Sharp gradient estimate for heat kernels on $RCD^*(K,N)$ metric measure spaces

0.17 MB·

by  Jia-Cheng Huang

SHARP GRADIENT ESTIMATE FOR HEAT KERNELS ON RCD (K,N) METRIC
∗
MEASURE SPACES
JIA-CHENGHUANGANDHUI-CHUNZHANG
7 Abstract. Inthispaper,wewillestablishanellipticlocalLi-Yaugradientestimateforweaksolutionsof
1 theheatequationonmetricmeasurespaceswithgeneralizedRiccicurvatureboundedfrombelow. Oneof
0 itsmainapplicationsisasharpgradientestimateforthelogarithmofheatkernels.Theseresultsseemsnew
2 evenforsmoothRiemannianmanifolds.

n
a
J

0 1. Introduction
1
TheLi-YaugradientestimateforevolutionequationsiscertainlycentralingeometricanalysisonRie-

] mannianmanifolds. Oneofthefundamentalresultsisthefollowinggradientestimatesforheatequations.
G
D Theorem 1 (Li-Yau [26]). Let Mn be an n-dimensional complete non-compact Riemannian manifold
. withRic(Mn) > k,k > 0. Let B beageodesic ball withradius R. Ifuisasmooth positive solution of
h − R
t theheatequation∆u = ∂tuon BR (0,T),0 < T 6 ,then
a × ∞
m C α2 α2 nα2k nα2
(1.1) sup f 2 α ∂ f (x,t) 6 n· + √kR + +
[ |∇ | − · t R2 α2 1 2(α 1) 2t
x∈BR/2(cid:16) (cid:17) (cid:16) − (cid:17) −
2 whereα > 1, f := lnu,andC isaconstant depending onlyonn.
n
v
3 In another direction, Hamilton established an elliptic gradient estimate for bounded solutions of the
0
heatequationoncompactmanifolds.
8
1 Theorem2(Harmilton[19]). LetMnbeann-dimensionalcompactRiemannianmanifoldwithoutbound-
0
. aryandwithRic(Mn) > k,k > 0. LetubeasmoothpositivesolutionoftheheatequationonMn (0, ).
1 − × ∞
Suppose thatu 6 M on Mn (0, ). Then,bysetting f := lnu,wehave
0
× ∞
7 1 M
1 (1.2) f 2 6 +2k ln .
|∇ | t u
: (cid:16) (cid:17)
v
In[39],SoupletandZhangprovedalocalizedellipticgradientforheatequationonnoncompactman-
i
X ifolds.
r
a Theorem3(Souplet-Zhang[39]). Let(Mn,g)beann-dimensional completeRiemannianmanifoldwith
Ric(Mn) > kwithk > 0. AssumethatuisasmoothpositivesolutionoftheheatequationonB (0,T).
R
− ×
Suppose alsothatu6 M on B (0,T). Thenwehave
R
×
1 1 M 2
(1.3) f 2 6C + +k 1+ln , (x,t) B (T/2,T),
|∇ | n· R2 T u ∈ R/2×
(cid:16) (cid:17)(cid:16) (cid:17)
where f := lnuandtheconstantC depends onlyonthedimensionn.
n
Thereisarich literature onextensions andimprovements ofthese Li-Yau’sgradient estimates. Here,
werefersomerecentniceworksandsurveysonthistopic,[24,36,7,25,9,28,34,27]andsoon.
InthepioneeringworksofStrum[42,43]andLott-Villani[30,31],ansyntheticnotionoflowerRicci
bounds on metric spaces has been introduced. Up to now, many improvements were given along this
direction(see,forexample,[4,1,5,3,6,12,14]andsoon). Inparticular, asatisfactorynotion,so-called
Riemanniancurvature-dimension condition(denotebyRCD (K,N)),wasgivenin[12,5]. Theconstants
∗
K and N play the role of “Ricci curvature > K and dimension 6 N”. Let (X,d,µ) be a metric measure
1
2 JIA-CHENGHUANGANDHUI-CHUNZHANG
space(ametricspaceequipped aRadonmeasure)satisfying RCD (K,N)isageneralized notionfor“an
∗
Riemannian manifoldwithRicci > K anddim 6 N”.
Let(X,d,µ)beametricmeasurespacesatifying RCD (K,N),forsome K Rand N [1, ). Given
∗
∈ ∈ ∞
any domain Ω X, according to [10, 38], the Sobolev spaces W1,p(Ω), 1 6 p 6 , are well defined.
⊂ ∞
Moreover, the space W1,2(Ω)isaHilbert space [4]. Then, theweak solutions ofthe heat equation onΩ
are well defined. That is, given an interval I R, a function u(x,t) W1,2(Ω I) is called a (locally)
⊂ ∈ loc ×
weaksolution fortheheatequation onΩ I ifitsatisfies
×
∂u
u, φ dµdt = φdµdt
−Z Z h∇ ∇ i Z Z ∂t ·
I Ω I Ω
forallLipschitz functions φwithcompactsupport inΩ I,where u, φ dµistheinnerproduct of
× Ωh∇ ∇ i
W1,2(Ω). The local boundedness and the Harnack inequality for anRy such locally weak solutions of the
heatequationhavebeenestablished in[44,45,32].
In the case when Ω = X and I = [0, ), the heat flow (Htf)t>0 with initial data f L2(X) provides
∞ ∈
a globally weak solution of the heat equation on X. By an abstract Γ -calculus for (H f), some global
2 t
versionsofLi-Yautypegradientestimatesfor(H f)havebeenobtained(see[37,13,22,23]). However,
t
the locally weak solutions u(x,t) do not form a semi-group in general, and hence the method of Γ -
2
calculus doesnotworkingeneral.
Ourmain result inthis paper is the following local gradient estimate on RCD (K,N) metric measure
∗
spaces. ThisisnewevenforsmoothRiemannianmanifolds!
Theorem1.1. GivenK > 0andN (1, ),let(X,d,µ)beametricmeasurespacesatisfyingRCD ( K,N).
∗
LetT (0, )andletB Xbea∈geode∞sicballofradiusR. Assumethatu(x,t) W1,2 B isalo−cally
∈ ∞ R ⊂ ∈ loc R,T
weak solution of the heat equation on BR,T := BR (0,T). Suppose also that there exi(cid:0)st M,(cid:1)m > 0 such
×
thatm 6 u6 M on B . Thenwehavethelocalgradientestimate:
R,T
ln(M/m) 1 M
(1.4) f 2(x,t)6C + +K ln
|∇ | N · R2 T · u(x,t)
(cid:16) (cid:17)
foralmostevery(x,t) B (T/2,T),where f = lnuandtheconstantC depends onlyon N.
R/2 N
∈ ×
ThesecondauthorandZhuin[47]haveextended(1.1)togeneralRCD (K,N)metricmeasurespaces.
∗
Comparing with the argument in [47], there is a new technical difficulty in the proof of Theorem 1.1.
Indeed, recalling the proof of the elliptic Li-Yau gradient estimates (1.2) and (1.3) in the smooth case,
weneedasimplealgebrainequality: foranyC2-functions f,φon(Mn,g),itholds
(1.5) f +φ g 2 > 1 trace f +φ g 2 = 1 ∆f +nφ 2.
ij ij ij ij
· n · n
(cid:12) (cid:12) h (cid:16) (cid:17)i (cid:16) (cid:17)
(cid:12) (cid:12)
We have not an appropri(cid:12)ate analogo(cid:12)us of (1.5) on general RCD (K,N) metric measure spaces. In this
∗
paper, wefindthatthis lack of(1.5)can becompensated byanimprovement ofBochner inequality, see
alsoRemark2.7.
Oneofthemainapplication ofTheorem1.1isthefollowingsharpgradientestimateforthelogarithm
oftheheatkernel.
Theorem1.2. GivenN (1, )andK > 0,let(X,d,µ)beametricmeasurespacesatisfyingRCD ( K,N).
∗
∈ ∞ −
Let H(x,y,t) be the heat kernel on X. Then there exists a constant C , depending only on N and K,
N,K
suchthat,foralmostevery(x,y,t) X X (0, ),wehave
∈ × × ∞
1 d2(x,y)
lnH(x,y,t)2 6C +K 1+ +t .
N,K
|∇ | t · t
(cid:16) (cid:17) (cid:16) (cid:17)
Moreover, inthecasewhereK = 0,thereexistsaconstantC ,depending onlyon N,suchthat
N
C d2(x,y)
lnH(x,y,t)2 6 N 1+ .
|∇ | t · t
(cid:16) (cid:17)
SHARPGRADIENTESTIMATEFORHEATKERNEL... 3
Wegivetheresultinthecaseofsmoothmanifolds, whichmaybeofindependent interest.
Corollary 1.3. Let (Mn,g) be an n-dimensional complete Riemannian manifold with Ric(Mn) > k,
−
k > 0. LetH(x,y,t)betheheatkernelon Mn. Then,wehave
1 d2(x,y)
lnH(x,y,t)2 6 c +k 1+ +t .
n,k
|∇ | t · t
(cid:16) (cid:17) (cid:16) (cid:17)
forevery(x,y,t) Mn Mn (0, ),wherec isaconstant depends onlyonnandk.
n,k
∈ × × ∞
Moreover, inthecasewhereRic(Mn) > 0,wehave,forsomeconstant c ,depending onlyonn,that
n
c d2(x,y)
lnH(x,y,t)2 6 n 1+ .
|∇ | t · t
(cid:16) (cid:17)
Remark1.4. (1) Thisresultissharp,sinceonehas,onRn,that
1 x y2
lnH(x,y,t)2(x,t) = | − | .
|∇ | t · 4t
(2) In the case of compact manifolds, this result has been proved in [20, 40, 11] via Malliavin’s cal-
culus. Inthe caseofnon-compact manifolds, thisresult improves theprevious estimates ofSouplet and
Zhang in [39], by using their elliptic Li-Yau gradient estimate Theorem3. Very recently, under to add
anassumption thatthetimeisbounded, asimalarresult hasbeenobtained onnon-compact Riemannian
manifolds withappropriate Bakry-Emeryconditions byLi[29].
ThesecondapplicationofTheorem1.1istheLipschitzregularityoflocallyweaksolutionsoftheheat
equation on RCD (K,N) metric measure spaces. Let u be a locally weak solution of the heat equation
∗
on B := B (0,T). Recallingthatthelocalboundedness andtheHarnackinequality foruhavebeen
R,T R
×
established in [44, 45, 32]. In particular, u(, ) must be locally Ho¨lder continuity in B . On the other
R,T
· ·
hand, inthecasewhereu(x,t) = H f isaglobalheat flowon X,theLipschitz continuity ofu(,t)in B ,
t R
·
foranyt (0,T),comesfromtheBakry-Emerycondition, see[1,3]. Here,fromTheorem 1.1,wehave
∈
thefollowinglocallyLipschitz continuity foru.
Corollary1.5. LetK,N,XandB beasintheaboveTheorem1.1. Assumeubealocallyweaksolution
R,T
oftheheatequationon B . Then,foranyt (0,T),thefunctionu(,t)isLipschitzcontinuous on B .
R,T R/2
∈ ·
At last, two immediate consequences of Theorem 1.1 is the following Hamilton’s gradient estimates
onnon-compact caseandaLiouville’stheorem forancientsolutions oftheheatequation.
Corollary 1.6. Let X,K,N be as the above Theorem 1.1 and let T (0, ). Assume that u(x,t)
W1,2 X (0,T) is a weak solution of the heat equation on X (0,T∈). Su∞ppose also that there exis∈t
loc × ×
M,m(cid:0)> 0suchth(cid:1)atm6 u6 M on X (0,T). Thenwehave
×
1 M
f 2(x,t)6C +K ln
N
|∇ | · T · u(x,t)
(cid:16) (cid:17)
foralmostevery(x,t) X (T/2,T), where f = lnuandtheconstantC depends onlyon N.
N
∈ ×
Remark1.7. ThisisanextensionofTheorem2tonon-compactspaces. Indeed,inTheorem2,sinceuis
boundedfromaboveandpositive,theHarnackinequalityimpliesthatumustbeboundedfrombelowby
a positive number. Very recently, Theorem 2 has been extended to non-compact Riemannian manifolds
withappropriate Bakry-Emeryconditions byLi[29].
Corollary 1.8. Let (X,d,µ) be a metric measure space satisfying RCD (0,N) for some N [1, ).
∗
∈ ∞
Assumethatu(x,t)isanancientsolution oftheheatequation on X ( ,0]. If
× −∞
sup u
(1.6) liminf BR×(−R2,0)| | = 0,
R R
→∞
thenuisaconstant.
4 JIA-CHENGHUANGANDHUI-CHUNZHANG
Organization of the paper. In Section 2, we will provide some necessary materials on RCD (K,N)
∗
metricmeasurespaces. InSection3,wewillproveTheorem1.1,Theorem1.2andtheCorollary1.5,1.6
and1.8.
Acknowledgements. H.C.Zhangispartially supported byNSFC11521101.
2. Preliminaries
Let (X,d) be a proper (i.e., closed balls of finite radius are compact) complete metric space and µ be
aRadonmeasure on X withsupp(µ) = X.Denoteby B (x)theopenballcentered at xwithradius r. For
r
anyopensubsetΩ X andany p [1, ],wedenotebyLp(Ω):= Lp(Ω,µ).
⊂ ∈ ∞
2.1. Riemanniancurvature-dimension conditionsRCD*(K,N).
Thecurature-dimension conditionon(X,d,µ)hasbeenintroducedbySturm[42,43]andLott-Villani
[30]. Given two constants K R and N [1, ], the curvature-dimension condition CD(K,N) is a
∈ ∈ ∞
syntheticnotionfor“generalizedRiccicurvature> Kanddimension6 N”on(X,d,µ). Bacher-Sturm[6]
introducedthereducedcurvature-dimensionconditionCD (K,N),andAmbrosioetal. [1]introducedthe
∗
Riemannian curvature-dimension condition RCD(K, ). Very recently, Erbar et al. [12] and Ambrosio
∞
et al. [5] introduced a dimensional version of Riemannian curvature-dimension condition RCD (K,N).
∗
InthecaseofRiemannian geometry, thenotion RCD (K,N)coincides withtheoriginal Riccicurvature
∗
> K and dimension 6 N. Inthesetting ofAlexandrov geometry, itisimplied bygeneralized (sectional)
curvature bounded belowinthesenseofAlexandrov[35,46].
We denote by P (X,d) the L2-Wasserstein space over (X,d), i.e., the set of all Borel probability
2
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
1 2
×
with marginals ν and ν . Such a coupling q realizes the L2-Wasserstein distance is called an optimal
0 1
coupling ofν andν .
0 1
Givenameasureν P (X,d),itsrelativeentropyisdefinedby
2
∈
Ent(ν) := ρlnρdµ,
Z
X
if ν = ρ µ is absolutely continuous w.r.t. µ and (ρlnρ) is integrable. Otherwise weset Ent(ν) = + .
+
LetP (·X,d,µ) P (X,d)bethesubsetofallmeasuresνsuchthatEnt(ν)< . ∞
2∗ ⊂ 2 ∞
Definition 2.1([12]). Given K Rand N [1, ). Ametric measure space (X,d,µ)iscalled tosatisfy
the entropy curvature-dimensio∈n condition∈CDe∞(K,N) if any only if for each pair ν ,ν P (X,d,µ)
thereexistaconstantspeedgeodesic(νt)06t61inP2∗(X,d,µ)connectingν0toν1suchth0at1fo∈rallt2∗∈ [0,1]:
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) andthefunction
N − N
(cid:0) (cid:1)
sin(√k·tθ), 0 < kθ2 < π2,
sin(√kθ)
σ(kt)(θ) := tss,iinnhh((√√−·kk·tθθ)), kkθθ22 =< 00,,
∞, − · kθ2 > π2.
SHARPGRADIENTESTIMATEFORHEATKERNEL... 5
Givenafunction f C(X),thepointwiseLipschitzconstant ([10])of f at xisdefinedby
∈
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
→ →
where we put Lipf(x) = 0 if x is isolated. Clearly, Lipf is aµ-measurable function on X. The Cheeger
energy, denotedbyCh: L2(X) [0, ],isdefined([4])by
→ ∞
1
Ch(f) := inf liminf (Lipf )2dµ ,
n j 2 ZX j o
→∞
where the infimum is taken over all sequences of Lipschitz functions (f ) converging to f in L2(X).
j j N
Ingeneral, Chisaconvexandlowersemi-continuous functional onL2(X).∈
Definition2.2([4,12]). Ametricmeasurespace(X,d,µ)iscalledinfinitesimallyHilbertianiftheassoci-
atedCheegerenergy isquadratic. Moreover, (X,d,µ)issaidtosatisfyRiemannian curvature-dimension
condition RCD (K,N), for K R and N [1, ), if it is infinitesimally Hilbertian and satisfies the
∗
∈ ∈ ∞
CDe(K,N)condition.
Let(X,d,µ)beametricmeasure space satisfying RCD (K,N). Foreach f D(Ch) := f L2(X) :
∗
∈ { ∈
Ch(f) < ,ithas
∞}
1
Ch(f) = f 2dµ,
2 Z |∇ |
X
where f istheso-calledminimalrelaxedgradientof f (see 4in[4]). Itwasprovedin[4,Lemma4.3]
|∇ | §
thatLipschitz functions aredensein D(Ch)inthesense that,foreach f D(Ch),there existasequence
∈
ofLipschitzfunctions(f ) suchthat f f inL2(X)and (f f) 0inL2(X). SincetheCheeger
j j N j j
∈ → |∇ − | →
energy Chisaquadratic form,bythepolarization, theminimalrelaxed gradients bringaninnerproduct
asfollowing: given f,g D(Ch),itwasprovedin[15]thatthelimit
∈
(f +ǫ g)2 f 2
f, g := lim |∇ · | −|∇ |
h∇ ∇ i ǫ 0 2ǫ
→
exists in L1(X). The inner product is bi-linear and satisfies Cauchy-Schwarz inequality, Chain rule and
Leibnizrule(seeGigli[15]).
2.2. SobolevspacesandtheweakLaplacian.
Given K Rand N [1, ). Let(X,d,µ)beanRCD (K,N)metricmeasurespace. Severaldifferent
∗
∈ ∈ ∞
notions of Sobolev spaces on (X,d,µ) have been established in [10, 38, 2, 17, 16]. They coincide each
otheronRCD (K,N)metricmeasurespaces(see,forexample,[2]).
∗
LetΩ X beadomain. Wedenote by Lip (Ω)thesetoflocally Lipschitz continuous functions on
loc
⊂
Ω, and by Lip(Ω) (resp. Lip (Ω)) the set of Lipschitz continuous functions on Ω (resp, with compact
0
support inΩ).
Forany1 6 p 6 + and f Lip (Ω),itsW1,p(Ω)-normisdefinedby
loc
∞ ∈
f := f + Lipf .
W1,p(Ω) Lp(Ω) Lp(Ω)
k k k k k k
TheSobolevspacesW1,p(Ω)isdefinedbytheclosureoftheset
f Lip (Ω) : f < +
loc W1,p(Ω)
∈ k k ∞
(cid:8) (cid:9)
under the W1,p(Ω)-norm. The space W1,p(Ω) is defined by the closure of Lip (Ω) under the W1,p(Ω)-
0 0
norm. Wesaythatafunction f W1,p(Ω)if f W1,p(Ω )foreveryopensubsetΩ Ω.
∈ loc ∈ ′ ′ ⊂⊂
Since (X,d,µ) is assumed to be infinitesimally Hilbertian, it is well known that D(Ch) = W1,2(X),
see, for example, [47, Lemma 2.5]. Given any function f W1,2(X), the W1,2-norm of f, f =
W1,2(X)
∈ k k
f +2Ch(f).
L2(X)
k k
Fix any open set Ω X and p (1, ). According to [15, 4.1], the space W1,2(Ω) is still a Hilbert
⊂ ∈ ∞ §
space, and for any f,g W1,2(Ω), the function f 2 and f, g are well defined in L1 (Ω). In the
∈ loc |∇ | h∇ ∇ i loc
6 JIA-CHENGHUANGANDHUI-CHUNZHANG
sequel of this paper, we will always denote by H1(Ω) := W1,2(Ω), H1(Ω) := W1,2(Ω) and H1 (Ω) :=
0 0 loc
W1,2(Ω).
loc
Definition 2.3. Foreach f H1 (Ω),thedistribution L f isafunctional definedonH1(Ω) L (Ω)by
∈ loc 0 ∩ ∞
L f(φ) := f, φ dµ φ H1(Ω) L (Ω).
−Z h∇ ∇ i ∀ ∈ 0 ∩ ∞
Ω
Foranyg H1(Ω) L (Ω),thedistribution g L f isdefinedby
∞
∈ ∩ ·
g L f(φ) := L f(gφ) φ H1(Ω) L (Ω).
· ∀ ∈ 0 ∩ ∞
Bythelinearityofinnerproduct f, g ,thisdistributional Laplacianislinear.
h∇ ∇ i
If,given f H1 (Ω),thereexistsafunction g L1 (Ω)suchthat
∈ loc ∈ loc
L f(φ) > g φdµ 06 φ H1(Ω) L (Ω),
Z · ∀ ∈ 0 ∩ ∞
Ω
then we say that “L f > g in the sense of distributions”. In this case, L f is a signed Radon measure,
and hence, wealso denote “L f > g µinthesenseofmeasures”. Itissimilar whenwereplace “>”by
·
“=”orby“6”.
L satisfiesthefollowingChainruleandLeibnizrule[15],seealso[47,Lemma3.2].
Lemma2.4([15,47]). GivenK RandN [1, ). LetΩbeanopendomainofanRCD (K,N)metric
∗
∈ ∈ ∞
measurespace(X,d,µ).
(i)(Chainrule) Let f H1(Ω) L (Ω)andη C2(R). Thenwehave,inthesenseofdistributions,
∞
∈ ∩ ∈
L[η(f)] = η (f) L f +η (f) f 2.
′ ′′
· ·|∇ |
(ii)(Leibnizrule) Let f,g H1(Ω) L (Ω). Thenwehave,inthesenseofdistributions,
∞
∈ ∩
L(f g) = f Lg+g L f +2 f, g .
· · · h∇ ∇ i
LetΩ X beadomain. GivenT > 0,wedenotebyΩ := Ω (0,T].
T
⊂ ×
Definition 2.5. A function u(x,t) H1(Ω ) (= W1,2(Ω )) is called a locally weak solution of the heat
T T
∈
equation onΩ ifforany[t ,t ] (0,T)andanygeodesic ball B Ω,wehave
T 1 2 R
⊂ ⊂⊂
t2
∂u φ+ u, φ dµdt = 0
Z Z t · h∇ ∇ i
t1 BR(cid:16) (cid:17)
forallφ(x,t) Lip (B (t ,t ) .Hereandinthesequel,wedenotealways∂u := ∂u.
∈ 0 R× 1 2 t ∂t
(cid:1)
ThelocalboundednessandtheHarnackinequalityforsuchweaksolutionshavebeenprovedbySturm
[44, 45] in the setting of abstract local Dirichlet form and by Marola and Masson [32] in the setting of
metricmeasurespacewithastandardvolumedoublingpropertyandsupportingaL2-Poincare´ inequality.
Ofcourse, itisvividformetricmeasurespacessatisfying RCD (K,N)forsome K Rand N [1, ).
∗
∈ ∈ ∞
2.3. Thelocalized Bochnerformula.
Combiningwiththeworks[12,5]onaglobalversionofBochnerformulaandagoodcut-offfunction
in[5,33,18],onecanobtainthefollowinglocalizedBochnerformula,see[47,Corollary3.6]fordetails.
Theorem 2.6 ([47]). Let (X,d,µ) be a metric measure space satisfying RCD (K,N) for K R and
∗
∈
N [1, ). Let B beageodesic ballwithradiusRandcenteredatafixedpoint x .
R 0
∈Assum∞e that f H1(B ) satisfies L f = g on B in the sense of distributions with the function
R R
g H1(B ) L (B∈ ). Thenwehave f 2 H1(B ) L (B )andthatthedistribution L( f 2)is
R ∞ R R/2 ∞ R/2
∈ ∩ |∇ | ∈ ∩ |∇ |
asignedRadonmeasureon B . IfitsRadon-Nikodym decomposition w.r.t. µisdenotedby
R/2
L( f 2)= Lac( f 2) µ+Lsing( f 2),
|∇ | |∇ | · |∇ |
SHARPGRADIENTESTIMATEFORHEATKERNEL... 7
thenwehaveLsing( f 2) > 0and,forµ-a.e. x B ,
R/2
|∇ | ∈
1 g2
Lac( f 2)> + f, g +K f 2.
2 |∇ | N h∇ ∇ i |∇ |
Furthermore, if N > 1,wehave,forµ-a.e. x B y : f(y) , 0 ,animprovement estimate
R/2
∈ ∩ |∇ |
(cid:8) (cid:9)
1 g2 N f, f 2 g 2
(2.1) Lac( f 2) > + f, g +K f 2 + h∇ ∇|∇ | i .
2 |∇ | N h∇ ∇ i |∇ | N 1 · 2 f 2 − N
(cid:16) (cid:17)
− |∇ |
Remark 2.7. In thispaper, the key fact isthat thelast term in theimprovement estimate (2.1)isenough
to compensate for the lack of (1.5) in general RCD (K,N) metric measure spaces. In the smooth case,
∗
thisideaisusedin[8].
2.4. Thepointwisemaximumprinciples.
The pointwise maximum principle states that, given a C2-function f defined on a smooth manifold
(M,g),if f achievesoneofitslocalmaximumatpoint x M,thenwehave f(x ) = 0and∆f(x ) 6 0.
0 0 0
∈ ∇
Itisapowerfultool ingeometric analysis. However, itdoes notmakesenseonsingular metricmeasure
spaceingeneral.
To compensate for the lack of the pointwise maximum principle, the second author and Zhu in [47,
Theorem1.3]provedthefollowinganalogous toolongeneral RCD (K,N)-spaces.
∗
Theorem 2.8 ([47]). LetΩ be a bounded domain in a metric measure space (X,d,µ) with RCD (K,N)
∗
for some K Rand N > 1. Let f(x) H1(Ω) L (Ω)such that L f is asigned Radon measure with
∈ ∈ ∩ ∞loc
Lsingf > 0,whereLsingf isthesingularpartwithrespecttoµ. Supposethat f achievesoneofitsstrict
maximuminΩinthesensethat: thereexistsaneighborhood U Ωsuchthat
⊂⊂
sup f > sup f.
U Ω U
\
Then,givenanyw H1(Ω) L (Ω),thereexistsasequenceofpoints x U suchthattheyarethe
∞ j j N
approximate contin∈uity poin∩tsofLacf and f, w ,andthat { } ∈ ⊂
h∇ ∇ i
f(x ) > sup f 1/j and Lacf(x )+ f, w (x )6 1/j.
j j j
− h∇ ∇ i
Ω
Remark 2.9. The assumption that singular part Lsingf > 0 is necessary. Let us consider a simple
example: f(t) = t defined on ( 1,1). Then f (t) = δ(0), the Dirac measure with center at s = 0.
′′
−| | − −
By choosing w = f, then, at each the approximate continuity points of f and f w = f 2, we have
′′ ′ ′ ′
·
f + f w = f + f 2 = 1.
′′ ′ ′ ′′ ′
Wewillneedalsothefollowingparabolic version(see[47,Theorem4.4]):
Theorem 2.10 ([47]). Let Ω be a bounded domain and let T > 0. Let f(x,t) H1(Ω ) L (Ω )
T ∞ T
∈ ∩
and suppose that f achieves one of its strict maximum in Ω (0,T] in the sense that: there exists a
×
neighborhood U Ωandaninterval (δ,T] (0,T]forsomeδ > 0suchthat
⊂⊂ ⊂
sup f > sup f.
U×(δ,T] ΩT\(U×(δ,T])
Assume that, for almost every t (0,T), L f(,t) is a signed Radon measure with Lsingf(,t) > 0. Let
∈ · ·
w H1(Ω ) L (Ω ) with ∂w(x,t) 6 C for some constant C > 0, for almost all (x,t) Ω . Then,
T ∞ T t T
∈ ∩ ∈
thereexistsasequenceofpoints (x ,t ) U (δ,T]suchthatevery x isanapproximatecontinuity
j j j N j
pointofLacf(,t )and f, w{(,t ),a}n∈dt⊂hat ×
j j
· h∇ ∇ i ·
∂
f(x ,t ) > sup f 1/j and Lacf(x ,t )+ f, w (x ,t ) f(x ,t ) 6 1/j.
j j j j j j j j
− h∇ ∇ i − ∂t
Ω
T
8 JIA-CHENGHUANGANDHUI-CHUNZHANG
3. TheproofofTheorem1.1anditsconsequences
Given K R and N [1, ). Let(X,d,µ)be ametric measure space satisfying RCD (K,N). In this
∗
∈ ∈ ∞
section, wewillproveTheorem1.1anditsconsequences.
Thefollowingisthemainlemmainthispaper.
Lemma3.1. GivenK > 0andN (1, ). Let(X,d,µ)beametricmeasurespacesatisfyingRCD ( K,N).
∗
∈ ∞ −
Let f(x,t)beafunctiononB withall f,∂ f, f 2 H1(B ) L (B ). Assumethat,foralmostall
R,T t R,T ∞ R,T
|∇ | ∈ ∩
t (0,T),thefunction f(,t)satisfies
∈ ·
(3.1) L f = ∂ f f 2 on B
t R
−|∇ |
inthesenseofdistributions. Letη C2(R)withη(f) > 0. Weput
∈
F := η(f) f 2.
·|∇ |
Then, we have that F H1(B ) L (B ) and that, for almost all t (0,T), the function F(,t)
R,T ∞ R,T
∈ ∩ ∈ ·
satisfies thatLF isasignedRadonmeasureon B withLsingF > 0and
R/2
η (η +η ) 2(η )2
(3.2) LacF ∂ F +2 f, F > · ′′ ′ − ′ F2 2KF,
− t h∇ ∇ i η3 · −
µ a.e.on B . Hereandinthesequel, wedenotebyη:= η(f),η := η (f)andη := η (f).
R/2 ′ ′ ′′ ′′
−
Proof. From the assumption f, f 2 H1(B ) L (B ), we know that F H1(B ) L (B ).
R,T ∞ R,T R,T ∞ R,T
|∇ | ∈ ∩ ∈ ∩
Set
g(x,t) := ∂ f f 2 H1(B ) L (B ).
t R,T ∞ R,T
−|∇ | ∈ ∩
The Fubini Theorem implies that, for almost all t (0,T), the function g(,t) H1(B ) L (B ).
R ∞ R
Therefore, by using Theorem 2.6 to RCD ( K,N)-∈spaces, we get that the d·istri∈bution L(∩ f 2) is a
∗
signed Radonmeasureon B ,whichsatisfie−sthatLsing( f 2) > 0and,forµ-a.e. x B ,|t∇hat|
R/2 R/2
|∇ | ∈
2g2
(3.3) Lac( f 2)> +2 f, g 2K f 2,
|∇ | N h∇ ∇ i− |∇ |
andmoreover, forµ-a.e. x B y : f(y) , 0 ,wehave
R/2
∈ ∩ |∇ |
2g2 (cid:8) (cid:9) 2N f, f 2 g 2
(3.4) Lac( f 2) > +2 f, g 2K f 2 + h∇ ∇|∇ | i .
|∇ | N h∇ ∇ i− |∇ | N 1 · 2 f 2 − N
(cid:16) (cid:17)
− |∇ |
FromLemma2.4,weget,foralmostallt (0,T),thatthefunction F(,t)isasignedRadonmeasureon
∈ ·
B with
R/2
LF = f 2 L(η)+η L( f 2)+2 f 2, η µ
|∇ | · · |∇ | h∇|∇ | ∇ i·
= η f 2 L(f)+η f 4 µ+η L( f 2)+2η f 2, f µ
′ ′′ ′
·|∇ | · ·|∇ | · · |∇ | h∇|∇ | ∇ i·
inthesenseofmeasures. Thisimpliesthat
LsingF = η f 2 Lsingf +η Lsing( f 2) > 0,
′
·|∇ | · · |∇ |
sinceη(f) > 0andLsingf = 0,andthat
(3.5) LacF = η f 2 g+η f 4+η Lac( f 2)+2η f 2, f
′ ′′ ′
·|∇ | · ·|∇ | · |∇ | ·h∇|∇ | ∇ i
foralmostall x B . From F H1(B ),weget
R/2 R,T
∈ ∈
(3.6) ∂ F = η ∂ f f 2+2η ∂ f, f
t ′ t t
· ·|∇ | ·h∇ ∇ i
foralmostall x B .
R
∈
Case1: Letusconsider pointsin B y : f(y) , 0 . Denoteby,atpointswhere f , 0,
R/2
∩ |∇ | |∇ |
(cid:8) (cid:9) f, f 2
A := f 2 and B:= h∇ ∇|∇ | i.
|∇ | f 2
|∇ |
SHARPGRADIENTESTIMATEFORHEATKERNEL... 9
Bycombining (3.4),(3.5),(3.6)andg = ∂ f A,weconcludes that
t
−
LacF ∂ F = η A (g ∂ f)+η A2+η Lac( f 2)+2η AB 2η f, (g+A)
t ′ t ′′ ′
− · · − · · |∇ | · − h∇ ∇ i
> ( η +η ) A2+2η AB 2η f, (g+A)
′ ′′ ′
− · · − h∇ ∇ i
g2 N B g
+2η + f, g KA+ 2
N h∇ ∇ i− N 1 2 − N
(cid:16) − (cid:0) (cid:1) (cid:17)
= ( η +η ) A2+2η AB 2η f, (F/η)
′ ′′ ′
− · · − h∇ ∇ i
g2 N B2 Bg
+2η KA+ ·
N 1 − 4(N 1) − N 1
(cid:16) (cid:17)
− − −
η B2
> ( η +η ) A2+2η AB 2 f, F +2η A2 2ηKA+ ·
′ ′′ ′ ′
− · · − h∇ ∇ i · − 2
foralmostall x B y : f(y) , 0 ,wherewehaveused
R/2
∈ ∩ |∇ |
(cid:8) (cid:9) N B2 (N 1) B2
F = ηA and g2 Bg+ · > − · .
− 4 4
Noticingthatη > 0and
η B2 2(η )2
· +2η AB> ′ A2,
′
2 · − η ·
wehave,µ-a.e. on B y: f(y) , 0 ,that(since F = ηA,)
R/2
∩ |∇ |
(cid:8) (cid:9) 2(η )2
LacF ∂ F > (η +η ) A2 ′ A2 2 f, F 2ηKA
t ′ ′′
− · − η · − h∇ ∇ i−
(3.7)
η (η +η ) 2(η )2
= · ′′ ′ − ′ F2 2 f, F 2KF.
η3 · − h∇ ∇ i−
Case2: LetusconsiderpointsinB y : f(y) = 0 . AtpointwhereA = f 2 = 0,thecombination
R/2
∩ |∇ | |∇ |
of(3.3),(3.5)and(3.6)impliesthat (cid:8) (cid:9)
LacF ∂ F = η Lac( f 2)+2η f 2, f 2η f, (g+A)
t ′
− · |∇ | ·h∇|∇ | ∇ i− h∇ ∇ i
g2
> 2η A, f 2η f, (g+A) +2η + f, g KA
′
·h∇ ∇ i− h∇ ∇ i N h∇ ∇ i−
(cid:16) (cid:17)
2η g2
= 2(η η) f, A + · > 2(η η) f, A .
′ ′
− h∇ ∇ i N − h∇ ∇ i
Noticingthat,µ a.e.on y: f (y) = 0 ,andthat f, A 6 f A,weconcludethat f, A =
− |∇ | |h∇ ∇ i| |∇ |·|∇ | |h∇ ∇ i|
0foralmostall x y : (cid:8) f (y) = 0 . Th(cid:9)esameholdsfor f, F .
∈ |∇ | |h∇ ∇ i|
(3.8) Lac(cid:8)F ∂ F > 0 =(cid:9) 2 f, F , µ a.e. on B y: f(y) = 0 .
t R/2
− − h∇ ∇ i − ∩ |∇ |
(cid:8) (cid:9)
Thecombination of(3.7)inCase1and(3.8)inCase2givestheresult(3.2). (cid:3)
Wefirstlyshowthemainresultunderanaddedassumption thatu(x,t)satisfies
(3.9) u H1(B ) L (B ) and ∂u H1(B ) L (B ).
R,T ∞ R,T t R,T ∞ R,T
∈ ∩ ∈ ∩
Lemma3.2. LetK > 0andN (1, ),andlet(X,d,µ)beametricmeasurespacesatisfyingRCD ( K,N).
∗
∈ ∞ −
Let T (0, ) and let B be a geodesic ball of radius R, B = B (0,T), and let u(x,t)
R R,T R
∈ ∞ × ∈
H1 B L (B ) be a positive locally weak solution of the heat equation on B . Suppose that
R,T ∞ R,T R,T
∩
ther(cid:0)eexi(cid:1)sts δ,δ′ (0,1/2) suchthatδ 6 u 6 1 δ′. Suppose also∂tu H1(BR,T) L∞(BR,T). Then,we
∈ − ∈ ∩
havethefollowinglocalgradient estimate
f 2 ln(1/δ) 1
(3.10) sup |∇ | (x,t)6C + +K ,
f N · R2 T
BR/2×(T/2,T) − (cid:16) (cid:17)
10 JIA-CHENGHUANGANDHUI-CHUNZHANG
where f = lnu.
Proof. From the assumption of u, we know f,∂ f = ∂u/u H1(B ) L (B ). It was proved
t t R,T ∞ R,T
∈ ∩
in [47, Lemma 5.5] that u2 H1(B ) L (B ). Hence, by 1 δ > u > δ > 0, we have
3R/4,T ∞ 3R/4,T ′
|∇ | ∈ ∩ −
f 2 = u2/u2 H1(B ) L (B ).ByFubiniTheorem,itisclearthat,foralmostallt (0,T),
3R/4,T ∞ 3R/4,T
t|∇he|func|t∇ion| u(,t∈) satisfies Lu∩= ∂u on B in the sense of distributions. From Lemma2.4, fo∈r almost
t R
·
allt (0,T),thefunction f(,t)satisfies(3.1). Therefore, f satisfiesallofassumptions inLemma3.1.
∈ ·
Let η(s) = 1/s and consider F(x,t) := η(f) f 2 on B . Since lnδ 6 f 6 ln(1 δ ) < 0, we
3R/4,T ′
− ·|∇ | −
haveη(f) > 1 > 0. AccordingtoLemma3.1and
ln(1/δ)
η(η +η ) 2(η )2 = η3,
′′ ′ ′
−
we conclude that F H1(B ) L (B ) and that, for almost all t (0,T), the function F(,t)
3R/4,T ∞ 3R/4,T
∈ ∩ ∈ ·
satisfiesthatLF isasigned Radonmeasureon B withLsingF > 0and
5R/8
(3.11) LacF ∂ F +2 f, F > F2 2KF, µ a.e. on B .
t 5R/8
− h∇ ∇ i − −
We can assume that sup F > 0, otherwise, we are done. Fix a constant ǫ > 0 to be suf-
BR/2 (T/2,T)
ficiently small, (for example, w×e can choose ǫ := min 1/10,sup F/10 ). We now choose a
modified cutoff in the spatial direction φ(x) = φ(r(x)) to{be a funcBtiRo/2n×(oTf/2t,hTe) dista}nce r to the center of
B withthefollowingproperties that
R
ǫ 6 φ6 1 on B , φ= 1 on B , φ= ǫ on B B ,
R R/2 R 9R/16
\
and
C C
φ21 6 φ′(r) 6 0 and φ′′(r) 6 r (0,R)
−R | | R2 ∀ ∈
for some universal constant C (which is independent of N,K,R). Then, according to the Laplacian
comparison theorem [15,Corollary5.15]onRCD ( K,N)spaces, wehavethat
∗
−
φ2 C √K 1
(3.12) |∇ | 6 1 and Lφ> C ( + )
φ R2 − 2 R R2
on B . (A detailed calculation for this can be found on Page 22 in [47].) Here and in the sequel of this
R
proof, wedenote C ,C ,C , the various constants which depend only on N. Hence, the distribution
1 2 3
···
Lφisasigned Radon measure with (Lφ)ac > C (√K/R+1/R2)a.e. x B ,and (Lφ)sing > 0. Let
2 R
− ∈
ξ(t)beafunctionon(0,T)suchthat
C
(3.13) ξ(t)= ǫ on (0,T/4), ξ(t)= 1 on (T/2,T) and 0 6 ξ (t)6 3 on (0,T).
′
T
Letψ(x,t) := φ(x)ξ(t)definedon B .
R,T
PutG(x,t) := ψF. ThenwehaveG H1(B ) L (B ). According toLemma2.4,wehave,
3R/4,T ∞ 3R/4,T
∈ ∩
foralmosteveryt (0,T),thatthefunctionG(,t)satisfiesthat
∈ ·
LG = FLψ+ψLF +2 ψ, F
h∇ ∇ i
in the sense of distributions. Fix arbitrarily a such t (0,T). Then LG is a signed Radon measure on
∈
B with
5R/8
(3.14) (LG)sing = F(Lψ)sing+ψ(LF)sing > 0,
since F > 0,andthat (LG)ac = F(Lψ)ac +ψ(LF)ac +2 ψ, F a.e. x B . Wehave, foralmost
5R/8
h∇ ∇ i ∈
all x B ,
5R/8
∈
(LG)ac ∂G+2 f, G =ψ (LF)ac ∂ F +2 f, F
t t
(3.15) − h∇ ∇ i − h∇ ∇ i
(cid:16) (cid:17)
+F(Lψ)ac +2 ψ, F +2 f, ψ F F ∂ψ.
t
h∇ ∇ i h∇ ∇ i − ·