Gradient estimates for positive harmonic functions by stochastic analysis

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Gradient estimates for positive harmonic functions by stochastic analysis

Marc Arnaudon

, Bruce K. Driver

, Anton Thalmaier

aD´epartement de math´ematiques, Universit´e de Poitiers, T´el´eport 2 – BP 30179, 86962 Futuroscope Chasseneuil Cedex, France

bDepartment of Mathematics-0112, University of California at San Diego, La Jolla, CA 92093-0112, USA cMathematics Laboratory, University of Luxembourg, 162A, Avenue de la Fa¨ıencerie, 1511 Luxembourg, Luxembourg

Received 6 February 2006; received in revised form 9 July 2006; accepted 17 July 2006 Available online 4 August 2006

Abstract

We prove Cheng–Yau type inequalities for positive harmonic functions on Riemannian manifolds by using methods of Stochastic Analysis. Rather than evaluating an exact Bismut formula for the differential of a harmonic function, our method relies on a Bismut type inequality which is derived by an elementary integration by parts argument from an underlying submartingale. It is the monotonicity inherited in this submartingale which allows us to establish the pointwise estimates.

c

2006 Elsevier B.V. All rights reserved.

MSC:58J65; 60H30

Keywords:Harmonic function; Curvature; Gradient estimate; Harnack inequality

1. Introduction

The effect of curvature on the behavior of harmonic functions on a Riemannian manifoldM is a classical problem. A quantitative measurement of this behavior is encoded most directly in terms of gradient estimates and Harnack inequalities involving constants depending only on a lower bound of the Ricci curvature onM, the dimension ofM, and the radius of the ball on which the harmonic function is defined. Such estimates in global form, i.e., for positive harmonic

∗Corresponding author. Tel.: +352 46 66 44 6676; fax: +352 46 66 44 6237.

E-mail addresses:marc.arnaudon@math.univ-poitiers.fr(M. Arnaudon),driver@math.ucsd.edu(B.K. Driver), anton.thalmaier@uni.lu(A. Thalmaier).

0304-4149/$ - see front matter c2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.spa.2006.07.002

functions on Riemannian manifolds, are due to Yau [13]; local versions have been established by Cheng and Yau [3].

The classical proof of gradient estimates for a harmonic functionu relies on two ingredients.

The first ingredient is a comparison theorem for the Laplacian of the Riemannian distance function, which allows one to bound the mean curvature of geodesic spheres from above in terms of a lower bound on the Ricci curvature. The second ingredient is Bochner’s formula which is used to give a lower bound on1|gradu|² in terms of the lower bound on the Ricci curvature.

The gradient estimate itself then relies on a clever use of the maximum principle; see [9].

From a probabilistic point of view, it seems tempting to work with Bismut type representation formulas for the differential of a harmonic function.Theorem 1.1 below is taken from [11], [12] and gives a typical gradient formula; for related and more general results the reader may consult [4]. See [10] for additional background on Bismut formulas.

Notation 1. Throughout this paper,M is ann-dimensional Riemannian manifold withh·,·i,∇, div = ∇·, grad ,1= ∇ ·grad , and Ric denoting (respectively) the Riemannian metric, the Levi- Civita covariant derivative, the divergence operator, the gradient operator, the Laplacian, and the Ricci tensor onM. Forv∈ T_mM, let|v| := hv, vi^1/2, and forx,y∈ M, letd(x,y)denote the Riemannian distance betweenxandy.

For a relatively compact open subset, D ⊂ M, let∂D denote the boundary of Dand for x ∈Dlet

rD(x):=d(x, ∂D)= inf

y∈∂Dd(x,y)

be the distance ofxto∂D. (By convention, if the boundary∂Dis empty we setrD(x)= ∞.) LetK =K(D)≥0 be the smallest non-negative constant such that Ric(v, v)≥ −K|v|²for all v∈T D⊂T Mand letk=k(D)≥0 be defined by the equationK =(n−1)k².

Theorem 1.1 (Stochastic Representation of the Gradient).Let M be a complete Riemannian manifold, D⊂M be a relatively compact open domain, and u: D→Rbe a bounded harmonic function. Then, for anyv∈TxM and x ∈D,

(du)xv= −E

u(X_τ)Z _τ

0

Θ_s`˙s,dB_s

, (1.1)

where:

(1) X is a Brownian motion on M, starting at x , and τ =inf{t>0:Xt 6∈ D}

its time of first exit from D; the stochastic integral is taken with respect to the Brownian motion B in T_xM, related to X by the Stratonovich equation dB_t = //⁻¹t δX_t, where //_t:T_xM →T_XtM denotes the stochastic parallel transport along X .

(2) The processΘ takes values in the group of linear automorphisms of T_xM and is defined by the pathwise covariant ordinary differential equation

dΘ_t = −1 2Ric_//

t(Θ_t)dt, Θ₀=id_TxM,

where Ric_//t = //⁻¹t ◦Ric^]_X

t ◦ //t (a linear transformation of T_xM), and hRic^]_zu, wi = Ricz(u, w)for any u, w∈TzM, z ∈M.

(3) Finally,`t may be any adapted finite energy process taking values in T<sub>x</sub>M such that`0=v,

`τ =0and Z _τ

0

e^{K t}| ˙`t|²dt 1/2

∈ L^1+ε for someε >0, where K =K(D)is as inNotation1.

By making a clever choice for`t, it is possible to estimate the right hand side of Eq.(1.1)so as to obtain sharp estimates of the form

|gradu|(x)≤C(r_D(x) ,n,K)sup

x∈D

|u(x)|,

whereC(r,n,K)is a certain function ofr>0. See [12, Cor. 5.1] for details.

For positive harmonic functions however, one would like to estimate|gradu|(x)in terms of u(x)only. Such estimates provide elliptic counterparts to the famous parabolic Li–Yau estimates for solutions of the heat equation; see [6]. It is an intriguing problem how to gain such estimates in probabilistic terms from Bismut type formulas.

In this paper we deal with pointwise estimates of gradu(x)in terms of u(x) for positive harmonic functionsu. We show that such estimates may indeed be derived by stochastic analysis methods involving certain basic submartingales. In particular, we give a stochastic proof of the following gradient estimate due to Cheng and Yau [3]. In addition, our approach provides an explicit value for the constantc(n)appearing in(1.2); see(4.11)and(4.12).

Theorem 1.2. Let M be a complete Riemannian manifold of dimension n≥2and let D⊂M be a relatively compact domain. Let u:D→]0,∞[be a strictly positive harmonic function. Then

|gradu(x)| ≤c(n) k+ 1

rD(x)

·u(x), (1.2)

where k=k(D)and r_D(x)are as inNotation1.

Our method of proof is inspired by the stochastic approach to gradient estimates used in [12]

for harmonic functions, and in [1] for harmonic maps, where one represents, as described, the differential by a Bismut type mean value formula which may then be evaluated in explicit terms.

However in this paper, we do not use the mean value formula directly. Roughly speaking, Bismut type formulas are derived from certain underlying martingales by taking expectation. We sharpen this approach by constructing analogous submartingales (seeTheorem 3.1) which after taking expectation provide Bismut type inequalities (see Eq.(3.6)). From these probabilistic inequalities we are able to establish the pointwise estimates; seeTheorem 3.2andCorollary 3.5. Finally, as in [12], explicit constants depend on a reasonable choice of a finite energy process which is used for integration by parts on path space; seeTheorems 4.1–4.3.

2. Some elementary geometric calculations

Let M be a (not necessarily complete) Riemannian manifold of dimension n ≥ 2, and u:M → R be a harmonic function. For x ∈ M let ϕ(x) = |gradu|(x). For x ∈ M with ϕ(x)6=0, letn(x)=ϕ(x)⁻¹gradu(x).

If f:M →Ris a smooth function, then we have the well-known formula

grad f =grad1f +Ric^]gradf, (2.1)

where=trace∇²denotes the rough Laplacian onΓ(T M), and where by definition, hRic^]X,Yi =Ric(X,Y), X,Y ∈Γ(T M).

Eq.(2.1)applied tougives

gradu=Ric^]gradu. (2.2)

In the following lemma we calculate gradϕand1ϕin terms ofn.

Lemma 2.1. Let u:M →Rbe a harmonic function,ϕ = |gradu|, andn=(gradu)/ϕwhere it is defined. Then on M, whereϕdoes not vanish,

1ϕ=ϕ[Ric(n,n)+ |∇n|²

H.S.], (2.3)

grad logϕ= ∇_nn−(divn)n, (2.4)

and

|grad logϕ|²≤(n−1)|∇n|²

H.S.. (2.5)

Proof. (i) We start by proving Eq.(2.3). To this end, fixx∈ Msuch thatϕ(x)6=0 and choose an orthonormal frame(e_i)1≤i≤natxsuch that(∇_e

ie_j)(x)=0 for alli,j. Then, we have atx, (ϕn)=(1ϕ)n+2∇_e

igradu,n∇_e

in+ϕn. (2.6)

Since hn,ni = 1, we have 0 = vhn,ni = 2h∇_vn,ni for any v ∈ TxM. Thus, taking scalar product withnmakes the second term of the right hand side of(2.6)vanish and yields

1ϕ= h(ϕn),ni −ϕhn,ni. It is easily seen that

hn,ni = −|∇n|²_H.S.,

so that with Eq.(2.2)the claimed equality for1ϕfollows.

(ii) To establish(2.4), note that1u=0 can be written as 0=div(ϕn)=dϕ (n)+ϕdivn.

Letn^[= hn,·i =ϕ⁻¹hgradu,·i =ϕ⁻¹du. Then on one hand, ιndn^[ =ιn(−ϕ⁻²dϕ∧du)= −ϕ⁻²(dϕ (n)du−du(n)dϕ)

= −ϕ⁻²(−ϕdivndu−ϕ|n|²dϕ)

=divnhn,·i + hgrad logϕ,·i, while on the other hand,

ιndn^[= h∇_nn,·i − h∇_.n,ni = h∇_nn,·i. Comparing these last two equations proves Eq.(2.4).

(iii) Finally, to establish(2.5), note first that, as a consequence of(2.4),

|grad logϕ|²=(divn)²+ |∇_nn|².

Next, fixx ∈ M such thatϕ(x)6=0 and choose an orthonormal frame(e⁰_i)1≤i≤natxsuch that e_n⁰ =n. Then

|grad logϕ|² =

n−1

X

j=1

h∇_e0 jn,e⁰_ji

!2

+ |∇_nn|²

≤ (n−1)

n−1

X

j=1

h∇_e0

jn,e⁰_ji²+ |∇_e0 ne⁰_n|²

≤ (n−1)

n

X

j=1

|∇_e0

jn|²=(n−1)|∇n|²

H.S.. 3. Gradient estimates for positive harmonic functions

The following theorem gives the submartingale property which will be crucial for our estimates; see Bakry [2] for related analytic results.

Theorem 3.1. Let M be a (not necessarily complete) Riemannian manifold of dimension n≥2.

Let X be a Brownian motion on M and let u:M→Rbe a harmonic function. For anyα≥ ⁿ⁻²

n−1, the process

Y_t := |gradu|^α(X_t)exp

−α 2

Z t 0

Ric(n,n)(X_r)dr

(3.1) is a local submartingale, where by convention,Ric(n,n)(x) := 0 at points x wheregradu(x) vanishes.

Proof. First assume that gradudoes not vanish onM. Then, making use of Eq.(2.3), we have 1ϕ^α =αϕ^α−11ϕ+α(α−1)ϕ^α−2|gradϕ|²

=αϕ^α(Ric(n,n)+ |∇n|²_H.S.+(α−1)|grad logϕ|²). (3.2) Since our assumption onαis equivalent toα−1≥ −1/(n−1), it now follows from the estimate in Eq.(2.5)that

|∇n|²_H.S.+(α−1)|grad logϕ|²≥ |∇n|²_H.S.− 1

n−1|grad logϕ|²≥0. Combining this estimate with Eq.(3.2)shows

1ϕ^α ≥αϕ^αRic(n,n). (3.3)

An application of Itˆo’s lemma now impliesYt =Nt+At where dN_t =exp

−α 2

Z t 0

Ric(n,n)(X_r)dr

h//⁻¹_t gradϕ^α(X_t),dB_ti, and

dAt = 1 2

1ϕ^α

ϕ^α (Xt)−αRic(n,n)(Xt) Ytdt.

(Here//t is stochastic parallel translation along X and B is theT_xM-valued Brownian motion introduced in Theorem 1.1.) By the inequality (3.3), dAt ≥ 0 and therefore, Yt is a local

submartingale, which completes the proof under the assumption thatϕ never vanishes on M. This assumption however is easily removed by letting Ric(n,n)(x) = 0 in (3.1)at points x where gradu(x)=0.

Indeed, let[0, ζ[be the maximal interval on which our Brownian motionXis defined. Fixing ε >0, we consider the partition

0=τ0≤σ1≤τ1≤σ2≤τ2≤ · · · of[0, ζ[defined by

σi =inf{t ≥τi−1:Yt ≤ε/2} and τi =inf{t ≥σi :Yt ≥ε}, i≥1.

Now considerY_t^ε := Y_t ∨εwhich is seen to be a local submartingale on each of the sub- intervals of our partition. Indeed, on[σi, τi[the process is constant,Y^ε≡ε, while on[τi−1, σi[ it is a local submartingale, since thereY itself is a local submartingale by Itˆo’s formula, as shown above, using the fact that X|[τi−1, σi[takes its values in{x ∈ M: gradu(x) 6= 0}. Now since eachY^εis a local submartingale,Y_t =lim_ε↓0Y_t^εitself is a local submartingale.

Remark 2. In(3.1)we adopted the convention that Ric(n,n)(x)=0 at pointsxwhere gradu(x) vanishes. It should be noted that any other convention also gives a local submartingale as well.

Remark 3. InAppendixwe provide a generalization of Eq.(3.3), along with a unified proof of the submartingale property ofYt in(3.1). The argument there directly takes care of the possible vanishing of the gradient of u and does not require the case distinction made in the proof of Theorem 3.1.

Theorem 3.2. Let M be a Riemannian manifold of dimension n ≥ 2and let u:M → Rbe a harmonic function. Further let α ∈ [ⁿ⁻²

n−1,2[ withα > 0, and p > 1, q > 1 such that p⁻¹+q⁻¹ =1. For x ∈ M, let X be a Brownian motion on M starting at x , and denote byτ the time of first exit of X from some relatively compact neighbourhood D of x . Further assume thatρis a bounded stopping time withρ ≤τ and that`t is a real-valued decreasing, adapted process with C<sup>1</sup>paths such that`0=1,`_ρ =0. Then

|gradu|(x)≤ I1(α,p)·I2(α,p) (3.4)

where

I1(α,p)=E

"

Z _ρ

0

|gradu|²(Xs)ds

αp/2#1/αp

and

I2(α,p)=E

"Z _ρ

0

exp −α

2−α Z s

0

Ric(n,n)(Xr)dr

| ˙`s|^2/(2−α)ds

₍2−α)q/2#1/αq

. Proof. LetY_tbe the process defined in Eq.(3.1). Under our assumptionsYˆ_t :=Y_ρ∧tis a bounded non-negative local submartingale which according to Doob and Meyer may be decomposed as Yˆ_t = ˆY₀+ ˆN_t+ ˆA_t whereNˆ₀= ˆA₀ =0;Nˆ is the local martingale part, andAˆis the drift part.

We further assert thatYˆ_t is aL²-submartingale in the sense that k ˆN∞k²

2=E[h ˆN,Nˆi_∞]<∞ and k ˆA∞k

2<∞. (3.5)

To verify the estimates in (3.5), let(T_n)n≥0 be an increasing sequence of stopping times converging almost surely to+∞, such that each stopped processYˆⁿ:= ˆY^Tn is a submartingale.

WritingNˆⁿ= ˆN^Tn,Aˆⁿ= ˆA^Tn, we have by Itˆo’s formula h ˆNⁿ,Nˆⁿi_t =(Yˆ_tⁿ)²−(Yˆ₀ⁿ)²−2

Z t 0

Yˆ_sⁿdYˆ_sⁿ ≤R²−2 Z t

0

Yˆ_sⁿdNˆ_sⁿ

whereRis an upper bound forYˆ. (Here we already used the fact thatYˆ ≥0 and that Aˆis non- decreasing.) Taking expectation yieldsE[h ˆNⁿ,Nˆⁿi_∞] ≤ R², from which the first estimate in (3.5)follows by monotone convergence. To bound theL²-norm of the total variation ofA, sinceˆ Aˆ is non-decreasing, it suffices to bound k ˆA∞k

2. But since Yˆ∞ := lim_t→∞Yˆ_t exists almost surely, it follows that

k ˆA∞k

2≤ k ˆY∞− ˆY₀k

2+ k ˆN∞k

2≤ R+ k ˆN∞k

2<∞. Let

St :=Yt`t− Z t

0

Ys`˙sds.

SinceS0=Y0=ϕ^α(x)and dSt =`tdYt, S_t =ϕ^α(x)+

Z t 0

`sdY_s

is a local submartingale. Moreover, since`is bounded and S_ρ∧t =ϕ^α(x)+

Z t 0

`sdYˆs,

it is clear thatS_ρ∧t is also aL²-submartingale. In particular we have ϕ^α(x)=S0≤E

S_ρ

= −E Z _ρ

0

Ys`˙sds

=E Z _ρ

0

Ys| ˙`s|ds

.

Combining this inequality with the definition ofY in Eq.(3.1)implies ϕ^α(x)≤E

Z _ρ

0

ϕ^α(X_s)exp

−α 2

Z s 0

Ric(n,n)(X_r)dr

| ˙`s|ds

. (3.6)

Assumingα <2, an application of H¨older’s inequality shows ϕ(x)≤ E

"

Z _ρ

0

ϕ²(X_s)ds α/2

× Z _ρ

0

exp

− α 2−α

Z s 0

Ric(n,n)(X_r)dr

| ˙`s|^2/(2−α)ds

(2−α)/2#1/α

. (3.7) One more application of H¨older’s inequality to Eq.(3.7)then gives Eq.(3.4).

To estimateI1(α,p)we use the following lemma.

Lemma 3.3. Forβ ∈]0,1[, let γ_β =2^−1/2



 Γ_1−β

2

Γ

1 2





1/β

=2^−1/2



 Γ_1−β

2

√π





1/β

. (3.8)

Then for every positive local martingale Z with infinite lifetime and deterministic starting point Z0=z, we have

E[hZi^β/_∞²]^1/β ≤γβz,

wherehZi_∞=lim_t↑∞hZi_t andhZi_t is the quadratic variation process associated with Z . Proof. Without loss of generality we may assume that z = 1. Moreover, applying the Dambis–Dubins–Schwarz Theorem (cf. [7] or [8]), by “enriching” the filtered probability space if necessary, we may assume there exists a Brownian motionBstarting at 0 such that

Z_t =1+B_hZi_t.

LetT :=inf{t ≥0, 1+B_t =0}. By the reflection principle, P{B_t ≤ −1} = 1

2P{T ≤t} for allt≥0,

and the scaling property of Brownian motion, we conclude thatT has the same law as 1/B₁². Consequently,

E[T^β/2]^1/β =E[|B1|^−β]^1/β=γβ. Moreover, we havehZi_∞≤T a.s., so that

E[hZi^β/_∞²]^1/β ≤E[T^β/2]^1/β =γ_β. To exploit estimate(3.4)we now chooseα∈ [ⁿ⁻²

n−1,1[ifn ≥3,α∈]0,1[ifn =2, and p >1 such thatβ :=αp<1.

Proposition 3.4 (Gradient Estimate; General Form). Let M be a Riemannian manifold of dimension n ≥ 2and D ⊂M be a relatively compact domain. For x ∈ D, let X be Brownian motion starting at x , τ be the time of first exit of X from D, and ρ be a bounded stopping time such that ρ ≤ τ. Assume that u:M →]0,∞[ is a positive harmonic function and let n=gradu/|gradu|whengradu6=0andn=0otherwise. Further, letα∈ [ⁿ⁻²

n−1,1[∩]0,1[and q >1be such thatβ:= ^q

q−1α <1. Then, for each x∈ D, the following estimate holds:

|grad logu|(x)

≤γβE

"Z _ρ

0

exp

− α 2−α

Z s 0

Ric(n,n)(Xr)dr

| ˙`s|^2/(2−α)ds

₍2−α)q/2#1/αq

(3.9) whereγβ is given by(3.8)and the process`sis chosen as inTheorem3.2.

Proof. Lemma 3.3applied toZ_t :=u(X^ρ_t)gives

E

"

Z _ρ

0

ϕ²(Xs)ds

_β/2#1/β

≤γ_βu(x). (3.10)

Bounding the term I1(α,p)in estimate (3.4)by means of (3.10), and dividing by u(x), the claimed inequality follows from(3.4).

Corollary 3.5. Keeping the assumptions of Proposition 3.4, if u:M →]0,∞[ is a positive harmonic function, then

|grad logu|(x)≤γ_βE

"

Z _ρ

0

exp αK s

2−α

| ˙`s|^2/(2−α)ds

(2−α)q/2#1/αq

, (3.11)

whereRic≥ −K = −K(D)as inNotation1.

4. Explicit constants

In order to estimate the right hand sides in Eqs. (3.9) and (3.11), we are going to use the methods developed in [12]; see especially [12, Corollary 5.1]. Throughout this section the assumptions onαandqfromProposition 3.4will be preserved.

Fixx∈ Mand letD⊂Mbe a relatively compact open neighbourhood ofxinMwith smooth boundary. Let f ∈C²(D¯)be a positive function onDwhich is bounded by 1 and vanishing on

∂Dand letXt be a Brownian motion onM starting atx∈ D. Define T(s):=

Z s 0

f⁻²(X_t)dt, s< τD, and

ρ(t):=inf{s≥0:T(s)≥t}, (4.1)

whereτDis the time of first exit ofXfromD. Alternatively we may expressρas ρ (t)=

T⁻¹(t) ift ≤T (τD)

∞ ift >T(τD) , from which we see that

ρ (˙ t)= 1

T⁰(ρ (t)) = f²(X_ρ(t)) fort<T(τD) .

In particular it follows thatρ(t) ≤ t for t < T(τD). By [12, Proposition 2.3], the process X_t⁰:=X_ρ(t)is a diffusion with generatorL⁰:= ¹

2f²1, and infinite lifetime and as a consequence, T(τD)= ∞a.s.

The idea now is to use the fact thatT(t) ↑ ∞, as t ↑ τD, to construct the required finite energy process`s meeting the crucial conditions`0 = 1 and`ρ = 0. More precisely, we fix t>0 (ρ(t)will be ourρinCorollary 3.5) and let

h₀(s):=

Z s 0

f⁻²(X_r)1_{r<ρ(t)}dr=T(ρ (t)∧s) .

Hence h₀(s) = h₀(ρ(t)) = T(ρ(t)) = t for s ≥ ρ(t). Further let h₁ ∈ C¹([0,t],R) be a function with non-positive derivative such that h₁(0) = 1 and h₁(t) = 0, and define

`s := (h<sub>1</sub> ◦h<sub>0</sub>)(s). Since `s has non-positive derivative, | ˙`s|ds is a probability measure on [0, ρ(t)].

Theorem 4.1 (Gradient Estimate; Specific Form). Let M be a Riemannian manifold of dimension n ≥ 2, and D be a relatively compact open domain in M with smooth boundary

∂D. Let f ∈C²(D¯)be strictly positive on D and vanish on∂D, and K =K(D)≥0be chosen so thatRic≥ −K as inNotation1. Further suppose that0< α∈ [ⁿ⁻²

n−1,1[and q>1/ (1−α). Then, letting p=q/ (q−1)be the conjugate exponent of q, for any positive harmonic function u:M →]0,∞[and any x∈ D, we have

|grad logu|(x)≤ 1 f(x)



 Γ

1−αp 2

Γ

1 2





1/αp

pC(α,K,q,f) (4.2)

where

C(α,K,q, f):=sup

y∈D

{K f²(y)−(f1f)(y)+(αq+1)|grad f|²(y)}.

Proof. Our assumptions onαimply that pα <1 andq >2/ (2−α). Since the inequality in Eq. (4.2)is invariant under scaling f by a positive constant, we may assume without loss of generality that f is bounded above by 1. We want to estimateI^1/αqwhere

I :=E



 Z _ρ(t)

0

exp αK s

2−α

| ˙`s|^2/(2−α)ds

!(2−α)q/2

.

By means of Jensen’s inequality, we get I =E



 Z _ρ(t)

0

exp αK

2−αs

| ˙`s|<sup>α/(2−α)</sup>| ˙`s|ds

!₍2−α)q/2



≤ E

"

Z _ρ(t) 0

exp αK q

2 s

| ˙`s|<sup>qα/2</sup>| ˙`s|ds

#

=E

"

Z _ρ(t) 0

exp αK q

2 s

| ˙`s|^(qα+2)/2ds

#

=E

"

Z _ρ(t) 0

exp αK q

2 s

| ˙h₁(h₀(s))|^(qα+2)/2| ˙h₀(s)|^(qα+2)/2ds

#

=E

"

Z _ρ(t)

0

exp αK q

2 s

| ˙h₁(h₀(s))|^(qα+2)/2f^−qα−2(X_s)ds

#

=E Z _t

0

exp αK q

2 ρ(r)

| ˙h₁(r)|^(qα+2)/2 f^−qα(X⁰_r)dr

= Z t

0

| ˙h1(r)|^(qα+2)/2E

exp αK q

2 ρ(r)

f^−qα(X⁰_r)

dr. (4.3)

LetZ_s =e^{αK qρ(s)/2} f^−αq(X⁰_s). Writing=^mfor equality up to a differential of a local martingale, we have

dZ_s =^m 1

2Z_s(αK qρ⁰(s)+ f²(X⁰_s)(1f^−αq)(X_s⁰))ds

=m 1

2Zs(αK q f²(X_s⁰)−αq(f1f)(X_s⁰)+αq(αq+1)|gradf|²(X⁰_s))ds

which implies

dZ_s ≤dM_s+C₁Z_sds (4.4)

whereMis a local martingale and C1:= 1

2 sup

x∈D

{αK q f²(x)−αq(f1f)(x)+αq(αq+1)|gradf|²(x)}.

LetD_n⊂Dbe an increasing sequence of relatively compact open subsets ofDsuch thatx∈ D_n and∪D_n=D. Ifσnis the time of first exit ofX⁰fromD_n, thenM^σnis a martingale,E[Z_s^σn]<∞ for alls, and from Eq.(4.4),

E[Z^σ_sⁿ] ≤C₁ Z s

0

E[Z_t^σn]dt.

It now follows by an application of Gronwall’s lemma along with Fatou’s lemma that E[Z_s] ≤lim inf

n→∞ EZ_s^σn ≤ Z0e^C1s = f^−αq(x)e^C1s. Using this estimate in Eq.(4.3)gives

I^1/αq ≤ Z t

0

f^−αq(x)e^C1s| ˙h₁(s)|^(qα+2)/2ds 1/αq

= 1 f(x)

Z t 0

e^C1s| ˙h₁(s)|^(qα+2)/2ds 1/αq

=: 1

f(x)J¯(t,h₁) . Fora >0, let J(t,a):= ¯J(t,h₁)whereh₁(s):=1−^1−e−as

1−e^−at fors∈ [0,t]. It then follows by Corollary 3.5(withρ =ρ (t)≤τ ∧t≤τ) that

|grad logu|(x)≤γβ 1

f(x)J(t,a) (4.5)

where, by a simple computation, J(t,a)=

a 1−e^−at

₍qα+2)/(2αq) 1−e^{(C1−(qα+2)a/2)t} (qα+2)a/2−C1

!1/αq

.

Now suppose thata>2C₁/(qα+2). Then inf

t>0J(t,a)≤ lim

t→∞J(t,a)= a^(qα+2)/2 (qα+2)a/2−C1

!1/αq

and the minimum ina>2C₁/(qα+2)of the latter expression is√

2C₁/αq=√

C(α,K,q, f) which is attained ata =2C₁/αq. Consequently we have shown

inf

a>0inf

t>0J(t,a)≤p

C(α,K,q,f)

which combined with Eq.(4.5)(recall thatβ = αp andγ_β is given as in Eq.(3.8)) gives the estimate in Eq.(4.2).

Remark 4. Note thatC(α,K,q, f)inTheorem 4.1differs from the constantc1(f)/2 in [12]

only by the coefficient,αq+1, which in [12] was the number “3”.

Remark 5. LetMbe a complete Riemannian manifold,x∈ M,R(y):=d(x,y), Cut(x)⊂M denote the cut-locus ofx, and forr >0 letD =Br(x)(the open ball aboutxof radiusr) and define f:D→Rby

f(y):=cos π

2 r R(y)

. (4.6)

It is shown in the proof of corollary 5.1 in [12] that

|gradf| ≤ π

2r and −1f ≤ π√

K(n−1) 2r +π²n

4r² onD\Cut(x) . Hence it follows that onD\Cut(x),

K f²−(f1f)+(αq+1)|grad f|²≤C(α,q,r) where

C(α,q,r)= π² 4

n+αq+1

r² +π

2

√

K(n−1)

r +K

= π² 4

n+αq+1

r² +π

2

k(n−1)

r +k²(n−1) . HereK =(n−1)k²is as inNotation 1.

Now suppose thatr ∈]0, ιx[, whereιx =d(x,Cut(x))is the distance fromx to Cut(x). In this caseDis precompact (becauseM is complete),∂Dis smooth, and f ∈C² D¯

with f >0 onDand f =0 on∂D. Hence byTheorem 4.1, ifu is a positive harmonic function defined on a neighbourhood ofD, then for all¯ y∈D,

|grad logu|(y)≤

cos

πd(x,y) 2r

−1



Γ_1−αp

2

Γ

1 2





1/αp

pC(α,q,r),

whereα,q, andpare as inTheorem 4.1. In particular, takingy=xin this inequality shows

|grad logu|(x)≤





Γ_1−αp

2

Γ

1 2





1/αp

pC(α,q,r). (4.7)

Whenr ≥ ιx, the functiond²(x,·)is no longer smooth on Br(x)and the boundary of Br(x) need not be smooth either. Hence, it is not permissible to applyTheorem 4.1directly to obtain the estimate in(4.7). Nevertheless, using the methods in the proof of [12, Corollary 5.1] the estimate in Eq.(4.7)still holds for arbitraryr >0.

Theorem 4.2. Let α,q, and p be as in Theorem4.1and continue with the notation used in Remark5. Then for any r >0and positive harmonic function u:B_r(x)→]0,∞[, the estimate in Eq.(4.7)holds.

Proof. Fix x ∈ M and let R(y) := d(x,y). By Kendall [5, Corollary 1.2], there exists a continuous adapted increasing process,L_t, such that

d(R(Xt))= hgradR(Xt) , //tdBti + 1

21R(Xt)dt−dLt (4.8)

where//t andB_t are as inTheorem 1.1, and gradR(y)and1R(y)are interpreted as being zero at pointsy ∈ M whereRfails to be smooth. In particular, ifg ∈ C²(R), then by Eq.(4.8)and Itˆo’s formula we have

d(g(R(X_t)))=g⁰(R(X_t))

hgradR(X_t) , //_tdB_ti +1

21R(X_t)dt−dL_t

+1

2g⁰⁰(R(X_t))|gradR(X_t)|²dt. (4.9) Now at points whereRis smooth,

1 (g(R))= ∇ ·(g⁰(R)∇R)=g⁰(R) 1R+g⁰⁰(R)|gradR|². Hence Eq.(4.9)may be written as

d(g(R(Xt)))= hgrad(g◦R) (Xt) , //tdBti +1

21(g◦R)(Xt)dt−g⁰(R(Xt))dLt

with the convention that1(g◦R)=0 and grad(g◦R)=0 at points whereRis not smooth.

Let f be defined as in Eq.(4.6),ρ (s)be as in Eq.(4.1), andX_s⁰ =X_ρ(s)be as above, then d(g(R(X⁰_s)))=^m 1

2 f²(X⁰_s)1 (g(R)) (X_ρ(s))ds−g⁰(R(X_s⁰))dL_ρ(s). (4.10) Letg(τ):=(cos(_2r^πτ))^−αqand, as in the proof ofTheorem 4.1, let

Z_s =e^{αK qρ(s)/2} f^−αq(X_s⁰)=e^{αK qρ(s)/2}g(X⁰_s).

Then, using Eq.(4.10)and the convention that1f (y)=0 and grad f(y)=0 at pointsy∈ D whereRis not smooth, we have

dZ_s =^m 1

2 Z_s(αK qρ⁰(s)+ f²(X⁰_s)(1f^−αq)(X_s⁰))ds−e^{αK qρ(s)/2}g⁰(R(X_s⁰))dL_ρ(s)

≤m 1

2 Z_s(αK q f²(X_s⁰)−αq(f1f) (X_s⁰)+αq(αq+1)|gradf(X_s⁰)|²)ds,

where we have used −e^{αK qρ(s)/2}g⁰(R(X⁰_s)) ≥ 0 and L_ρ(s) is increasing in s. With this observation, the remainder of the proof ofTheorem 4.1 starting with Eq. (4.4)goes through without any further modification. Hence the statement ofTheorem 4.1holds forD=Br(x)and f =cos(πR/ (2r)). Therefore the argument used to arrive at Eq.(4.7)is still valid provideduis a positive harmonic function defined on a neighbourhood ofB¯_r(x). A simple limiting argument shows that Eq.(4.7)is still valid even whenu is a positive harmonic function defined only on B_r(x).

Theorem 4.3 (Gradient Estimate).Let Mⁿbe a complete Riemannian manifold, n ≥2, and let D⊂ M be a relatively compact domain, k=k(D)≥0be chosen so thatRic≥ −(n−1)k², and r_D(x) = dist(x, ∂D)as in Notation1. If u: D →]0,∞[is a strictly positive harmonic function, then

|gradu(x)| ≤c(n)

k+ 1 rD(x)

u(x) for all x ∈D, where

c(2):= π 2

r3 2 exp

−Γ⁰ 1

2 Γ

1

2 , (4.11)

and for n≥3,

c(n):= π 2



 Γ

1 2(2n−3)

Γ

1 2





(n−3/2)/(n−2)

p3(n−1) /2. (4.12)

As already mentioned, Theorem 4.3 is originally due to Cheng and Yau [3]. Our approach however gives an explicit value for the constantc(n).

Proof. Letr =rD(x). Ifn=2, letα↓0 in Eq.(4.7)to conclude that

|grad logu|(x)≤ 1

√ 2 exp

−Γ⁰ 1

2 Γ

1 2

sπ² 4

3 r²+π

2 k r +k². Ifn≥3, letα= ⁿ⁻²

n−1, andq = ²

1−α =2(n−1). Then again from Eq.(4.7)we find

|grad logu|(x)≤γαp

s (n−1)

3π² 4

1 r² +π

2 1 rk+k²

where

γ_αp= 1

√ 2



 Γ

1 2(2n−3)

Γ

1 2





(n−3/2)/(n−2)

.

This completes the proof of the theorem in view of the following simple estimate:

3π² 4

1 r² +π

2 k

r +k²≤ 3π² 4

1 r +k

2

.

Corollary 4.4 (Gradient Estimate on Geodesic Balls). Let M be a complete Riemannian manifold withRic ≥ −(n−1)k², k ≥ 0. If u is a positive harmonic function on a geodesic ball Br(x)⊂M, then

sup

Br/2(x)

|gradu|

u ≤c(n) k+2

r

.

In particular, if Ric≥0then any positive harmonic function u on M is constant.

Corollary 4.5 (Elliptic Harnack Inequality). Let M be a complete Riemannian manifold with Ric≥ −(n−1)k². Suppose that u is a positive harmonic function on a geodesic ball B_r(x)⊂M.

Then sup

B_r/2(x)u≤exp(c(n) (2+kr)) inf

B_r/2(x)u. (4.13)

Proof. By the gradient estimate, we have sup_B

r/2(x)|gradu|/u ≤ c(n) (k+2/r). Letx1,x2 ∈ B¯r/2(x) be such that sup_Br/2(x)u = u(x1) and infB_r/2(x)u = u(x2). Then let σ: [0,1] →

B¯_r/2(x) be a parametrization of the path following a length minimizing geodesic from x2 to xconcatenated with a length minimizing geodesic joiningxtox1. Then

logu(x₁) u(x₂) =

Z 1 0

d logu(σ (s))

ds ds

≤ Z 1

0

|grad logu|(σ (s))|σ⁰(s)|ds

≤ c(n) k+2

r Z 1

0

|σ⁰(s)|ds

≤ c(n) k+2

r

·r from which Eq.(4.13)follows.

Acknowledgements

The authors are grateful to the referee whose careful reading and detailed comments have significantly improved the paper. B.K. Driver was partially supported by NSF Grant no. DMS- 0504608.

Appendix. Expansion on the submartingale proof

We start by generalizing inequality(3.3).

Lemma A.1. Letε≥0, u:M →Rbe a harmonic function, M⁰= {x∈ M :gradu(x)6=0}, ϕ_ε =

q

|gradu|²+ε², and

n_ε(x):=ϕ_ε⁻¹gradu =(|gradu|²+ε²)^−1/2gradu,

with the convention thatn₀(x):=0if x 6∈M⁰. Then forα∈ [ⁿ⁻²

n−1,1],

1ϕ_ε^α ≥αϕ^α_εRic(n_ε,n_ε) (A.1)

where(A.1)holds for all x∈ M if ε >0and for all x∈ M⁰if ε=0.

Proof. Suppose eitherε >0 orε=0 and gradu(x)6=0. We begin by observing that 1ϕ_ε^α =αϕ_ε^α−11ϕ_ε+α (α−1) ϕ^α_ε⁻²|gradϕ_ε|²

=αϕ_ε^α1ϕε

ϕε

+(α−1)|grad logϕ_ε|²

. (A.2)

Set f_ε(s)=(ε²+s)^1/2, so thatϕ_ε(x)= f_ε |gradu(x)|² . Then f_ε⁰(s)= 1

2(ε²+s)^−1/2 and f_ε⁰⁰(s)= −1

4(ε²+s)^−3/2

and forv∈T M,

vϕε =2f_ε⁰(|gradu|²)h∇_vgradu,gradui

=ϕ_ε⁻¹h∇_vgradu,gradui

= h∇_vgradu,n_εi = h∇_n

εgradu, vi (A.3)

where in the last equality we have used the fact that∇ has zero torsion. From this equation it follows that

gradϕ_ε = ∇_n

εgradu = ∇_n

ε(ϕ_εn_ε)=n_εϕ_ε·n_ε+ϕ_ε∇_n

εn_ε, (A.4)

and in particular that

grad logϕ_ε =n_εlogϕ_ε·n_ε+ ∇_n

εn_ε. (A.5)

Since

divn_ε =div(ϕ⁻¹_ε gradu)= −ϕ_ε⁻²hgradϕ_ε,gradui +ϕ_ε⁻¹1u

= − hgrad logϕ_ε,n_εi = −n_εlogϕ_ε, Eq.(A.5)may be written as

grad logϕ_ε = ∇_n

εn_ε−(divn_ε)n_ε. (A.6)

From Eq.(A.3)we also have

∇_v²_⊗vϕ_ε = h∇_v²_⊗vgradu,n_εi + h∇_vgradu,∇_vn_εi

= h∇_v²_⊗vgradu,n_εi + h∇_v(ϕ_εn_ε) ,∇_vn_εi

= h∇_v²_⊗vgradu,n_εi +vϕ_ε· hn_ε,∇_vn_εi +ϕ_εh∇_vn_ε,∇_vn_εi which upon summing onvrunning through an orthonormal frame shows

1ϕε = hgradu,n_εi + hn_ε,∇_gradϕεn_εi +ϕε|∇n_ε|²

H.S.

= hgrad1u+Ric^#gradu,n_εi + hn_ε,∇_gradϕεn_εi +ϕ_ε|∇n_ε|²_H.S.

= hRic^#∇u,n_εi + hn_ε,∇_gradϕ

εn_εi +ϕ_ε|∇n_ε|²

H.S.

=ϕε{Ric(n_ε,n_ε)+ hn_ε,∇_{grad logϕε}n_εi + |∇n_ε|²_H.S.}. (A.7) Whenε6=0 and gradu(x)=0, we see from Eq.(A.7)that

1ϕε(x)=ϕε(x)|∇n_ε|²_H.S.(x)

and from Eq.(A.4)that(grad logϕ_ε) (x)=0. Combining these identities with Eq.(A.2)gives (1ϕ_ε^α) (x)=αϕ_ε^α(x)|∇n_ε|²

H.S.(x)≥αϕ_ε^α(x)Ric(n_ε,n_ε) (x)=0. (A.8) This shows that Eq.(A.1)is valid forx 6∈ M⁰. To finish the proof it suffices to show that Eq.

(A.1)is valid for allx∈ M⁰.

So for the rest of the proof we will assume thatx∈M⁰. Sincehn₀,n₀i =1 onM⁰,

0=v1=2h∇_vn₀,n₀i for allv∈T_xM and x ∈M⁰. (A.9) Takingε=0 in Eq.(A.7)gives

1ϕ0=ϕ0(Ric(n₀,n₀)+ |∇n₀|²_H.S.) (A.10)

and from Eqs.(A.6)and(A.9)we find

|grad logϕ0|²= |∇_n

0n₀|²+(divn₀)² (A.11)

onM⁰. LetF_ε(s):=(s²+ε²)^α/2so thatϕ_ε^α =F_ε(ϕ0)and by elementary calculus, F_ε⁰(s)=αs(s²+ε²)⁻¹F_ε(s) and

F_ε⁰⁰(s)=α(s²+ε²)⁻²(ε²−(1−α)s²)F_ε(s).

Therefore we find

1ϕ_ε^α =1F_ε(ϕ0)=div(F_ε⁰(ϕ0)gradϕ0)=F_ε⁰(ϕ0) 1ϕ0+F_ε⁰⁰(ϕ0)|gradϕ0|²

=αϕ0(ϕ₀²+ε²)⁻¹F_ε(ϕ0) 1ϕ0+α(ϕ₀²+ε²)⁻²(ε²−(1−α) ϕ²0)F_ε(ϕ0)|gradϕ0|²

≥ αϕ_ε^α{ϕ0(ϕ₀²+ε²)⁻¹1ϕ0−(ϕ₀²+ε²)⁻²(1−α) ϕ0²|gradϕ0|²}

=αϕ_ε^α ϕ²₀ ϕ²₀+ε²

(1ϕ0

ϕ0

−(1−α) ϕ₀²

ϕ₀²+ε²|grad logϕ0|² )

≥ αϕ_ε^α ϕ²₀ ϕ²₀+ε²

1ϕ0

ϕ0

−(1−α)|grad logϕ0|²

=αϕ_ε^α ϕ²₀

ϕ²₀+ε²{Ric(n0,n0)+ |∇n0|²_H.S.−(1−α)|grad logϕ0|²} (A.12) where in the last equality we have used Eq.(A.10).

Letting{e_i}ⁿ

i=1be an orthonormal frame such thate_n=n₀shows (divn0)² =

n

X

i=1

∇_e

in0,e_i

!2

=

n−1

X

i=1

∇_e

in0,e_i

!2

≤ (n−1)

n−1

X

i=1

∇_e

in₀,ei2

≤(n−1)

n−1

X

i=1

|∇_e

in₀|² and therefore, using Eq.(A.11),

|grad logϕ0|² = |∇_n

0n₀|²+(divn₀)²

≤ (n−1)

n−1

X

i=1

|∇_e

in₀|²+ |∇_e

nn₀|²≤(n−1)|∇n₀|²

H.S.. Combining this estimate with Eq.(A.12)shows

1ϕ_ε^α ≥αϕ^α_ε ϕ₀²

ϕ₀²+ε²{Ric(n₀,n₀)+(1−(1−α) (n−1))|∇n₀|²

H.S.}. (A.13) Takingα≥ ⁿ⁻²

n−1(which is equivalent to 1−(1−α) (n−1)≥0) in Eq.(A.13)implies 1ϕ_ε^α ≥αϕ^α_ε ϕ₀²

ϕ₀²+ε²Ric(n₀,n₀)=αϕ_ε^αRic(n_ε,n_ε) .

Combining this estimate with that in Eq.(A.8)proves inequality(A.1).