LOCAL LI-YAU’S ESTIMATES ONRCD (K,N) METRIC MEASURE SPACES

HUI-CHUN ZHANG AND XI-PING ZHU

Abstract. In this paper, we will study the (linear) geometric analysis on metric measure
spaces. We will establish a local Li-Yau’s estimate for weak solutions of the heat equation
and prove a sharp Yau’s gradient gradient for harmonic functions on metric measure spaces,
under the Riemannian curvature-dimension conditionRCD^{∗}(K,N).

Contents

1. Introduction 1

2. Preliminaries 6

2.1. Reduced and Riemannian curvature-dimension conditions 6 2.2. Canonical Dirichlet form and a global version of Bochner formula 7

2.3. Sobolev spaces 8

3. The weak Laplacian and a local version of Bochner formula 9

4. The maximum principle 12

4.1. The Kato’s inequality 12

4.2. Maximum principles 14

5. Local Li-Yau’s gradient estimates 16

6. A sharp local Yau’s gradient estimate 24

References 26

1. Introduction

In the field of geometric analysis, one of the fundamental results is the following Li-Yau’s local gradient estimate for solutions of the heat equation on a complete Riemannian manifold.

Theorem 1.1(Li-Yau [34]). Let(M^{n},g)be an n-dimensional complete Riemannian manifold,
and let B_{2R} be a geodesic ball of radius 2R centered at O ∈ M^{n}. Assume that Ric(M^{n}) >

−k with k > 0. If u(x,t) is a smooth positive solution of the heat equation∆u = ∂tu on
B_{2R}×(0,∞), then for anyα >1, we have the following gradient estimate in BR:

(1.1) sup

x∈B_{R}

|∇f|^{2}−α·∂tf(x,t)6 Cα^{2}
R^{2}

α^{2}
α^{2}−1 + √

kR

+ nα^{2}k

2(α−1)+ nα^{2}
2t
where f :=lnu and C is a constant depending only on n.

By lettingR→ ∞in (1.1), one gets a global gradient estimate, for anyα >1, that
(1.2) |∇f|^{2}−α·∂tf 6 nα^{2}k

2(α−1)+ nα^{2}
2t .

There is a rich literature on extensions and improvements of the Li-Yau inequality, both the local version (1.1) and the global version (1.2), to diverse settings and evolution equations,

1

for example, in the setting of Riemannian manifolds with Ricci curvature bounded below [15, 9, 47, 33, 32], in the setting of weighted Riemannian manifolds with Bakry-Emery Ricci curvature bounded below [12, 35, 43, 7] and some non-smooth setting [10, 44], and so on.

Let (X,d, µ) be a complete, proper metric measure space with supp(µ)= X.Thecurature- dimension conditionon (X,d, µ) has been introduced by Sturm [48] and Lott-Villani [36].

Given K ∈ R andN ∈ [1,∞], the curvature-dimension conditionCD(K,N) is a synthetic
notion for “generalized Ricci curvature> Kand dimension6N” on (X,d, µ). Bacher-Sturm
[6] introduced thereducedcurvature-dimension conditionCD^{∗}(K,N), which satisfies a local-
to-global property. On the other hand, to rule out Finsler geometry, Ambrosio-Gigli-Savar´e
[1] introduced theRiemannian curvature-dimension condition RCD(K,∞), which assumes
that the heat flow onL^{2}(X) is linear. Remarkably, Erbar-Kuwada-Sturm [16] and Ambrosio-
Mondino-Savar´e [5] introduced a dimensional version of Riemannian curvature-dimension
conditionRCD^{∗}(K,N) and proved that it is equivalent to a Bakry-Emery’s Bochner inequality
via an abstractΓ2-calculus for semigroups. In the case of Riemannian geometry, the notion
RCD^{∗}(K,N) coincides with the original Ricci curvature > K and dimension 6 N, and for
the case of the weighted manifolds (M^{n},g,e^{φ}·volg), the notionRCD^{∗}(K,N) coincides with
the corresponding Bakry-Emery’s curvature-dimension condition ([48, 36]). In the setting of
Alexandrov geometry, it is implied by generalized (sectional) curvature bounded below in the
sense of Alexandrov [42, 52].

Based on the Γ2-calculus for the heat flow (Htf)t>0 on L^{2}(X), many important result-
s in geometric analysis have been obtained on a metric measure space (X,d, µ) satisfying
RCD^{∗}(K,N) condition. For instance, Li-Yau-Hamilton estimates for the heat flow (Htf)t>0

[17, 28, 30] and spectral gaps [37, 44, 31] for the infinitesimal generator of (Htf)t>0.

In this paper, we will study thelocallyweak solutions of the heat equation on a metric
measure space (X,d, µ). Let Ω ⊂ X be an open set. The RCD^{∗}(K,N) condition implies
that the Sobolev space W^{1,2}(Ω) is a Hilbert space. Given an interval I ⊂ R, a function
u(x,t) ∈ W^{1,2}(Ω×I) is called a locally weak solution for the heat equation on Ω× I if it
satisfies

(1.3) −

Z

I

Z

Ωh∇u,∇φidµdt= Z

I

Z

Ω

∂u

∂t ·φdµdt

for all Lipschitz functionsφwith compact support inΩ×I, where the inner producth∇u,∇φi
is given by polarization inW^{1,2}(Ω).

Notice that the locally weak solutionsu(x,t) do not form a semi-group in general. The method ofΓ2-calculus for the heat flow in the previous works [17, 28, 31] is no longer be suitable for the problems on locally weak solutions of the heat equation.

To seek an appropriate method to deal with the locally weak solutions for the heat equation, let us recall what is the proof of Theorem 1.1 in the smooth context. There are two main ingredients: the Bochner formula and a maximum principle. The Bochner formula states that

(1.4) 1

2∆|∇f|^{2} > (∆f)^{2}

n +h∇f,∇∆fi+K|∇f|^{2}

for anyC^{3}-function f on M^{n} with Ricci curvature Ric(M^{n}) > K for some K ∈ R. The
maximum principle states that if f(x) is ofC^{2} on M^{n} and if it achieves its a local maximal
value at pointx_{0} ∈M^{n}, then we have

(1.5) ∇f(x0)=0 and ∆f(x0)60.

For simplification, we only consider the special case thatu(x,t) is a smooth positive solution
for heat equation on a compact manifold M^{n} with Ric(M^{n}) > 0. By using the Bochner
formula to lnu, one deduces a differential inequality

∆− ∂

∂t

F>−2h∇f,∇Fi+ 2

ntF^{2}− F
t,

where f =lnuandF= t|∇f|^{2}−∂tf.Then by using the maximum principle toFat one of
its maximum points (x0,t_{0}), one gets the desired Li-Yau’s estimate

maxF =F(x0,t_{0})6 n
2.

In this paper, we want to extend these two main ingredients to non-smooth metric measure
spaces. Firstly, let us consider the Bochner formula in non-smooth context. Let (X,d, µ) be a
metric measure space withRCD^{∗}(K,N). Erbar-Kuwada-Sturm [16] and Ambrosio-Mondino-
Savar´e [5] proved thatRCD^{∗}(K,N) condition is equivalent to a Bakry-Emery’s Bochner in-
equality for the heat flow (Htf)t>0onX. This provides a global version of Bochner formula
for the infinitesimal generator of the heat flow (Htf)t>0(see Lemma 2.3). On the other hand,
a good cut-offfunction has been obtained in [5, 40, 24]. By combining these two facts and
an argument in [24], one can localize the global version of Bochner formula in [16, 5] to a
local one.

To state the local version of Bochner formula, it is more convenient to work with a notion
of the weak Laplacian, which is a slight modification from [18, 20]. LetΩ⊂ Xbe an open
set. Denote byH^{1}(Ω) := W^{1,2}(Ω) andH_{0}^{1}(Ω) := W_{0}^{1,2}(Ω). Theweak LaplacianonΩis an
operatorL onH^{1}(Ω) defined by: for each function f ∈H^{1}(Ω),L f is a functional acting on
H_{0}^{1}(Ω)∩L^{∞}(Ω) given by

L f(φ) :=− Z

Ω

h∇f,∇φidµ ∀φ∈H_{0}^{1}(Ω)∩L^{∞}(Ω).

In the case when it holds L f(φ)>

Z

Ωh·φdµ ∀06φ∈H_{0}^{1}(Ω)∩L^{∞}(Ω)

for some functionh∈L^{1}_{loc}(Ω), then it is well-known [23] that the weak LaplacianL fcan be
extended to a signed Radon measure onΩ. In this case, we denote by

L f >h·µ onΩin the sense of distributions.

Now, the local version of Bochner formula is given as follows.

Theorem 1.2([5, 24]). Let(X,d, µ)be a metric measure space with RCD^{∗}(K,N) for some
K∈Rand N >1. Assume that f ∈H^{1}(BR)such thatL f is a signed measure on BRwith the
density g∈H^{1}(B_{R})∩L^{∞}(BR). Then we have|∇f|^{2} ∈H^{1}(BR/2)∩L^{∞}(BR/2)and thatL(|∇f|^{2})
is a signed Radon measure on B_{R/2}such that

1

2L(|∇f|^{2})>hg^{2}

N +h∇f,∇gi+K|∇f|^{2}i

·µ
on B_{R/2}in the sense of distributions.

Next, we consider to extend the maximum principle (1.5) from smooth Riemannian man-
ifolds to non-smooth metric measure spaces (X,d, µ). A simple observation is that the maxi-
mum principle (1.5) on a smooth manifoldM^{n}has the following equivalent form:

Suppose that f(x) is of C^{2}on M^{n}and that it achieves its a local maximal value at point
x_{0}∈M^{n}. Given any w∈C^{1}(U)for some neighborhood U of x_{0}. Then we have

∆f(x0)+h∇f,∇wi(x_{0})60.

In the following result, we will extend the observation to the non-smooth context. Technical- ly, it is our main effort in the paper.

Theorem 1.3. LetΩbe a bounded domain in a metric measure space(X,d, µ)with RCD^{∗}(K,N)
for some K ∈ Rand N > 1.Let f(x) ∈ H^{1}(Ω)∩L^{∞}_{loc}(Ω) such thatL f is a signed Radon
measure withL^{sing}f > 0, where L^{sing}f is the singular part with respect to µ. Suppose
that f achieves one of its strict maximum inΩin the sense that: there exists a neighborhood
U⊂⊂Ωsuch that

sup

U

f >sup

Ω\U

f.

Then, given any w∈H^{1}(Ω)∩L^{∞}(Ω), there exists a sequence of points{x_{j}}j∈N⊂U such that
they are the approximate continuity points ofL^{ac}f andh∇f,∇wi, and that

f(xj)>sup

Ω f −1/j and L^{ac}f(xj)+h∇f,∇wi(xj)61/j.
Here and in the sequel of this paper,sup_{U} f meansess sup_{U} f .

This result is close to the spirit of the Omori-Yau maximum principle [41, 51]. It has also some similarity with the approximate versions of the maximum principle developed, for instance by Jensen [26], in the theory of second order viscosity solutions.

A similar parabolic version of the maximum principle, Theorem 4.4, will be given in§4.

After obtaining the above Bochner formula and the maximum principle (Theorem 1.2 and Theorem 4.4), we will show the following Li-Yau type gradient estimates for locally weak solutions of the heat equation, which is our main purpose in this paper.

Theorem 1.4. Let K > 0and N ∈ [1,∞), and let (X,d, µ) be a metric measure space sat-
isfying RCD^{∗}(−K,N). Let T∗ ∈ (0,∞]and let B_{2R} be a geodesic ball of radius2R centered
at p∈X, and let u(x,t)∈W^{1,2} B_{2R}×(0,T∗)

be a positive locally weak solution of the heat
equation on B_{2R}×(0,T∗). Then, given any T ∈(0,T∗), we have the following local gradient
estimate

sup

B_{R}×(β·T,T]

|∇f|^{2}−α· ∂

∂tf

(x,t)6max 1,1

2+ KT

2(α−1)

· Nα^{2}
2T · 1

β^{2}
+ C_{N}·α^{4}

R^{2}(α−1)· 1

(1−β)β^{2} +C_{N}·α^{2}
β^{2} ·

√ K R + 1

R^{2}
(1.6)

for anyα >1and anyβ∈(0,1), where f =lnu, and CN is a constant depending only on N.

Here and in the sequel of this paper,sup_{B}_{R}_{×[a,b]}g meansess sup_{B}_{R}_{×[a,b]}g for a function g(x,t).

The local boundedness and the Harnack inequality for locally weak solutions of the heat
equation have been established by Sturm [49, 50] in the setting of abstract local Dirichlet
form and by Marola-Masson [39] in the setting of metric measure with a standard volume
doubling property and supporting aL^{2}-Poincare inequality. Of course, they are available on
metric measure spaces (X,d, µ) satisfyingRCD^{∗}(K,N) for someK ∈ RandN ∈ [1,∞). In
particular, any locally weak solutions for the heat equation must be locally H¨older continuous.

As a consequence of Theorem 1.4, lettingR→ ∞andβ→1, we get the following global gradient estimates.

Corollary 1.5. Let(X,d, µ)and K,N,T∗ be as in the Theorem 1.4. Let u(x,t)is a positive solution of the heat equation on X×(0,T∗). Then, for almost all T ∈ (0,T∗), the following gradient estimate holds

sup

x∈X

|∇f|^{2}−α· ∂

∂tf

(x,T)6max 1,1

2+ KT

2(α−1)

· Nα^{2}
2T 6

1+ KT 2(α−1)

· Nα^{2}
2T
for anyα >1,where f =lnu.

As another application of the maximum principle, Theorem 1.3, and the Bochner formula,
we will deduce asharp Yau’s gradient estimate for harmonic functions on metric measure
spaces satisfyingRCD^{∗}(−K,N) forK >0 andN >1.

Let us recall the classical local Yau’s gradient estimate in geometric analysis (see [14, 51,
38]). Let M^{n} be ann(> 2)-dimensional complete non-compact Riemannian manifold with
Ric(M^{n})>−kfor somek>0. The local Yau’s gradient estimate asserts that for any positive
harmonic functionuonB_{2R}, then

(1.7) sup

BR

|∇lnu|6 p

(n−1)k+C(n) R .

In particular, ifuis positive harmonic onM^{n}andRic> −(n−1) onM^{n}then it follows that

|∇logu| 6 n−1 onM^{n}. This result is sharp, in fact the equality case was characterized in
[38]. This means that fork=n−1 in (1.7) the factor √

n−1 on the right hand side is sharp.

Let (X,d, µ) be a metric measure space satisfying RCD^{∗}(−K,N) for some K > 0 and
N ∈(1,∞). It was proved in [27] the following form of Yau’s gradient estimate that, for any
positive harmonic functionuonB_{2R}⊂X, it holds

(1.8) sup

BR

|∇lnu|6C(N,K,R).

In the setting of Alexandrov spaces, by using a Bochner formula and an argument of Nash- Moser iteration, it was proved in [53, 25] the following form of Yau’s gradient estimate holds:

given ann-dimensional Alexandrov spaceMand a positive harmonic functionuonB_{2R}⊂ M,
if the generalized Ricci curvature onB_{2R} ⊂ M has a lower bound Ric > −k, k > 0, in the
sense of [52], then

sup

B_{R}

|∇lnu|6C_{1}(n)

√

k+ C_{2}(n)
R .

Indeed, by applying Theorem 1.2, the same argument in [53, 25] implies this estimate still
holds for harmonic functionuon a metric measure space (X,d, µ) withRCD^{∗}(−k,n).Howev-
er, it seems hopeless to improve the factC_{1}(n) to the sharp √

n−1 in (1.7) via a Nash-Moser iteration argument.

The last result in this paper is to establish a sharp local Yau’s gradient estimate on metric measure spaces with Riemannian curvature-dimension condition.

Theorem 1.6. Let K >0and N∈(1,∞), and let(X,d, µ)be a metric measure space satisfy-
ing RCD^{∗}(−K,N). Let B_{2R}be a geodesic ball of radius2R centered at p∈X, and let u(x)be
a positive harmonic function on B_{2R}. Then the following local Yau’s gradient estimate holds

(1.9) sup

B_{R}

|∇lnu|6 s

1+β

1−β ·(N−1)K+ C(N) pβ(1−β)·R for anyβ∈(0,1).

Acknowledgements. H. C. Zhang is partially supported by NSFC 11571374. X. P. Zhu is partially supported by NSFC 11521101.

2. Preliminaries

Let (X,d) be a complete metric space andµbe a Radon measure onX with supp(µ) = X.

Denote byB_{r}(x) the open ball centered atxand radiusr. Throughout the paper, we assume
thatXis proper (i.e., closed balls of finite radius are compact). Denote byL^{p}(Ω) :=L^{p}(Ω, µ)
for any open setΩ⊂Xand any p∈[1,∞].

2.1. Reduced and Riemannian curvature-dimension conditions.

LetP2(X,d) be theL^{2}-Wasserstein space over (X,d), i.e., the set of all Borel probability
measuresνsatisfying

Z

X

d^{2}(x0,x)dν(x)<∞

for some (hence for all)x_{0} ∈ X. Given two measuresν_{1}, ν_{2} ∈ P2(X,d), theL^{2}-Wasserstein
distance between them is given by

W^{2}(ν_{0}, ν_{1}) :=inf
Z

X×X

d^{2}(x,y)dq(x,y)

where the infimum is taken over all couplingsqofν_{1}andν_{2}, i.e., Borel probability measures
qonX×Xwith marginalsν_{0}andν_{1}.Such a couplingqrealizes theL^{2}-Wasserstein distance
is called anoptimal couplingofν_{0} andν_{1}.LetP2(X,d, µ) ⊂ P2(X,d) be the subspace of
all measures absolutely continuous w.r.t. µ.Denote byP∞(X,d, µ) ⊂ P2(X,d, µ) the set of
measures inP2(X,d, µ) with bounded support.

Definition 2.1. Let K ∈ R andN ∈ [1,∞). A metric measure space (X,d, µ) is called to
satisfy the reduced curvature-dimension condition CD^{∗}(K,N) if any only if for each pair
ν_{0}=ρ_{0}·µ, ν_{1}=ρ_{1}·µ∈P∞(X,d, µ) there exist an optimal couplingqof them and a geodesic
(νt:=ρt·µ)t∈[0,1]inP∞(X,d, µ) connecting them such that for allt∈[0,1] and allN^{0}> N:

Z

X

ρ^{−}_{t}^{1/N}^{0}dνt >

Z

X×X

hσ^{(1−t)}_{K/N}0 d(x_{0},x_{1})ρ^{−}_{0}^{1/N}^{0}(x_{0})+σ^{(t)}_{K/N}0 d(x_{0},x_{1})ρ^{−}_{1}^{1/N}^{0}(x_{1})i

dq(x_{0},x_{1}),
where the function

σ^{(t)}_{k} (θ) :=

sin(

√k·tθ) sin(√

k·θ), 0<kθ^{2}< π^{2},

t, kθ^{2}=0,

sinh(√

−k·tθ) sinh(√

−k·θ), kθ^{2}<0,

∞, kθ^{2}>π^{2}.

Given a function f ∈C(X), thepointwise Lipschitz constant([13]) of f atxis defined by Lipf(x) :=lim sup

y→x

|f(y)− f(x)|

d(x,y) =lim sup

r→0

d(x,y)6rsup

|f(y)− f(x)|

r ,

where we put Lipf(x)=0 ifxis isolated. Clearly, Lipfis aµ-measurable function onX.The
Cheeger energy, denoted by Ch : L^{2}(X)→[0,∞], is defined ([4]) by

Ch(f) :=infn lim inf

j→∞

1 2

Z

X

(Lipf_{j})^{2}dµo
,

where the infimum is taken over all sequences of Lipschitz functions (f_{j})j∈Nconverging to f
inL^{2}(X).In general, Ch is a convex and lower semi-continuous functional onL^{2}(X).

Definition 2.2. A metric measure space (X,d, µ) is called infinitesimally Hilbertian if the
associated Cheeger energy is quadratic. Moreover, (X,d, µ) is said to satisfy Riemannian
curvature-dimension condition RCD^{∗}(K,N), forK∈RandN ∈[1,∞), if it is infinitesimally
Hilbertian and satisfies theCD^{∗}(K,N) condition.

Let (X,d, µ) be a metric measure space satisfyingRCD^{∗}(K,N). For each f ∈ D(Ch), i.e.,
f ∈L^{2}(X) and Ch(f)<∞, it has

Ch(f)= 1 2

Z

X

|∇f|^{2}dµ,

where |∇f| is the so-called minimal relaxed gradient of f (see §4 in [4]). It was proved,
according to [4, Lemma 4.3] and Mazur’s lemma, that Lipschitz functions are dense inD(Ch),
i.e., for eachf ∈D(Ch), there exist a sequence of Lipschitz functions (f_{j})j∈Nsuch that f_{j} → f
inL^{2}(X) and|∇(fj− f)| →0 inL^{2}(X). Since the Cheeger energy Ch is a quadratic form, the
minimal relaxed gradients bring an inner product as following: given f,g ∈ D(Ch), it was
proved [18] that the limit

h∇f,∇gi:= lim

→0

|∇(f+·g)|^{2}− |∇f|^{2}
2

exists inL^{1}(X).The inner product is bi-linear and satisfies Cauchy-Schwarz inequality, Chain
rule and Leibniz rule (see Gigli [18]).

2.2. Canonical Dirichlet form and a global version of Bochner formula.

Given an infinitesimally Hilbertian metric measure space (X,d, µ), the energyE := 2Ch
gives a canonical Dirichlet form onL^{2}(X) with the domainV := D(Ch). Let K ∈ R and
N ∈ [1,∞), and let (X,d, µ) be a metric measure space satisfyingRCD^{∗}(K,N). It has been
shown [1, 3] that the canonical Dirichlet form (E,V) is strongly local and admits a Carr´e du
champΓwith Γ(f) = |∇f|^{2} of f ∈ V. Namely, the energy measure of f ∈ Vis absolutely
continuous w.r.t. µwith the density|∇f|^{2}. Moreover, the intrinsic distance d_{E} induced by
(E,V) coincides with the original distancedonX.

It is worth noticing that if a metric measure space (X,d, µ) satisfyingRCD^{∗}(K,N) then its
associated Dirichlet form (E,V) satisfies the standard assumptions: the local volume dou-
bling property and supporting a localL^{2}-Poincare inequality (see [48, 45]).

Let ∆_{E},D(∆_{E})

and (Htf)t>0denote the infinitesimal generator and the heat flow induced from (E,V). Let us recall the Bochner formula (also called the Bakry-Emery condition) in [16] as following.

Lemma 2.3. Let(X,d, µ)be a metric measure space satisfying RCD^{∗}(K,N)for K ∈Rand
N ∈ [1,∞), and let(E,V) be the associated canonical Dirichlet form. Then the following
properties hold.

(i) ([16, Theorem 4.8]) If f ∈ D(∆_{E}) with∆_{E}f ∈ Vand ifφ ∈ D(∆_{E})∩L^{∞}(X) with
φ>0and∆_{E}φ∈L^{∞}(X), then we have the Bochner formula:

1 2

Z

X

∆Eφ· |∇f|^{2}dµ> 1
N

Z

X

φ(∆E f)^{2}dµ+
Z

X

φh∇(∆Ef),∇fidµ+K Z

X

φ|∇f|^{2}dµ.

(2.1)

(ii) ([5, Theorem 5.5]) If f ∈D(∆_{E})with∆_{E}f ∈L^{4}(X)∩L^{2}(X)and ifφ∈Vwithφ>0,
then we have|∇f|^{2}∈Vand the modified Bochner formula:

Z

X

− 1

2h∇|∇f|^{2},∇φi+∆_{E}f· h∇f,∇φi+φ·(∆_{E}f)^{2}
dµ

>

Z

X

K|∇f|^{2}+ 1

N(∆_{E}f)^{2}

·φdµ.

(2.2)

We need the following result on the existence of good cut-offfunctions onRCD^{∗}(K,N)-
spaces from [40, Lemma 3.1]; see also [19, 5, 24].

Lemma 2.4. Let(X,d, µ)be a metric measure space satisfying RCD^{∗}(K,N)for K ∈Rand
N ∈ [1,∞). Then for every x0 ∈ X and R > 0 there exists a Lipschitz cut-off function
χ:X→[0,1]satisfying:

(i) χ=1on B_{2R/3}(x0)andsupp(χ)⊂ BR(x0);

(ii) χ∈D(∆E)and∆Eχ∈V∩L^{∞}(X), moreover|∆Eχ|+|∇χ|6C(N,K,R).

2.3. Sobolev spaces.

Several different notions of Sobolev spaces on metric measure space (X,d, µ) have been
established in [13, 46, 22, 21]. They are equivalent to each other onRCD^{∗}(K,N) metric
measure spaces (see, for example, [2]).

Let (X,d, µ) be a metric measure space satisfyingRCD^{∗}(K,N) for someK ∈ RandN ∈
[1,∞). Fix an open setΩinX. We denote byLip_{loc}(Ω) the set of locally Lipschitz continuous
functions onΩ, and byLip(Ω) (resp. Lip_{0}(Ω)) the set of Lipschitz continuous functions on
Ω(resp, with compact support inΩ).

LetΩ ⊂ X be an open set. For any 1 6 p 6 +∞and f ∈ Lip_{loc}(Ω), itsW^{1,p}(Ω)-norm is
defined by

kfk_{W}1,p(Ω):=kfk_{L}p(Ω)+kLipfk_{L}p(Ω).
The Sobolev spacesW^{1,p}(Ω) is defined by the closure of the set

f ∈Lip_{loc}(Ω)| kfk_{W}1,p(Ω) <+∞

under theW^{1,p}(Ω)-norm. Remark that W^{1,p}(Ω) is reflexive for any 1 < p < ∞(see [13,
Theorem 4.48]). SpacesW_{0}^{1,p}(Ω) is defined by the closure of Lip_{0}(Ω) under theW^{1,p}(Ω)-
norm. We say a function f ∈W_{loc}^{1,p}(Ω) if f ∈W^{1,p}(Ω^{0}) for every open subsetΩ^{0} ⊂⊂Ω.

The following two facts are well-known for experts. For the convenience of readers, we include a proof here.

Lemma 2.5. (i) For any1< p<∞, we have W^{1,p}(X)=W_{0}^{1,p}(X).

(ii) W^{1,2}(X)=D(Ch).

Proof. If Xis compact, the assertion (i) is clear. Without loss of generality, we can assume
thatXis non-compact. Given a function f ∈Lip(X)∩W^{1,p}(X), in order to prove (i), it suffices
to find a sequence (fj)j∈Nof Lipschitz functions with compact supports inXsuch that fj → f
inW^{1,p}(X).

Consider a family of Lipschitz cut-offχjwith, for each j∈N,χj(x)=1 forx∈Bj(x_{0}) and
χj(x)=0 forx<Bj+1(x0), and 06χj(x)61,|∇χj|(x)61 for allx∈X. Now f·χj ∈Lip_{0}(X)
and f ·χj(x) → f(x) forµ-almost all x ∈ X. Notice that|f ·χj| 6 |f| ∈ L^{p}(X) for all j, the
dominated convergence theorem implies that f ·χj → f inL^{p}(X) as j → ∞. On the other
hand, since

|∇(f·χj)|6|∇f| ·χj+|f| · |∇χj|6|∇f|+|f| ∈L^{p}(X)

for all j ∈ N, we obtain that the sequence (f ·χj)j∈N is bounded in W^{1,p}(X). By noticing
thatW^{1,p}(X) is reflexive (see [13, Theorem 4.48]), we can see that f ·χj converges weakly
to f inW^{1,p}(X) as j→ ∞.Hence, by Mazurs lemma, we conclude that there exists a convex
combination of f ·χj converges strongly to f in W^{1,p}(X) as j → ∞. The proof of (i) is
completed.

Let us prove (ii). It is obvious thatW^{1,2}(X)⊂D(Ch), sinceLip(X)∩W^{1,2}(X)⊂D(Ch) and

|∇fn| ≤ Lip(fn). We need only to showD(Ch)⊂W^{1,2}(X). This follows immediately from the
fact that Lipschitz functions are dense inD(Ch). The proof of (ii) is completed.

3. The weakLaplacian and a local version ofBochner formula

Let (X,d, µ) be a metric measure space satisfyingRCD^{∗}(K,N) for someK ∈ RandN ∈
[1,∞). Fix any open setΩ ⊂ X. We will denote by the Sobolev spacesH^{1}_{0}(Ω) := W_{0}^{1,2}(Ω),
H^{1}(Ω) :=W^{1,2}(Ω) andH_{loc}^{1} (Ω) :=W_{loc}^{1,2}(Ω).

Definition3.1 (Weak Laplacian). LetΩ⊂ Xbe an open set, theLaplacianonΩis an operator
L on H^{1}(Ω) defined as the follows. For each function f ∈ H^{1}(Ω), its LapacianL f is a
functional acting onH_{0}^{1}(Ω)∩L^{∞}(Ω) given by

L f(φ) :=− Z

Ω

h∇f,∇φidµ ∀φ∈H_{0}^{1}(Ω)∩L^{∞}(Ω).

For anyg∈H^{1}(Ω)∩L^{∞}(Ω), the distributiong·L fis a functional acting onH_{0}^{1}(Ω)∩L^{∞}(Ω)
defined by

(3.1) g·L f(φ) :=L f(gφ) ∀φ∈H_{0}^{1}(Ω)∩L^{∞}(Ω).

This Laplacian (onΩ) is linear due to that the inner producth∇f,∇giis linear. The strongly local property of the inner productR

Xh∇f,∇gidµimplies that if f ∈H^{1}(X) and f =constant
onΩthenLf(φ)=0 for anyφ∈H_{0}^{1}(Ω)∩L^{∞}(Ω).

If, given f ∈H^{1}(Ω), there exists a functionu_{f} ∈L^{1}_{loc}(Ω) such that

(3.2) L f(φ)=Z

Ωuf ·φdµ ∀φ∈H_{0}^{1}(Ω)∩L^{∞}(Ω),

then we say that “Lf is a function in L^{1}_{loc}(Ω)” and write as “L f = uf in the sense of
distributions”. It is similar to say that “L f is a function in L_{loc}^{p} (Ω) or W_{loc}^{1,p}(Ω) for any
p∈[1,∞]”, and so on.

The operatorL satisfies the following Chain rule and Leibniz rule, which is essentially due to Gigli [18].

Lemma 3.2. LetΩbe an open domain of a metric measure space(X,d, µ)satisfying RCD^{∗}(K,N)
for some K∈Rand N∈[1,∞).

(i) (Chain rule) Let f ∈H^{1}(Ω)∩L^{∞}(Ω)andη∈C^{2}(R). Then we have
(3.3) L[η(f)]=η^{0}(f)·L f+η^{00}(f)· |∇f|^{2}.
(ii) (Leibniz rule) Let f,g∈H^{1}(Ω)∩L^{∞}(Ω). Then we have
(3.4) L(f·g)= f·Lg+g·L f+2h∇f,∇gi.

Proof. The proof is given essentially in [18]. For the completeness, we sketch it. We prove only the Chain rule (3.3). The proof of Leibniz rule (3.4) is similar.

Given anyφ∈H^{1}_{0}(Ω)∩L^{∞}(Ω), we have
L[η(f)](φ)=−

Z

Ωh∇[η(f)],∇φidµ=− Z

Ωη^{0}(f)· h∇f,∇φidµ,

where we have used that (see [18,§3.3]) the inner producth∇f,∇φisatisfies the Chain rule,
i.e.,h∇[η(f)],∇φi=η^{0}(f)· h∇f,∇φi.

On the other hand, by (3.1), we obtain
hη^{0}(f)·L f +η^{00}(f)· |∇f|^{2}i

(φ)=L f η^{0}(f)·φ+
Z

Ωη^{00}(f)· |∇f|^{2}·φdµ

=− Z

Ωh∇f,∇ η^{0}(f)·φidµ+
Z

Ωη^{00}(f)· |∇f|^{2}·φdµ

=− Z

Ωh∇f,∇φi ·η^{0}(f)dµ,

where we have used thatη^{0}(f)·φ∈H_{0}^{1}(Ω)∩L^{∞}(Ω) and that (see [18,§3.3]) the inner product
h∇f,∇gisatisfies the Chain rule and Leibniz rule, i.e.,

h∇f,∇ η^{0}(f)·φ

i=h∇f,∇φiη^{0}(f)+h∇f,∇ η^{0}(f)iφ

=h∇f,∇φiη^{0}(f)+h∇f,∇fi ·η^{00}(f)φ.

The combination of the above two equations implies the Chain rule (3.3). The proof is com-

pleted.

To compare the above Laplace operatorL on Xwith the generator ∆_{E} of the canonical
Dirichlet form (E,V), it was shown [18] that the following compatibility result holds.

Lemma 3.3(Proposition 4.24 in [18]). The following two statements are equivalent:

i) f ∈H^{1}(X)andL f is a function in L^{2}(X),
ii) f ∈D(∆_{E}).

In each of these cases, we haveL f = ∆Ef.

The following regularity result for the Poisson equation has been proved under a Bakry-
Emery type heat semigroup curvature condition, which is implied by the Riemannian curvature-
dimension conditionRCD^{∗}(K,N) (see [16, Theorem 7] and [5, Theorem 7.5]).

Lemma 3.4([27, 29]). Let(X,d, µ)be a metric measure space satisfying RCD^{∗}(K,N)for K ∈
Rand N ∈[1,∞). Let g∈L^{∞}(BR), where BRis a geodesic ball with radius R and centered at
a fixed point x_{0}. Assume f ∈H^{1}(BR)andL f = g on B_{R}in the sense of distributions. Then
we have|∇f| ∈L^{∞}_{loc}(BR),and

k|∇f|kL^{∞}(BR/2)6C(N,K,R)· 1

µ(BR)kfk_{L}1(BR)+kgkL^{∞}(BR)

.

Proof. In the case ofg=0, i.e., f is harmonic onBR, the assertion is proved in [27, Theorem
1.2] (see also [19, Theorem 3.9]). In the general case g ∈ L^{∞}(Ω), this is proved in [29,
Theorem 3.1]. The assertion of the constantC(N,K,R) depending only on N,K,R comes
from the fact that both the doubling constant and L^{2}-Poincare constant on a ball BR of a

RCD^{∗}(K,N)-space depend onN,KandR.

Now we will give a local version of the Bochner formula, Theorem 1.2, by combining the modified Bochner formula (2.2) and a similar argument in [24, 28].

Theorem 3.5([5, 24]). Let (X,d, µ) be a metric measure space satisfying RCD^{∗}(K,N) for
K∈Rand N∈[1,∞). Let BRbe a geodesic ball with radius R and centered at a point x_{0}.

Assume that f ∈ H^{1}(BR) satisfies Lf = g on BR in the sense of distributions with the
function g∈H^{1}(BR)∩L^{∞}(BR). Then we have|∇f|^{2} ∈H^{1}(BR/2)∩L^{∞}(BR/2)and

(3.5) 1

2L(|∇f|^{2})>hg^{2}

N +h∇f,∇gi+K|∇f|^{2}i

·µ on B_{R/2}
in the sense of distributions, i.e.,

−1 2

Z

BR/2

h∇|∇f|^{2},∇φidµ>

Z

BR/2

φ·g^{2}

N +h∇f,∇gi+K|∇f|^{2}
dµ

for any06φ∈H_{0}^{1}(B_{R/2})∩L^{∞}(B_{R/2}).

Proof. From Lemma 3.4 and thatg∈L^{∞}(BR), we know|∇f| ∈L_{loc}^{∞}(BR).

We take a cut-offχsatisfying (i) and (ii) in Lemma 2.4. Let ef(x) :=

f ·χ if x∈BR

0 if x∈X\B_{R}.

Then we have ef ∈Lip_{0}(BR). It is easy to check supp(L ef)⊂ BR. In fact, given anyψ∈H_{0}^{1}(X)
withψ=0 onB_{R}, the strongly local property implies thatR

Xh∇ef,∇ψidµ=0.

Now we want to calculateL ef onBR. By the Leibniz rule (3.4), we have, onBR, L ef =L(f ·χ)=χ·L f+ f·Lχ+2h∇f,∇χi

=χ·g+ f ·∆_{E}χ+2h∇f,∇χi ∈L^{∞}_{loc}(BR),

where we have usedg ∈ L^{∞}(BR) and |∇f| ∈ L^{∞}_{loc}(BR), and that χ,|∇χ|,∆_{E}χ ∈ L^{∞}(X) in
Lemma 2.4. Combining with supp(Lef)⊂BR, we haveL ef ∈L^{2}(X)∩L^{∞}(X). Therefore, by
Lemma 3.3, we get ef ∈D(∆E) and

(3.6) L^{2}(X)∩L^{∞}(X)3∆_{E}ef =L ef =

χ·g+ f ·∆_{E}χ+2h∇f,∇χi if x∈B_{R}

0 if x∈X\B_{R}.

According to Lemma 2.3(ii) and 06φ∈H^{1}_{0}(BR/2)⊂V, we conclude that|∇ef|^{2}∈Vand that
Z

X

− 1

2h∇|∇ef|^{2},∇φi+∆Eef· h∇ef,∇φi+φ·(∆Eef)^{2}
dµ

> 1 N

Z

X

φ·(∆Eef)^{2}dµ+K
Z

X

φ|∇ef|^{2}dµ.

Since ef = f onB_{R/2}, we have|∇f| =|∇ef|forµ-a.e. onB_{R/2}. Notice that|∇ef|^{2} ∈Vimplies
that|∇ef|^{2} ∈H^{1}(B_{R/2}).Then|∇f|^{2} ∈H^{1}(B_{R/2}),and|∇|∇f|^{2}|= |∇|∇ef|^{2}|inL^{2}(B_{R/2}).By (3.6)
and that|∇χ|= ∆Eχ= 0 onB_{R/2}(sinceχ =1 onB_{2R/3}), we have∆Eef =gonB_{R/2}. Hence,
we obtain

Z

BR/2

− 1

2h∇|∇f|^{2},∇φi+g· h∇f,∇φi+φ·g^{2}
dµ

> 1 N

Z

B_{R/2}

φ·g^{2}dµ+K
Z

B_{R/2}

φ|∇f|^{2}dµ.

(3.7)

Noticing thatg·φ ∈H_{0}^{1}(BR/2)∩L^{∞}(BR/2) andL f =gon BRin the sense of distributions,
we have

Z

B_{R/2}

g·gφdµ=L f(gφ)=− Z

B_{R/2}

h∇f,∇(gφ)idµ

=− Z

BR/2

h∇f,∇gi ·φdµ− Z

BR/2

h∇f,∇φi ·gdµ.

By combining this and (3.7), we get the desired inequality (3.5). The proof is finished.

By using the same argument of [8], one can get an improvement of the above Bochner formula. One can also consult a detailed argument given in [31, Lemma 2.3].

Corollary 3.6. Let(X,d, µ)be a metric measure space satisfying RCD^{∗}(K,N)for K ∈Rand
N∈[1,∞). Let BRbe a geodesic ball with radius R and centered at a fixed point x_{0}.

Assume that f ∈ H^{1}(BR) satisfies Lf = g on B_{R} in the sense of distributions with the
function g∈H^{1}(BR)∩L^{∞}(BR). Then we have|∇f|^{2}∈H^{1}(BR/2)∩L^{∞}(BR/2)and that the dis-
tributionL(|∇f|^{2})is a signed Radon measure on B_{R/2}. If its Radon-Nikodym decomposition
w.r.t.µis denoted by

L(|∇f|^{2})=L^{ac}(|∇f|^{2})·µ+L^{sing}(|∇f|^{2}),
then we haveL^{sing}(|∇f|^{2})>0and, forµ-a.e. x∈B_{R/2},

1

2L^{ac}(|∇f|^{2})> g^{2}

N +h∇f,∇gi+K|∇f|^{2}.
Furthermore, if N >1, forµ-a.e. x∈B_{R/2}∩

y: |∇f(y)|,0 ,

(3.8) 1

2L^{ac}(|∇f|^{2})> g^{2}

N +h∇f,∇gi+K|∇f|^{2}+ N

N−1·h∇f,∇|∇f|^{2}i
2|∇f|^{2} − g

N 2

. 4. The maximum principle

LetK∈RandN∈[1,∞) and let (X,d, µ) be a metric measure space satisfyingRCD^{∗}(K,N).

In this section, we will study the maximum principle on (X,d, µ). Let us begin from the Kato’s inequality forweighted measures.

4.1. The Kato’s inequality.

LetΩbe a bounded open set of (X,d, µ). Fix anyw ∈ H^{1}(Ω)∩L^{∞}(Ω), we consider the
weighted measure

µw :=e^{w}·µ on Ω.

Since, the densitye^{−kwk}^{L}^{∞}^{(}^{Ω}^{)} 6e^{w}6e^{kwk}^{L}^{∞}^{(}^{Ω}^{)} onΩ, we know that the associated the Lebesgue
spaceL^{p}(Ω, µw) and the Sobolev spacesW^{1,p}(Ω, µw) are equivalent to the originalL^{p}(Ω) and
W^{1,p}(Ω), respectively, for allp>1.Both the measure doubling property and theL^{2}-Poincare
inequality still hold with respect to this measureµw (the constants, of course, depend on
kwk_{L}^{∞}_{(}_{Ω}_{)}).

For this measureµw, we defined the associated LaplacianLwon f ∈H^{1}(Ω) by
Lwf(φ) :=−

Z

Ω

h∇f,∇φidµw

=− Z

Ω

h∇f,∇φie^{w}dµ
for anyφ∈H_{0}^{1}(Ω)∩L^{∞}(Ω).It is easy to check that

Lwf =e^{w}·L f +e^{w}· h∇w,∇fi

in the sense of distributions, i.e.,Lwf(φ)=L f(e^{w}·φ)+R

Ωh∇w,∇fie^{w}·φdµ.

When Ω be a domain of the Euclidean space R^{N} with dimension N > 1, the classical
Kato’s inequality states that given any function f ∈L^{1}_{loc}(Ω) such that∆f ∈L^{1}_{loc}(Ω), then∆f_{+}
is a signed Radon measure and the following holds:

∆f_{+}>χ[f >0]·∆f

in the sense of distributions, where f_{+} := max{f,0}.Here,χ[f > 0](x) = 1 for xsuch that
f(x)>0 andχ[f >0](x)=0 forxsuch that f(x)<0.In [11], the result was extended to the
case when∆f is a signed Radon measure.

In the following, we will extend the Kato’s inequality to the metric measure spaces (X,d, µw),
under assumption f ∈H^{1}(Ω).

Proposition 4.1(Kato’s inequality). LetΩ be a bounded open set of(X,d, µ) and let w ∈
H^{1}(Ω)∩L^{∞}(Ω). Assume that f ∈ H^{1}(Ω)such thatLwf is a signed Radon measure. Then
Lwf_{+}is a signed Radon measure and the following holds:

(4.1) Lwf_{+}>χ[f >0]·Lwf on Ω,

in the sense of distributions. In the sequel, we denote the Radon-Nikodym decomposition
Lwf =Lw^{ac}f·µ_{w}+Lw^{sing}f.

Proof. It suffices to prove the following equivalent property:

(4.2) Lw|f|>sgn(f)·Lwf,

where sgn(t)=1 fort>0, sgn(t)=−1 fort<0, and sgn(t)=0 fort=0.

Fix any >0 and let

f_{}(x) :=(f^{2}+^{2})^{1/2} >.
We have f_{}^{2}= f^{2}+^{2},

(4.3) |∇f_{}|= |f|

f_{}|∇f|6|∇f|
and

2f_{}·Lwf_{}+2|∇f_{}|^{2} =Lwf_{}^{2}=Lwf^{2}=2f·Lwf +2|∇f|^{2}.
Thus,

(4.4) Lwf_{} > f

f_{} ·Lwf,

Notice that|∇f_{}| 6 |∇f| and f_{} → |f| in L^{2}(Ω) implies that f_{} is bounded in H^{1}(Ω) and,
hence, there exists a subsequence f_{}_{j} converging weakly to|f|inH^{1}(Ω). Thus, the measures
Lw(f_{}_{j}) converges weakly toLw|f|.On the other hand, notice that f_{}(x) → |f(x)|for each
x∈Ωand that|f/f_{}|61 onΩ. Letting :=j →0 in (4.4), we conclude that

Lw(|f|)> f

|f| ·Lwf.

This is (4.2), and the proof is completed.

4.2. Maximum principles.

The above Kato’s inequality implies the maximum principle Theorem 1.3. Precisely, we have the following.

Theorem 4.2. LetΩbe a bounded domain. Let f(x) ∈ H^{1}(Ω)∩L_{loc}^{∞}(Ω) such thatL f is a
signed Radon measure withL^{sing}f > 0. Suppose that f achieves one of its strict maximum
inΩin the sense that: there exists a neighborhood U⊂⊂Ωsuch that

(4.5) sup

U

f >sup

Ω\U

f.

Here and in the sequel of the paper, the notionsup_{U} f means alwaysess sup_{U} f.Then, given
any w∈H^{1}(Ω)∩L^{∞}(Ω), for anyε >0, we have

(4.6) µn

x: f(x)>sup

Ω f −ε and L^{ac}f(x)+h∇f,∇wi(x)6εo

>0.

In particular, there exists a sequence of points{x_{j}}_{j∈}_{N}⊂U such that they are the approximate
continuity points ofL^{ac}f andh∇f,∇wi, and that

f(xj)>sup

Ω f −1/j and L^{ac}f(xj)+h∇f,∇wi(xj)61/j.

Proof. Suppose the first assertion (4.6) fails for some sufficiently smallε_{0}>0.Then we have
f−(sup_{Ω} f −ε_{0})

+∈H_{0}^{1}(Ω) (by the maximal property (4.5)) and
µn

x: f(x)>sup

Ω f −ε_{0} and L^{ac}f+h∇f,∇wi6ε_{0}o

=0.

Then for almostx∈ {y: f(y)>sup_{Ω} f −ε0}we have

Lw^{ac}f(x)·µw=e^{w(x)}·(L^{ac}f +h∇f,∇wi)(x)·µ >e^{−kwk}^{L}^{∞}ε_{0}·µ >0.

The assumptionL^{sing}f >0 implies thatLw^{sing}f >0. By applying the Proposition 4.1 to the
function f−(sup_{Ω}−ε_{0}), we have

Lw f −(sup

Ω f −ε_{0})

+>χ[f >sup

Ω f−ε_{0}]·Lw^{ac}f ·µw >0

onΩ,in the sense of distributions. Recall that the metric measure space (X,d, µw) satisfies a
doubling property and supports aL^{2}-Poincare inequality. Now the weak maximum principle
[13, Theorem 7.17] implies that f −(sup_{Ω} f −ε0)

+ =0 onΩ.Thus, sup_{Ω} f 6sup_{Ω} f −ε0

onΩ. This is a contradiction, and proves the first assertion (4.6).

The second assertion follows from the first one by takingε=1/j. Next, let us consider the parabolic version of the maximum principle. We need the follow- ing parabolic weak maximum principle.

Lemma 4.3. LetΩ be a bounded open subset and let T > 0. Let w ∈ H^{1}(ΩT)∩L^{∞}(ΩT)
with ∂tw(x,t) 6 C for some constant C > 0, for almost all (x,t) ∈ ΩT. Suppose that
f(x,t) ∈ H^{1}(ΩT)∩ L^{∞}(ΩT) with limt→0kf(·,t)k_{L}^{2}_{(Ω)} = 0 and, for almost all t ∈ (0,T),
that the functions f(·,t)∈H_{0}^{1}(Ω). Assume that, for almost every t∈(0,T), the function f(·,t)
satisfies

(4.7) Lw(·,t)f(·,t)− ∂

∂tf(·,t)·µ_{w(·,t)}>0 on Ω
Then we have

sup

Ω×(0,T) f(x,t)60.

Proof. The proof is standard via a Gaffney-Davies’ method (see also [49, Lemma 1.7]). We
include a proof here for the completeness. Since f_{+}meets all of conditions in this lemma, by
replacing f by f_{+}, we can assume that f >0.

Put

ξ(t) :=Z

Ωf^{2}(·,t)dµ_{w(·,t)}.

Sinceµ_{w(·,t)} =e^{w}·µ6e^{kwk}^{L}^{∞} ·µand f ∈H^{1}(ΩT), we have, for almost allt∈(0,T),
ξ^{0}(t)=Z

Ω∂t(f^{2})dµ_{w(·,t)}+Z

Ω f^{2}·∂tw·dµ_{w(·,t)}

6−2

Z

Ω|∇f|^{2}dµ_{w(·,t)}+C·ξ(t)6C·ξ(t),

where we have used∂tw 6 Cand that the functions f(·,t) ∈ H^{1}_{0}(Ω)∩L^{∞}(Ω) for almost all
t ∈ (0,T). By using limt→0ξ(t) = 0 (since ξ(t) 6 e^{kwk}^{L}^{∞} · kf(·,t)k_{L}^{2}_{(Ω)} and the assumption
limt→0kf(·,t)k_{L}2(Ω) = 0), one can obtain thatξ(t) 6 0. This implies f = 0 almost all inΩT.

The proof is finished.

By using the same argument as in Theorem 4.2, the combination of the Kato’s inequality and Lemma 4.3 implies the following parabolic maximum principle.

Theorem 4.4. LetΩbe a bounded domain and let T > 0. Let f(x,t) ∈ H^{1}(ΩT)∩L^{∞}(ΩT)
and suppose that f achieves one of its strict maximum inΩ×(0,T]in the sense that: there
exists a neighborhood U⊂⊂Ωand an interval(δ,T]⊂(0,T]for someδ >0such that

U×(δ,T]sup f > sup

ΩT\(U×(δ,T])f.

Heresup_{U×(δ,T]} f meansess sup_{U×(δ,T]} f . Assume that, for almost every t∈(0,T),L f(·,t)is
a signed Radon measure withL^{sing}f(·,t)>0. Let w∈H^{1}(ΩT)∩L^{∞}(ΩT)with∂tw(x,t)6C
for some constant C>0, for almost all(x,t)∈ΩT. Then, for anyε >0, we have

(µ× L^{1})n

(x,t) : f(x,t)>sup

ΩT

f −ε and L^{ac}f(x,t)+h∇f,∇wi(x,t)− ∂

∂tf(x,t)6εo

>0,
whereL^{1}is the 1-dimensional Lebesgue’s measure on(δ,T].

In particular, there exists a sequence of points{(xj,tj)}j∈N⊂U×(δ,T]such that every xj

is an approximate continuity point ofL^{ac}f(·,tj)andh∇f,∇wi(·,tj), and that
f(xj,tj)>sup

ΩT

f−1/j and L^{ac}f(xj,tj)+h∇f,∇wi(xj,tj)− ∂

∂tf(xj,tj)61/j.

Proof. We will argue by contradiction, which is similar to the proof of Theorem 4.2. Suppose
the assertion fails for some smallε_{0} > 0. Then, for almost all (x,t) ∈ {(y,s) : f(y,s) >

sup_{Ω}_{T} f −ε_{0}},we have

L^{ac}f(x,t)+h∇f,∇wi(x,t)− ∂

∂tf(x,t)>ε_{0}.
Thus, at such (x,t),

hLw^{ac}f(x,t)− ∂

∂tf(x,t)i

·µw

>h

L^{ac}f(x,t)+h∇f,∇wi(x,t)− ∂

∂tf(x,t)i

·e^{w}·µ>ε_{0}·e^{w}·µ>0.

The strictly maximal property of f gives that f_{ε}_{0} := f −(sup_{Ω}_{T} f −ε_{0})

+ ∈ H^{1}(ΩT) with
limt→0kf_{ε}_{0}(·,t)k_{L}^{2}_{(Ω)} = 0 and, for almost all t ∈ (0,T), that the functions f_{ε}_{0}(·,t) ∈ H_{0}^{1}(Ω).

Notice thatL_{w(·,t)}^{sing}f(·,t)>0 byL^{sing}f(·,t)>0. By using the Kato’s inequality, we have that,
for almost everyt∈(0,T),

Lw f−(sup

ΩT

f−ε0)

+>χ[f >(sup

ΩT

f−ε0)]·Lw^{ac}f

>χ[f >(sup

ΩT

f−ε_{0})]· ∂f

∂t ·µw= ∂

∂t f−(sup

ΩT

f−ε_{0})

+·µw.
Then Lemma 4.3 implies that f −(sup_{Ω}_{T} f −ε_{0})

+= 0 for almost all (x,t) ∈ΩT. This is a

contradiction.

5. LocalLi-Yau’s gradient estimates

LetK∈RandN∈[1,∞) and let (X,d, µ) be a metric measure space satisfyingRCD^{∗}(K,N).

In this section, we will prove the local Li-Yau’s gradient estimates–Theorem 1.3.

LetΩ⊂ Xbe a domain. GivenT >0, let us still denote ΩT := Ω×(0,T]

the space-time domain, with lateral boundaryΣand parabolic boundary∂PΩT : Σ:=∂Ω×(0,T) and ∂PΩT := Σ∪(Ω× {0}).

We adapt the following precise definition oflocally weak solutionfor the heat equation.

Definition5.1. LetT ∈ (0,∞] and let Ωbe a domain. A functionu(x,t) is called alocally
weak solutionof the heat equation onΩT ifu(x,t) ∈ H^{1}(ΩT) (= W^{1,2}(ΩT)) and if for any
subinterval [t1,t_{2}]⊂(0,T) and any geodesic ballB_{R} ⊂⊂Ω, it holds

(5.1)

Z t2

t1

Z

B_{R}

∂tu·φ+h∇u,∇φi

dµdt=0

for all test functionsφ(x,t) ∈ Lip_{0}(BR×(t_{1},t_{2}).Here and in the sequel, we denote always

∂tu:= ^{∂u}_{∂t}.

Remark5.2. The test functionsφin this definition can be chosen such that it has to vanish
only on the lateral boundary∂BR×(0,T). That is,φ∈ Lip(BR,T) withφ(·,t)∈ Lip_{0}(BR) for
allt∈(0,T).

The local boundedness and the Harnack inequality for locally weak solutions of the heat equation have been established by Sturm [49, 50] and Marola-Masson [39]. In particular, any locally weak solutions for the heat equation in Definition 5.1 must be locally H¨older continuous.

Letu(x,t) be a locally weak solution of the heat equation onΩ×(0,T).Fubini Theorem
implies, for a.e.t∈[0,T], that the functionu(·,t) ∈H^{1}(Ω) and∂tu ∈L^{2}(Ω). Hence, for a.e.

t∈(0,T), the functionu(·,t) satisfies, in the distributional sense,

(5.2) Lu=∂tu on Ω.

Conversely, if a functionu(x,t) ∈ H^{1} ΩT

and (5.2) holds for a.e. t ∈ [0,T], then it was shown [54, Lemma 6.12] thatu(x,t) is a locally weak solution of the heat equation onΩT.

In the case thatu(x,t) is a (globally) weak solution of heat equation onX×(0,∞) with ini-
tial value inL^{2}(X), the theory of analytic semigroups asserts that the functiont7→ kuk_{W}1,2(X)

is analytic. However, for a locally weak solution of the heat equation onΩT, we have not suf-
ficient regularity for the time derivative∂tu: in general,∂tuis only inL^{2}. This is not enough
to use Bochner formula in Theorem 3.5 to (5.2). For overcoming this difficulty, we recall the
so-calledSteklov average.

Definition 5.3. Given a geodesic ball B_{R} and a function u(x,t) ∈ L^{1}(BR,T), where B_{R,T} :=
BR×(0,T), theSteklov averageofuis defined, for everyε∈(0,T) and anyh∈(0, ε), by

(5.3) u_{h}(x,t) := 1

h Z h

0

u(x,t+τ)dτ, t∈(0,T−ε].

From the general theory of L^{p} spaces, we know that if u ∈ L^{p}(BR,T), then the Steklov
averageu_{h}converges touinL^{p}(BR,T−ε) ash→0, for everyε∈(0,T).

Lemma 5.4. If u∈H^{1}(B_{R,T})∩L^{∞}(B_{R,T}), then we have, for everyε∈(0,T), that
uh∈H^{1}(BR,T−ε)∩L^{∞}(BR,T−ε) and ∂tuh∈H^{1}(BR,T−ε)∩L^{∞}(BR,T−ε)
for every h∈(0, ε),and thatku_{h}k_{H}1(BR,T−ε)is bounded uniformly with respect to h∈(0, ε).

Proof. Sinceu ∈H^{1}(B_{R,T}), according to [22], there exists a functiong(x,t)∈ L^{2}(B_{R,T}) such
that

|u(x,t)−u(y,s)|6dP (x,t),(y,s)

·

g(x,t)+g(y,s) ,

for almost all (x,t),(y,s)∈B_{R,T} with respect to the product measuredµ×dt, whered_{P}is the
product metric onB_{R,T} defined by

d^{2}_{P} (x,t),(y,s):=d^{2}(x,y)+|t−s|^{2}.

Such a functiongis called a Hajłasz-gradient ofuonB_{R,T}(see [21,§8]). By the definition of
the Steklov averageuh, we have

|u_{h}(x,t)−uh(y,s)|6 1
h

Z h

0

g(x,t+τ)+g(y,s+τ)

·dP (x,t+τ),(y,s+τ)dτ

= 1 h

Z h

0

g(x,t+τ)+g(y,s+τ)

dτ·d_{P} (x,t),(y,s)

=

gh(x,t)+gh(y,s)

·dP (x,t),(y,s)

for almost all (x,t),(y,s)∈B_{R,T}. The factg(x,t)∈L^{2}(BR,T) implies thatgh(x,t)∈L^{2}(BR,T−ε)
for eachh ∈ (0, ε) and that the functionsg_{h} converges tog inL^{2}(BR,T−ε) as h → 0. Then
the previous inequality implies thatghis a Hajłasz-gradient ofuhon B_{R,T−ε}for allh∈(0, ε)
(see [21]). According to [21, Theorem 8.6], 2gh is a 2-weak upper gradient ofu_{h}. Thus we
conclude thatuh∈W^{1,2}(BR,T−ε) and

lim sup

h→0

Z

BR,T−ε

(|∇uh|^{2}+|∂tuh|^{2})dµdt6lim sup

h→0

Z

BR,T−ε

(2gh)^{2}dµdt64
Z

BR,T−ε

g^{2}dµdt.

Therefore, we get that kuhk_{H}1(BR,T−ε) is bounded uniformly with respect to h ∈ (0, ε) (by
combining withuh→uinL^{2}(BR,T−ε) ash→0).

Lastly, the assertionu_{h} ∈ L^{∞}(BR,T−ε) follows directly from the definition ofu_{h} andu ∈
L^{∞}(BR,T). The assertion of∂tufollows from that

∂tuh= u(x,t+h)−u(x,t)

h .

The proof is completed.