n µ ( B √ t ( x )) (cid:19) − ts θ x ( s, t ) ≤ (cid:0) (4 π ) n α (cid:1) ts Θ x (cid:16) α s (cid:17) . In particular, using Lemma 5.3, there is a positive constant η > depending only κ , γ , n such that for any x ∈ X and t, s > with t ≤ R and s ≤ α R then η µ ( B √ s ( x )) s n (cid:18) µ ( B √ t ( x )) t n (cid:19) − ts ≤ θ x ( s, t ) ≤ η − µ ( B √ s ( x )) s n (cid:18) µ ( B √ t ( x )) t n (cid:19) − ts . (72) TRUCTURE OF KATO LIMITS 47 The heat kernel upper bound implies similarly the continuity of the θ -volumeunder the pointed Mosco Gromov-Hausdorff convergence of PI( R )-Dirichlet space. Proposition 5.7. Let { ( X α , d α , µ α , o α , E α ) } α , ( X, d , µ, o, E ) be pointed PI κ , γ ( R ) -Dirichlet spaces such that ( X α , d α , µ α , o α , E α ) → ( X, d , µ, o, E ) in the pointed Mosco-Gromov-Hausdorff sense. Let x ∈ X and { x α } be such that x α ∈ X α for any α and x α → x . Then for any s, t > : lim α θ X α x α ( s, t ) = θ Xx ( s, t ) . Proof. Let s, t > , we know that the sequence f α ( y ) = H α ( t, x α , y ) convergeuniformly on compact set to f ( y ) = H ( t, x, y ) . Hence the same is true for theintegrand h α ( y ) = (4 πs ) − n (cid:18) (4 πt ) n H α ( t, x α , y ) (cid:19) ts = (4 πs ) − n e − UXα ( t,x,y )4 s uniformly on compact set to h ( y ) = (4 πs ) − n e − UX ( t,x,y )4 s . The uniform doubling and the uniform Poincaré inequality yields that there arepositive constants C, ν depending only on κ and γ such that H α ( t, x α , y ) ≤ Cµ ( B R ( x α )) max (cid:26) , R ν t ν (cid:27) e − d α ( xα,y ) Ct , for any y ∈ X . But lim α µ α ( B R ( x α )) = µ ( B R ( x )) , hence we find a constant suchthat for any α : h α ( y ) ≤ Ce − d α ( xα,y ) Cs . Hence the result follows also from Proposition B.3. (cid:3) A differential inequality. We now study the properties of the θ -volume onmanifolds satisfying a Dinkyin bound. Whenever this bound is improved to be anupper bound on the integral quantity ˆ T p k t ( M, g ) s d s ≤ Λ , for some T, Λ > , we obtain a monotone quantity that will be crucial in theremainder of this section. Proposition 5.8. Let ( M n , g ) be a closed Riemannian manifold satisfying k T ( M n , g ) ≤ n . for some T > . For τ ≤ T , set Γ τ ( M n , g ) := e √ n k τ ( M n ,g ) − . Then for any x ∈ M , t ∈ (0 , τ ) and s ≤ t/ (2Γ τ ( M n , g )) t ∂θ∂t + s ∂θ∂s + n Γ τ ( M n , g ) (cid:18) ts − st (cid:19) θ (73) has the same sign as t − s .Proof. Our proof will use the Li-Yau estimate ([Car19, Proposition 3.3]): when v : [0 , τ ] × M → R + is a solution of the heat equation then e − √ n k τ ( M n ,g ) | dv | v − v ∂v∂t ≤ e √ n k τ ( M n ,g ) n t . (74)For simplicity, let us write θ , ∂θ∂t , ∂θ∂s , U , H , k , Γ instead of θ x ( s, t ) , ∂θ∂t ( s, t ) , ∂θ∂s ( s, t ) , U ( t, x, y ) , H ( t, x, y ) , k τ ( M n , g ) , Γ τ ( M n , g ) , respectively. A direct computation implies ∂θ∂t = − s ˆ M ∂U∂t e − U s d ν g (4 πs ) n (75) ∂θ∂s = − n s θ + 14 s J (76)where J := ˆ M U e − U s d ν g (4 πs ) n · From (64) and the fact that H solves the heat equation, another computation gives − n t − ∂∂t (cid:18) U t (cid:19) − t ∆ U − t |∇ U | = 0 (77) − n − ∂U∂t + U t − 14 ∆ U − t |∇ U | = 0 (78)where here and in the rest of the proof Laplacian and gradient are taken withrespect to the y variable. Moreover the Li-Yau estimate provides (when t ≤ τ ): e − √ n k |∇ U | t + ∂∂t (cid:18) n t + U t (cid:19) ≤ e √ n k n t (79)Adding (77) and (79), together with the fact that e − √ n k − − e − √ n k Γ ≥ − Γ ,yields the estimate − ∆ U − Γ |∇ U | t ≤ n n . (80)Combine (75) and (78), to get ∂θ∂t = n s θ − ts J + 14 s ˆ M ∆ U e − U s d ν g (4 πs ) n + ˆ M |∇ U | ts e − U s d ν g (4 πs ) n (81)Integration by parts implies: s ˆ M ∆ U e − U s d ν g (4 πs ) n = − ˆ M |∇ U | s e − U s d ν g (4 πs ) n , Hence from (81) we get the two equalities: ∂θ∂t = n s θ − ts J + (cid:18) s − t (cid:19) ˆ M ∆ U e − U s d ν g (4 πs ) n , (82) ∂θ∂t = n s θ − ts J − (cid:18) s − t (cid:19) ˆ M |∇ U | s e − U s d ν g (4 πs ) n · (83)We use now in(82), the inequation (80) and we find a non negative function π : R + × [0 , τ ] × M → R + such that: ∂θ∂t = n t θ − ts J + (cid:18) s − t (cid:19) π − n (cid:18) s − t (cid:19) Γ θ − Γ (cid:18) s − t (cid:19) st ˆ M |∇ U | s e − U s d ν g (4 πs ) n · (84)We use now (83) to express in a different way the last term of the right handside of (84): ∂θ∂t = n t θ − ts J + (cid:18) s − t (cid:19) π − n (cid:18) s − t (cid:19) Γ θ + Γ st (cid:18) ∂θ∂t − n s θ + J ts (cid:19) · (85) TRUCTURE OF KATO LIMITS 49 Using (76) we get − n s θ + J ts = n (cid:18) t − s (cid:19) θ + st ∂θ∂s , hence ∂θ∂t = − st ∂θ∂s + (cid:18) s − t (cid:19) π − n (cid:18) s − t (cid:19) Γ θ + n t (cid:16) st − (cid:17) θ + Γ st (cid:18) ∂θ∂t + st ∂θ∂s (cid:19) · (86)And we get the following differential inequality: (cid:16) − Γ st (cid:17) (cid:18) t ∂θ∂t + s ∂θ∂s (cid:19) = (cid:18) ts − (cid:19) π − n (cid:18) ts − st (cid:19) θ (87)When we notice that s < t ⇒ ≤ − Γ st , we get that t ∂θ∂t + s ∂θ∂s + n Γ (cid:18) ts − st (cid:19) θ has the same sign as t − s . (cid:3) Remark 5.9. The same proof shows that, when the Ricci curvature is non-negative,the map λ θ x ( λs, λt ) is monotone non-increasing for s ≥ t and monotone non-decreasing for s ≤ t .The above proposition has the following immediate consequence under a strongerintegral bound. Corollary 5.10. Let ( M n , g ) be a closed manifold satisfying : ˆ T p k s ( M n , g ) s d s ≤ Λ · For any τ ∈ (0 , T ] we set Φ( τ ) = ˆ τ p k s ( M n , g ) s d s. Then there exists a positive constant c n , depending only on n , such that the followingholds. Let s, t ∈ (0 , T ] be such that s > t . Set λ := λ ( s, t ) = min n e − c n Λ st , e − √ n Λ o .Then the function λ ∈ [0 , λ ] → θ x ( λs, λt ) e c n Φ( λt ) ( ts − st ) is monotone:i) it is non decreasing if t ≥ s ,ii) it is non increasing if t ≤ s .Proof. Indeed we have for c ∈ R and λ, s > and t ∈ (0 , T ] : e − c Φ( λt ) ( ts − st ) ddλ (cid:16) θ x ( λs, λt ) e c Φ( λt ) ( ts − st ) (cid:17) = t ∂θ x ( λs, λt ) ∂t + s ∂θ ( λs, λt ) ∂s + c (cid:18) ts − st (cid:19) p k λt ( M n , g ) λ θ ( λs, λt ) . Now there is a constant b n such that if λ ∈ (0 , satisfies λ ≤ e − √ n Λ and λ ≤ e − b n st then k λt ( M n , g ) ≤ n and Γ λt ≤ b n p k λt ( M n , g ) ≤ b n /λ ) ˆ tλt p k s ( M n , g ) s d s ≤ b n Λlog(1 /λ ) ≤ t s · Under these conditions we get that s ≤ t λt and we can apply Proposition 5.8.Choosing c n = nb n yields the result. (cid:3) Consequences on non-collapsed strong Kato limits. We are now in po-sition to prove that tangent cones of strong non-collapsed Kato limits are measuremetric cones.Throughout this section, we fix constants T, Λ , v > and a non-decreasing func-tion f : (0 , T ] → R + such that ˆ T p f ( s ) s d s ≤ Λ . Without loss of generality, we assume that f ( T ) ≤ / (16 n ) and set Φ( τ ) = ˆ τ p f ( t ) t dt. According to Definition 2.17, a non-collapsed strong Kato limit ( X, d , µ, o ) is ob-tained as ( M α , d α , ν α , o α ) pmGH −→ ( X, d , µ, o ) , where { ( M α , g α ) } α are closed manifold satisfying the uniform estimatesfor all t ∈ (0 , T ] , sup α k t ( M α , g α ) ≤ f ( t ) , inf α ν g α ( B √ T ( o α )) ≥ vT n . Theorem 5.11. Let ( X, d , µ, o ) be a non-collapsed strong Kato limit in the senseof Definition 2.17, and x ∈ X . Then the following holds. (i) Any tangent cone of X at x is a measured metric cone. (ii) The volume density ϑ X ( x ) = lim r → µ ( B r ( x )) ω n r n is well defined. (iii) We have the following relationship between the behavior of the θ -volume andthe volume density : lim λ → θ x ( λs, λt ) = ϑ − ts X ( x ) . The proof of this theorem and Remark 5.1 (ii) will imply: Corollary 5.12. Let ( X, d , µ, o ) be a non-collapsed strong Kato limit. Then forany x ∈ X we have lim t → (4 πt ) n H ( t, x, x ) = 1 ϑ X ( x ) TRUCTURE OF KATO LIMITS 51 Moreover there is a positive constant η and a increasing function Φ : (0 , η T ] → R + satisfying lim t → Φ( t ) = 0 such that t ∈ (0 , η T ] exp (Φ( t )) (4 πt ) n H ( t, x, x ) is non decreasing.Proof of Theorem 5.11. Let ( X, d , µ ) be a non-collapsed strong Kato limit obtainedas above. According to Theorem 4.7, we know that we have Mosco convergence ofthe quadratic form u ˆ M α | du | g α d ν g α to the Cheeger energy Ch of ( X, d , µ ) . As a consequence, ( X, d , µ, Ch ) is a PI κ , γ ( √ T ) Dirichlet space. Therefore, the Θ and θ -volume are well defined on X . Moreoverwe know that if x α → x then Θ Xx ( s ) = lim α → + ∞ Θ M α x α ( s ) and θ Xx ( s, t ) = lim α → + ∞ θ M α x α ( s, t ) . According to Corollary 5.10, we also know that for any t, s > and t > with t ≤ T there is a constant ε > and a constant κ both depending on s and t , suchthat the function λ ∈ (0 , ε ] θ M α x α ( λs, λt ) e κ Φ( λt ) is monotone. Hence the same is true for the function λ ∈ (0 , ε ] θ Xx ( λs, λt ) e κ Φ( λt ) . Recall that, as we observed in Remark 2.18 and since ( X, d , µ ) is a non-collapsedstrong Kato limit, the measure µ has a local Ahlfors regularity and satisfies for all x ∈ X and ≤ r ≤ s ≤ √ T / µ ( B s ( x )) ≤ Cs n and µ ( B s ( x )) µ ( B r ( x )) ≤ C (cid:16) sr (cid:17) n . We also have the uniform lower bound: vs n e − λ d ( o,x ) √ T ≤ µ ( B s ( x )) . Hence by the estimate (72), there exist positive constants c, C depending on d ( o, x ) and on t/s such that c ≤ θ x ( λs, λt ) ≤ C. As a consequence, this monotone function has a well defined limit at λ = 0 , thatwe denote by ϑ x ( s, t ) . Moreover when t ≤ s : ϑ x ( s, t ) := lim λ → θ Xx ( λs, λt ) = sup λ ∈ (0 ,ε ) θ Xx ( λs, λt ) e κ Φ( λt ) , when t ≥ s : ϑ x ( s, t ) := lim λ → θ Xx ( λs, λt ) = inf λ ∈ (0 ,ε ) θ Xx ( λs, λt ) e κ Φ( λt ) · By construction the function ( s, t ) ϑ x ( s, t ) is -homogeneous : ∀ λ ∈ (0 , 1) : ϑ x ( λs, λt ) = ϑ x ( s, t ) . Let x ∈ X and ( Y, d Y , µ Y , y ) be a tangent cone of ( X, d , µ ) at x . Then thereexist a sequence { ε β } β ∈ (0 , + ∞ ) with ε β ↓ and a limit measure µ Y on Y suchthat ( X β = X, d β := ε − β d , µ β := ε − nβ µ, x ) pmGH −→ ( Y, d Y , µ, y ) . (88)Moreover the sequence (cid:16) ( X, d β := ε − β d , µ β := ε − nβ µ, Ch β = ε − nβ Ch , x ) (cid:17) β Mosco-Gromov-Hausdorff converges to ( Y, d Y , µ, Ch , y ) . Let H β be the heat kernel of the scaled Dirichlet spaces ( X β , d β , µ β , Ch β ) and U β the corresponding function suchthat H β ( t, x, y ) = (4 πt ) − n e − Uβ ( t,x,y )4 t . The scaling property of the heat kernel leads to U β ( t, x, y ) = ε − β U ( ε β t, x, y ) , so that we have θ X β x ( s, t ) = θ Xx ( ε β s, ε β t ) . Moreover, Proposition 5.7 ensures that for any s, t > we have θ Yy ( s, t ) lim β → + ∞ θ X β x ( s, t ) = lim β → + ∞ θ Xx ( ε β s, ε β t ) = ϑ x ( s, t ) . Hence the function ( s, t ) → θ Yy ( s, t ) is also -homogeneous, but according to Propo-sition 5.5, we get that Θ Yy ( s ) = lim t → ϑ X ( s, t ) . Hence s Θ Yy ( s ) is -homogeneous hence there is some c = Θ Yy (1) > such thatfor any s > : Θ Yy ( s ) = c. As a consequence, Lemma 5.3 implies for any r > µ Y ( B ( y, r )) = cω n r n . From Corollary 4.11, we know that ( Y, d Y , µ Y ) is a weakly non collapsed RCD(0 , n ) space, then according to Proposition 1.27, we know that ( Y, d Y , µ Y ) is a measuredmetric cone at y . This shows the first assertion (i).We have also shown that for any point x ∈ X and for any tangent cone ( Y, d Y , µ Y , y ) of X at x , the functions Θ Yy and θ Yy do not depend on the tangent cone Y .Recall that, as it was pointed out earlier in the proof, for any fixed x ∈ X thefunction r ∈ (0 , √ T / µ ( B r ( x )) /ω n r n is bounded above and below by positiveconstants, hence it admits limit points as r ↓ . Let ̟ be one of these limits pointsand ( r α ) α ⊂ (0 , + ∞ ) a sequence such that r α ↓ and ̟ = lim α µ ( B ( x, r α )) /ω n r nα .We can assume, extracting a subsequence, that the sequence of rescaled space ( X, d α := r − α d , µ α := r − nα µ, x ) converges for the pointed measure Gromov-Hausdorfftopology to some tangent cone ( Y, d Y , µ Y , y ) . In particular ̟ = µ Y ( B ( y )) ω n = Θ Yy (1) = lim t → ϑ x ( s, t ) . All limit points are hence equals and we obtain that the volume density is welldefined.It remains to show that lim λ → θ x ( λs, λt ) = ϑ − ts X ( x ) , that is to verify that ϑ x ( s, t ) = ϑ − ts X ( x ) . If ( Y, d Y , µ Y , y ) is a tangent cone of X at x , then we have shown that θ Yy ( s, t ) = ϑ x ( s, t ) . Since ( Y, d Y , µ Y , y ) is a measure metric cone at y , we get that for any z ∈ Y and t > : H Y ( t, y, z ) = 1 ϑ Y ( y )(4 πt ) n e − d Y ( y,z )4 t , TRUCTURE OF KATO LIMITS 53 where we recall that, since Y is a measure metric cone at y and a tangent cone of X at x , for any r > : ϑ Y ( y ) = µ ( B r ( y )) ω n r n = ϑ X ( x ) . Therefore the function U Y associated to H Y equals U Y ( t, y, z ) = d Y ( y, z ) + 4 tϑ Y ( y ) = d Y ( y, z ) + 4 tϑ X ( x ) . When using this equality in the definition of θ Yy ( s, t ) we obtain θ Yy ( s, t ) = ˆ Y e − UY ( t,y,z )4 s d µ Y ( z )(4 πs ) n = ϑ Y ( y ) − ts ˆ Y e − d Y ( y,z )4 s d µ Y ( z )(4 πs ) n = ϑ Y ( y ) − ts ˆ Y H Y ( s, y, z ) d µ Y ( z )= ϑ Y ( y ) − ts = ϑ X ( x ) − ts , where we have used the stochastic completeness of Y . (cid:3) This theorem has also the following useful consequence. Corollary 5.13. Let ( X, d , µ, o ) be a non-collapsed strong-Kato limit in the senseof Definition 2.17. Then at every point x ∈ X the volume density satisfies: ϑ X ( x ) ≤ . Proof. Let ( X, d , µ, o ) be a non-collapsed strong Kato limit defined as above andrecall that we defined for all t ∈ (0 , T ]Φ( τ ) = ˆ t p f ( s ) s d s. Let x ∈ X . We only need to show that lim t → θ Xx ( t/ , t/ 2) = ϑ X ( x ) − ≥ . Using Corollary 5.12 we know that for some η > , the function t ∈ (0 , η T ] exp (cid:18) Φ( t ) η (cid:19) (4 πt ) n H M α X ( t, x α , x α ) is non decreasing. But lim t → (4 πt ) n H M α X ( t, x α , x α ) = 1 hence for any t ∈ [0 , η T ] : θ M α x α ( t/ , t/ 2) = (4 πt ) n H M α X ( t, x α , x α ) ≥ exp (cid:18) − Φ( t ) η (cid:19) . By Proposition 5.7, we also have θ Xx ( t/ , t/ ≥ exp (cid:18) − Φ( t ) η (cid:19) . Hence the result. (cid:3) Our next result concerns the measure of balls in a non-collapsed strong Katolimit and the comparison between the n -Hausdorff measure and µ . Corollary 5.14. Let ( X, d , µ, o ) be a non-collapsed strong-Kato limit in the senseof 2.17, then for any ρ > and ε > there is some δ > such that r ≤ δ and x ∈ B ρ ( o ) : µ ( B r ( x )) ≤ ω n r n (1 + ǫ ) . As a consequence, µ is absolutely continuous with respect to the n -Hausdorff mea-sure and µ ≤ H n . To prove this corollary we will use, as in [Che01], the spherical Hausdorff measuredefined for any s > and any Borel set A in a metric space ( Z, d Z ) by H s ( A ) := lim δ → H sδ ( A ) , where for any δ ∈ (0 , + ∞ ] , H sδ ( A ) := inf (X i ω s r si : A ⊂ ∪ i B r i ( x i ) and ∀ i : r i < δ ) . Following [Sim83, Theorem 3.6] or [Mat95, Theorem 6.6], we have the followingresult : if H s ( A ) < ∞ then lim sup r → H s ( B r ( x ) ∩ A ) ω s r s ≤ (89)for H s − a.e. x ∈ A . Proof of Corollary 5.14. Indeed if the estimate were not true, then we would find ε > and sequences r α ↓ , x α ∈ B ρ ( o ) , such that the sequence of re-scaled spaces ( X, r − α d , r − nα µ, x α ) converges to some pointed metric measure space ( Z, d Z , µ Z , z ) with µ Z ( B ( z )) ≥ ω n (1 + ε ) . By Remark 2.19, we know that ( Z, d Z , µ Z , z ) is a non-collapsed strong Kato limitas well, then by Corollary 5.13 its volume density is smaller than 1. Moreover, ( Z, d Z , µ Z , z ) is obtained as a limit of re-scaled manifolds ( M α , ˜ g α ) such that for all t > α →∞ k t ( M α , ˜ g α ) = 0 . Then according to Remark 4.12, ( Z, d Z , µ Z , z ) is a weakly non collapsed RCD(0 , n ) space. Thus the Bishop-Gromov comparison theorem holds on ( Z, d Z , µ Z , z ) andwe get µ Z ( B ( z )) ≤ ϑ Z ( z ) ω n , hence a contradiction. The comparison with the Hausdorff measure is then straight-forward, because if A ⊂ B ρ ( o ) and ε > we find δ ∈ (0 , such that x ∈ B ρ +1 ( o ) and r < δ yields µ ( B r ( x )) ≤ ω n r n (1 + ǫ ) , so that µ ( A ) ≤ (1 + ǫ ) H n ( A ) . (cid:3) Remark 5.15. The above volume estimate can in fact be quantified on smoothclosed manifolds. Let v, T, Λ > and f : (0 , T ] → R + be a non-decreasing functionsuch that f ( T ) ≤ n and ˆ T p f ( s ) s d s ≤ Λ . Then for any ρ > and ε > , there exists δ > depending only on n , f , v , T suchthat if ( M n , g ) is a closed Riemannian manifold such that v ≤ ν g (cid:0) B √ T ( o ) (cid:1) T n and ∀ t ∈ (0 , T ] , k t ( M, g ) ≤ f ( t ) , TRUCTURE OF KATO LIMITS 55 then for any x ∈ B ρ ( o ) and r < δ : ν g ( B r ( x )) ≤ ω n r n (1 + ǫ ) . Stratification In this section, we prove a stratification theorem for non-collapsed strong Katolimit spaces. To state this result, we first give a useful definition. From now onwe equip R k with the classical Euclidean distance which we write d e , whatever k ∈ N \{ } . Definition 6.1. For k ∈ N \{ } , a pointed metric measure space ( Y, d Y , µ Y , y ) iscalled metric measure k -symmetric (mm k -symmetric for short) if there existsa metric measure cone ( Z, d Z , µ Z ) with vertex z such that ( Y, d Y , µ Y , y ) = ( R k × Z, d R k × Z , H k ⊗ µ Z , (0 k , z )) , where d R k × Z is the classical Pythagorean product distance, k is the origin of R k ,and the equality sign means that there exists an isometry ϕ : Y → R k × Z such that ϕ µ Y = H n ⊗ µ Z and ϕ ( y ) = (0 k , z ) . Let dim H A be the Hausdorff dimension of a subset A of a metric space ( X, d ) .Then our stratification theorem writes as follows. Theorem 6.2. Let ( X, d , µ, o ) be a non-collapsed strong Kato limit. We set forany x ∈ Xd ( x ) := sup { k ∈ N : one tangent cone at x is mm k -symmetric } ∈ { , . . . , n } and S k := { x ∈ X : d ( x ) ≤ k } for any k ∈ { , . . . , n } . Then the sets S k define afiltration of X S ⊂ S ⊂ . . . ⊂ S n − ⊂ S n , and the following holds. (i) The set S is countable. (ii) For any k ∈ { , . . . , n } we have dim H S k ≤ k. (iii) For µ -a.e. x ∈ X the set of tangent cones of ( X, d , µ ) at x is reduced to { ( R n , d e , ϑ X ( x ) H n , x ) } . Our proof of Theorem 6.2 is based on two intermediary results and an argumentoriginally due to B. White [Whi97]. The key point in this argument consists indealing with an appropriate upper or lower semi-continuous function, hence webegin with showing that the volume density ϑ X , that is well defined at any point ofa non-collapsed strong Kato limit thanks to Theorem 5.11, is lower semi-continuous.For the sake of clarity, like in the previous subsection, we fix constants T, Λ , v > and a non-decreasing function f : (0 , T ] → R + such that f ( T ) ≤ / (16 n ) and ˆ T p f ( τ ) d τ /τ ≤ Λ . Any non-collapsed strong Kato limit space ( X, d , µ, o ) considered in this sectionis the pmGH limit of a sequence of pointed Riemannian manifolds { ( M α , g α , o α ) } satisfying for all t ∈ (0 , T ] , k t ( M α , g α ) ≤ f ( t ) for all α, ν g α ( B √ T ( o α )) ≥ vT n . Proposition 6.3. Let ( X, d , µ, o ) be a non-collapsed strong Kato limit. Then thefunction ϑ X is lower semicontinuous. Moreover, if { ( X α , d α , µ α , o α ) } α is a pmGHconvergent sequence of non-collapsed strong Kato limit spaces with limit ( X, d , µ, o ) ,then for any x ∈ X and any sequence { x α } where x α ∈ X α for any α such that x α → x , ϑ X ( x ) ≤ lim inf α → + ∞ ϑ X α ( x α ) . Proof. Recall that the infimum of a family of continuous functions is upper semi-continuous. The proposition is then a consequence of the fact that ϑ − is the infin-imum of a family of continuous function. Indeed from Corollary 5.12 and Remark5.1 (ii) we know that there exists η > and an increasing function Φ : (0 , η T ] → R + satisfying lim t → Φ( t ) = 0 such that the function t ∈ (0 , η T ] exp (Φ( t )) (4 πt ) n H ( t, x, x ) is non decreasing. We also know that ϑ − X ( x ) = inf t ∈ (0 , η ] exp (Φ( t )) θ Xx ( t/ , t/ . The result then follows from Proposition 5.7. (cid:3) The next additional result deals with the volume density of weakly non-collapsed RCD(0 , n ) measure metric cones and was implicitly present in [DPG18, Lem. 2.9]. Proposition 6.4. Let ( Y, d , µ ) be a weakly non-collapsed RCD(0 , n ) space whichis a n -metric measure cone with vertex y ∈ Y . Then ϑ Y ( y ′ ) ≥ ϑ Y ( y ) for any y ′ ∈ Y . Moreover there exists k ∈ N such that the level set { ϑ Y ( · ) = ϑ Y ( y ) } isisometric to the Euclidean space R k and ( Y, d , µ, y ) is mm k -symmetric but not mm ( k + 1) -symmetric.Proof. By the Bishop-Gromov theorem for RCD(0 , n ) spaces, the volume ratio isnon-increasing, hence we know that for any y ′ ∈ Y and r > , ϑ Y ( y ′ ) ≥ µ ( B r ( y ′ )) ω n r n ≥ inf s> µ ( B s ( y ′ )) ω n s n = lim s → + ∞ µ ( B s ( y ′ )) ω n s n · The Bishop-Gromov theorem classically implies that the asymptotic volume ratio lim s → + ∞ µ ( B s ( y ′ )) ω n s n does not depend on y ′ ∈ Y. Thus ϑ Y ( y ′ ) ≥ lim s → + ∞ µ ( B s ( y )) ω n s n · Since ( Y, d , µ ) is a n -metric measure cone with vertex y , the function s µ ( B s ( y )) ω n s n is constantly equal to ϑ Y ( y ) . As a consequence, ϑ Y ( y ′ ) ≥ lim s → + ∞ µ ( B s ( y )) ω n s n = ϑ Y ( y ) . Since for any r > , ϑ Y ( y ′ ) ≥ µ ( B r ( y ′ )) ω n r n ≥ ϑ Y ( y ) , then ϑ Y ( y ′ ) = ϑ Y ( y ) if and only if the function r µ ( B r ( y ′ )) ω n r n is constantly equalto ϑ Y ( y ) , what occurs if and only if Y is a n -metric measure cone at y ′ thanks toProposition 1.27.If y ′ = y , this implies that Y is mm 1-symmetric along the geodesic connecting y and y ′ . Indeed, since Y is a metric cone at y ′ , it must be isometric to any tangentcone at y ′ . But Y is a metric cone at y , say ( Y, d , µ ) = ( C ( Z ) , d , µ ) : therefore, anytangent cone Y y ′ at y ′ = ( s, z ) = y is of the form ( R × Z z , d R × Z z , s n − dtdν z , y ′ ) , TRUCTURE OF KATO LIMITS 57 where s = d ( y, y ′ ) , and ( Z z , d z , ν z ) is a tangent cone of Z at z . (cid:3) Recall that, thanks to Corollary 4.11 and to Theorem 5.11, any tangent cone of anon-collapsed strong Kato limit space is a weakly non-collapsed RCD(0 , n ) n -metricmeasure cone. Then in our setting we directly obtain the following reformulationof Proposition 6.4. Corollary 6.5. Let ( X, d , µ ) be a non-collapsed strong Kato limit and x ∈ X . Let ( X x , d x , µ x , x ) be a tangent cone. Then ϑ X x ( z ) ≥ ϑ X x ( x ) for any z ∈ X x , and thereexists k ∈ N such that the level set { ϑ X x ( · ) = ϑ X x ( x ) } is isometric to the Euclideanspace R k and ( X x , d x , µ x , x ) is mm k -symmetric but not mm ( k + 1) -symmetric. Before proving Theorem 6.2, we recall the definition of H s ∞ from the previoussection and provide a couple of classical properties. For any s ∈ R + and any subset E of a metric space ( X, d ) , H s ∞ ( E ) := inf (X i ω s r si : E ⊆ [ i B r i ( x i ) ) . Lemma 6.6. The function H s ∞ satisfies the following properties.(1) dim H ( E ) = sup { s > H s ∞ ( E ) > } = inf { s > H s ∞ ( E ) = 0 } ;(2) if H s ∞ ( E ) > , then for H s -almost every x ∈ E :lim sup r → H s ∞ ( E ∩ B r ( x )) ω s r s ≥ − s ; (3) if E is a countable union of sets { E j } , then H s ∞ ( E ) > if and only if thereexists j such that H s ∞ ( E j ) > ;(4) H s ∞ is upper semi-continuous with respect to the Gromov-Hausdorff con-vergence of compact metric sets. We are now in a position to prove the existence of a well-defined stratificationfor non-collapsed strong Kato limits. Proof of Theorem 6.2. The proof is divided in three cases: first the case k = 0 ,then the case k ∈ { , . . . , n − } , and eventually the case k = n . Case I, k = 0 .Our argument to prove the assertion about S is inspired by [Whi97, Proposition3.3]. It suffices to prove the inclusion S ⊆ M := { x ∈ X, ϑ X ( x ) < lim inf y → x ϑ X ( y ) } . Indeed, M is countable as it can be written as the union over ℓ ∈ N \{ } of the sets M ℓ = (cid:26) x ∈ X : ∀ y ∈ B /ℓ ( x ) \ { x } , ϑ X ( x ) + 1 ℓ < ϑ X ( y ) (cid:27) which are all discrete and countable, since whenever two disjoint points x, y are in M ℓ they satisfy d ( x, y ) ≥ /ℓ .To show the inclusion S ⊆ M , let us take x 6∈ M . Then there exists a sequence { y ℓ } ℓ ⊂ X such that < r ℓ := d ( x, y ℓ ) < /ℓ and ϑ X ( x ) + 1 /ℓ ≥ ϑ X ( y ℓ ) for any ℓ ∈ N \{ } . In particular, y ℓ → x and lim ℓ ϑ X ( y ℓ ) = ϑ X ( x ) . Consider the sequence of rescalings { ( X, r − ℓ d , r − nℓ µ, x ) } ℓ . Since r ℓ ↓ , there existsa subsequence { ( X, r − ℓ ′ d , r − nℓ ′ µ, x ) } ℓ ′ which pmGH converges to a tangent cone ( X x , d x , µ x , x ) . Moreover, the points { y ℓ ′ } ℓ ′ converge to some y ∈ X x such that d x ( x, y ) = 1 . The lower semi-continuity of ϑ X through pmGH convergence, togetherwith the choice of { y ℓ } ℓ , ensures that ϑ X x ( y ) ≤ lim inf ℓ ′ ϑ X ( y ℓ ′ ) = ϑ X ( x ) = ϑ X x ( x ) . Thanks to Corollary 6.5, this implies that ϑ X x ( y ) = ϑ X x ( x ) and ( X x , d x , µ x , x ) ismm -symmetric. Hence d ( x ) ≥ and x S . Case II, k ∈ { , . . . , n − } .From (1) in Lemma 6.6, we only need to prove that for any s > , H s ∞ ( S k ) > implies s ≤ k . Thus we assume H s ∞ ( S k ) > . (90) Step 1. Let us write S k as a countable union of closed sets. From Remark 2.18,we know that there exists C, λ > such that for all x ∈ X and all < s < r ≤ R , ve − C d ( x,o ) R r n ≤ µ ( B r ( x )) ≤ Cr n and µ ( B r ( x )) µ ( B s ( x )) ≤ C (cid:16) rs (cid:17) n · (91)Arguing as in [Che01, Proof of Theorem 10.20], we write S k as the countable unionover j ∈ N \{ } of the closed sets S k,j := (cid:8) x ∈ X : D( B r ( x ) , B Zr ( z )) ≥ r/j, ∀ r ∈ (0 , /j ) , ∀ ( Z, d Z , µ Z , z ) ∈ Adm k +1 (cid:9) , where Adm k +1 is the set of mm ( k + 1) -symmetric spaces ( Z, d Z , µ Z , z ) satisfy-ing (91) and D( B r ( x ) , B Zr ( z )) is the sum of | µ ( B r ( x )) − µ Z ( B Zr ( x )) | and the L -transportation distance [Stu06a, p. 69] between the normalized metric measurespaces ( B r ( x ) , d , µ ( B r ( x )) − µ ¬ B r ( x )) and ( B Zr ( x ) , d Z , µ Z ( B Zr ( x )) − µ Z ¬ B Zr ( x )) .Moreover, for any j S k,j = [ N ∈ N S k,j ∩ B N ( o ) , so (3) in Lemma 6.6 ensures from (90) that there exists j, N ∈ N \{ } such that H s ∞ ( S k,j ∩ B N ( o )) > . Step 2. Let us write S k,j ∩ B N ( o ) as a countable union of closed sets. Take ε > . For any x ∈ X , since ϑ X ( x ) < + ∞ , there exists η ( x, ε ) > such that for all r ∈ (0 , η ( x, ε )] , (cid:12)(cid:12)(cid:12)(cid:12) µ ( B r ( x )) ω n r n − ϑ X ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε, and we can define δ ( x, ε ) := sup (cid:26) r > (cid:12)(cid:12)(cid:12)(cid:12) µ ( B σ ( x )) ω n σ n − µ ( B ρ ( x )) ω n ρ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε, ∀ σ, ρ ∈ (0 , r ] (cid:27) > . Then for all c > the set A ε,c ⊂ X defined by A ε,c := { x ∈ X : δ ( x, ε ) ≥ c } = \ <σ ≤ ρ ≤ c (cid:26) x ∈ X : (cid:12)(cid:12)(cid:12)(cid:12) µ ( B σ ( x )) ω n σ n − µ ( B ρ ( x )) ω n ρ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (cid:27) is closed. Hence for any p, q ∈ Q with q < p < q + ε , the set S ε,p,q := A ε, p − q ) ∩ (cid:26) x ∈ S k,j ∩ B N ( o ) : q ≤ µ ( B p − q ( x )) ω n ( p − q ) n ≤ p (cid:27) is compact since it is a closed subset of the compact set B N ( o ) . Observe that forany x ∈ S ε,p,q and ρ ∈ (0 , p − q )] we have q − ε ≤ µ ( B ρ ( x )) ω n ρ n ≤ p + ε, (92) TRUCTURE OF KATO LIMITS 59 and then q − ε ≤ ϑ X ( x ) ≤ p + ε as ρ ↓ . Finally, note that S k,j ∩ B N ( o ) = [ p,q ∈ Q q
Now let us consider the sequence { ε ℓ := 2 − ℓ } ℓ ∈ N \{ } . By (3) in Lemma6.6, for any ℓ there exist p ℓ , q ℓ ∈ Q with q ℓ < p ℓ such that H s ∞ ( S ε ℓ ,p ℓ ,q ℓ ) > , henceby (2) in Lemma 6.6 there exist x ℓ ∈ S ε ℓ ,p ℓ ,q ℓ and r ℓ > small such that H s ∞ ( S ε ℓ ,p ℓ ,q ℓ ∩ B r ℓ ( x ℓ )) ω s r sℓ ≥ − s . (93)As the pointed metric measure spaces { ( X, r − ℓ d , r − nℓ µ, x ℓ ) } ℓ all satisfy the volumeestimates (91), up to extracting a subsequence we can assume that they pmGHconverge as ℓ → + ∞ to a pointed metric measure space ( Z, d Z , µ Z , z ) . Sincethe sets { S ε ℓ ,p ℓ ,q ℓ } ℓ are compact, up to extracting another subsequence we canassume that the compact sets { S ε ℓ ,p ℓ ,q ℓ ∩ B r ℓ ( x ℓ ) } ℓ GH converge to some compactset K ⊂ B Z ( z ) containing z . Because of the upper semi-continuity of H s ∞ withrespect to GH convergence (i.e. (4) in Lemma 6.6) and because of (93), we have H s ∞ ( K ) ≥ ω s − s . In particular, dim H K ≥ s. Finally, up to extracting a further subsequence, we can assume that the boundedsequence of rational numbers { q ℓ } tends to some number Q > as ℓ → + ∞ . Step 4. Now we take y ∈ K and we let y ℓ ∈ S ε ℓ ,p ℓ ,q ℓ for any ℓ be such that y ℓ → y . Let ℓ be fixed. Take r ∈ [0 , ℓ ) and set ρ := rr ℓ . With no loss of generalitywe can assume r ℓ ≤ − ℓ +1 ( p ℓ − q ℓ ) , so that ρ ∈ (0 , p ℓ − q ℓ )] . Then the triangleinequality and (92) lead to (cid:12)(cid:12)(cid:12)(cid:12) µ ( B ρ ( y ℓ )) ω n ρ n − Q (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε ℓ + | q ℓ − Q | , which rewrites as (cid:12)(cid:12)(cid:12)(cid:12) µ ( B rr ℓ ( y ℓ )) ω n r nℓ − Qr n (cid:12)(cid:12)(cid:12)(cid:12) ≤ r n (2 ε ℓ + | q ℓ − Q | ) . (94)Since lim ℓ → + ∞ µ ( B r ℓ r ( y ℓ )) r nℓ = µ Z ( B ( y, r )) , inequality (94) yields µ Z ( B r ( y )) = ω n Qr n as l → + ∞ .Because of Remark 4.12, we know that ( Z, d Z , µ Z ) is a weakly non-collapsed RCD(0 , n ) metric measure space. In particular, its volume density ϑ Z is well definedat all points, and thanks to Proposition 1.27 the previous computation shows thatfor any y ∈ K , ( Z, d Z , µ Z ) is a metric measure cone at y and ϑ Z ( y ) = Q . ByProposition 6.4, this means that there exists an integer k ′ ≥ dim H K such that ( Z, d Z , µ Z , z ) is metric measure k ′ -symmetric. In particular, k ′ ≥ s. Step 5. To conclude, let us show that k ≥ k ′ . Since ( X, d , µ, o ) satisfiesthe volume estimates (91), so do the rescalings { ( X, r − ℓ d , r − nℓ µ, x ℓ ) } ℓ ∈ N \{ } . As ( Z, d Z , µ Z , z ) is the pmGH limit of these rescalings, this implies that ( Z, d Z , µ Z , z ) belongs to Adm k ′ . Since for ℓ large enough we have r ℓ < /j and D( B r ℓ ( x ℓ ) , B Zℓ ( z )) < r ℓ j , this means that x ℓ does not belong to S k ′ ,j . But x ℓ ∈ S k,j . As a consequence, bythe very definition of S k,j , the integer k ′ is necessarily smaller than k + 1 . Case III, k = n . Lebesgue differentiation theorem holds on locally doublingspaces (see [HKST15, Sect. 3.4]) so µ -a.e. x ∈ X is a Lebesgue point of the locallyintegrable function ϑ X . Thus it is enough to show that whenever x ∈ X is aLebesgue point of ϑ X , that is lim r → B r ( x ) ϑ X d µ = ϑ X ( x ) , then any tangent cone at x is equal to ( R n , d e , ϑ X ( x ) H n , n ) .Let x be a Lebesgue point of ϑ X , ( X x , d x , µ x , x ) be a tangent cone and { r α } α ⊂ (0 , + ∞ ) be such that r α ↓ and ( X, d α := r − α d , µ α := r − nα µ, x ) → ( X x , d x , µ x , x ) in the pmGH sense. According to Corollary 5.12, we know that if we set β X ( z, t ) := 1exp(Φ( t ))(4 πt ) n/ H ( t, z, z ) for any z ∈ X and any t small enough, then ϑ X ( z ) = lim t → β X ( z, t ) and t β X ( z, t ) is non increasing. The same is true if we define β X α (resp. β X x )in a similar way on the rescaled space ( X, d α , µ α , x ) (resp. on the tangent cone ( X x , d x , µ x , x ) ) for any α , and we have β X α ( · , t ) → β X x ( · , t ) uniformly on compactsets for any t small enough; this implies B d x ( x ) β X x ( z, t ) d µ x ( z ) = lim α B d α ( x ) β X α ( z, t ) d µ α ( z )= lim α B d rα ( x ) β X ( z, r α t ) d µ ( z ) ≤ lim α B d rα ( x ) ϑ X ( z ) d µ ( z ) = ϑ X ( x ) . By monotone convergence, letting t ↓ gives B d x ( x ) ϑ X x ( z ) d µ x ( z ) ≤ ϑ X ( x ) . By the first statement in Proposition 6.4, B d x ( x ) ϑ X x ( z ) d µ x ( z ) ≥ ϑ X x ( x ) = ϑ X ( x ) , hence ϑ X x is constantly equal to ϑ X ( x ) on B d x ( x ) . The second statement of Propo-sition 6.4 implies that X x is isometric to R n equipped with the Euclidean distanceand µ X x is given by c H n for some c ∈ (0 , . But since for all r > we have µ X x ( B r ( x )) = ϑ X ( x ) ω n r n , with H n ( B r ( x )) = ω n r n in R n , we get c = ϑ X ( x ) . (cid:3) Volume continuity This section is devoted to proving the following analog of volume continuity forRicci limit spaces [Col97, Theorem 0.1], [CC97, Theorem 5.9]. Theorem 7.1. Let ( X, d , µ, o, E ) be a non-collapsed strong Kato limit. Then µ coincides with the n -dimensional Hausdorff measure H n . The proof of the previous is a direct consequence of the next key result, of [Mat95,Theorem 6.9] and of the fact that we already know µ ≤ H n . TRUCTURE OF KATO LIMITS 61 Theorem 7.2. Let ( X, d , µ, o, E ) be a non-collapsed strong Kato limit and x ∈ X such that the set of tangent cones at x is reduced to { ( R n , d e , ϑ X ( x ) H n , x ) } . Then ϑ X ( x ) = 1 . As a consequence, we also obtain the following corollary, which generalizes[Che01, Theorem 9.31] for manifolds with Ricci curvature bounded below. Corollary 7.3. Let n ≥ , T, v, Λ > and f : (0 , T ] → R a function such that ˆ T p f ( s ) s d s ≤ Λ . Then for all ε > there exists δ = δ ( ε, n, Λ , v, T, f ) such that the following holds.Assume that for all t ∈ (0 , T ]k t ( M n , g ) ≤ f ( t ) , ν g ( B √ T ( x )) T n ≥ v, and for r ∈ (0 , δ √ T ] d GH ( B r ( x ) , B ( r )) ≤ δr. Then − ε ≤ ν g ( B r ( x )) ω n r n ≤ ε. Proof. Assume by contradiction that there exist ε such that for all δ the conclusionof the corollary is false. Than we can consider a sequence δ i tending to zero andmanifolds ( M ni , g i ) satisfying the assumptions above, for which there exist r i ∈ (0 , δ i √ T ] and x i ∈ M i such thatd GH ( B ( x i , r i ) , B ( r i )) ≤ δ i r i , (95)and (cid:12)(cid:12)(cid:12)(cid:12) ν g i ( B ( x i , r i )) ω n r ni − (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε . (96)The re-scaled sequence ( M ni , r − i g i , r − ni v g i , x i ) is a non-collapsing sequence satis-fying the strong Kato bound (SUK), with k t ( M i , ε − i ) tending to zero. As a con-sequence, up to a sub-sequence, it converges to a pointed metric measure space ( X, d , µ, x ) . Because of (95) and (96) the unit ball B ( x ) is isometric to the unitEuclidean ball B (1) and satisfies (cid:12)(cid:12)(cid:12)(cid:12) µ ( B ( x )) ω n − (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε . (97)But according to Theorem 7.1, µ = H n hence in particular µ coincides with theLebesgue measure on B ( x ) , this contradicting (96). (cid:3) In the remainder of this section, we prove Theorem 7.2. In order to do this,we start by proving the existence of GH-isometries with the appropriate regularityproperties.7.1. Existence of splitting maps. One of the most powerful tools in the studyof Ricci limit spaces and RCD spaces is given by ε -splitting maps, see for exampleDefinition 4.10 in [CJN18]. We are going to show that whenever a point x in anon-collapsed strong Kato limit admits a Eudlidean tangent cone, we can constructan ε -splitting map from a ball around x to a Euclidean ball. To this aim, weneed an approximation result for harmonic functions defined on PI Mosco-Gromov-Hausdorff limits, which is proven in the Appendix, together with the gradient andHessian estimates shown in Section 4.In the following, we denote a Euclidean ball of radius r centered at n as B ( r ) . Theorem 7.4. Let ( X, d , µ ) be a strong non-collapsed Kato limit obtained from asequence { ( M α , g α ) } α and x ∈ X a point that admits ( R n , d e , ϑ X ( x ) H n , n ) as atangent cone. Then there exist sequences { r α } , { ε α } ⊂ (0 , ∞ ) tending to zero, x α in M α and maps H α = ( h ,α , . . . , h n,α ) , H α : B r α ( x α ) → B ( r α ) , such that h i,α isharmonic on B r α ( x ) for all i = 1 , . . . , n . Moreover, the following holds. (i) H α is an ( ε α r α ) -GH isometry. (ii) H α is (1 + ε α ) -Lipschitz. (iii) B rα ( x ) | t dH α ◦ dH α − Id n | d ν g α ≤ ε α . (iv) r α B rα ( x ) |∇ dH α | d ν g α ≤ ε α . (v) lim α ν g α ( B tr α ( x )) ω n ( tr α ) n = ϑ X ( x ) for all t > . Before proving the previous theorem, we show an improvement of the Lipschitzconstant of Lipschitz harmonic functions whose gradient is suitably close to 1. Theargument we use is originally due to J. Cheeger and A. Naber, see [CN15, Lem. 3.34],and it relies on the existence of good cut-off functions, Bochner formula and theappropriate estimates for the heat kernel. Proposition 7.5. Let ( M n , g ) be a closed Riemannian manifold and u : B r ( x ) → R a κ -Lipschitz harmonic function, for some κ ≥ . Assume that there exists δ > such that k r ( M n , g ) ≤ δ ≤ n and B r ( x ) (cid:12)(cid:12) | du | − (cid:12)(cid:12) d ν g ≤ δ . Then u (cid:12)(cid:12) B r/ ( x ) is C ( n, κ ) δ -Lipschitz.Proof. Let χ ∈ C ∞ c ( M ) be a cut-off function as constructed in Proposition 3.1 suchthat:i) χ = 1 on B r/ ( x ) ,ii) χ = 0 on M \ B r ( x ) ,iii) | dχ | ≤ C ( n ) /r and | ∆ χ | ≤ C ( n ) /r on B r ( x ) \ B r/ .Apply Bochner’s formula on B r ( x ) to the κ -Lipschitz harmonic function u in orderto get |∇ du | + 12 ∆ (cid:0) | du | − (cid:1) = − Ric( ∇ u, ∇ u ) . Since |∇ du | ≥ and − Ric( ∇ u, ∇ u ) ≤ Ric - κ , this leads to 12 ∆ (cid:0) | du | − (cid:1) ≤ Ric - κ . Take y ∈ B r/ ( x ) and multiply the previous inequality evaluated at some z ∈ B r ( x ) by H ( t, y, z ) χ ( z ) , where t ∈ [0 , r ] , then integrate with respect to z and t : ¨ [0 ,r ] × B r ( x ) H ( t, y, z ) χ ( z )∆ (cid:0) | du | ( z ) − (cid:1) d ν g ( z ) d t ≤ ¨ [0 ,r ] × B r ( x ) H ( t, y, z ) χ ( z ) Ric - κ d ν g ( z ) d t. As immediately seen, the previous right-hand side is not greater than r ( M n , g ) κ which is not greater than δκ . Thus ¨ [0 ,r ] × B r ( x ) H ( t, y, z ) χ ( z )∆ (cid:0) | du | ( z ) − (cid:1) d ν g ( z ) d t ≤ δκ . (98) TRUCTURE OF KATO LIMITS 63 Use integration by parts to rewrite the left-hand side, with simplified notations, asfollows: ˆ H χ ∆ (cid:0) | du | − (cid:1) = ˆ ∆( H χ ) (cid:0) | du | − (cid:1) = ˆ (∆ H ) χ (cid:0) | du | − (cid:1) − ˆ h∇ H, ∇ χ i (cid:0) | du | − (cid:1) + ˆ H (∆ χ ) (cid:0) | du | − (cid:1) . Now ¨ [0 ,r ] × B r ( x ) ∆ z ( H ( t, y, z )) χ ( z ) (cid:0) | du | ( z ) − (cid:1) d ν g ( z ) d t = − ˆ B r ( x ) ˆ r ∂H ( t, y, z ) ∂t d t ! χ ( z ) (cid:0) | du | ( z ) − (cid:1) d ν g ( z )= − ˆ B r ( x ) H ( r , y, z ) χ ( z ) (cid:0) | du | ( z ) − (cid:1) d ν g ( z ) + χ ( y ) |{z} =1 (cid:0) | du | ( y ) − (cid:1) . Combining these three last estimates with the properties of the cut-off function χ ,we get | du | ( y ) − ≤ κ δ + I ( y ) + II ( y ) + III ( y ) where I ( y ) = ˆ M H ( r , y, z ) χ ( z ) (cid:0) | du | ( z ) − (cid:1) d ν g ( z ) ,II ( y ) = C ( n ) r ¨ [0 ,r ] × [ B r ( x ) \ B r/ ( x )] |∇ z H ( t, y, z ) | (cid:12)(cid:12) | du | ( z ) − (cid:12)(cid:12) d ν g ( z ) d t,III ( y ) = C ( n ) r ¨ [0 ,r ] × [ B r ( x ) \ B r/ ( x )] H ( t, y, z ) (cid:12)(cid:12) | du | ( z ) − (cid:12)(cid:12) d ν g ( z ) d t, and we are going to establish the following estimates: I ( y ) ≤ C ( n ) δ , II ( y ) ≤ C ( n ) δ p κ , III ( y ) ≤ C ( n ) δ , which are enough to complete the proof.We recall the upper bound for the heat kernel, with ν = e n and t ≤ r , y, z ∈ M : H ( t, y, z ) ≤ C ( n ) ν g ( B r ( y )) r ν t ν e − d y,z )5 t . Moreover, the doubling condition implies for y ∈ B r/ ( x ) H ( t, y, z ) ≤ C ( n ) ν g ( B r ( x )) r ν t ν e − d y,z )5 t . (99)Therefore, for any z ∈ B r ( x ) \ B r/ ( x ) and y ∈ B r/ ( x ) we obtain H ( r , y, z ) ≤ C ( n ) ν g ( B r ( x )) . Using this inequality and the assumption on | du | leads to the estimate for I ( y ) : I ( y ) ≤ C ( n ) ν g ( B r ( x )) ˆ B r ( x ) || du | ( z ) − | d ν g ( z ) ≤ C ( n ) δ . We now obtain the estimate for III ( y ) . Consider z ∈ B r ( x ) \ B r/ ( x ) and y ∈ B r/ ( x ) as above. Inequality (99) and the fact that ˆ r r ν t ν e − r t d t = r ˆ t ν e − t d t imply that ˆ r H ( t, y, z ) d t ≤ C ( n ) ν g ( B r ( x )) ˆ r r ν t ν e − r t d t = C ( n ) r ν g ( B r ( x )) , and as a consequence III ( y ) ≤ C ( n ) B r ( x ) (cid:12)(cid:12) | du | ( z ) − (cid:12)(cid:12) d ν g ( z ) ≤ C ( n ) δ . As for II ( y ) we use the Cauchy-Schwarz inequality twice, first in d ν g and then dt ,together with the result of Lemma 3.3: ˆ M |∇ z H ( t, y, z ) | H ( t, y, z ) d ν g ( z ) ≤ C ( n ) t , thus we obtain II ( y ) = C ( n ) r ˆ r ˆ B r ( x ) \ B r/ ( x ) |∇ z H ( t, y, z ) | p H ( t, y, z ) p H ( t, y, z ) (cid:12)(cid:12) | du | ( z ) − (cid:12)(cid:12) d ν g ( z ) d t ≤ C ( n ) r ˆ r √ t ˆ B r ( x ) \ B r/ ( x ) H ( t, y, z ) (cid:12)(cid:12) | du | ( z ) − (cid:12)(cid:12) d ν g ( z ) ! d t ≤ C ( n ) ˆ [0 ,r ] × ( ( B r ( x ) \ B r/ ( x ) ) H ( t, y, z ) t (cid:12)(cid:12) | du | ( z ) − (cid:12)(cid:12) d ν g ( z ) d t ! Using ˆ r r ν t ν e − r t d t = ˆ t ν e − t d t, we have ˆ r H ( t, z, y ) t d t ≤ ˆ r C ( n ) ν g ( B r ( x )) r ν t ν e − r t d t ≤ C ( n ) ν g ( B r ( x )) . We then obtain II ( y ) ≤ C ( n )(1 + κ ) (cid:18) B (cid:12)(cid:12) | du | ( z ) − (cid:12)(cid:12) d ν g ( z ) (cid:19) ≤ C ( n )(1 + κ ) δ. This allows us to obtain the desired bound on | dh | over B r/ ( x ) . (cid:3) We are now in a position to prove Theorem 7.4. Proof of Theorem 7.4. Let x ∈ X and assume that ( R n , d e , ϑ X ( x ) H n , n ) is a tan-gent cone at x . Then by definition of tangent cones of strong Kato limit spaces,there exist sequences ( r α ) α ⊂ (0 , + ∞ ) , r α ↓ and x α ∈ M α such that ( M α , r − α d g α , r − nα d ν g α , x α ) −→ ( R n , d e , ϑ X ( x ) H n , n ) , and property (v) holds. Denote by ˜ g α = r − α g α and µ α = r − nα d ν g α . Balls withrespect to ˜ g α are denoted by ˜ B s ( y ) . Notice that is is enough to prove the existenceof a map H α : ˜ B ( x α ) → B n (1) satisfying properties (i) to (iv) with respect to there-scaled metric ˜ g α and with r α replaced by . Then the map on B r α ( x α ) is simplyobtained by re-scaling H α by a factor r α . As a consequence, in the rest of the proof,we only work with the re-scaled manifolds ( M α , ˜ g α ) .Consider the coordinate maps x i : R n → R for all i = 1 , . . . , n . Then x i are har-monic and we can apply Proposition E.11: for all α there exist harmonic functions h i,α : ˜ B ( x α ) → B (1) such that(i) h i,α → x i | B (1) uniformly; TRUCTURE OF KATO LIMITS 65 (ii) For all s ≤ α →∞ ˆ ˜ B s ( x α ) | dh i,α | g α d µ α = ˆ B ( s ) | dx i | ϑ X ( x ) d H n = ϑ X ( x ) ω n . Define H α = ( h ,α , . . . , h n,α ) : ˜ B ( x α ) → B (1) . Since H α converges uniformly to the identity Id n = ( x , . . . , x n ) , it is not difficult toshow that H α is an ε α -GH isometry where ( ε α ) α ⊂ (0 , + ∞ ) is a sequence tendingto zero. In the continuation of the reasoning, we will take the freedom of modifyingthis sequence tending to zero while keeping its notation.Since µ α ( ˜ B ( x α )) tends to ϑ X ( x ) ω n , the second property implies that lim α →∞ ˜ B ( x α ) | dh i,α | g α d µ α = 1 . Using the first estimate in Lemma 3.6 we then deduce that there exists C ( n ) > such that sup ˜ B / ( x α ) | dh i,α | ˜ g α ≤ C ( n ) , that is h i,α is C ( n ) -Lipschitz on ˜ B / ( x α ) . We can then apply Proposition 3.5 andget some uniform estimates ˜ B / ( x α ) |∇ dh i,α | g α d µ α ≤ C n Then, Remark E.6 in the Appendix ensures that | dh i,α | ˜ g α tends to 1 in L . Thenby additionally using the fact that ( M α , ˜ g α ) is doubling, we have lim α →∞ ˜ B / ( x α ) || dh i,α | ˜ g α − | d µ α = 0 . This, together with h i,α being Lipschitz, implies lim α →∞ ˜ B / ( x α ) || dh i,α | g α − | d µ α = 0 . Recall that, as observed in Remark 2.10, a sequence of re-scaled manifolds con-verging to a tangent cone of a strong Kato limit is such that for all t > α →∞ k t ( M α , ˜ g α ) = 0 . Modifying the sequence ( ε α ) α if necessary, we have k / ( M α , ˜ g α ) < ε α , ˜ B / ( x α ) || dh i,α | g α − | d µ α < ε α . This means that the assumptions of Proposition 7.5 are satisfied, therefore for α large enough h i,α is (1 + ε α ) -Lipschitz on the ball ˜ B / ( x α ) .Applying Proposition 3.5 and using again the doubling property, we also obtain ˜ B / ( x α ) |∇ dh i,α | g α d µ α ≤ C n ˜ B / ( x α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | dh i,α | g α − ˜ B ( x α ) | dh i,α | g α d µ α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d µ α < ε α . Here we used that k r − α T ( M α , ˜ g α ) ≤ / n , thus for α large enough we have min { / , r − α T } = 1 / . As a consequence, up to replacing r α by r α / , h i,α is (1 + ε α ) -Lipschitz on ˜ B ( x α ) and satisfies ˜ B ( x α ) || dh i,α | g α − | d µ α < ε α , ˜ B ( x α ) |∇ dh i,α | g α d µ α < ε α . In order to obtain properties (iii) and (iv) for H α , one can consider the function x i + x j . Since we know that h i,α + h j,α converges uniformly to x i + x j , by arguingas above we get lim α →∞ ˜ B ( x α |h dh i,α , dh j,α i ˜ g α − δ ij | d µ α = 0 . Then the same argument that we used for h i,α finally leads to properties (iii) and(iv) for H α . (cid:3) Remark 7.6. The same argument as above shows that if ( X, d , µ ) is a strong non-collapsed Kato limit and x ∈ X admits an mm k -symmetric tangent cone, thenthere exist harmonic ε -splitting maps from a ball around x to a Euclidean ball ofthe same radius in R k .7.2. Proof of Theorem 7.2. Our proof of Theorem 7.2 is inspired by the argu-ment illustrated in [Che01, Theorem 9.31] and [Gal98, Theorem 1.6]. Both proofsare based on degree theory. Proof. Let x be a point in X that admits an Euclidean tangent cone ( R n , d e , ϑ X ( x ) H n , n ) .Let r α , ε α and H α : B r α ( x ) → R n be as in Theorem 7.4. Let ρ α : B r α ( x ) → R be defined by ρ α ( x ) = || H α ( x ) || . Fix τ α ∈ (cid:0) r α − ε α r α , r α − ε α r α (cid:1) . Since ρ α is smooth, by Sard’s theorem τ α can be chosen so that τ α is a regular value of ρ α .Now define the compact set Ω α = { x ∈ B r α ( x ) , || H α ( x ) || ≤ τ α } . Since H α is an ( ε α r α ) -GH isometry, we have H α ( B rα − r α ε α ( x )) ⊂ B (cid:16) r α − ε α r α (cid:17) . Also, for any x such that || H α ( x ) || ≤ r α / − ε α r α , x ∈ B r α / ( x ) . Then with ourchoice of interval and τ α we have B rα − ε α r α ( x ) ⊂ Ω α ⊂ B rα ( x ) . (100)We claim that if for α large enough, H α : Ω α → B ( τ α ) is surjective, then ϑ X ( x ) = 1 .Indeed, if H α is surjective, the estimates on dH α and on the Lipschitz constant of H α imply H n ( B ( τ α )) ≤ (1 + ε α ) n ν g α (Ω α ) , which, together with the inclusion above leads to ω n (cid:16) (1 − ε α ) r α (cid:17) n ≤ (1 + ε α ) n ν g α (cid:16) B rα ( x ) (cid:17) . Together with (6) in the previous Theorem, this shows that ϑ X ( x ) ≥ . But since ϑ X is lower semi-continuous, we already know that ϑ X ( x ) ≤ , then ϑ X ( x ) = 1 .In the rest of the proof we show by contradiction that H α : Ω α → B ( τ α ) issurjective for α large enough. We first assume that M α is oriented and H α is notsurjective. We let Θ α be the unit volume form of ( M α , g α ) We Then, since Ω α iscompact the set B ( τ α ) \ H α (Ω α ) is open and there exists an open ball B ( p, η ) ⊂ B ( τ α ) \ H α (Ω α ) . We claim that there exists an ( n − -form γ α on the set B ( τ α ) \ B ( p, η ) such that d γ α equals the n -volume form ω α = d x ∧ . . . ∧ d x n and ι ∗ γ α = 0 ,where ι : ∂ B ( p, η ) → B ( p, η ) is the inclusion map.Now observe that H α ( ∂ Ω α ) = ∂ B τ α : if x ∈ ∂ Ω α , then || H α ( x ) || = τ α and H α ( x ) ∈ ∂ B ( τ α ) because τ α is a regular value of ρ α = || H α || . Then Stokes’theorem implies ˆ Ω α H ∗ α ( ω α ) = ˆ Ω α d H ∗ α ( γ α ) = ˆ ∂ Ω α H ∗ α γ α = 0 . TRUCTURE OF KATO LIMITS 67 But we also know H ∗ α ( ω α ) = H ∗ α (d x ∧ . . . ∧ d x n ) = d h ,α ∧ . . . ∧ d h n,α = f α Θ α , where for x ∈ Ω α , f α ( x ) = det (d x H α ( e ) , . . . , d x H α ( e n )) , where ( e , . . . , e n ) is a direct orthonormal basis of ( T x M α , g α ( x )) . Then ˆ Ω α f α d ν g α = 0 . Denote by B α the ball B (cid:0) x, r α (cid:1) . Let x ∈ B α and λ , . . . λ n the eigenvalues of t d x H α ◦ d x H α . Then we have | f α ( x ) | = p λ · . . . · λ n , and | λ i − | ≤ | t d x H α ◦ d x H α − Id n | := D α . Observe that property (iii) of H α implies that for α large enough D α < . Thereforewe have | f α | ≥ (1 − D α ) n ≥ − n D α . As a consequence and using the doubling condition B α | f α | d ν g α ≥ − n B ( x, rα ) D α d ν g α ≥ − C B rα ( x ) D α d ν g α . Then for α large enough we have m α := B α | f α | d ν g α ≥ . (101)We also have for x ∈ B α |∇ f α | ≤ n (1 + ε α ) n − |∇ dH α | . We then use Cauchy-Schwartz and Poincaré’s inequalities to get B α | f α − m α | d ν g α (cid:18) B α | f α − m α | d ν g α (cid:19) ≤ Cr α (cid:18) B α |∇ f α | d ν g α (cid:19) Then using the doubling condition and property (iv) in the previous Propositionwe get B α | f α − m α | d ν g α ≤ Cr α (cid:18) B α |∇ dH α | d ν g α (cid:19) ≤ C ′ √ ε α , and for α large enough B α | f α − m α | d ν g α ≤ . Then combining this last inequality with (101) we obtain m α ≥ . (102)We are going to contradict the previous lower bound by using the properties of H α and Poincaré’s inequality. We can write | m α | = (cid:12)(cid:12)(cid:12)(cid:12) m α − ˆ Ω α f α (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Ω α × B α ( f α ( x ) − f α ( y )) d ν g α ( x ) d ν g α ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ν g α ( B α ) ν g α (Ω α ) B α × B α | f α ( x ) − f α ( y ) | d ν g α ( x ) d ν g α ( y ) ≤ ν g α ( B α ) ν g α (Ω α ) (cid:18) B α × B α | f α ( x ) − f α ( y ) | d ν g α ( x ) d ν g α ( y ) (cid:19) Thanks to the first inclusion in (100), for α large enough B r α / ( x ) ⊂ Ω α . Then theratio ν g α ( B α ) /ν g α (Ω α ) has an upper uniform bound, since ( M α , g α ) is doubling.Moreover B α × B α | f α ( x ) − f α ( y ) | d ν g α ( x ) d ν g α ( y ) = 2 B α | f α − m α | d ν g α , then thanks to the estimate on the hessian of H α we can conclude | m α | ≤ Cr α B α |∇ dH α | d ν g α ≤ C ′ ε α . Since ε α tends to zero, this contradicts for α large enough inequality (102). Thenif M α is oriented, H α is surjective from Ω α to B ( τ α ) .If M α is not oriented, we consider the two-fold orientation covering ˆ π α : ˆ M α toM α and choose ˆ x ∈ ˆ π − α ( x ) . We observe that ˆ M α endowed with the pull-back metric ˆ g α = ˆ π ∗ α g α satisfies for all s > s ( ˆ M α , ˆ g α ) = k s ( M α , g α ) . Then ( ˆ M α , ˆ g α ) is PI at the same scale as ( M α , g α ) . Moreover, the map ˆ H α = H α ◦ ˆ π α : B (ˆ x, r α ) → B ( τ α ) is (1+ ε α ) -Lipschitz and satisfies properties (iii) and (iv) of the previous Proposition.Then we can apply the same argument as above and show that ˆ H α : ˆ π − α (Ω α ) → B ( τ α ) is surjective. It finally follows that H α : Ω α → B ( τ α ) is also surjective. (cid:3) Appendix In this appendix, we provide a proof of Theorem 1.17 and of several other usefulconvergence results.For the two next subsections, we put ourselves in the following setting: we let { ( X α , d α , µ α , o α ) } α , ( X, d , µ, o ) be proper geodesic pointed metric measure spacessuch that • ( X α , d α , µ α , o α ) pmGH −→ ( X, d , µ, o ) , and we use the sequences { R α } , { ε α } and { Φ α } given by Characterization 1, • there exists κ ≥ and R > such that the spaces { ( X α , d α , µ α ) } α are all κ -doubling at scale R , hence so is ( X, d , µ ) .A. Approximation of functions. The following result is known by experts. Itsays that the space C c ( X ) is somehow the limit of the spaces {C c ( X α ) } , in the sensethat any ϕ ∈ C c ( X ) can be nicely approximated by functions ϕ α ∈ C c ( X α ) . Proposition A.1. For any r > and any α large enough, we can build a linearmap A α : C c ( B r ( o )) → C c ( X α ) such that the following holds for any ϕ ∈ C c ( B r ( o )) . TRUCTURE OF KATO LIMITS 69 i) If ϕ ≥ , then A α ϕ ≥ for any α .ii) If ≤ ϕ ≤ L , then ≤ A α ϕ ≤ L for any α .iii) The convergence A α ϕ C c → ϕ holds.iv) The functions {A α ϕ } α are uniformly equicontinuous.v) There exists a constant ¯ C > depending only on κ such that if ϕ is Λ -Lipschitz,then A α ϕ is ¯ C Λ -Lipschitz.Proof. Let r > . With no loss of generality we assume that sup α ε α ≤ r/ andthat r ≤ R α ; if this is not true, we let A α be the zero map for all α such that ε α > r/ or r > R α . Let ϕ ∈ C c ( B r ( o )) Step 1. [Construction of A α ϕ ] Let α be arbitrary. Let D α ⊂ X α be a maximal ε α -separated set of points, i.e. a maximal set such that X α = S p ∈D α B ε α ( p ) andfor any p, q ∈ D α , p = q ⇒ B ε α ( p ) ∩ B ε α ( q ) = ∅ . For any x ∈ X , we set V ( x ) := D α ∩ B ε α ( x ) and we point out that V ( x ) ≤ κ . (103)Indeed we have ∪ p ∈V ( x ) B ε α ( p ) ⊂ B ε α ( x ) and µ α ( B ε α ( x )) ≤ µ α ( B ε α ( p )) ≤ κ µ α ( B ε α ( p )) for any p ∈ V ( x ) .Let us consider the -Lipschitz function χ : [0 , + ∞ ) → [0 , defined by χ ( t ) = if t ≤ , − t if t ∈ [1 , , if t ∈ [2 , + ∞ ) . (104)For any p ∈ D α , we define ˆ ξ αp , σ α , ξ αp : X α → R by ˆ ξ αp ( x ) = χ (cid:18) d α ( p, x )2 ε α (cid:19) , σ α ( x ) = X p ∈D α ˆ ξ αp ( x ) , ξ αp ( x ) = ˆ ξ αp ( x ) /σ α ( x ) , for any x ∈ X . By construction, ≤ σ α ≤ κ (the upper bound actually followsfrom (103)), the function ξ αp is (1 + κ )(2 ε α ) − -Lipschitz, and X p ∈D α ξ αp = 1 . Then we define ϕ α = A α ϕ := X p ∈D α ϕ (Φ α ( p )) ξ αp . As a linear combination of compactly supported Lipschitz functions, ϕ α ∈ Lip c ( X α ) .Moreover, supp ϕ α ⊂ B r +5 ε α ( o α ) . Linearity of the map A α is clear from the con-struction. Finally, properties ii) and iii) are trivially respected. Step 2. [Convergence ϕ α C c → ϕ ] Let us show that if ω ϕ is the modulus ofcontinuity of ϕ , defined by ω ϕ ( δ ) := sup d ( x,y ) ≤ δ | ϕ ( x ) − ϕ ( y ) | for any δ > , then k ϕ ◦ Φ α − ϕ α k L ∞ ( X α ,µ α ) ≤ κ ω ϕ (5 ε α ) (105) for any α . Take x ∈ X α . Then | ϕ ◦ Φ α ( x ) − ϕ α ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X p ∈D α ( ϕ (Φ α ( x )) − ϕ (Φ α ( p ))) ξ αp ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X p ∈V ( x ) ω ϕ ( d (Φ α ( x ) , Φ α ( p ))) ξ αp ( x ) ≤ κ ω ϕ (5 ε α ) . Hence (105) is proved, and consequently ϕ α C c → ϕ . Step 3. [Equicontinuity and Lipschitz estimate] Take x, y ∈ X α . Then ϕ α ( x ) − ϕ α ( y ) = X p ∈D α [ ϕ (Φ α ( p )) − ϕ (Φ α ( x ))] (cid:0) ξ αp ( x ) − ξ αp ( y ) (cid:1) = X p ∈V ( x ) ∪V ( y ) [ ϕ (Φ α ( p )) − ϕ (Φ α ( x ))] (cid:0) ξ αp ( x ) − ξ αp ( y ) (cid:1) . Observe that when p ∈ V ( x ) ∪ V ( y ) , then d α ( x, p ) ≤ ε α + d α ( x, y ) , hence we have | ϕ (Φ α ( p )) − ϕ (Φ α ( x )) | ≤ ω ϕ ( d α ( x, p ) + ε α ) ≤ ω ϕ (5 ε α + d α ( x, y )) . Using X p ∈D α (cid:12)(cid:12) ξ αp ( x ) − ξ αp ( y ) (cid:12)(cid:12) ≤ min (cid:26) , κ ε α d α ( x, y ) (cid:27) , we obtain the estimate | ϕ α ( x ) − ϕ α ( y ) | ≤ ( (1+ κ ) ω ϕ (6 ε α ) ε α d α ( x, y ) if d α ( x, y ) ≤ ε α , ω ϕ (6 d α ( x, y )) if d α ( x, y ) ≥ ε α . In particular, if ϕ is Λ -Lipschitz, then ϕ α is (cid:0) κ (cid:1) Λ -Lipschitz. This estimatealso implies the equicontinuity of the sequence: if δ ∈ (0 , and d α ( x, y ) ≤ δ then | ϕ α ( x ) − ϕ α ( y ) | ≤ ω ϕ (cid:16) √ δ (cid:17) + 2(1 + κ ) k ϕ k L ∞ √ δ. (cid:3) B. Convergence of integrals. In this subsection, we prove two results aboutconvergence of integrals under pmGH convergence that are used repeatedly in thisarticle. We recall our setting: { ( X α , d α , µ α , o α ) } α , ( X, d , µ, o ) are κ -doubling atscale R proper geodesic pointed metric measure spaces such that ( X α , d α , µ α , o α ) → ( X, d , µ, o ) in the pmGH sense, and we use the notations of Characterization 1.It is easy to prove the first convergence result. Proposition B.1. Let u ∈ C ( X ) , v ∈ L ( X, µ ) and u α ∈ C ( X α ) , v α ∈ L ( X α , µ α ) for any α be such that: • sup α k u α k L ∞ < ∞ , • u α → u uniformly on compact subsets, • v α → v strongly in L .Then lim α ˆ X α u α v α d µ α = ˆ X α uv d µ. Proof. The result follows from establishing the weak convergence u α v α L ⇀ uv . Re-mark first that the hypotheses of the Proposition imply sup α k u α v α k L < + ∞ . TRUCTURE OF KATO LIMITS 71 Moreover, when ϕ α C c → ϕ , then obviously ϕ α u α C c → ϕu , and as v α L ⇀ v we get lim α → + ∞ ˆ X α ϕ α u α v α d µ α = ˆ X ϕuv d µ. (cid:3) We make now a few useful remarks. The first point is that for any r > : x α ∈ X α → x ∈ X = ⇒ lim α → + ∞ µ α ( B r ( x α )) = µ ( B r ( x )) . (106)In full generality, this convergence result holds when µ ( ∂B r ( x )) = 0 [Bil99, Theo-rem 2.1], and this condition is guaranteed by the doubling condition [1.2-v)]. Wealso have that for any r > : ϕ α C c → ϕ, x α ∈ X α → x ∈ X = ⇒ lim α → + ∞ ˆ B r ( x α ) ϕ α d µ α = ˆ B r ( x ) ϕ d µ. (107)Even better, the convergence result takes place as soon as ϕ α ∈ C ( X α ) convergesuniformly on compact set to ϕ ∈ C ( X ) .The above convergence results (106) and (107) imply, by definition, that when r > , p > and x α ∈ X α → x ∈ X , then B r ( x α ) L p → B r ( x ) . (108)This implies the following criterion for L p weak convergence: Lemma B.2. For p ∈ (1 , ∞ ) , let u α ∈ L p ( X α , µ α ) for any α and u ∈ L p ( X, µ ) begiven. Then u α L p ⇀ u if and only if sup α k u α k L p < ∞ x α ∈ X α → x ∈ X, r > ⇒ lim α B r ( x α ) u α d µ α = B r ( x ) u d µ. (109) Proof. The direct implication follows from (108). The converse one is a consequenceof the fact that if B ⊂ A is such that { u β } β ∈ B converges weakly in L p to v then ´ B r ( x ) u d µ = ´ B r ( x ) v d µ for any x ∈ X and any r > , and so B r ( x ) u d µ = B r ( x ) v d µ. By Lebesgue differentiation theorem (true on any doubling space), this implies that u = v µ -a.e. (cid:3) Our second convergence result is the following. Proposition B.3. Let u ∈ C ( X ) and u α ∈ C ( X α ) for any α be such that • u α → u uniformly on compact subsets, • there exists C, β > such that for any α and µ α -a.e. x ∈ X α , | u α ( x ) | ≤ C e − β d α ( o α ,x ) . (110) Then the functions u α and u are L p -integrable for any p ≥ and(1) ´ X α u α d µ α → ´ X u d µ ,(2) u α → u strongly in L p when p > . For the proof of this proposition, we use the following lemma which is a conse-quence of the ideas of the proof of (67). Lemma B.4. Let ( X, d , µ ) be κ -doubling at scale R . Then for any c > thereexists A > depending only on c , κ and R such that for any o ∈ X , ˆ X e − c d ( o,x ) d µ ( x ) ≤ Aµ ( B R ( o α )) . (111) Moreover, there exists β : (0 , + ∞ ) → (0 , + ∞ ) depending only on c , C and R suchthat β ( ρ ) → when ρ → + ∞ and for any ρ > , ˆ X \ B ρ ( o ) e − c d ( o,x ) d µ ( x ) ≤ β ( ρ ) µ ( B R ( o )) . (112) Proof. We have ˆ X e − c d ( o,x ) d µ ( x ) ≤ µ ( B R ( o )) + ˆ X \ B R ( o ) e − c d ( o,x ) d µ ( x ) . Moreover, using Cavalieri’s formula and Proposition 1.2-ii) we get that for any ρ ≥ R , ˆ X \ B ρ ( o ) e − c d ( o,x ) d µ ( x ) = ˆ + ∞ ρ cre − cr ( µ ( B r ( o )) d r ≤ µ ( B R ( o )) ˆ + ∞ ρ cre − cr + λ rR d r. (cid:3) We can now prove Proposition B.3. Proof of Proposition B.3. As a consequence of the previous lemma, for any p ≥ ,we get ˆ X α \ B ρ ( o α ) | u α | p d µ α ≤ β ( ρ ) µ α ( B R ( o α )) (113)where β depend only on p, β, κ . The discussion above implies that: u α L ploc → u. With the estimate (113), we get that that the sequence {k u α k L p } is bounded hence u α L p ⇀ u. But the estimate (113) implies the validity of the intervention on limits : lim ρ → + ∞ lim α ˆ B ρ ( o α ) | u α | p d µ α = lim α lim ρ → + ∞ ˆ B ρ ( o α ) | u α | p d µ α ; that is to say lim α ˆ X α | u α | p d µ α = ˆ X | u | p d µ. Thus u α L p → u. The statement Proposition B.3-(1) is proven in the same way. (cid:3) Remark B.5. When the function u α are only assumed to be measurable, theconclusion Proposition B.3-(1) holds assuming u α L loc ⇀ u in place of the uniformconvergence on compact sets. Indeed this hypothesis implies that for any R > : lim α ˆ B R ( o α ) u α d µ α = ˆ B R ( o ) u d µ. And the proof of Proposition B.3 can be applied. But we won’t need this refinementhere. TRUCTURE OF KATO LIMITS 73 C. Heat kernel characterization of PI -Dirichlet spaces. In this subsection,we provide a set of conditions on the heat kernel of a metric Dirichlet space ( X, d , µ, E ) for him to be regular, strongly local and with d E being a distancebi-Lipschitz equivalent to d . We use this result in the next subsection to proveTheorem 1.17. Wde let R > be fixed throughout this subsection.C.1. Heat kernel bound. We need an important statement about regular, stronglylocal Dirichlet spaces. It is the combination of several well-known theorems [Gri91,SC92, Stu96]. If ( X, d , µ, E ) is a metric measure space equipped with a Dirichletform with associated operator L , we call local solution of the heat equation anyfunction u satisfying ( ∂ t + L ) u = 0 in the sense of [Stu94] (see also [CT19, Def. 2.3]). Theorem C.1. Let ( X, T , µ, E ) be a regular, strongly local Dirichlet space with d E being a distance compatible with T . Then the following are equivalent: (c1) ( X, d E , µ, E ) is a PI κ , γ ( R ) Dirichlet space, (c2) E admits a heat kernel H satisfying Gaussian bounds: there exists β > such that β − µ ( B √ t ( x )) e − β d E ( x,y ) t ≤ H ( t, x, y ) ≤ β µ ( B √ t ( x )) e − d E ( x,y ) β t (114) for all x, y ∈ X and t ∈ (0 , R ] , (c3) the local solutions of the heat equation satisfy a uniform Hölder regularityestimate: there exist constants α ∈ (0 , , A > such that if B is a ball ofradius r ≤ R and u : (0 , r ) × B → (0 , ∞ ) is a local solution of the heatequation then for any s, t ∈ (cid:0) r / , r / (cid:1) and x, y ∈ B , | u ( s, x ) − u ( t, y ) | ≤ Ar α (cid:16)p | t − s | + d E ( x, y ) (cid:17) α sup (0 ,r ) × B | u | . (115)As a corollary, the heat kernel of a regular, strongly local PI Dirichlet spacesatisfies the following properties: Proposition C.2. Let ( X, d E , µ, E ) be a regular, strongly local, PI κ , γ ( R ) Dirichletspace for some κ ≥ and γ > . Let d be a distance on X bi-Lipschitz equivalentto d E . Then E admits a heat kernel H such that the following holds: (a) H is stochastically complete (12) , (b) there exists β ≥ such that the following Gaussian double-sided boundshold: β − µ ( B √ t ( x )) e − β d x,y ) t ≤ H ( t, x, y ) ≤ β µ ( B √ t ( x )) e − d x,y ) β t (116) for all x, y ∈ X and t ∈ (0 , R ] , (c) there exists α ∈ (0 , and A > such that for any x, y, z ∈ X and s, t ∈ (0 , R ) such that | t − s | ≤ t/ and d ( y, z ) ≤ √ t , | H ( s, x, z ) − H ( t, x, y ) | ≤ A p | t − s | + d ( y, z ) √ t ! α H ( t, x, y ) . (117)The next theorem is our key statement to establish Theorem 1.17. Theorem C.3. Let ( X, d , µ, E ) be a metric Dirichlet space such that ( X, d ) isgeodesic and for which a heat kernel H exists and satisfies ( a ) , ( b ) and ( c ) in theprevious proposition C.2. Then ( X, d E µ, E ) is a PI κ , γ ( R ) Dirichlet space for some κ ≥ and γ > depending only on the constants from ( b ) and ( c ) , and the distance d is bi-Lipschitz equivalent to the intrinsic distance d E . C.2. Domain characterization. In order to prove Theorem C.3, we start by showingthe following crucial proposition. It is a generalization of a similar result of A.Grigor’yan, J. Hu and K-S. Lau [GHL03, Theorem 4.2] (see also [Gri10, Corollary4.2] and references therein) where the mesure is additionally assumed uniformlyAhlfors regular. Proposition C.4. Under the assumptions of Theorem C.3 , the domain of E coin-cides with the Besov space B , ∞ ( X ) , consisting of the functions u ∈ L ( X, µ ) suchthat N ( u ) := lim sup r → + r ˆ X B r ( x ) ( u ( x ) − u ( y )) d µ ( y ) d µ ( x ) < ∞ . Moreover, there is a constant C depending only on β such that for any u ∈ D ( E ) , C N ( u ) ≤ E ( u ) ≤ CN ( u ) . Proof. For any function u ∈ L ( X, µ ) , define the decreasing function t 7→ E t ( u ) where for any t > , E t ( u ) := 1 t h u − e − tL u, u i = ˆ X × X H ( t, x, y )( u ( x ) − u ( y )) d µ ( x ) d µ ( y )2 t · We first observe that, because of assumption (a), a function u belongs to D ( E ) ifand only if sup t E t ( u ) < ∞ . Moreover, if u ∈ D ( E ) , then E ( u ) = lim t → + E t ( u ) . This is explained in [Gri10, Sect. 2.2], for instance. Step 1. We begin with showing the easiest inclusion, namely D ( E ) ⊂ B , ∞ ( X ) .Take u ∈ D ( E ) . For t > , set I ( t ) := ˆ { ( x,y ) ∈ X × X : d ( x,y ) ≤√ t } H ( t, x, y )( u ( x ) − u ( y )) d µ ( x ) d µ ( y )2 t , and observe that I ( t ) ≤ E t ( u ) ≤ E ( u ) . The lower bound for the heat kernel givenby assumption ( b ) implies I ( t ) ≥ ˆ { ( x,y ) ∈ X × X : d ( x,y ) ≤√ t } β − µ ( B √ t ( x )) e − β d x,y ) t ( u ( x ) − u ( y )) dµ ( x ) dµ ( y )2 t ≥ β − e − β t ˆ X B √ t ( x ) ( u ( x ) − u ( y )) dµ ( x ) dµ ( y ) , hence letting t tend to shows that N ( u ) ≤ β e β E ( u ) . Step 2. In order to prove the converse inclusion, we need some volume estimates.Our assumptions imply that the measure µ is doubling at scale R . Indeed for all x ∈ X and r ≤ R , thanks to assumptions (a) and (b) we have β e − β µ ( B r ( x )) µ ( B r ( x )) ≤ ˆ B r ( x ) H ( r , x, y ) dµ ( y ) ≤ ˆ X H ( t, x, y ) dµ ( y ) = 1 , and therefore µ ( B r ( x )) ≤ βe β µ ( B r ( x )) . Because of the doubling condition at scale R , we obtain for s ≤ r ≤ R , x ∈ Xµ ( B r ( x )) ≤ C (cid:16) rs (cid:17) ν µ ( B s ( x )) , where ν and C depends only on β (see Proposition 1.2). The Gaussian estimate ofthe heat kernel (116) implies that if < t ≤ τ ≤ R then H ( t, x, y ) ≤ H ( τ, x, y ) C (cid:16) τt (cid:17) ν/ e − d ( x,y ) ( β t − β τ ) (118) TRUCTURE OF KATO LIMITS 75 We introduce Ω r = { ( x, y ) ∈ X × X : d ( x, y ) ≥ r } and I λ ( t ) := ˆ ( X × X ) \ Ω λ √ t H ( t, x, y )( u ( x ) − u ( y )) d µ ( x ) d µ ( y )2 t . The same reasoning as in Step 1 implies that for λ ≥ and t > such that λ √ t < R tehn I λ ( t ) ≤ Cλ ν t ˆ X B λ √ t ( x ) ( u ( x ) − u ( y )) d µ ( y ) d µ ( x ) . (119)Using the estimate (118) with τ = λt and assuming λ ≥ and λ t ≤ R , weestimate: E t ( u ) − I λ ( t ) ≤ Cλ ν/ e − λ t ( β t − β λt ) ˆ Ω λ √ t H ( λt, x, y )( u ( x ) − u ( y )) d µ ( x ) d µ ( y )2 t ≤ Cλ ν/ e − λ ( β − β λ ) E λt ( u ) ≤ Cλ ν/ e − λ ( β − β λ ) E t ( u ) , where we have used that t 7→ E t ( u ) is non increasing. If we choose λ = λ ( β ) sufficiently large so that Cλ ν +1 e − λ ( β − β λ ) ≤ , then we get E t ( u ) ≤ Cλ ν t ˆ X B λ √ t ( x ) ( u ( x ) − u ( y )) d µ ( y ) d µ ( x ) . Hence the result. (cid:3) Remark C.5. The above reasoning implies that if U : X × X → R + is a non nega-tive integrable function such that the limit lim t → ´ X × X H ( t, x, y ) C ( x, y ) d µ ( x ) d µ ( y )2 t exist and is finite then lim t → ˆ X × X H ( t, x, y ) C ( x, y ) d µ ( x ) d µ ( y )2 t ≤ C ( β ) lim sup r → + ˆ X B r ( x ) C ( x, y ) d µ ( y ) d µ ( x ) r . We are now in position to prove Theorem C.3. Proof of Theorem C.3. Regularity. We start by showing that ( X, d , µ, E ) is regular, that is we provethat the space C c ( X ) ∩ D ( E ) is dense in ( C c ( X ) , || · || ∞ ) and in ( D ( E ) , | · | D ( E ) ) .Observe that Lip c ( X ) is contained in D ( E ) . Indeed, for any u ∈ Lip c ( X ) thereexists Λ such that for all x, y ∈ X | u ( x ) − u ( y ) | ≤ Λ d ( x, y ) , and there exists ρ > such that the support of u is included in the ball B ρ ( o ) .Therefore for any r > and x ∈ X we have e r ( x ) := B r ( x ) ( u ( x ) − u ( y )) d µ ( y ) ≤ Λ r and moreover e r ( x ) = 0 if x / ∈ B ρ +1 ( o ) . As a consequence, for any u ∈ Lip c ( X ) ,there exists ρ such that N ( u ) ≤ µ ( B ρ +1 ( o ))Λ , thus Lip c ( X ) ⊂ B , ∞ ( X ) and by the previous theorem Lip c ( X ) ⊂ D ( E ) . SinceLip c ( X ) is dense in ( C c ( X ) , || · || ∞ ) , this implies that C c ( X ) ∩ D ( E ) is also dense in ( C c ( X ) , || · || ∞ ) .In order to prove that C c ( X ) ∩D ( E ) is dense in D ( ε ) , we follow the same argumentas in the proof of [CT19, Prop. 3.8] i.e. we show that if t ∈ (0 , R ) and u ∈ L ( X, µ ) , then f = e − tL u belongs to C ( X ) . To see that f tends to zero at infinity, noticethat the upper bound for the heat kernel implies | f ( x ) | ≤ β µ ( B √ t ( x )) e − d x, supp u ) β t ˆ X | u | d µ for any x ∈ X ; from Proposition 1.2-i), if o ∈ supp u we obtain | f ( x ) | ≤ Cµ ( B √ t ( o )) e − d x, supp u ) β t + λ d ( o,x ) √ t ˆ X | u | d µ, therefore f is bounded and tends to zero as d ( o, · ) goes to infinity.As for the continuity of f , assumption (c) ensures that for any x, x ′ ∈ X suchthat d ( x, x ′ ) ≤ √ t we have | f ( x ) − f ( x ′ ) | ≤ A (cid:18) d ( x, x ′ ) √ t (cid:19) α ( e − tL | u | )( x ) . Since e − tL | u | is also bounded, this shows that f is continuous. Strong locality. We aim to prove that if u, v ∈ D ( E ) have compact supportsand if u is constant in a neighbourhood of supp ( v ) , then E ( u, v ) = 0 . Assume thatboth u and v are supported in B ρ ( o ) and denote by K the support of v . Thereexist η > and c ∈ R such that if d ( x, K ) ≤ η , then u ( x ) = c . Let us introduce K r = S x ∈ K B r ( x ) . Then for any r ≤ η , u is constantly equal to c on K r .As in the previous theorem, we can define E t ( u, v ) = ˆ X × X H ( t, x, y )( u ( x ) − u ( y ))( v ( x ) − v ( y )) d µ ( x ) d µ ( y )2 t , and we have E ( u, v ) = lim t → + E t ( u, v ) . From Remark C.5, there exists a constant C > such that |E ( u, v ) | ≤ C lim sup r → + r ˆ X B r ( x ) | u ( x ) − u ( y ) | | v ( x ) − v ( y ) | d µ ( y ) ! d µ ( x ) . Now observe that for any r > , if x / ∈ K r and y ∈ B r ( x ) , the triangle inequalityensures that d ( y, K ) > , thus both v ( x ) and v ( y ) are equal to zero. We are thenleft with considering lim sup r → + r ˆ K r B r ( x ) | u ( x ) − u ( y ) | | v ( x ) − v ( y ) | d µ ( y ) ! d µ ( x ) . But the same arguing implies that when r ≤ η/ , x ∈ K r and y ∈ B r ( x ) , then y ∈ K r ; as a consequence both x and y belong to K η , so we have u ( x ) = u ( y ) = c .Finally, for r ≤ η/ we get ˆ X B r ( x ) | u ( x ) − u ( y ) | | v ( x ) − v ( y ) | d µ ( y ) ! dµ ( x ) = 0 . This ensures that E ( u, v ) = 0 and thus ( X, d , µ, E ) is strongly local. Equivalence between the distance and the intrinsic distance. Let usbegin with proving the existence of C > such that d E ≥ C d . Again from Remark C.5, there exists a constant C such that for any u ∈ D ( E ) and φ ∈ C c ( X ) ∩ D ( E ) with φ ≥ then ˆ X φ dΓ( u ) ≤ C lim sup r → + r ˆ X φ ( x ) B r ( x ) ( u ( x ) − u ( y )) d µ ( y ) ! d µ ( x ) . TRUCTURE OF KATO LIMITS 77 If u ∈ Lip c ( X ) then ˆ X φ dΓ( u ) ≤ C Lip( u ) ˆ X φ d µ. Hence dΓ( u ) ≤ C Lip( u ) d µ. Take x, y ∈ X and set r := d ( x, y ) and u ( z ) := χ (cid:18) d ( x, z )2 r (cid:19) d ( x, z ) for any z ∈ Z , where χ is defined as in (104). Then u ∈ Lip c ( X ) and u ( y ) − u ( x ) = d ( x, y ) . Moreover, Lip( u ) ≤ . Thus, testing u/ (9 C ) in the definition of d E , we get d E ( x, y ) ≥ (3 √ C ) − d ( x, y ) . Now let us prove d E ≤ p β / d . We act as in the proof of [CT19, Prop. 3.9]. We consider a bounded function v ∈ D loc ( E ) ∩ C ( X ) such that Γ( v ) ≤ µ . For any a ≥ , t ∈ (0 , R ) and x ∈ X weset ξ a ( x, t ) := av ( x ) − a t/ . Take x, y ∈ X and assume with no loss of generalitythat v ( y ) − v ( x ) > . From [CT19, Claim 3.10] applied to f = 1 B √ t ( y ) , one gets ˆ B √ t ( x ) ˆ B √ t ( y ) H ( t, z , z ) d µ ( z ) ! e ξ a ( z ,t ) d µ ( z ) ≤ ˆ B √ t ( y ) e av d µ which leads to µ ( B √ t ( x )) µ ( B √ t ( y )) exp (cid:18) aδ t ( x, y ) − a t (cid:19) inf B √ t ( x )) × B √ t ( y ) H ( t, · , · ) ≤ , (120) where we define δ t ( x, y ) := inf B √ t ( y ) v − sup B √ t ( x ) v. Observe that sup B √ t ( x ) × B √ t ( y ) d ( · , · ) ≤ d ( x, y ) + 2 √ t, so that the Gaussian lower bound in (116) yields, for any ( z , z ) ∈ B √ t ( x ) × B √ t ( y ) , H ( t, z , z ) ≥ β − µ ( B √ t ( z )) exp (cid:18) − β ( d ( x, y ) + 2 √ t ) t (cid:19) ; the doubling condition implies µ ( B √ t ( z )) ≤ µ ( B √ t ( x ) ≤ κ µ ( B √ t ( x )) , then we get (cid:18) inf B √ t ( x )) × B √ t ( y ) H ( t, · , · ) (cid:19) ≥ ( βκ ) − µ ( B √ t ( x ) exp (cid:18) − β ( d ( x, y ) + 2 √ t ) t (cid:19) . The continuity of v yields lim t → δ t ( x, y ) = v ( y ) − v ( x ) , hence we can take t smallenough to ensure δ t ( x, y ) > and choose a = δ t ( x, y ) /t . Then (120) implies µ ( B √ t ( y )) µ ( B √ t ( x )) exp − β (cid:0) d ( x, y ) + 2 √ t (cid:1) t + δ t ( x, y )2 t ! ≤ ( βκ ) . Thanks to Proposition 1.2-i), this leads to exp − λ d ( x, y ) √ t − β (cid:0) d ( x, y ) + 2 √ t (cid:1) t + δ t ( x, y )2 t ! ≤ C ( βκ ) . Letting t → , we get ( v ( y ) − v ( x )) ≤ β d ( x, y ) . Since v is arbitrary, we finally obtain d E ( x, y ) ≤ p β / d ( x, y ) . (cid:3) D. Proof of Theorem 1.17. Proof. We assume that the spaces { ( X α , d α = d E α , µ α , o α , E α ) } α ∈ A are PI κ , γ ( R ) Dirichlet spaces and that for any α , η − ≤ µ α ( B R ( o α )) ≤ η. (121)The existence of ( X, d , µ, o ) and a subsequence B ⊂ A such that ( X β , d β , µ β , o β ) pmGH −→ ( X, d , µ, o ) follow from Proposition 1.7. Moreover ( X, d ) is complete and geodesic, and ( X, d , µ ) is κ -doubling at scale R .Furthermore, Proposition C.2 ensures that any E β admits a stochastically com-plete heat kernel H β satisfying the Gaussian bounds (116) and the estimate (117)with constants β , α and A depending only on κ and γ . Let t ∈ (cid:0) , R (cid:1) and ρ > .By Proposition 1.2-i), we get that for any x, y ∈ B ρ ( o β ) , H β ( t, x, y ) ≤ Ce λ ρ √ t µ β (cid:0) B √ t ( o β ) (cid:1) , from which the doubling condition and the non-collapsing condition (121) yield theuniform estimate : H β ( t, x, y ) ≤ C (cid:18) R √ t (cid:19) ν exp (cid:18) λ ρ √ t (cid:19) η, (122)where ν, C, λ depend only on κ , γ . Hence for any t ∈ (cid:0) , R (cid:1) and ρ > , there is aconstant Λ depending only on t, ρ, κ , γ , R, η such that for x, x ′ , y, y ′ ∈ B ρ ( o β ) , weget the Hölder estimate: | H β ( t, x, y ) − H β ( t, x ′ , y ′ ) | ≤ Λ min (cid:8) t α ; [ d ( x, x ′ ) + d ( y, y ′ )] α (cid:9) . (123)Thanks to this local Hölder continuity estimate and the uniform estimate (122), theArzelà-Ascoli theorem with respect to pGH convergence (see e.g. [Vil09, Prop. 27.20])implies that, up to extracting another subsequence, the functions H β ( t, · , · ) convergeuniformly on compact sets to some function H ( t, · , · ) ∈ C ( X × X ) , where t > isfixed from now on. A priori this subsequence may depend on t , but for the moment t ∈ (cid:0) , R (cid:1) is fixed.Let S : L c ( X, µ ) → L ∞ loc ( X, µ ) be the integral operator on defined by setting Su ( x ) := ˆ X u ( y ) H ( t, x, y ) d µ ( y ) for any u ∈ L c ( X, µ ) and x ∈ X .We claim that S has a bounded linear extension S : L ( X, µ ) → L ( X, µ ) .Firstly, thanks to the uniform convergence on compact sets H β ( t, · , · ) → H ( t, · , · ) ,the symmetry with respect to the two space variables of H β transfers to H . More-over, PI Dirichlet spaces are stochastically complete hence for any x ∈ X β : ˆ X β H β ( t, x, y ) d µ β ( y ) = 1 . Using the uniform Gaussian estimate (116) and Proposition B.3, we have similarly ˆ X H ( t, x, y ) d µ ( y ) = 1 , TRUCTURE OF KATO LIMITS 79 for any x ∈ X . The Schur test implies that for any p ∈ [0 , + ∞ ] , S extends to abounded operator S : L p ( X, µ ) → L p ( X, µ ) with operator norm satisfying: k S k L p → L p ≤ . (124)The symmetry with respect to the two space variables of H implies that S : L ( X, µ ) → L ( X, µ ) is self-adjoint. Hence there exists a non-negative self-adjoint operator L with densedomain D ( L ) ⊂ L ( X, µ ) such that S = e − tL . Moreover we have: f ≥ ⇒ Sf ≥ .Let us show now the strong convergence of bounded operators e − tL β → e − tL . (125)The operator are all self-adjoint hence it is enough to show the weak convergence ofbounded operators and this amounts to showing that if u β L ⇀ u , then e − tL β u β L ⇀e − tL u . Note that sup β k u β k L < + ∞ . Since the operators e − tL β have all anoperator norm less than , then sup β k e − tL β u β k L < + ∞ . In particular, the functions e − tL β u β are bounded in L . Now take x β → x . Theuniform Gaussian estimate (116) and Proposition B.3 ensures that the functions f β = H β ( t, x β , · ) converge strongly in L to the functions f = H ( t, x, · ) . Then h f β , u β i L ( X β , d µ β ) → h f, u i L ( X, d µ ) , that is to say e − tL β u β ( x β ) = ˆ X β H β ( t, x β , y ) u β ( y ) d µ β ( y ) → ˆ X H ( t, x, y ) u ( y ) d µ ( y ) = e − tL u ( x ) . The same argument can be used with f β,r ( y ) = ´ B r ( x β ) H β ( t, z, y ) d µ β ( z ) and f r ( y ) = ´ B r ( x ) H ( t, z, y ) d µ ( z ) for any r > , hence Lemma B.2 eventually implies e − tL β u β L ⇀ e − tL u. Using now [KS03, Theorem 2.1], we get that for any τ > , the strong conver-gence of bounded operators e − τL β → e − τL . But the above argumentation shows that for any τ ∈ (0 , R ) , the function H β ( τ, · , · ) has a unique limit (for the uniform convergence on compact set of X × X ) and thislimit is the Schwartz kernel of the operator e − τL . Moreover ≤ f ≤ ⇒ ≤ e − τL f ≤ , hence the quadratic form E ( u ) := ˆ X ( Lu ) u d µ define a Dirichlet form E on ( X, d , µ ) .From Proposition 1.15, the strong convergence (125) implies the Mosco conver-gence E β → E . As a consequence, the functions H β : (0 , R ] × X β × X β uniformlyconverge on compact sets to the heat kernel ˜ H of E restricted to (0 , R ] × X β × X β .Then the dominated convergence theorem ensures that ˜ H satisfy the assumptions ( a ) , ( b ) and ( c ) in Theorem C.3, thus ( X, d , µ, E ) is regular, strongly local, and withintrinsic distance d E bi-Lipschitz equivalent to d so that ( X, d , µ, E ) is a PI κ , γ ′ ( R ) Dirichlet space. It remains to show that d ≤ d E . According to (21), the heat kernel of E β satisfiesthe uniform upper Gaussian estimate H β ( t, x, y ) ≤ Cµ β ( B R ( x )) R ν t ν d E β ( x, y ) t ! ν +1 exp − d E β ( x, y )4 t ! valid for any x, y ∈ X β and t ∈ (cid:0) , R (cid:1) . By uniform convergence on compact set H β → H and d E β → d , we get the same estimate for the heat kernel of E : H ( t, x, y ) ≤ Cµ ( B R ( x )) R ν t ν (cid:18) d ( x, y ) t (cid:19) ν +1 exp (cid:18) − d ( x, y )4 t (cid:19) . From there it is easy to conclude that d ≤ d E thanks to Varadhan’s formula (22). (cid:3) E. Further convergence results. In this last subsection we assume that ( X, d , µ, E , o ) is a PI κ , γ ( R ) Dirichlet space that is a pointed Mosco-Gromov-Hausdorff limit ofa sequence { ( X α , d α , µ α , E α , o α ) } α of PI κ , γ ( R ) Dirichlet spaces, and we use thenotations { ε α } , { R α } , { Φ α } of Characterization 1.We begin with a technical result. Proposition E.1. Let { u α } be such that u α ∈ L ( B r ( o α ) , µ α ) for any α , for some r > . Assume that:(1) there exists u ∈ L ( B r ( o ) , µ ) such that u α L ⇀ u (2) sup α ´ B r ( o α ) dΓ α ( u α ) < + ∞ .Then lim α ˆ B r ( o α ) u α d µ α = ˆ B r ( o ) u d µ. (126) Proof. We first prove that for any s < r , lim α ˆ B s ( o α ) u α d µ α = ˆ B s ( o ) u d µ. (127)For ε < ( r − s ) / , we introduce u α,ε ( x ) = ˆ B ε ( x ) u α d µ α . The Poincaré inequality implies the Pseudo Poincaré inequality [CSC93, Lemme inpage 301]: k u α − u α,ε k L ( B s ( o α )) ≤ Cε and k u − u ε k L ( B s ( o )) ≤ Cε, where C depends only on the doubling constant, the Poincaré constant and of sup α ´ B r ( o α ) dΓ α ( u α ) . The Hölder inequality and the doubling property [1.2-v]imply that for fixed ε > , the sequence { u α,ε } α is equicontinuous on B ( s + r ) / ( o α ) ,hence u α,ε → u ε uniformly in B s ( o α ) . Hence we get the strong convergence in L ( B s ( o )) and the convergence (127).Since the spaces are uniformly PI, they satisfy a same local Sobolev inequality[Stu96, Th. 2.6] meaning that there exists C > , ν > and δ ∈ (0 , independanton α such that ˆ B r ( o α ) | ψ | νν − d µ α ! − /ν ≤ C ˆ B r ( o α ) d Γ α ( ψ ) + ˆ B r ( o α ) | ψ | d µ α ! for any ψ ∈ D ( E α ) . In particular, we get the a priori bound sup α k u α k L νν − B r ( o α ) ≤ C. TRUCTURE OF KATO LIMITS 81 With Hölder’s inequality and the doubling property [1.2-v], this yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B s ( o α ) u α d µ α − ˆ B r ( o α ) u α d µ α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B r ( o α ) u α ( B s ( o α ) − 1) d µ α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ B r ( o α ) | ψ | νν − d µ α ! − ν µ ( B r ( o α ) \ B s ( o α )) ν ≤ C ( r − s ) δ/ν which justifies the intervention lim s → r lim α ˆ B s ( o α ) u α d µ α = lim α lim s → r ˆ B s ( o α ) u α d µ α . (cid:3) E.1. Convergence of the core C c ∩ D ( E ) . The next result indicates that in a certainsense the space C c ( X ) ∩ D ( E ) is the limit of the spaces C c ( X α ) ∩ D ( E α ) . Proposition E.2. Let ϕ ∈ C c ( X ) ∩ D ( E ) . Then there exists { ϕ α } , with ϕ α ∈C c ( X α ) ∩ D ( E α ) for any α , such that ϕ α C c → ϕ and ϕ α E → ϕ. Moreover if ϕ is non negative then each ϕ α can be chosen to be also non negative.Proof. Step 1. We construct ψ α ∈ C ( X α ) ∩ D ( E α ) such that ψ α → ϕ uniformlyon compact sets and such that ψ α E → ϕ. Proposition A.1 allows us to build f α ∈ C c ( X α ) such that f α C c → ϕ . Moreover weknow that the sequence { f α } is uniformly equicontinuous: there is ω : R + → R + non decreasing, bounded and satisfying ω ( δ ) → when δ → such that for any α | f α ( x ) − f α ( y ) | ≤ ω ( d α ( x, y )) for any α and any x, y ∈ X α . In addition, if it turns out that ϕ is non negative, sois f α . As f α C c → ϕ , we also have f α L → ϕ and for any ε > : P αε f α E → P ε ϕ . And ϕ being in D ( E ) we get that D ( E ) − lim ε → P ε ϕ = ϕ. Let us show now that if ε α ↓ then we have P αε α f α → ϕ uniformly. To do so, itis sufficient to demonstrate that lim ε → sup α k P αε f α − f α k L ∞ = 0 . Using the stochastic completeness, we know that for any x ∈ X α : P αε f α ( x ) − f α ( x ) = ˆ X α H α ( ε, x, y ) ( f α ( y ) − f α ( x )) d µ α ( y ) . Hence for any κ > , we have | P αε f α ( x ) − f α ( x ) | ≤ ˆ X α H α ( ε, x, y ) ω ( d α ( x, y )) d µ α ( y ) ≤ ω (cid:0) κ √ ε (cid:1) + k ω k L ∞ ˆ X α \ B κ √ ε ( x ) H α ( ε, x, y ) d µ α ( y ) ≤ ω (cid:0) κ √ ε (cid:1) + k ω k L ∞ γ µ (cid:0) B √ ε ( x ) (cid:1) ˆ X α \ B κ √ ε ( x ) e − d α ( x,y ) γ ε d µ α ( y ) ω (cid:0) κ √ ε (cid:1) + k ω k L ∞ γ µ (cid:0) B √ ε ( x ) (cid:1) ˆ ∞ κ √ ε e − r γ ε r γ ε µ ( B r ( x )) dr We use the doubling condition [1.2-iii)] to deduce that if r ≥ κ √ ε > R > √ ε then µ ( B r ( x )) ≤ e λ r √ ε µ (cid:0) B √ ε ( x ) (cid:1) , Hence if κ √ ε > R > √ ε then | P αε f α ( x ) − f α ( x ) | ≤ ω (cid:0) κ √ ε (cid:1) + k ω k L ∞ ˆ ∞ κ √ ε e − r γ ε + λ r √ ε rε dr ≤ ω (cid:0) κ √ ε (cid:1) + k ω k L ∞ ˆ ∞ κ e − r γ + λr r dr ≤ ω (cid:0) κ √ ε (cid:1) + C ( λ, γ ) e − κ γ k ω k L ∞ . We then choose κ = ε − and we get for ε small enough: k P αε f α − f α k L ∞ ≤ ω (cid:16) ε (cid:17) + C ( λ, γ ) e − γ √ ε k ω k L ∞ . Remark E.3. The same estimate leads to the following decay estimate for P αε f α .Assume that R > and L are such that supp f α ⊂ B R ( o α ) and that k f α k L ∞ ≤ L .Then for x ∈ X α \ B R ( o α ) : | P αε f α ( x ) | ≤ CL e − d α ( oα,x )4 γ ε + λ R √ ε . To build ψ α we use U. Mosco’s argument for the proof of the implication ⇒ in Proposition 1.15. We find a decreasing sequence η ℓ ↓ and a increasing sequence α ℓ ↑ + ∞ such that < ε ≤ η ℓ = ⇒ (cid:12)(cid:12) k P ε ϕ k L − k ϕ k L (cid:12)(cid:12) + |E ( P ε ϕ ) − E ( ϕ ) | ≤ − ℓ α ≥ α ℓ = ⇒ (cid:12)(cid:12) k P αη ℓ f α k L − k P η ℓ ϕ k L (cid:12)(cid:12) + (cid:12)(cid:12) E α (cid:0) P αη ℓ f α (cid:1) − E ( P η ℓ ϕ ) (cid:12)(cid:12) ≤ − ℓ Then if α ∈ [ α ℓ , α ℓ +1 ) , we define ε α = η ℓ and δ α = 2 − ℓ and we let ψ α = P αε α f α . Then we have lim α δ α = 0 and ψ α → ϕ uniformly and (cid:12)(cid:12) k ψ α k L − k ϕ k L (cid:12)(cid:12) + |E α ( ψ α ) − E ( ϕ ) | ≤ δ α . We necessarily have ψ α L ⇀ ϕ and the above estimate implies the strong convergence ψ α E → ϕ . Step 2. We modify each ψ α with appropriate cut-off functions. Let R > suchthat that for any α : supp f α ⊂ B R ( o α ) and supp ϕ ⊂ B R ( o ) and let L such thatfor any α k f α k L ∞ ≤ L. We let χ α : X α → R be defined by χ α ( x ) = ξ ( d α ( o α , x ) / R ) . TRUCTURE OF KATO LIMITS 83 where ξ is defined by (104). And we let ϕ α = χ α ψ α . It is easy to check that ϕ α C c → ϕ . In order to verify that ϕ α E → ϕ , we need tocheck that lim α →∞ E α ( (1 − χ α ) ψ α ) = 0 . The chain rule implies that E α ( (1 − χ α ) ψ α ) = E α (cid:0) ψ α , (1 − χ α ) ψ α (cid:1) + ˆ X α ψ α dΓ α ( χ α ) . (128)We have ψ α E → ϕ and (1 − χ α ) ψ α E ⇀ hence the first term in the right hand sideof (128) tends to when α → ∞ . The function χ α are uniformly (1 / r ) -Lipschitzhence by (19) there is a constant C independant of α such that ˆ X α ψ α dΓ α ( χ α ) ≤ Cµ α ( B R ( o α )) sup B R ( o α ) \ B R ( o α ) | ψ α | . Using the Remark (E.3),we can conclude that lim α →∞ ˆ X α ψ α dΓ α ( χ α ) = 0 . (cid:3) E.2. Energy convergence and convergence of the carré du champ. We can now easilydeduced the following convergence result for the carré du champ under convergencein energy. Proposition E.4. Assume that ϕ ∈ C ( X ) ∩ D ( E ) , u ∈ D loc ( E ) , and ϕ α ∈ C ( X α ) ∩D ( E α ) , u α ∈ D loc ( E α ) for any α , are such that ϕ α C c → ϕ , ϕ α E → ϕ , u α E loc → u and sup α k u α k L ∞ < ∞ . Then lim α →∞ ˆ X α ϕ α dΓ( u α ) = ˆ X ϕ dΓ( u ) . (129) Moreover if for each ρ > there is some p > such that sup α ˆ B ρ ( o α ) (cid:12)(cid:12)(cid:12)(cid:12) dΓ( u α )d µ α (cid:12)(cid:12)(cid:12)(cid:12) p d µ α < ∞ then for each ρ > α →∞ ˆ B ρ ( o α ) dΓ( u α ) = ˆ B ρ ( o ) dΓ( u ) . (130) Proof. We give a proof only in case u α E → u , the demonstration in the stated casebeing identical up to a few immediate but cumbersome justifications.To prove (129) we use the definition of the carré du champ, ˆ X α ϕ α dΓ( u α ) = E α ( ϕ α u α , u α ) − E α (cid:0) ϕ α , u α (cid:1) , (131)together with the following observation: the chain rule implies that the sequences { ϕ α u α } α and { u α } α are bounded in energy and thus have weak limit in E . Howeverwhen ψ α E → ψ we get ψ α ϕ α E → ψϕ and then ´ X α ψ α ϕ α u α d µ α = ´ X ψϕu d µ , so that ϕ α u α E ⇀ ϕu. Moreover, when ψ α E → ψ we get ψ α u α → ψu and then ´ X α ψ α u α d µ α = ´ X ψu d µ ,so that u α E ⇀ u . Thus (131) converges to ´ X ϕ dΓ( u ) .To prove (130), take ε > . Acting as in the proof of Proposition E.1, withHölder’s inequality and the doubling condition [1.2-v] we can find τ ∈ (0 , ρ ) suchthat for any α , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B ρ − τ ( o α ) dΓ( u α ) − ˆ B ρ ( o α ) dΓ( u α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . Moreover, by regularity of the Radon measure Γ( u ) , we an assum that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B ρ − τ ( o ) dΓ( u ) − ˆ B ρ ( o ) dΓ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . We set ϕ ( x ) := if x ∈ B ρ − τ ( o ) , (2 ρ − τ − d ( o, x )) /τ if x ∈ B ρ − τ/ ( o ) \ B ρ − τ ( o ) , if x B ρ − τ/ ( o ) . We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B ρ ( o ) dΓ( u ) − ˆ X ϕ dΓ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε · (132)Thanks to Proposition E.2, we can choose ϕ α ∈ C c ( X α ) ∩ D ( E α ) non-negative forany α such that ϕ α C c → ϕ and ϕ α E → ϕ . Then there is some sequence δ α ↓ suchthat • | ϕ α − | ≤ δ α on B ρ − τ ( o α ) , • ϕ α ≤ δ α , • ϕ α ≤ δ α outside B ρ ( o α ) .We easily get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B ρ ( o α ) dΓ( u α ) − ˆ X α ϕ α dΓ( u α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ δ α E α ( u α ) + (1 + δ α ) ε . (133)Using (132) and (133), we find α such that α ≥ α = ⇒ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B ρ ( o α ) dΓ( u α ) − ˆ B ρ ( o ) dΓ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε. (cid:3) The above argument also implies that the lower semi-continuity of the Carré duchamp under weak convergence in energy. Proposition E.5. In the setting of the previous proposition, we also have lim inf α →∞ ˆ X α ϕ α dΓ( u α ) ≥ ˆ X ϕ dΓ( u ) , and for any ρ > , lim inf α →∞ ˆ B ρ ( o α ) dΓ( u α ) ≥ ˆ B ρ ( o ) dΓ( u ) . Proof. We also only prove the result under the stronger hypothesis u α E ⇀ u .Once we have shown that when v α E → v and sup α k v α k L ∞ < ∞ , then lim α →∞ ˆ X α ϕ α dΓ( u α , v α ) = ˆ X ϕ dΓ( u, v ) , TRUCTURE OF KATO LIMITS 85 it is classical to conclude. We use the formula ˆ X α ϕ α dΓ( u α , v α ) = E α ( ϕ α u α , v α ) + E α ( ϕ α v α , u α ) − E α ( ϕ α , u α v α ) . The above arguments show that the first and the last term in the right handside are converging to the right data. For the second term, we need to show that ϕ α v α E → ϕ v . We have strong convergence in L and E α ( ϕ α v α ) = E α (cid:0) ϕ α , v α ϕ α (cid:1) + ˆ X α ϕ α dΓ( v α ) . Using Proposition E.4, we deduce that lim α ˆ X α ϕ α dΓ( v α ) = ˆ X ϕ dΓ( v ) , and as ϕ α E → ϕ and v α ϕ α E ⇀ v α ϕ , so that lim α E α (cid:0) ϕ α , v α ϕ α (cid:1) = E (cid:0) ϕ, v ϕ (cid:1) so that lim α E α ( ϕ α v α ) = E ( ϕ v ) and ϕ α v α E → ϕ v and lim α E α ( ϕ α v α , u α ) = E α ( ϕ v, u ) . (cid:3) Remark E.6. The above result can be localized, meaning that if we assume thatfunctions u α ∈ D ( B ρ ( o α ) , E α ) satisfy • u α E → u , • sup α k u α k L ∞ < ∞ , • sup α ´ B ρ ( o α ) (cid:12)(cid:12)(cid:12) dΓ( u α )d µ α (cid:12)(cid:12)(cid:12) p d µ α < ∞ for some p > ,then lim α →∞ ˆ B ρ ( o α ) dΓ( u α ) = ˆ B ρ ( o ) dΓ( u ) . Using Proposition E.1, we also get the following strong convergence result forthe energy measure density. Proposition E.7. Assume that functions u α ∈ D ( B ρ ( o α ) , E α ) satisfy • u α E → u , • sup α k u α k L ∞ < ∞ , • sup α ´ B ρ ( o α ) dΓ( ρ α ) < ∞ , where ρ α = (cid:12)(cid:12)(cid:12) dΓ( u α )d µ α (cid:12)(cid:12)(cid:12) for any α ,then (cid:12)(cid:12)(cid:12)(cid:12) dΓ( u α )d µ α (cid:12)(cid:12)(cid:12)(cid:12) L → (cid:12)(cid:12)(cid:12)(cid:12) dΓ( u )d µ (cid:12)(cid:12)(cid:12)(cid:12) . Proof. Indeed Proposition E.1 implies that up to extracting a subsequence we canassume that ρ α L → f and we want to show that f = (cid:12)(cid:12)(cid:12) dΓ( u )d µ (cid:12)(cid:12)(cid:12) b µ a.e. on B ρ ( o ) .Following the proof of Proposition E.1 (using the Sobolev inequality), we havesome p > such that sup α k ρ α k L p < ∞ , hence Remark E.6 implies that if x β ∈ B ρ ( o β ) → x ∈ B ρ ( o ) , for any r > such that r + d ( o, x ) < ρ we have lim α →∞ ˆ B r ( x α ) dΓ( u α ) = ˆ B r ( x ) dΓ( u ) . But the strong L -convergence also yields that lim α →∞ ˆ B r ( x α ) dΓ( u α ) = lim α →∞ ˆ B r ( x α ) ρ α d µ α = ˆ B r ( x ) f d µ. Hence for any x ∈ B ρ ( o ) and r > such that r + d ( o, x ) < ρ : ˆ B r ( x ) f d µ = ˆ B r ( x ) dΓ( u ) . By Lebesgue differentiation theorem (that hold true on any doubling space), thisimplies that f = dΓ( u )d µ µ -a.e. (cid:3) E.3. Convergence of harmonic functions. Proposition E.8. Let u α : B ρ ( o α ) → R such that sup α ( k u α k L ∞ + k L α u α k L ) < ∞ . Then there is a sub-sequence B ⊂ A and u : B ρ ( o ) → R in D loc ( B ρ ( o ) , E ) such that Lu ∈ L and u β L → u and L β u β L loc ⇀ Lu. Moreover, if ϕ β ∈ C c ( B ρ ( o β )) ∩ D ( E β ) and ϕ ∈ C c ( B ρ ( o )) ∩ D ( E ) are such that ϕ β C c → ϕ and ϕ β E → ϕ, then lim β →∞ ˆ X β ϕ β dΓ( u β ) = ˆ X ϕ dΓ( u ) . Remark E.9. Looking at the proof of Proposition E.2, we remark that for any ϕ ∈ C c ( B ρ ( o )) ∩ D ( E ) , we can find a sequence ϕ α ∈ C c ( B ρ ( o α )) ∩ D ( E α ) such that ϕ α C c → ϕ and ϕ α E → ϕ. Proof. We can find a sub-sequence B ⊂ A , u ∈ L ( B ρ ( o )) ∩ L ∞ ( B ρ ( o )) and f ∈∈ L ( B ρ ( o )) such that u β L ∩ L ⇀ u and f β := L β u β ⇀ f. For any r < ρ , we consider the function χ β ( x ) = ξ (cid:16) d β ( o β ,x ) − r − ρ ρ − r (cid:17) where ξ isdefined by (104), this function has the following properties: • χ β = 1 on B r ( o β ) , • χ β = 0 outside B ( ρ + r ) / ( o β ) , • χ β is / ( ρ − r ) -Lipschitz.We have the estimates ˆ B r ( o β ) dΓ( u β ) ≤ ˆ B ρ ( o β ) dΓ( χ β u β ) but ˆ B ρ ( o β ) dΓ( χ β u β ) = ˆ B ρ ( o β ) u β dΓ( χ β ) + E β ( χ β u β , u β )= ˆ B ρ ( o β ) u β dΓ( χ β ) + ˆ B ρ ( o β ) u β χ β f β d µ β ) . Using the comparison (19), we get ˆ B r ( o β ) dΓ( u β ) ≤ ρ − r ) k u β k L ∞ µ β ( B ρ ( o β )) + k u β k L k f β k L . Hence u β is locally bounded in D loc ( E α ) and u ∈ D loc ( E ) and u β E loc ⇀ u and also u β E loc ⇀ u . The formula ˆ B ρ ( o β ) ϕ β dΓ( u β ) = ˆ B ρ ( o β ) ϕ β u β f β d µ β − E β (cid:0) ϕ β , u β (cid:1) TRUCTURE OF KATO LIMITS 87 and the fact that from Proposition E.1, we have u β L loc → u implies the claimed resultof convergence. (cid:3) In case where we have a sequence of harmonic functions, this result can be slightlyimproved. Proposition E.10. Let { h α } be such that h α : B ρ ( o α ) → R is L α -harmonic forany α and sup α k h α k L ∞ < ∞ , Then there is a subsequence B ⊂ A and a harmonic function h : B ρ ( o ) → R suchthati) h β L → h ii) For each r < ρ : h β | B r ( o β ) → h | B r ( o ) uniformly.iii) For each r < ρ : lim β →∞ ´ B r ( o β ) dΓ( h β ) = ´ B r ( o ) dΓ( h ) .Proof. Proposition E.8 implies the existence of a subsequence B ⊂ A and of aharmonic function h : B ρ ( o ) → R such that we have the strong convergence in L .The uniform convergence follows from the fact that each ( X β , d β , µ β , E β ) satisfiesuniform Parabolic/Elliptic Harnack inequality and hence uniform local Hölder es-timate for harmonic function ([SC02, Lemma 2.3.2] or [GT12, Lemma 5.2]. In ourcases, there is a constant θ ∈ (0 , and C such that for any α and x, y ∈ B r ( o α ) | h α ( x ) − h α ( y ) | ≤ C (cid:18) d α ( x, y ) ρ − r (cid:19) θ k h α k L ∞ . The last point is a consequence of a uniform reverse Hölder inequality for the energydensity of harmonic function. There is some p > and some constant C such thatif B ⊂ X α is a ball of radius r ( B ) ≤ R and f : B → R is harmonic then B (cid:12)(cid:12)(cid:12)(cid:12) dΓ( f )d µ α (cid:12)(cid:12)(cid:12)(cid:12) p d µ α ! p ≤ C B dΓ( f ) . This is explained in [AC05, subsection 2.1], it relies on a self-improvement of the L -Poincaré inequality to a L − ε -Poincaré inequality [KZ08] and of the Gehringlemma [Gia83, Chapter V]. (cid:3) E.4. Approximation of harmonic functions. Let us conclude with an approximationresult for harmonic functions. Proposition E.11. Let h : B ρ ( o ) → R be a harmonic function and let r < ρ . Thenthere exists { h α } , with h α : B r ( o α ) → R harmonic for any α , such thati) h α → h | B r ( o ) uniformly on compact sets,ii) ´ B s ( o α ) dΓ( h α ) → ´ B s ( o ) dΓ( h ) for any s ≤ r .Proof. Set δ := ( ρ − r ) / and ξ ( x ) := χ (cid:18) d ( o, x ) − ( r + 2 δ ) δ (cid:19) for any x ∈ X , where χ is as in (104). Set ϕ := ξh ∈ C c ( X ) ∩ D ( E ) . Let { ϕ α } be given by Proposition E.2, i.e. ϕ α ∈ C c ( X α ) ∩ D ( E α ) for any α and ϕ α C c → ϕ , ϕ α E → ϕ. For any α , let h α be the harmonic replacement of ϕ α on B r + δ ( o α ) that is to say h α ∈ D ( E α ) is the unique solution of ( L α h α = 0 on B r + δ ( o α ) ,h α = ϕ α outside B r + δ ( o α ) , which is characterized by E α ( h α ) = inf {E α ( f ) : f ∈ D ( E α ) and f = ϕ α outside B r + δ ( o α ) } . In particular E α ( h α ) ≤ E α ( ϕ α ) for any α , hence we can find a subsequence B ⊂ A and f ∈ D ( E ) such that h β E ⇀ f . 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Carron, Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, Univer-sité de Nantes, 2, rue de la Houssinière, B.P. 92208, 44322 Nantes Cedex 3, France. Email address : [email protected] I. Mondello, Université Paris Est Créteil, Laboratoire d’Analyse et Mathéma-tiques appliqués, UMR CNRS 8050, F-94010 Creteil, France. Email address : [email protected] D. Tewodrose, Université Libre de Bruxelles, Service d’Analyse, CP 218, Boule-vard du Triomphe, B-1050 Bruxelles, Belgique. Email address ::