Weyl's law on $RCD^*(K,N)$ metric measure spaces

0.27 MB·

by  Hui-Chun Zhang

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 f˜and
\
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. ∞ ∞ ∞ ∞
∗
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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 con-
verging 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)