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WEYL’S LAW ON RCD (K,N) METRIC MEASURE SPACES ∗ HUI-CHUNZHANGANDXI-PINGZHU Abstract. Inthis paper, we will prove the Weyl’slaw for the asymptotic formula of Dirichlet eigenvalues on 7 metricmeasurespaceswithgeneralizedRiccicurvatureboundedfrombelow. 1 0 2 r p 1. Introduction A One of most fundamental theorems in spectral geometry is the Weyl’s law [14], which states that, on any 4 closedn-dimensional Riemannianmanifold(Mn,g),wehavealeadingasymptotic ] N(λ) ω G lim = n vol (Mn), λ λn/2 (2π)n g D →∞ whereλ ,1 6 j < ,aretheeigenvalues ofLaplace-Beltrami operator ∆on(Mn,g),and N(λ)isthespectral . j h ∞ counting function t a N(λ) := # λ Spec(∆), λ 6 λ , m { j ∈ j } andω isthevolumeofunitballinRn. IfΩ Mnisadomainin(Mn,g)withsmoothboundary,thenthesame [ n ⊂ asymptotic formulaholdsfortheDirichlet(orNeumann)eigenvalues, byreplacing vol (Mn)byvol (Ω). g g 3 It has a wide range of interests about the extensions of Weyl’s law, (see, for examples, [40, 41, 37] and a v survey [27]). In particular, on a weighted Riemannian manifold with Bakry-Emery Ricci curvature bounded 7 6 from below, ifthe density µ := f vol is smooth, and is bounded awayfrom 0 and ,then it wasshown by g 9 E.Milmanin[37]thattheclassica·l Weyl’slawstillholdsforweightedLaplacian∆ ∞:= ∆+ ln f, . µ 1 h∇ ∇i In this paper, we will extend this classical result to non-smooth settings. To formulate our main result, 0 . we need to introduce some notations. Let (X,d,µ) be a metric measure space (a metric space equipped a 1 Radon measure). A synthetic notion of lower Ricci bounds on (X,d,µ) was introduced in the pioneering 0 7 works of Sturm [47, 48] and Lott-Villani [35, 36]. Nowadays, many important developments were given in 1 thisfield(see[4,1,8,3,9,16,12,17,39,26,34,28]andsoon). Inparticular,toruleouttheFinslerspaces,an : v improvementnotion,RCD(K, )-condition,wasintroducedbyAmbrosio-Gigli-Savare´ in[3, 5]. Thefinitely i dimensional case, RCD(K,N)∞, was given by Gigli in [17, 18, 4.3], and a splitting theorem§for RCD(0,N)- X § spacewasprovedbyGigli[17]. TheconstantsKandN playtheroleof“Riccicurvature> Kanddimension6 r a N”. Veryrecently, Ambrosio-Gigli-Savare´ [4],Erbar-Kuwada-Sturm[16]andAmbrosio-Mondino-Savare´ [8] introduced aBakry-Emerycondition BE,whichisaweakformulation ofBochnerinequality. Theyprovedin [8,16]thatthecondition BE(K,N)isequivalent tothe(reduced) Riemannian curvature-dimension condition RCD (K,N) for constants K R and N > 1. In [11, Theorem 1.1], Cavalletti-Milman showed that the ∗ ∈ condition RCD (K,N)isequivalent tothecondition RCD(K,N)providedthetotalmeasureµ(X)< . ∗ Let (X,d,µ) be a metric measure space satisfying RCD (K,N) for some K R and N (1, )∞. For any ∗ ∈ ∈ ∞ bounded domain Ω X, according to [13, 45], the Sobolev spaces W1,p(Ω), 1 6 p 6 , are well defined. ⊂ ∞ Moreover, thespaceW1,2(Ω)isaHilbertspace([4]). TheCheegerenergyoverΩ Ch(f) = f 2dµ Z |∇ |w Ω providesaclosedquadraticformactingontheSobolevspaceW1,2(Ω),where f istheweakuppergradient 0 |∇ |w of f ([4]). The Dirichlet form (Ch,W1,2(Ω)) is associated with a self-adjoint operator ∆Ω. If diam(Ω) 6 0 1 2 HUI-CHUNZHANGANDXI-PINGZHU diam(X)/a for some a > 1, then the Rellich’s compactness theorem holds (see [23] or [10, Eq.(5.2)]), and hence the operator (Id ∆Ω) 1 is compact. Theclassical spectral theorem implies that Dirichlet spectrum is − − discrete, denoted by 0 < λΩ 6 λΩ 6 6 λΩ 6 , j N. 1 2 ··· m ··· ∈ OurmainresultinthispaperisthefollowingWeylasymptotic formulafortheseDirichleteigenvalues: Theorem 1.1. Let (X,d,µ) be a metric measure space (X,d,µ) satisfying RCD (K,N) for some K R and ∗ ∈ some N > 1. Assume that the Hausdorff dimension of X is N. Suppose that µ is absolutely continuous with respect to H N with the density e V, i.e. µ = e V H N. We also suppose the V L (X). Let Ω X be a − − · ∈ ∞loc ⊂ bounded domain of X such thatdiam(Ω) 6 diam(X)/sfor some s > 1. Then N isanintegral and there holds theasymptotic formula: N (λ) ω H N(Ω) (1.1) lim Ω = N · , λ λN/2 (2π)N →∞ where N (λ) := # λΩ : λΩ 6 λ . Ω { j j } Remark that the RHS of (1.1) dose not depend on the weight of the measure µ. Theorem 1.1 is a conse- quenceofTheorem4.6,amoregeneralresultonRCD -spaces. InthecaseofasmoothRiemannianmanifold ∗ (M,g)ofn-dimension withtheRiemannian volume µ := vol ,therelation (1.1)recovers the classical Weyl’s g law. Letuslookatthecaseofann(> 2)-dimensional Alexandrovspace(X,d)withtheHausdorffmeasureH n, and with curvature > k for some k R. It was proved [42, 51] that (X,d,H n) satisfies RCD ((n 1)k,n). ∗ ∈ − FromTheorem1.1,wehavethefollowingconsequence. Corollary 1.2. LetΩbeaboundeddomaininann-dimensional Alexandrov space(X,d,H n). Thenwehave theWeyl’slaw N (λ) ω H n(Ω) (1.2) lim Ω = n· . λ λn/2 (2π)n →∞ Another consequence is that the Weyl’s law also holds for noncollapsing limit spaces in the sense of Cheeger-Colding. More precisely, if (X,d,µ) is a Gromov-Hausdoff limit space of a sequence of pointed Riemannianmanifolds (M ,g ,p )with j j j Ric > K, dim(M ) = n, vol (B (p )) > v > 0, Mj j gj 1 j 0 thentheWeyl’slaw(1.2)stillholds. Thiscasehasbeenalready provenbyDingin[15]. Recalling that in the proof of the Weyl’s law on smooth setting, a key ingredient is that a uniformly small timeasymptoticbehaviourofheattraceH(t,x,x)viatheparametrixofheatkernels. However,theconstruction of the parametrix on smooth manifolds does not work on singular metric measure spaces. To deal with this lackoftheparametrix, weshallgetthesmalltimeasymptoticbehaviorviathe(locally)uniformconvergce of Dirichlet heat kernels along a pointed measured Gromov-Hausdorff converging sequence of metric measure spaces, asin[15,50,19]. As a byproduct, we show a local spectral convergence on RCD (K,N)-spaces, which is of independent ∗ interesting (SeeTheorem3.8). Proposition 1.3. Let pointed metric measure spaces (Xj,dj,µj,pj)j N converge to (X ,d ,µ ,p ) in the senseofpointedmeasureGromov-Hausdorff. Supposethatall(X ,d ,∈µ )satisfyRCD (K∞,N∞)for∞som∞eK R j j j ∗ and some N > 1. Let R > 0 with R (0,diam(X )/a) for some a > 2, j N. Assume that ∂B (p ∈) = j R ∂ X B (p ) .1 ∈ ∀ ∈ ∞ R ∞\ ∞ (cid:0) (cid:1) 1WeremarkthatthisassumptioncanbereplacedbyCap ∂B (p ) ∂ X B (p ) =0. 2 R ∞ \ ∞\ R ∞ (cid:0) (cid:0) (cid:1)(cid:1) WEYL’SLAWONMM-SPACES 3 For each j N, we denote by λ(R) the m th Dirichlet eigenvalues of ∆(R) on ball B (p ). Then we have ∈ m,j − j R j thatthespectralconvergence lim λ(R) = λ(R) . j m,j m, →∞ ∞ Remark1.4. Aspectralconvergence theoremforeigenvalues λ (X),differentfromthelocalDirichleteigen- m,j valuesinProposition1.3,onasequenceofconvergentcompactmetricmeasurespaces(X ,d ,µ )wasproved j j j byGiglie.t. in[19]. The following example shows that the assumption ∂B (p ) = ∂ X B (p ) is necessary in Proposition R R ∞ ∞\ ∞ 1.3. WewouldliketothankProf. S.Hondafortellingussuchanex(cid:0)ample. (cid:1) Example 1.5. Let X := [ 1 + 1,1 1] equip the Euclidean distance d and the 1-dimensional Lebesgue j − j − j E measure 1 andlet p = 0. Thenthepointed measuremeasurespaces j L pmGH X ,d , 1,p X := [ 1,1],d , 1,p := 0 . j E j E L −→ ∞ − L ∞ (cid:0) (cid:1) (cid:0) (cid:1) NowletusconsidertheballsB (p )(= X ). ItisclearthatB (p )convergetoB (p ) = ( 1,1) inthesense 1 j j 1 j 1 ofGromov-Hausdorff, as j . However,weremarkthat∂B (p )= 1,1 and∞tha(cid:0)t∂ X− B(cid:1)(p ) = ∅. 1 1 Consider the first Dirich→let∞eigenvalue λ1,j of on B1(pj). Beca∞use ∂{B−1(pj)} = ∅, we h(cid:0)av∞e\Lip0 ∞B1(cid:1)(pj) = Lip B1(pj) . So the function f = 1 is in Lip0 B1(pj) . This implies that λ1,j = 0. On the other(cid:0)hand, i(cid:1)t is obvi(cid:0)ous tha(cid:1)tthefirstDirichleteigenvalue of B (cid:0)(p )is(cid:1)λ = π2/4(witheigenfunction f(t) = cos(πt/2)). 1 1, ∞ ∞ Remark 1.6. (1)Veryrecently, inanindependent work[7]byL.Ambrosio, S.Honda andD.Tewodrose, they show that the Weyl’s law for eigenvalues λ (X) of a whole compact RCD (K,N)-space (and the Neumann j ∗ eigenvalues), different fromthelocalDirichleteigenvalues inthispaper,holdsifandonlyif rk rk lim dµ = lim dµ, r 0ZX µ(Br(x)) ZXr 0µ(Br(x)) → → wherekisthelargestinteger ksuchthatµ( ) > 0,andthe isthepiecesinthedecomposition in[38]. See k k R R alsotheconstant k inTheorem4.6. 1 (2) In another independent work [6] by L. Ambrosio and S. Honda, they get that the same local spectral convergence resultinProposition 1.3holdsifandonlyifthefollowinganalytic condition holds: W1,2(B (p )) = W1,2(B (p )). 0 R ∞ ∩ǫ>0 0 R+ǫ ∞ Organizationofthepaper. InSection2,wewillprovidesomenecessarymaterialsaboutRCD (K,N)metric ∗ measure spacesandheatkernels onmetricmeasurespaces. InSection3,wewillprovethelocally uniformly convergence of heat kernels for a sequence of converging metric measure spaces. The main result Theorem 1.1willbeprovedinthelastsection. Acknowledgements. In the previous version we overlooked the condition ∂B (p ) = ∂ X B (p ) in R R ∞ ∞\ ∞ Proposition 1.3. We appreciate Prof. S. Honda for showing us Example 1.5. We are also(cid:0)grateful to P(cid:1)rof. Ambrosio and S. Honda for sharing us their interesting manuscripts [6, 7]. We would like to thank Prof. D. G.Chen,B.B.HuaandZ.Q.Wang,andDr. X.T.Huangfortheirinteresting inthepaper. Thefirstauthoris partially supported byNSFC11571374, andthesecondauthorispartially supported byNSFC11521101. 2. Preliminaries Let (X,d) be a complete metric measure space and µ be a Radon measure on X with supp(µ) = X. Given any p X andR > 0,wedenoteby B (p)theballcentered at pwithradiusR. R ∈ 4 HUI-CHUNZHANGANDXI-PINGZHU 2.1. Sobolevspaces,infinitesimallyHilbertianspacesandlocalDirichletheatkernels. Let(X,d,µ)be ametric measure space. Severaldifferent notions ofSobolev spaces on(X,d,µ)have been givenin[13,45,2,23,22]. Theycoincide eachother onametricmeasurespaces having ameasuredoubling property andsupporting L2-Poincare´ inequality (see,forexample,[2]). A metric measure space (X,d,µ) is called to have a (local) measure doubling property if, for any R > 0, thereexistsaconstantC :=C (R)suchthat D D (2.1) µ B (x) 6 2CD µ B (x) , ball B (x)withr (0,R). 2r r r · ∀ ∈ The (X,d,µ) is called to (cid:0)support(cid:1)a (local) (cid:0)L2-Poi(cid:1)ncare´ inequality if, for any R > 0, there exists a constant C :=C (R)suchthat P P (2.2) f f¯ dµ 6C r2 (Lipf)2dµ ball B (x)withr (0,R), Z | − Br(x)| P· Z ∀ r ∈ Br(x) B2r(x) for all f Lip (B (x)), the locally Lipschitz continuous functions on B (x), where f¯ =: 1 fdµ, and ∈ loc 2r 2r B µ(B) B Lipf(x)isthepointwiseLipschitzconstant ([13])of f at x,i.e., R f(y) f(x) f(y) f(x) Lipf(x) := limsup | − | = limsup sup | − |, d(x,y) r y x r 0 d(x,y)6r → → andLipf(x) = 0if xisisolated. Inthesequel ofthis subsection, wealwaysassume that(X,d,µ)hasalocally doubling property andthatit supportsalocalL2-Poincare´ inequality. ForanopensubsetΩ X,wedenotebyLip (Ω)(andLip (Ω)),the loc 0 ⊂ set of all locally Lipschitz continuous functions on Ω (and the set of all Lipschitz continuous functions with compactsupportinΩ,respectively). Let p [1, ]andΩ X beanopensubset, andlet f Lp(Ω) Lip (Ω). TheW1,p(Ω)-norm, f ,is loc 1,p ∈ ∞ ⊂ ∈ ∩ k k givenby f := f + Lipf , 1,p p p k k k k k k here and in the sequel, wedenote f := f . TheSobolev spaces W1,p(Ω)is defined tobe the closure of p Lp k k k k alloflocally Lipschitz continuous, f, forwhich f < ,under thenorm f . Given p (1, ), itwas 1,p 1,p k k ∞ k k ∈ ∞ proved [13, 2] , for each f W1,p(Ω), that there exists a function f Lp(Ω), the so-called minimal weak ∈ |∇ | ∈ uppergradient, suchthat f = f + f . 1,p p p k k k k k|∇ |k For a locally Lipschitz function f W1,p(Ω), it was showed [13] that f = Lipf a.e. in Ω. We say that a ∈ |∇ | function f W1,p(Ω)if f W1,p(Ω )for every open subset Ω Ω.Werefer the readers to[13,45,2]for ∈ loc ∈ ′ ′ ⊂⊂ thefurthermore information oftheseSobolevspaces. For1 < p < ,letusrecallfrom[24]thattheSobolevp-capacity oftheset E X: ∞ ⊂ Cap (E):= inf f p : f W1,p(X)suchthat f > 1onaneighborhood of E . p k kW1,p(X) ∈ (cid:8) (cid:9) Ifthereisnosuchafunction f,wesetCap (E) = .Itisclear thatCap (E) = Cap (E).Moreover, wehave p ∞ p p [24]that Cap (E) = inf Cap (O) : O E, Oopen . p p ⊃ Anequivalent definitionisgivenin[45],seefor(cid:8)instance [31,Theorem3.4](cid:9)and[46]. A property holds p-q.e. (p-quasi everywhere), if it holds except of a set Z with Cap (Z) = 0. Since p Cap (Z) = Cap (Z), we may assume that the except set Z is closed. A function f : X [ , ] is called p p → −∞ ∞ p-quasicontinuousinX ifforeachǫ > 0,thereisasetF suchthatCap (F )< ǫ andtherestriction f is ǫ p ǫ |X\Fǫ continuous. Wemayalsoassumethat F isclosed. ǫ It is well-known that any W1,p-function f has a p-quasi continuous representative (see [24]). We will always use such a reprensentative in this paper. In [31, Theorem 3.2], it is proved that, for any two p-quasi continuous functions f andg,if f = gµ-a.e. inanopensetO,then f = g p-q.e. inO. WEYL’SLAWONMM-SPACES 5 Definition 2.1 ([31]). Let 1 < p < and E X, a function f on E is called to belong to the Sobolev space withzeroboundaryvalues,denoted∞by f W⊂1,p(E),ifthereexistsa p-quasicontinuousfunction f˜ W1,p(X) ∈ 0 ∈ suchthat f˜= f µ-a.e. in E and f˜= 0 p-q.e. in X E. \ If Ω X is an open sebset, it was shown [13, 2] that the space W1,p(Ω) is reflexive, for 1 < p < . ⊂ ∞ According to [31, Remark 5.10] (see also [46, Theorem 4.8]), the space W1,p(Ω) = H1,p(Ω), which is the 0 0 closureof Lip (Ω)undertheW1,p(Ω)-norm. 0 Given any open set Ω X, it is clear that W1,p(Ω) W1,p(Ω). However, generally speaking, W1,p(Ω) , ⊂ 0 ⊂ 0 0 W1,p(Ω). 0 Lemma2.2. LetO X beanopen setand let1 < p < . Suppose that f isa p-quasi continuous in X and ⊂ ∞ that f = 0 p-q.e. inO. Thenwehavethat f = 0 p-q.e. inO. Proof. Fromthedefinition,weknowthat f = 0 p-q.e. inO.Soitsufficestoshowthat f = 0 p-q.e. in∂O.We canassumeCap (∂O)> 0. Otherwise,itisnothing todo. p Arguebycontradiction, suppose thatthereisasubset A ∂Osuch thatCap (A) > 0andthat f(x) , 0for ⊂ p any x A. ∈ Taken arbitrarily ǫ (0,Cap A/2), since f is a p-quasi continuous in X, we can find a closed set F with ∈ p ǫ Cap (F ) < ǫ andtherestriction f iscontinuous. Notingthat f = 0 p-q.e. inO,i.e.,thereexistsaclosed setZpwiǫthCap (Z)= 0suchthat f|X=\F0ǫ onO Z. Wehavethattherestriction f 0. By Cap (Fp Z) < ǫ < Cap A/2, we k\now that A (F Z) , ∅. Let |xO\(Fǫ∪AZ)(≡F Z). There exists p ǫ ∪ p \ ǫ ∪ 0 ∈ \ ǫ ∪ a sequence x O with lim x = x , since x ∂O. Noting that F Z is closed, we have that x < F Z{ fjo}r∞j=a1ll⊂sufficiently lajr→ge∞ j.j By c0ombining0th∈e facts that f ǫ ∪is continuous at x and that f(jx ) =ǫ0∪for all large j(since x O (F Z)for all large j), weconc|lXu\d(Feǫ∪thZa)t f(x ) = 0. This c0ontradicts j j ǫ 0 ∈ \ ∪ with x A,andhencewefinishtheproof. (cid:3) 0 ∈ Corollary 2.3. LetΩ X beanopensetandlet1 < p < . If∂Ω = ∂(X Ω),thenW1,p(Ω)= W1,p(Ω). ⊂ ∞ \ 0 0 Proof. Itsuffices toshow W1,p(Ω) W1,p(Ω). Given any f W1,p(Ω), there exists afunction f˜isa p-quasi 0 ⊃ 0 ∈ 0 continuous in X such that f˜= f µ-a.e. in Ωand that f˜= 0 p-q.e. in X Ω. Byapplying Lemma2.2 to fãnd \ O := X Ω,wehavethat f˜= 0 p-q.e. inX Ω.Bytheassumption ∂Ω= ∂(X Ω),wehave \ \ \ X Ω = (X Ω) ∂(X Ω)= (X Ω) ∂Ω = Ω ∂Ω= Ω. \ \ ∪ \ \ ∪ ∪ Therefore, we get that f˜ = 0 p-q.e. in Ω. Noting that f˜ = f µ-a.e. in Ω Ω, we have f W1,2(Ω), by ⊂ ∈ 0 Definition2.1. Theproofisfinished. (cid:3) Remark2.4. (1)Infact,inCorollary 2.3,weonlyneedtoassumethat Cap ∂Ω ∂(X Ω) = 0. p \ \ (cid:0) (cid:1) (2)ThespaceW1,p(Ω)isequivalent tothespace Hˆ1,p(Ω)givenin[6]byAmbrosio-Honda. 0 0 Ingeneral, thespaceW1,2(Ω)isnotaHilbertspace. Definition 2.5 ([4, 18]). A metric measure space (X,d,µ) is called infinitesimally Hilbertian if the Sobolev spaceW1,2(X)isaHilbertspace. Equivalently, theassociated Cheegerenergyover X,Ch(f) := f 2dµis X|∇ | quadratic. R Let (X,d,µ) be an infinitesimally Hilbertian and proper metric measure space. Given any bounded open set Ω X and p (1, ), according to [18, 4.3], the space W1,2(Ω) is still a Hilbert space, and for any ⊂ ∈ ∞ § f,g W1,2(Ω),theinnerproduct f, g iswelldefinedinL1 (Ω). ∈ loc h∇ ∇ i loc 6 HUI-CHUNZHANGANDXI-PINGZHU In the sequel of the paper, for an infinitesimally Hilbertian and proper metric measure space (X,d,µ), we willalwaysdenotethat H1(Ω):= W1,2(Ω), H1(Ω):= W1,2(Ω) and H1 (Ω):= W1,2(Ω). 0 0 loc loc Fixanybounded domainΩ X,weconsider thecanonical Dirichletform(E ,H1(Ω)),where ⊂ Ω 0 (2.3) E (f) := f 2dµ, f H1(Ω). Ω Z |∇ | ∈ 0 Ω It is well-known (see for example [49]) that the canonical Dirichlet form is strongly local and regular, the associated infinitesimal generator, denoted by ∆ with domain D(∆ ), is a non-positive definite self-adjoint Ω Ω operator, andtheassociated analytic semi-groupisgivenby H f(x) = HΩ(t,x,y)f(y)dµ, t > 0 t Z Ω for any f L2(Ω), where HΩ(t,x,y) isthe (local) Dirichlet heat kernel (the fundamental solution ofthe heat ∈ equation withDirichletboundary value). Ifdiam(Ω) 6 diam(X)/sforsomes> 1,acompactembeddingofH1(Ω)intoL2(Ω)wasprovedin[23](see 0 also [10, Eq.(5.2)]), and hence that the operator (Id ∆ ) 1 is compact. Thus the spectral theorem implies Ω − − thatspectrum isdiscrete(see,forexample[14]). Wedenoteby 0 < λΩ 6 λΩ 6 6 λΩ 6 , j N, 1 2 ··· m ··· ∈ the(Dirichlet)eigenvalues of∆ . ForeachλΩ,theassociated eigenfunction isφΩ,i.e., Ω m m (2.4) ∆ φΩ = λΩφΩ. Ω m − 1 m WcoemnpolermteablaizseistohfemL2s(Ωo)t,haantdkφthΩmakt2th=e1lofcoarleDaicrhichmle∈thNea.tIkteirsnweleclla-nknboewwnritthteant tahse sequence {φm}m∈N forms a (2.5) HΩ(t,x,y) = e−λΩmφΩm(x)φΩm(y), (x,y,t) Ω Ω (0, ). X ∀ ∈ × × ∞ m>1 Theweak maximum principle implies the monotonicity of Dirichlet heat kernels withrespect to domains. Namely,giventwodomainsΩ Ω X,wehave ′ ⊂ ⊂ HΩ(t,x,y) 6 HΩ′(t,x,y), (x,y,t) Ω Ω (0, ). ∀ ∈ × × ∞ TheexistenceandGaussianboundsoftheglobalheatkernelhasbeenestablished in[49]onametricmeasure space having a measure doubling property and supporting a L2-Poincare´ inequality. Thus for a sequence of balls {BRj(x0)} with Rj ր ∞, the heat kernels HBRj(x0)(x,y,t) converge to a global heat kernel H(x,y,t) on X X (0, ),asR . j × × ∞ ր ∞ LetusrecallthedefinitionofthedistributionalLapalacian. Givenafunction f H1 (Ω),thedistributional LaplacianL f isdefinedasafunctional ∈ loc (2.6) L f(φ) := f, φ dµ, φ H1(Ω) L (Ω). −Z h∇ ∇ i ∀ ∈ 0 ∩ ∞ Ω If f H1(Ω),thenL f can beextended toafunctional on H1(Ω).Itisclear thatif f D(∆ )and∆ f = g, ∈ 0 ∈ Ω Ω thenL f = g µinthesenseofdistributions. Conversely, itwasproved[18]thatany f H1(Ω),ifthereisa · ∈ 0 function g L2(Ω)suchthatL f = g µinthesenseofdistributions, then f D(∆ )and∆ f = g. Ω Ω ∈ · ∈ WEYL’SLAWONMM-SPACES 7 2.2. Riemanniancurvature-dimension conditionsRCD*(K,N). Let(X,d,µ)beametric measure space. Wedenote by P (X,d) the L2-Wasserstein space over (X,d), i.e., 2 thesetofallBorelprobability measuresνwith d2(x ,x)dν(x) < Z 0 ∞ X forsome(henceforall) x X. Givenν ,ν P (X,d),their L2-Wasserstein distance isdefinedby 0 1 2 2 ∈ ∈ W2(ν ,ν ) := inf d2(x,y)dq(x,y) 2 0 1 Z X X × wheretheinfimumistakenoverallcouplings qofν andν ,i.e.,Borelprobability measuresqonX X with 1 2 × marginals ν andν .Suchacoupling qrealizing the L2-Wasserstein distance iscalled anoptimalcoupling of 0 1 ν andν .Givenameasureν P (X,d),itsrelativeentropy isdefinedby 0 1 2 ∈ Ent(ν) := ρlnρdµ, Z X if ν = ρ µ is absolutely continuous w.r.t. µ and (ρlnρ) is integrable. Otherwise we set Ent(ν) = + . Let + P (X,d,·µ) P (X,d)bethesubsetofallmeasuresνsuchthatEnt(ν)isfinite. ∞ 2∗ ⊂ 2 Wesetthefunction sin(√k·tθ), 0 < kθ2 < π2, sin(√kθ) σ(kt)(θ) := tss,iinnhh((√√−·kk·tθθ)), kkθθ22 =< 00,, SeveralequivalentdefinitionsforRiemanni∞an,cur−va·ture-dimkθe2ns>ioπn2c.onditionwereintroducedin[16,8,17, 18]. Inthispaper,weadaptthefollowingnotionsfortheconvenience. Definition 2.6 ([16]). Let K R and N [1, ). A metric measure space (X,d,µ) is called to satisfy the entropy curvature-dimension ∈condition CD∈e(K,∞N)ifanyonly iffor eachpair ν ,ν P (X,d,µ)there exist aconstant speedgeodesic (νt)06t61 inP2∗(X,d,µ)connecting ν0 toν1 suchthat0for1a∈llt ∈2∗[0,1]: (2.7) UN(νt) > σ(K1/−Nt) W2(ν0,ν1) ·UN(ν0)+σ(Kt)/N W2(ν0,ν1) ·UN(ν1), (cid:0) (cid:1) (cid:0) (cid:1) whereU (ν):= exp 1Ent(ν) . MoreoNver, (X,d,µ(cid:0))−isNsaid to(cid:1)satisfy Riemannian curvature-dimension condition RCD∗(K,N), for K R ∈ and N [1, ),ifitisinfinitesimally Hilbertianandsatisfies theCDe(K,N)condition. ∈ ∞ Let N > 1, the generalized Bishop-Gromov inequality for RCD (K,N) space (by a combination of [21, ∗ Corollaryof1.5]and[48,Remark5.3])statesthatforany p X andany0 < r < R, ∈ µ B (p) µ B (p) (2.8) R 6 r , 0R(cid:0)sNK−1(t)(cid:1)dt 0rs(cid:0)NK−1(t)(cid:1)dt wherethefunctions (t)isgivenby R N−1 R N−1 k sin(√k·τ) if k > 0,  √k (2.9It)was proved [48, 16] that a metricskm(τe)as=ureτsisnpha(√√c−e−kk(·τX),d,iiffµ) wkki=<th00R,.CD (K,N) for K R and N [1, ) ∗ ∈ ∈ ∞ satisfiesthefollowingproperties: (X,d)isalocally compactlengthspace, i.e.,forany p,q X,thereisashortestcurvejoinedthem. • ∈ 8 HUI-CHUNZHANGANDXI-PINGZHU (X,d,µ)hasalocalmeasuredoublingproperty oneachball B (x) X. Moreover,by(2.8),wehave,forall R • ⊂ 0 < r < R,that (2.10) µ BR(p) 6 R N exp (N 1)K 0 R :=C R N. µ(cid:0)B (p)(cid:1) r · − | ∧ |· N,K,R· r r (cid:16) (cid:17) (cid:0)p (cid:1) (cid:16) (cid:17) (cid:0) (cid:1) (X,d,µ) supports a local L2-Poincare´ inequality on each ball B (x) X. Moreover, the Poincare´ constant R • ⊂ C (R)depends onlyon N and √K 0R. P | ∧ | SupposeK 6 0. ThereexistsapositiveconstantC ,dependingonlyonN andK,suchthattheheatkernel N,K • H(x,y,t)onX satisfies,(see[29,Theorem1.2]) C d2(x,y) (2.11) H(x,y,t) 6 N,K exp +C t . N,K µ B (x) − 5t · √t (cid:16) (cid:17) (cid:0) (cid:1) 2.3. PointedmeasuredGromov-Hausdorffconvergence. Inthispaper,wewillpayusattentionstotheRCD (K,N)-spaces. Itisconvenienttoconsideronlythecase ∗ whereallmetricmeasurespacesarelocallycompactlengthspace. Definition 2.7. Let (X ,d ,µ ), for j N , be a sequence of compact metric measure spaces. It is said j j j ∈ ∪{∞} that (Xj,dj,µj)j N converge to (X ,d ,µ ) in the sense of measured Gromov-Hausdorff topology, denoted ∈ ∞ ∞ ∞ by mGH (X ,d ,µ ) (X ,d ,µ ), j j j −→ ∞ ∞ ∞ ifthereexistasequence ofǫ -GH-approximations Φ : X X withǫ 0suchthat,foreach j N, j j j j (1) forall x,y X , d Φ (x),Φ (y) d (x,y) 6 ǫ ,→ ∞ → ∈ j j j j j (2) forany x ∈ X ,|th∞e(cid:0)reexistsan x(cid:1)j− Xj such| thatd x ,Φj(xj) 6 ǫj, ∞ ∈ ∞ ∈ ∞ ∞ (3) lim g Φ dµ = gµ foranyg C (X ).(cid:0) (cid:1) j→∞RX∞ ◦ j j RX∞ ∞ ∈ 0 ∞ Definition 2.8. Let(X ,d ,µ ,p ),for j N ,beasequence ofpointed metricmeasure spaces. Assume j j j j ∈ ∪{∞} that they arelocally compact length spaces. Wesay (X ,d ,µ ,p )converge to(X ,d ,µ ,p )inthe sense j j j j of pointedmeasuredGromov-Hausdorff topology, denoted by ∞ ∞ ∞ ∞ pmGH (X ,d ,µ ,p ) (X ,d ,µ ,p ), j j j j −→ ∞ ∞ ∞ ∞ if for each R > 0, the pointed metric measure spaces (B (p ),d ,µ )converge to the pointed metric measure R j j j space (B (p ),d ,µ ) in the sense of measured Gromov-Hausdorff topology with a sequence of ǫ -GH- R j,R approximatio∞ns Φ∞ s∞uchthatΦ (p )= p . j,R j,R j ∞ We also consider the uniform-convergence of functions on a pointed measured Gromov-Hausdorff con- vergeing sequence ofpointed metricmeasurespaces. pmGH Definition2.9. Letthepointedmetricmeasurespaces(Xj,dj,µj,pj)j N satisfy(Xj,dj,µj,pj) (X ,d ,µ ,p ). ∈ ∪{∞} −→ ∞ ∞ ∞ ∞ GH Let K X beaconverging subsets K K withasequence ofǫ -GH-approximations Φ ,whereǫ 0 j j j j j j as j→⊂∞.Supposethat{fj}j∈N∪{∞} arefu−n→ction∞sonKj andthat f∞ iscontinuousonK∞. If fj◦Φ−j1 conve→rge to f uniformlyon K ,thenitissaidthat f f uniformlyover K . j j ∞ ∞ → ∞ We remark that the Arzela-Ascoli theorem can be generalized to the case where the functions live on differentspaces(see,forexample,[35]orProposition 2.12in[38]). Lemma2.10(Lowersemi-continuity oftheenergy,Proposition2.13in[38]). Letthepointedmetricmeasure pmGH spaces (Xj,dj,µj,pj)j N satisfy that (Xj,dj,µj,pj) (X ,d ,µ ,p ), and let R > 0. Assume that theyareRCD (K,N)s∈pa∪c{e∞s}forsome K Randsome N−>→1. ∞ ∞ ∞ ∞ ∗ ∈ WEYL’SLAWONMM-SPACES 9 If f is asequence of locally Lipschitz functions on B (p ), for each j N , and f f uniformly j R j j over B (p ),thenwehave ∈ ∪{∞} → ∞ R j liminf f 2dµ > f 2dµ , r (0,R). j j j→∞ ZBr(pj)|∇ | ZBr(p )|∇ ∞| ∞ ∀ ∈ ∞ Itisclearthatif,inaddedthat f L2(B (p )),thenwehaveliminf f 2dµ > f 2dµ . |∇ ∞|∈ R ∞ j→∞RBR(pj)|∇ j| j RBR(p∞)|∇ ∞| ∞ Lemma2.11([13,Lemma10.7]). LetR > 0andlet(X ,d ,µ ,p ) betwopointedmetricmeasurespaces j j j j j=1,2 with RCD (K,N) for some K R and some N > 1. Assume that d B (p ),B (p ) < ε, for some ε > 0, ∗ GH R 1 R 2 ∈ withanε-GH-approximation Φ: BR(p1) BR(p2). (cid:0) (cid:1) → If f isaLipschitzfunctionon B (p )with f 6 L,thenthereexistsaLipschitzfunction f on 1 R 1 k|∇ 1|kL∞(BR(p1)) 2 B (p )suchthat R 2 f f Φ 1 6 κ(ε), k 1− 2◦ − kL∞(BR(p1)) f 6 L+κ(ε), k|∇ 2|kL∞(BR(p2)) f 2dµ 6 f 2dµ +κ(ε), Z |∇ 2| 2 Z |∇ 1| 1 BR(p2) BR(p1) whereκ(ε) := κ (ε)isapositivefunction, depending on N,K,Rand L,withlim κ(ε) = 0. N,K,R,L ε 0 → 3. TheconvergeforDirichletheatkernels In this section, we will discuss the convergence of the local Dirichlet heat kernels on a sequence of converging pointedmetricmeasurespaces. 3.1. ConvergenceoffunctionslivingonpmGH-convergingspaces. Wefixasequence of pointed metric measure spaces (Xj,dj,µj,pj)j N in this subsection. Assumethat theysatisfyRCD (K,N)forsomeK Randsome N > 1,andthat ∈ ∪{∞} ∗ ∈ pmGH (X ,d ,µ ,p ) (X ,d ,µ ,p ). j j j j −→ ∞ ∞ ∞ ∞ LetusfirstintroducethenotionsofL2-convergenceandH1-convergenceforfunctionslivingonvaryingspaces X . Wewilladaptanintrinsicpointofviewforthedefinitions,similarasin[15,32,25]. Wereferalsoreaders j to[19,5]forsomesimilarconcepts ofconvergence viaanextrinsic pointofview. Definition3.1. LetR > 0. (1) Suppose that f L2(B (p )) for each j N . We say that f f in L2(B (p )) if for any j j R j j R j { } ∈ ∈ ∪{∞} → ∞ GH ε > 0, there is a sequence of subsets K B (p ), with K K B (p ), such that f f uniformly j R j j R j ⊂ → ∞ ⊂ ∞ → ∞ over K and j (3.1) f 2dµ 6 ε, j N . j j Z | | ∀ ∈ ∪{∞} BR(pj) Kj \ (2) Suppose that f H1(B (p )) := W1,2(B (p )) for each j N . We say that f f in j j R j R j j H1(B (p ))ifithold{s f} ∈ f in L2(B ((cid:0)p ))and (cid:1) ∈ ∪{∞} → ∞ R j j R j → ∞ (3.2) lim f 2dµ = f 2dµ . j→∞ZBR(pj)|∇ j| j ZBR(p )|∇ ∞| ∞ ∞ Itisnot hardtoseethat if f f in L2(B (p ))intheabove definition 3.1(i), then their zeroextensions j R j f˜ (that is, f˜ = f in B (p ) a→nd f∞˜ = f in X B (p )) strongly L2-converge to f˜ in the sense of [19] j j j R j j j j R j (see also [5]). Indeed, by using that the weak co\mpactness of f˜ in L2(X ) (see, p∞age 1115 on [19]), we j j { } get that f˜ weakly L2-converge to f˜ in the sense of [19]. From the definition 3.1 (i), we have also that j f˜ f˜ . ∞ k jkL2(Xj) →k ∞kL2(X∞) 10 HUI-CHUNZHANGANDXI-PINGZHU Letussumupsomebasisproperties ontheseconvergences. Proposition 3.2. LetR> 0. (i) Assumethat∂B (p )= ∂ X B (p ) . Iffunctionsg H1(B (p ))with g 6CforsomeC > 0, forall j N,andifgR ∞g in(cid:0)L2∞(B\ (Rp )∞),t(cid:1)henwehaveg j ∈ H01(BR(pj )). k|∇ j|k2 (ii) If∈g H1(B j(p→))∞and f RLipj(B (p ))withg ∞ ∈f 0 HR1(B∞(p )),then, given any sequence of Lipschitz fu∞nc∈tions fR ∞Lip(B (p∞))∈, j N,Rsuc∞h that f ∞ −f ∞un∈ifor0mlyRov∞er B (p ), there exists functions j R j j R j g H1(B (p ))sucht∈hatg g in H∈1(B (p ))andg →f ∞ H1(B (p ))forall j N. j I∈n partiRculajr, if g H1j(→B (p∞ )), thenRtherje exists jfu−nctjio∈ns 0g R Hj1(B (p )), f∈or all j N such that g g in H1(B (p∞))∈. 0 R ∞ j ∈ 0 R j ∈ j R j → ∞ Proof. (i). Fromthedenseness ofthe Lip (B (p ) H1(B (p )),wecanassumethatg Lip (B (p ))for each j N. 0 R j ⊂ 0 R j j ∈ 0 R j Letg˜∈ bethezeroextensionofg inX foreach j N. Namely,g˜ = g inB (p )andg˜ = 0inX B (p ). j j j j j R j j j R j ∈ \ Noticingthatg˜ weakly L2-converge tog˜ inthesenseof[19](seealso[5])andthat j ∞ g˜ = g 6C, k|∇ j|kL2(Xj) k|∇ j|kL2(BR(pj)) weobtainthatg˜ H1(X )andthatg˜ = g µ -a.e. in B (p ),andthatg˜ = 0µ -a.e. in X B (p ). R R Now we wan∞t t∈o show∞g H1(B∞ (p ∞)). ∞Noting that g˜∞ is a 2-qua∞si conti∞nuous func\tion a∞nd that ∞ ∈ 0 R ∞ ∞ X B (p ) is an open set, we conclude, by [31, Theorem 3.2], that g˜ = 0 2-q.e. in X B (p ). By R R us∞in\g the∞fact that g˜ = g µ -a.e. in B (p ) and that µ (∂B (p )) =∞ 0, we have g˜ =∞g\ µ ∞-a.e. in R R B (p ). Hence,byD∞efiniti∞on2∞.1,wehaveg ∞ H1(B (p )∞). ∞ ∞ ∞ ∞ R ∞ ∞ ∈ 0 R ∞ Atlast,byusingtheassumption∂B (p ) = ∂ X B (p ) andCorollary2.3,weconcludeg H1(B (p )). (ii). FromthedensenessoftheLipR (B∞(p )(cid:0) ∞H\1(RB (∞p (cid:1))),wecanassumethatg f ∞Li∈p (B0 (pR ))∞, 0 R ∞ ⊂ 0 R ∞ ∞− ∞ ∈ 0 R ∞ and hence g Lip(B (p )). We can use Lemma 2.11 to lift a sequence of functions gˆ Lip(B (p )) so R j R j ∞ ∈ ∞ ∈ that: (3.3) gˆ g Φ 6 κ(ε ), j j L j k − ∞◦ k ∞ (3.4) gˆ 6 g +κ(ε )6C, j L L j k|∇ |k ∞ k|∇ ∞|k ∞ (3.5) gˆ 6 g +κ(ε ), j 2 2 j k|∇ |k k|∇ ∞|k whereκ(ε )dependsonK,N,Rand g . By(3.3)andthefactsthat f f uniformlyoverB (p )and j L j R j k|∇ ∞|k ∞ → ∞ thatg f Lip (B (p )),wecanget 0 R ∞ − ∞ ∈ ∞ gˆ (x) f (x) 6 κ (ε ) forall x isclosenear ∂B (p ), j j 1 j R j | − | foreach j N,andforsomepositivefunctionκ withlim κ (t)= 0. Weshallmodifygˆ slightly to 1 t 0 1 j ∈ → gˆ κ (ε ) if gˆ f > κ (ε ), j 1 j j j 1 j − − (3.6) gj := gˆfj +κ (ε ) iiff |gˆgˆj −ffj|66 κκ1((εεj)),. j 1 j j j 1 j Thenwehave,foreach j N,thatgj fj Lip0(BR(pj))an−dthat,−by(3.4)and(3.5), ∈ − ∈ (3.7) g 6C and limsup g 6 g . j L j 2 2 k|∇ |k ∞ j k|∇ |k k|∇ ∞|k →∞ From(3.6),wehave g gˆ 6 κ (ε ). j j L 1 j k − k ∞ The combination of this and (3.3) implies that g g uniformly over B (p ). By using the lower semi- j R j continuity of energy, Lemma 2.10, wehave liminf→ ∞ g > g . Thus we finish the proof of (ii). j j 2 2 Theproofiscompleted. →∞k|∇ |k k|∇ ∞|k (cid:3)