BOUNDS FOR THE FUNDAMENTAL SOLUTION OF

ELLIPTIC AND PARABOLIC EQUATIONS

IN NONDIVERGENCE FORM

Luis Escauriaza

Universidad del Paıs VascoApartado 644, 48080 Bilbao, Spain.

email: mtpeszul@lg.ehu.esIn memory of Eugene Fabes

Abstract. It is shown that any elliptic or parabolic operator in nondiver-

gence form with measurable coefficients has a global fundamental solution

verifying certain pointwise bounds.

1. Introduction

Given operators in divergence form

Eu =n∑

i,j=1

Di(aij(x)Dju) and Pu =n∑

i,j=1

Di(aij(X)Dju)−Dtu

where x ∈ Rn, t ∈ R , X = (x, t), and under the assumptions that thecoefficient matrices of both operators are measurable functions verifying forsome positive constant ν ∈ (0, 1](1.1)

ν|ξ|2 ≤n∑

i,j=1

aij(x)ξiξj ≤ ν−1|ξ|2 , ν|ξ|2 ≤n∑

i,j=1

aij(X)ξiξj ≤ ν−1|ξ|2

for all x, ξ ∈ Rn and t ∈ R , it is well known that both operators have aglobal Green’s function or fundamental solution defined in the whole space.In particular, for both the elliptic operator E and the parabolic operator P

Typeset by AMS-TEX

1

2

and all x ∈ Rn and X ∈ Rn+1, there exist functions g(x, y) and G(X,Y ),Y = (y, s), verifying

(1.2) ϕ(x) = −∫

Rn

g(x, y)Eϕ(y) dy

(1.3) ψ(X) = −∫ t

−∞

∫Rn

G(X,Y )Pψ(Y ) dY

for all ϕ ∈ C∞0 (Rn) and ψ ∈ C∞0 (Rn+1).The basic facts about the fundamental solution for selfadjoint elliptic op-

erators in divergence form were proved by W. Littman, G. Stampachia andH. Weinberger in [32], where they showed that when the dimension n ≥ 3,there is a constant N = N(n, ν) verifying

(1.4) N−1|x− y|2−n ≤ g(x, y) ≤ N |x− y|2−n

for all x, y ∈ Rn. In the two-dimensional setting, the existence and boundsof the global fundamental solution were obtained by C.E. Kenig and Wei-Ming Ni in [27], where it is shown that the fundamental solution necessarilychanges sign and for some constant N as above

|g(x, y)| ≤ N (1 + | log |x− y||) .

In [11], S. Chanillo and Li Yanyan derived that g(x, y) is a function ofbounded mean oscillation in the plane whose BMO norm can be estimated interms of ν. Finally, the study of the fundamental solution for nonselfadjointelliptic operators in divergence form was carried out by M. Gruter and K.O.Widman [26], where it is shown that (1.4) also holds in this setting.

The first author establishing Gaussian bounds for the fundamental solu-tion G(X,Y ) of a parabolic operator in divergence form with measurablecoefficients was D.G. Aronson [2]. He proved that for some constant N asabove,

(1.5) N−1(t− s)−n2 e−

N|x−y|2t−s ≤ G(X,Y ) ≤ N(t− s)−

n2 e−

|x−y|2N(t−s)

when X,Y ∈ Rn+1 and t > s.The proofs of these results relied on the fundamental works of Nash, Moser

and De Giorgi on the Holder continuity and Harnack’s inequality for solutionsof elliptic and parabolic equations in divergence form [12],[33],[34], [37], andsome of them, for instance, Aronson’s proof of the Gaussian lower bound,depend strongly on Harnack’s inequality. On the other hand, it is known

3

that the bounds (1.4) and (1.5) for the fundamental solution imply the cor-responding Harnack’s inequality for nonnegative solutions, and thus it wasinteresting to obtain these estimates independently of Harnack’s inequality.In this direction, we should mention the works [21], [22] by E.B. Fabes andD.W. Stroock.

The purpose of this work is to extend the above results to the setting ofelliptic and parabolic equations in nondivergence form with measurable co-efficients, and to obtain the corresponding“analog”of the pointwise bounds.The arguments we use rely on a Harnack inequality, but a Harnack inequal-ity for normalized adjoint solutions (See the definition below). We believethat the concept of normalized adjoint solution in the elliptic case was firstintroduced by P. Bauman in [5], where the corresponding Harnack inequality(Theorem 2.3) was proved. Later, these estimates were of great help in workslike [9], [14], [17] and [13]. The normalized adjoint solutions in the paraboliccase were first used in [16], here E.B. Fabes, N. Garofalo and S. Salsa proveda Harnack inequality for normalized adjoint solutions but only applying tonormalized adjoint solutions vanishing on the lateral boundary of a cylinder.In this work we prove a general Harnack inequality for them and use it toobtain estimates for the fundamental solution.

In the elliptic setting we consider an operator

(1.6) Eu =n∑

i,j=1

aij(x)Diju

whose coefficient matrix is symmetric, defined in Rn, and satisfying (1.1) forsome ν ∈ (0, 1]. Also, in what follows N will denote a constant dependingon n ≥ 2 and ν, Br a ball of radius r centered at the origin and Br(x) a ballof radius r centered at x ∈ Rn.

The following definitions are given for operators E with measurable coef-ficients, but the functions involved should be regarded as smooth (except forthe natural singularities) when the coefficients of E are smooth.

Definitions. A function g(x, .) ∈ L1loc(Rn) is called a fundamental solution

for E, with pole at x ∈ Rn if (1.2) holds for all ϕ ∈ C∞0 (Rn). A functionw ∈ L1

loc(Ω) is called an adjoint solution for E in an open set Ω, if for everyϕ ∈ C∞0 (Ω) ∫

Ω

w(y)Eϕ(y) dy = 0 .

LetW be a fixed nonnegative and nontrivial adjoint solution for E, a functionw defined in Ω is a normalized adjoint solution for E relative to W in Ω, ifwW ∈ L1

loc(Ω) is an adjoint solution for E in Ω. A nonnegative function win Ω is a Muckenhoupt weight in the reverse Holder class B n

n−1(Ω) if there is

4

a constant N such that

(1.7)

(—∫

Br(x)

wn

n−1 dy

)n−1n

≤ N—∫

Br(x)

w dy

for all balls with B2r(x) ⊂ Ω. When w is defined in Rn and (1.7) holds forall the balls Br(x) we say w ∈ B n

n−1.

In the statements of theorems 1.1, 1.2 and 1.3 and until we say the contrarywe assume that the coefficients of the operators involved are smooth in thewhole space except possibly at infinity.

Theorem 1.1. There exists a unique nonnegative adjoint solution W definedin Rn such that

(1.8) W (B1) =∫

B1

Wdy = |B1| .

Moreover, W ∈ B nn−1

with a constant N = N(n, ν) and E has a globalfundamental solution g with pole at the origin verifying

|g(y)| ≤ N

(1 + |

∫ |y|

1

s

W (Bs)ds|

)W (y) .

Moreover,i. E has a nonnegative fundamental solution if and only if

∫∞1

sW (Bs) ds <

+∞, and in this case there is a fundamental solution g such that for ally ∈ Rn

N−1

∫ ∞

|y|

s

W (Bs)ds W (y) ≤ g(y) ≤ N

∫ ∞

|y|

s

W (Bs)ds W (y) .

ii. E has a nonpositive fundamental solution if and only if∫ 1

0s

W (Bs) ds <

+∞, and in this case there is a fundamental solution g such that for ally ∈ Rn

N−1

∫ |y|

0

s

W (Bs)ds W (y) ≤ − g(y) ≤ N

∫ |y|

0

s

W (Bs)ds W (y) .

Finally, if∫ 1

0s

W (Bs) ds and∫∞1

sW (Bs) ds are both infinite, any fundamental

solution necessarily changes sign.

Before dealing with all the generality in the parabolic case we explain whathappens when the coefficients are time independent. Again, we are assumingthat the coefficients are smooth.

5

Theorem 1.2. Let P denote the parabolic operator Pu = Eu−Dtu. Then,P has a fundamental solution G(X,Y ) = Γ(x, t−s, y) where for all x, y ∈ Rn

and t > 0

N−1 max 1W (B√t(x)) ,

1W (B√t(y)) e

−N|x−y|2t W (y) ≤ Γ(x, t, y)

Γ(x, t, y) ≤ N min 1W (B√t(x)) ,

1W (B√t(y)) e

− |x−y|2Nt W (y) .

Let P denote a parabolic operator

(1.9) Pu =n∑

i,j=1

aij(X)Diju−Dtu

with symmetric coefficients matrix satisfying (1.1). For Z = (z, τ) ∈ Rn+1

define Qr(Z) = Br(z)× (τ −r2, τ +r2), and for function W defined on Rn+1,F ⊂ Rn+1, E ⊂ Rn set

W (F ) =∫

F

W dY , W (E, τ) =∫

E

W (y, τ) dy

Definitions. A nonnegative function G(X, .) ∈ L1loc(Rn+1) is called a fun-

damental solution for P with pole at X if for all α < β∫ β

α

∫Rn

G(X,Y ) dY < +∞

and (1.3) holds for all ψ ∈ C∞0 (Rn+1). A function w ∈ L1loc(Ω) is called and

adjoint solution for P in an open set Ω ⊂ Rn+1 if for all ψ ∈ C∞0 (Ω)∫Ω

w(Y )Pψ(Y ) dY = 0 .

A nonnegative function w is called a parabolic Muckenhoupt weight in thereverse Holder class Bn+1

n(Ω) if for all Q2r(Z) ⊂ Ω

(1.10)

(—∫

Qr(Z)

wn+1

n dY

) nn+1

≤ N—∫

Qr(Z)

w dY,

and when w is defined in Rn+1 and (1.10) holds for all Qr(Z), we say w ∈Bn+1

n.

6

Theorem 1.3. Let P be as above, then there exists a unique nonnegativeadjoint solution W defined on Rn+1 verifying W (B1 × (0, 1)) = |B1| and(1.11)W (B2r(z), τ +θr2) ≤ N W (Br(z), τ) , W (B2r(z), τ) ≤ N W (Br(z), τ +θr2)

for all Z ∈ Rn+1, r > 0 and 0 ≤ θ ≤ 1. Moreover, W ∈ Bn+1n

with constantN = N(n, ν) and P has a global fundamental solution G verifying

N−1 max 1W (B√t−s(x),s) ,

1W (B√t−s(y),s) e

−N|x−y|2t−s W (Y ) ≤ G(X,Y )

G(X,Y ) ≤ N min 1W (B√t−s(x),s) ,

1W (B√t−s(y),s) e

− |x−y|2N(t−s) W (Y )

for all X,Y ∈ Rn+1 with t > s. Also, G(X, .) ∈ Bn+1n

(Ω(X,K, ε)) for allK > 0 and 0 < ε < 1 with constant N = N(n, ν,K, ε) , where

Ω(X,K, ε) = Y ∈ Rn+1 : | x− y | ≤ K√t− s , s ≤ εt .

When the operators E and P have measurable coefficients, the usual com-pactness arguments and the above estimates show that the claims in theorems1.1 and 1.2 still hold, though (1.11) should be understood as if the associ-ated nonnegative adjoint solution W for P had a restriction [18] to each timeτ ∈ R as a measure W (dy, τ) verifying (1.11) and

(1.12)∫ +∞

τ

∫Rn

W (Y )Pψ(Y ) dY =∫

Rn

ψ(y, τ) W (dy, τ)

for all ψ ∈ C∞0 (Rn+1).Recall that an equivalent formulation of what has been called the “weak

uniqueness property”for elliptic equations is the following: The weak unique-ness property holds for the operator E on a smooth domain Ω ⊂ Rn whenfor all f ∈ L∞(Ω), there is a unique generalized or viscosity solution (in thesense of [9], [10], [29], [36] and [39]) for the problem Eu = −f in Ω, u = 0on ∂Ω.

A simple corollary of the above results and along the lines of the resultsin [29] is the following.

Theorem 1.4. The weak uniqueness property holds for E on all boundedsmooth domains if and only if there is a unique nonnegative adjoint solutionW for E on Rn verifying (1.8).

On the other hand N.S. Nadirashvili [36], has constructed an elliptic op-erator E for which the weak uniqueness fails in the unit ball, and as a conse-quence this operator or any of its possible extensions to Rn has at least twodifferent nonnegative and global adjoint solutions verifying (1.8).

Finally, ifW is one of the global adjoint solutions associated to an operatorE the following holds

7

Theorem 1.5. If the coefficients of E are continuous at infinity, then forall ε > 0 there is Nε such that for r ≥ 1

Nε−1rn−ε ≤W (Br) ≤ Nεr

n+ε .

If the coefficients are continuous at zero, then for all ε > 0 there is Nε suchthat for r ≤ 1

Nε−1rn+ε ≤W (Br) ≤ Nεr

n−ε .

Therefore, if n ≥ 3 and the coefficients are continuous at infinity thereis a nonnegative fundamental solution, and can not exist a nonpositive fun-damental solution if the coefficients are continuous at zero. On the otherhand, when n = 2 and even if the coefficients are both continuous at zeroand at infinity the information in the above theorem does not imply that afundamental solution must change sign, and in fact we show examples whereany of the possible cases occur. When n ≥ 3 it is possible to give examples ofoperators were again all possibilities take place, but of course these operatorsfail to have nice coefficients at some of the points.

In section 2 we prove theorems 1.1, 1.4, 1.5 and give examples of ellipticoperators showing the different behaviors of the fundamental solution. Insection 3 we prove the results related to parabolic operators.

2. The elliptic case

Recall that a Green’s function for E on a smooth domain Ω with pole atx ∈ Ω, is a nonnegative function gΩ(x, .) ∈ L1(Ω) verifying

ϕ(x) = −∫

Ω

gΩ(x, y)Eϕ(y) dy

for all ϕ ∈ C2(Ω) with ϕ = 0 on ∂Ω.Then, under the assumption that the coefficients of E are smooth on Rn

except possibly at infinity the following results are well known.

Theorem 2.1. For any bounded smooth domain Ω and all x ∈ Ω there isa unique Green’s function for E and Ω with pole at x. Moreover, gΩ(x, .) ∈B n

n−1(Ω) with a constant N and

(∫Ω

gΩ(x, y)n

n−1 dy

)n−1n

≤ Nd(Ω)

where d(Ω) denotes the diameter of Ω.

8

Theorem 2.2. (Muckenhoupt properties of adjoint solutions) Let w be anonnegative adjoint solution for E in Ω. Then, w ∈ B n

n−1(Ω) with a constant

N = N(n, ν).

Theorem 2.3. (Harnack Inequality for normalized adjoint solutions) LetW be a nonnegative and nonzero adjoint solution for E on Ω and w a non-negative normalized adjoint solution for E relative to W on B5r(x) ⊂ Ω.Then,

supBr(x)

w ≤ N infBr(x)

w.

Theorem 2.4. (Strong maximum principle for normalized adjoint solutions)Let W be as above and w a normalized adjoint solution for E, relative to Win Ω. Then, w can not attain a maximum or minimum in the interior of Ωunless it is constant.

Theorem 2.5. (Holder continuity of normalized adjoint solutions) Let Wbe as above and w be a normalized adjoint solution in a ball Br(x) ⊂ Ω.Then, there exists α = α(n, ν) ∈ (0, 1] such that for all y ∈ Br(x)

|w(y)− w(x)| ≤ N

(|y − x|r

)α

supBr(x)

|w|

Theorem 2.6. (Bounds for normalized adjoint solutions) Let w be a nor-malized adjoint solution relative to W in a ball Br(x) ⊂ Ω. Then

supBr/2(x)

|w| ≤ N—∫

Br(x)

|w| Wdy

where the average is taken with respect to Wdx.

The first claim in theorem 2.1 follows from the fundamental works byA.D. Aleksandrov [1] and C. Pucci [38], the second claim and theorem 2.2were proved by E.B. Fabes and D.W. Stroock [22] and theorem 2.3 by P.Bauman [5, Theorem 4.4]. Theorems 2.4 and 2.5 are standard consequencesof Harnack’s inequality and the fact that constants are normalized adjointsolutions. A proof of theorem 2.6 can be found in [13, Theorem 2.3].

Remark 2.1. Before we proceed we make the following remarks which willhelp the reader to understand in what sense are true the results in theorem1.1 for operators with measurable coefficients:

i. The weak uniqueness property holds for E in Ω if and only if there is aunique Green’s function for E in Ω [29].

ii. If w ∈ L1loc(Ω) is a weak nonnegative adjoint solution for E in Ω,

defining wε = w ∗ θε + ε, where θ is a smooth regularization of the identity

9

supported in the unit ball, one easily verifies that wε is a nonnegative adjointsolution in Ωε = x ∈ Ω : d(x, ∂Ω) > ε , for the operator Eε withcoefficients given by aε

ij = ((aijw) ∗ θε + εδij) /wε. Then, letting ε → 0 itfollows from theorem 2.2 that w ∈ B n

n−1(Ω) with constant N = N(n, ν).

iii. Reviewing the proof of theorem 2.3, if W and w are as in theorem 2.3with B5r ⊂ Ω, the integration by parts argument used in [5, Theorem 4.4]gives

N−1gε1(2ren, y)

∫B4r

W ε dx ≤W ε(y) ≤ Ngε1(2ren, y)

∫B4r

W ε dx

N−1gε2(2ren, y)

∫B4r

(wW )εdx ≤ (wW )ε (y) ≤ Ngε

2(2ren, y)∫

B4r

(wW )εdx

for all y ∈ Br, and where gε1, g

ε2 are respectively the Green’s functions for

the smooth operators associated as before to W and wW in B5r. Passing tothe limit as ε→ 0 gives

N−1g1(2ren, y)∫

B4r

W dx ≤W (y) ≤ Ng1(2ren, y)∫

B4r

W dx

N−1g2(2ren, y)∫

B4r

wW dx ≤ wW (y) ≤ Ng2(2ren, y)∫

B4r

wW dx

for almost every y ∈ Br, and where both g1, g2 are Green’s functions forE in B4r with pole at 2ren. Hence, if the weak uniqueness holds for E onsubsets of Ω , g1 = g2, and the above inequalities imply that theorem 2.3still remain true for weak normalized adjoint solutions. Of course, the samehappens with theorem 2.4.

Proof of theorem 1.1. Letting gR denote the Green’s function for E on BR

with pole at the origin, define

WR(y) =g2R(y)− gR(y)

α(R)where α(R) = —

∫B1

(g2R − gR) dy

The maximum principle and theorem 2.2 imply that WR is a nonnegativeadjoint solution in BR, WR ∈ B n

n−1(BR) with constants depending on n and

ν. Also, WR(B1) = |B1|, and the Muckenhoupt property of WR [35] impliesthere exists θ = θ(n, ν) > 0, possibly large, such that for 0 < s ≤ t ≤ R/2(

|Bs||Bt|

)θ

≤ NWR(Bs)WR(Bt)

,

10

and in particular WR(Br) ≤ N |Br|θ for 1 ≤ r ≤ R/2. The last bound andcompactness imply the existence of a sequence Rk tending to infinity suchthat WRk

converges weakly in Ln

n−1loc (Rn) to a global nonnegative adjoint

solution W defined in Rn verifying W ∈ B nn−1

with constants depending onn and ν, W (B1) = |B1|.

If W1, W2 are two global nonnegative adjoint solutions verifying (1.8),W1/ (W1 +W2) is a bounded normalized adjoint solution relative to W1+W2

in Rn, and the identity W1 = W2 follows from the Liouville-type theoremimplied by theorem 2.5 and the normalization condition (1.8).

Recall that the normalized Green’s function for a bounded domain Ω isdefined as gΩ(x, y) = gΩ(x, y)/W (y), gΩ ∈ C

(Ω× Ω\ (x, y) : x = y

),

gΩ(x, y) = 0 for y ∈ ∂Ω and x ∈ Ω. The normalized Green’s function forBR with pole at the origin will be denoted gR. The following is proved by P.Bauman in [7, Theorem 2.3]

Theorem 2.7. There is a constant N = N(n, ν) such that for x, y ∈ BR/2(z)with |x− y| ≤ R/2

N−1

∫ R

|x−y|

s

W (Bs(x))ds ≤ gBR(z)(x, y) ≤ N

∫ R

|x−y|

s

W (Bs(x))ds .

The second claim in theorem 1.1 follows by compactness and the followingestimate.

Lemma 2.1. There is a constant N = N(n, ν) such that for |y| ≤ R/8

|gR(y)−(∫

B1

gR dx

)W (y)| ≤ N

(1 + |

∫ |y|

1

s

W (Bs)ds|

)W (y) .

Proof Lemma 2.1. Recall that the Muckenhoupt property of W [35] impliesthere is N such that

(2.1) W (B2r(x)) ≤ N W (Br(x)) for all x ∈ Rn, r > 0

The maximum principle (theorem 2.4) implies gR − gr ≤ max∂BrgR on Br

for r < R, and in particular

0 ≤ max∂Br

gR −(gR − gr/2

)+(gr − gr/2

)on Br/2 .

From (2.1) and theorem 2.7, gr − gr/2 ≤ N r2

W (Br) on Br/2. Hence, on thisball

gR − gr/2 ≤ Nr2

W (Br)+ max

∂Br

gR ,

11

and taking the maximum of the function in the left hand side of the lastinequality over Br/2

(2.2) max∂Br/2

gR −max∂Br

gR ≤ Nr2

W (Br)for r < R .

Again, by the maximum principle, gR ≤ max∂Br/2 gR on BR\Br/2, and from(2.2)

Nr2

W (Br)+ max

∂Br

gR − gR ≥ 0 on BR\Br/2 .

Applying the Harnack inequality to the above nonnegative normalized adjointsolution in B2r\Br/2 gives

sup∂Br

(N

r2

W (Br)+ max

∂Br

gR − gR

)≤ N inf

∂Br

(N

r2

W (Br)+ max

∂Br

gR − gR

)when r ≤ R/2. Thus,

(2.3) max∂Br

gR −min∂Br

gR ≤ Nr2

W (Br)for r ≤ R/2.

Taking averages with respect to the measure Wdy we have

—∫

Br

|gR −min∂Br

gR| Wdy = —∫

Br

(gR − gr −min∂Br

gR) Wdy + —∫

Br

gr Wdy

and from the maximum principle for normalized adjoint solutions, (2.3) andtheorem 2.1

—∫

Br

|gR −min∂Br

gR| Wdy ≤ Nr2

W (Br)for r ≤ R/2 .

Using standard arguments for functions of bounded mean oscillation and(2.1) the last inequality imply that

(2.4)

—∫

Br

|gR −—∫

Br

gR Wdx| Wdy ≤ Nr2

W (Br)

|—∫

B1

gR Wdy −—∫

B2k

gR Wdy| ≤ N |∫ 2k

1

s

W (Bs)ds|

when r ≤ R/2 and k ∈ Z with 2k ≤ R/8. Then, if 2k−1 ≤ |y| ≤ 2k ≤ R/8,theorem 2.6, (2.1) and the first inequality in (2.4) imply

|gR(y)−—∫

B2k+1

gR Wdx| ≤ N—∫

B2k+1

|gR−—∫

B2k+1

gR Wdx|Wdy ≤ N 22k

W (B2k ) ,

12

and from the second inequality in (2.4) and (2.1)

|gR(y)−—∫

B1

gR Wdx| ≤ N

(22k

W (B2k)+ |∫ 2k

1

s

W (Bs)ds|

)proving the lemma.

The existence of a nonnegative fundamental solution when∫∞1

sW (Bs) ds

is finite is obvious by compactness and theorem 2.7. Also, if there is anonnegative fundamental solution g, from theorem 2.4 we have g ≥ gR onBR for all R > 0, where g = g/W , and again the estimate in theorem 2.7shows that the integral is bounded by N max∂B1 g.

If E has a nonpositive fundamental solution g, g1−g is a normalized adjointsolution in B1 with boundary values −g, and then g1 − g ≤ −min∂B1 g onB1. In particular g1(y) ≤ −min∂B1 g for all y ∈ B1, and the finiteness of theintegral follows from theorem 2.7 with R = 1.

On the other hand, if∫ 1

0s

W (Bs) ds is finite and R > 0, theorem 2.7 impliesthat lim supy→0 gR(y) = gR(0) < +∞ and

(2.5) N−1

∫ R

0

s

W (Bs)ds ≤ gR(0) ≤ N

∫ R

0

s

W (Bs)ds .

The maximum principle implies that in fact there exists limy→0 gR(y) =gR(0) and gR(0) − gR ≥ 0 in BR. For r ≤ R/2, the function gR(0) − gR −gr(0)+gr is a normalized adjoint solution in Br vanishing at zero, hence thereis xr ∈ ∂Br such that gR(0) − gR(xr) − gr(0) + gr(xr) = gR(0) − gR(xr) −gr(0) = 0. Thus, gR(0) − gR(xr) = gr(0), and from Harnack’s inequalitymax∂Br

(gR(0)− gR) ≤ N min∂Br(gR(0)− gR). Therefore, from (2.5)

N−1

∫ |y|

0

s

W (Bs)ds ≤ gR(0)−gR(y) ≤ N

∫ |y|

0

s

W (Bs)ds for all y ∈ BR/2 ,

and the existence and bounds of a nonpositive fundamental solution followfrom the above estimate, compactness and as a limit when R → +∞ ofgR − gR(0) W .

Proof of theorem 1.5. Assuming that the coefficients matrix a(x) = (aij(x))is continuous at infinity, we may assume a(∞) = I, where I is the identitymatrix, and given ε > 0 there is rε > 0 such that |a(y) − I| ≤ ε for |y| ≥rε. Also (2.1) implies there is α = α(n, ν) ∈ (0, 1] such that W (Brε

) ≤N(rε/r)

αW (Br) for r > rε. Then, integration by parts and the identity

E(r2 − |y|2

)= −2 trace(a) give

r

∫Br

trace(a) Wdy =∫

∂Br

a(y)y · y Wdy ,

13

and setting f(r) = W (Br), the above identity and previous estimates give

(n−Nεα) f(r) ≤ rf ′(r) ≤ (n+Nεα) f(r) for r ≥ rε/ε

which implies the first claim. The case of continuity at zero follows in asimilar way.

Some examples.For ϕ ∈ C((0,+∞)) with 1 + ϕ > 0 set

Eϕ =n∑

i,j=1

aij(x)Dij , aij = δij + ϕ(|x|)xixj

|x|2.

These operators [24] have continuous coefficients in Rn\0 and the weakuniqueness holds in all domains ([9], [10], [39]). When W (x) = W (|x|) ,w = w(|x|) a calculation shows that Dj(aijW ) = 0 for i = 1, . . . , n andDij(aijwW ) = 0 on Rn\0 provided that

W ′(1 + ϕ) +W (ϕ′ + n−1r ϕ) = 0 , (1 + ϕ)w′′ + n−1

r w′ = 0

Also, if limε→0W (ε)εn = 0, W verifies in the distribution sense, Dj(aijW ) =0 for i = 1, . . . , n on Rn. For a constant β > −1 and n ≥ 2 the correspondingglobal nonnegative adjoint solution for ϕ ≡ β is given by c(n, β)|x|−

(n−1)β1+β .

In these cases the integral over (1,+∞) of s/W (Bs) is finite when β < n−2,the integral over (0, 1) is finite when β > n−2 and both integrals are infinitewhen β = n − 2, and fundamental solutions with pole at zero are givenrespectively by

1ωn(n−2−β) |x|

2−n, 1

ωn(n−2−β) |x|2−n

, − 1ωn(n−1) |x|

2−n log |x|

where ωn is the surface area of the unit ball in Rn.For α ∈ R the function | log |x||α satisfies Dj(aijW ) = 0 for i = 1, . . . , n

when ϕ solves(rn−1| log r|αϕ

)′ = −αrn−2| log r|α/ log r. If n = 2, α, β > 0,setting

gα =α

r| log r|α

∫ r

0

| log t|α−1 dt

fβ = − β

r(log r)β

∫ r

2

(log t)β−1 dt+A

r(log r)β

it is possible to choose ε small, R large and A positive such that 1 + gα,1 + fβ are respectively positive and bounded for r ≤ ε and r ≥ R, andgα(ε) = fβ(R) = γ. Defining

ϕ =

ga r ≤ ε

γ ε ≤ r ≤ R

fβ r ≥ R

, W (x) =

| log |x||α r ≤ ε

αB|x|−γ

1+γ ε ≤ r ≤ R

C| log |x||β r ≥ R

14

where B,C > 0 are chosen so that W is continuous on R2 \ 0, it turnsout that a positive multiple of W is the global nonnegative adjoint solutionassociated to Eϕ which is an operator with continuous coefficients in theplane and at infinity, and choosing different values of α, β one can make atwill the integrals of s/W (Bs) over the intervals (0, 1) and (1,∞) be eitherfinite or infinite.

Proof of theorem 1.4. If the weak uniqueness holds the Green’s function forany bounded smooth domain is unique, and the remark after theorem 2.6imply that the Harnack inequality does hold for nonnegative weak-normalizedadjoint solutions and as a consequence, a Liouville-type theorem holds for aglobal and bounded normalized adjoint solution for E obtained as the ratio oftwo nonnegative and nonzero weak-adjoint solutions. Then, the uniquenessfor W follows as in the case of smooth coefficients.

To prove the reciprocal we need to recall the following results which holdfor operators E with smooth coefficients and where normalized adjoint solu-tions are defined with respect to the global adjoint solution W associated toE in theorem 1.1.

Theorem 2.8. Let Ω be a Lipschitz domain in Rn, ϕ ∈ C(∂Ω). Then,there is a unique normalized adjoint solution wϕ in Ω verifying wϕ ∈ C(Ω) ,wϕ = ϕ on ∂Ω. The moduli of continuity of wϕ in Ω can be controlled interms of n , ν , the Lipschitz character of Ω and the moduli of continuity ofϕ. Moreover, for y ∈ Ω there is a probability measure dωy on ∂Ω, called thenormalized harmonic measure, such that

wϕ(y) =∫

∂Ω

ϕ dωy for all ϕ ∈ C(∂Ω) .

Theorem 2.9. Let w1 , w2 be nonnegative normalized adjoint solutions inBR\BR/4(z) vanishing continuously on ∂BR(z). Then, if yR = z + R

2 en

w1(y)w2(y)

≤ Nw1(yR)w2(yR)

for all y ∈ BR(z)\BR/2(z) .

Theorem 2.8 is nowhere explicitly stated but it is an standard consequenceof Harnack’s inequality and the results by E.B. Fabes, N. Garofalo, S. MarınMalave y S. Salsa in [17, Theorem I.1.6], theorem 2.9 follows from [17, The-orem I.3.7].

Now, if E has a unique global and nonnegative adjoint solutionW verifying(1.8), given any sequence of operators Ek with smooth coefficients verifying(1.1) and whose coefficient matrices converge pointwise to the coefficients

15

of E, necessarily the corresponding sequence of global adjoint solutions Wk

converges weakly in Ln

n−1loc (Rn) to W . This, theorem 2.7 and (2.1) imply

there is a constant N such that if g1, g2 are two Green’s functions for Eon B4r(z) generated by compactness and regularizations of the coefficients,then g1(x, y) ≤ Ng2(x, y) for x, y ∈ B2r(z). This and the third remarkafter theorem 2.6 imply that the Harnack inequality in theorem 2.3 holds forweak-nonnegative normalized adjoint solutions relative to W . These clearlyimplies that the normalized harmonic measure associated to E and W bymeans of compactness on any bounded smooth domain Ω is unique, for ifw ∈ C(Ω) is a normalized adjoint solution in Ω relative to W vanishing on∂Ω, w must attain its maximum at some point z ∈ Ω, and w(z) − w is aweak-nonnegative normalized adjoint solution for E in Ω, and by Harnack’sinequality w = w(z) = 0 in Ω.

On the other hand, it is shown by N.V. Krylov in [29, Lemma 3.1] that if gis a weak-Green’s function with pole at zero for E in BR, there is a sequenceEk of operators with smooth coefficients verifying (1.1) and converging point-wise almost everywhere to the coefficients of E such that the correspondingsequence of Green’s functions with pole at zero converges weakly in L

nn−1 (BR)

to g. It is also proved in the last work that suffices to know that the weakuniqueness holds for E on BR at zero in order to conclude that the weakuniqueness holds in BR.

Now, if g1, g2 are two Green’s functions for E in BR the above remarksand theorem 2.7 imply that for some N , g1(y) ≤ Ng2(y) for y ∈ BR/2. Also,since normalized harmonic measure relative to W is unique in BR\BR/4,compactness and theorem 2.8 give

gj(y) =∫

∂BR/4

gj dωy ,

for y ∈ BR\BR/4 , j = 1, 2 , and where dωy is the normalized harmonicmeasure for E on BR\BR/4, and this and theorem 2.9 and 2.7 imply theexistence of a constant N such that g1(y) ≤ Ng2(y) for all y ∈ BR. If N > 1,setting A = (N − 1)−1 , g1 + A(g1 − g2) is a weak-nonnegative Green’sfunction for E in BR with pole at zero, thus g2 ≤ N (g1 +A(g1 − g2)) in BR

and iterating g1 +A(g1−g2)+A[(g1−g2)+A(g1−g2)] is also a weak-Green’sfunction for E on BR with pole at zero. Continuing in this manner, we findthere is Ak , Ak → +∞ such that g1 +Ak(g1− g2) ≤ Ng1 and this is onlypossible if g1 ≤ g2. Similarly g2 ≤ g1. Finally, from [29, Theorem 2.1] theweak uniqueness holds for E if and only if it does hold for all balls, and thisis what was just shown.

3. The Parabolic case

Let P denote the parabolic operator in (1.9) verifying (1.1). The letters

16

X, Y, Z will denote points in Rn+1 with X = (x, t), Y = (y, s), and Z =(z, τ). For R, r > 0 set QR = BR × R, Qr(Z) = Br(z) × (τ − r2, τ + r2)and Cr(Z) = Br(z)× (τ − r2, τ). Also, in the next theorems and proofs anduntil we say the opposite we assume that the coefficients of P are smooth inRn+1. Then, under these hypothesis the following results are well known.

Theorem 3.1. (Harnack Inequality) Let u be a nonnegative solution toPu = 0 on Q2r(Z). Then,

supBr(z)×(τ−3r2,τ−r2)

u ≤ N infBr(z)×(τ,τ+4r2)

u

Theorem 3.2. (Interior elliptic-type Harnack inequality) Suppose u is anonnegative solution to Pu = 0 in B2 × (τ, τ + T ] vanishing continuously on∂B2 × (τ, τ + T ) , and 0 < δ ≤ 1

2 min 1,√T. Then

supB2−δ×(τ+δ2,τ+T ]

u ≤ N infB2−δ×(τ+δ2,τ+T ]

u

where the constant N = N(n, ν, δ, T ).

Theorem 3.3. (Bounds for the normal derivative) Let Z lie in the lateralboundary of Q1, 0 < r ≤ 1/2 and u be a nonnegative solution to Pu = 0in Q2r(Z) ∩ Q1 vanishing continuously on Q2r(Z) ∩ (∂B1 × R). Then, forX ∈ Qr(Z) ∩Q1

u((1− r)z, τ − 2r2)(4− |x|2)Nr

≤ u(X) ≤ Nu((1− r)z, τ + 2r2)(4− |x|2)r

andu((1− r)z, τ − 2r2)

Nr≤ ∂u

∂ν(X) ≤ Nu((1− r)z, τ + 2r2)

r,

where ∂u∂ν (X) = −

∑ni,j=1 aij(X)Diu(X)xj is the conormal derivative of u at

X in Qr(Z) ∩ (∂B1 × R).

Recall that a Green’s function for P on QR with pole at X ∈ QR is anonnegative function gR(X, .) ∈ L1 (BR × (α, β)) for all α < β and verifying

ψ(X) = −∫ t

τ

∫BR

gR(X,Y ) Pψ dY +∫

BR

gR(X, y, τ)ψ(y, τ) dy

for all τ < t and ψ ∈ C2,1(QR) vanishing on ∂BR × R.

17

Theorem 3.4. Given R > 0 there is a unique Green’s function gR(X,Y )for P on QR with pole at X ∈ QR. Moreover, gR(X,Y ) = 0 for t < s and(∫

QR

gR(X,Y )n+1

n dY

) nn+1

≤ NRn

n+1(3.1) (∫ β

α

∫BR

gR(X,Y )n+1

n dY

) nn+1

≤ N(β − α)n

2(n+1) ,(3.2)

and P has a unique global fundamental solution G(X,Y ) in Rn+1 with G(X,Y ) =0 for t < s and verifying

(3.3)

(∫ β

α

∫Rn

G(X,Y )n+1

n dY

) nn+1

≤ N(β − α)n

2(n+1) .

Theorem 3.5. (Doubling property of caloric measure) There is a constantN verifying

(3.4)∫

B2r(z)

G(X, y, τ) dy ≤ N

∫Br(z)

G(X, y, τ) dy

for all r > 0, X, Z ∈ Rn+1 with |x− z| ≤√t− τ .

Theorem 3.6. There is a constant N verifying

N

∫B2r(z)

G(X, y, τ) dy ≥ 1

for all Z ∈ Rn+1, r > 0 and X ∈ Br(z)× [τ, τ + r2] .

Theorem 3.1 was proved by N.V. Krylov and M.V. Safonov [30], andtheorems 3.2 and 3.3 by N. Garofalo [23]. The estimate (3.1) is called bysome authors the Krylov-Bakelman-Aleksandrov-Tso inequality ([28], [42])and the estimate (3.2) is due to X. Cabre [8, Theorem 1.15 and Remark1.14]. It is not clear that there is any place where the bound (3.3) is explicitlystated, but it follows from (3.2). Finally, theorems 3.5 and 3.6 are due toM.V. Safonov and Yu Yuan [41, Theorem 1.1], [41, Lemma 4.1].

Before proving theorem 1.3 we need the following lemmas.

Lemma 3.1. For all r > 0, X, Z ∈ Rn+1 with t ≥ τ∫Br(z)

G(X, y, τ) dy ≤(

1 +t− τ

r2

)−nν24

exp

(ν

4

(1− |x− z|2

t− τ + r2

)).

Proof. The function u(X) on the left hand side in the above inequality is thesolution to Pu = 0 on Rn × (τ,+∞) with data given by χBr(z). A simplecalculation shows that the function v(X) on the right hand side, verifiesPv ≤ 0 on Rn × (τ,+∞) and v ≥ u on the parabolic boundary, and thelemma follows from the maximum principle.

18

Lemma 3.2. Given 0 ≤ θ ≤ 1 and r > 0, the inequalities∫B2r(z)

G(X, y, τ + θr2) dy ≤ N

∫Br(z)

G(X, y, τ) dy(3.5) ∫B2r(z)

G(X, y, τ) dy ≤ N

∫Br(z)

G(X, y, τ + θr2) dy(3.6)

hold respectively when |x− z| ≤√t− τ and t ≥ τ + θr2, and when |x− z| ≤√

t− τ − θr2.

Proof. Given Z ∈ Rn+1, r > 0 and 0 ≤ θ ≤ 1 theorem 3.5 gives∫B4r(z)

G(X, y, τ) dy ≤ N

∫Br(z)

G(X, y, τ) dy when |x− z| ≤√t− τ

and from theorem 3.6

N

∫B4r(z)

G(X, y, τ) dy ≥ 1 on B2r(z)× [τ, τ + 4r2] .

Thus, to prove (3.5) suffices to show that for t ≥ τ + θr2 and x ∈ Rn∫B2r(z)

G(X, y, τ + θr2) ≤ N

∫B4r(z)

G(X, y, τ) ,

but this is implied by the previous inequality and the maximum principle.To prove (3.6), we let u(X) denote the solution to Pu = 0 in Rn×(τ,+∞)

with initial data χB2r(z). Since for t > σ ≥ τ we have

u(X) =∫

Rn

G(X, y, σ)u(y, σ)dy ,∫B2r(z)

G(X, y, τ) dy =∫

Rn

G(X, y, τ + θr2)u(y, τ + θr2) dy .(3.7)

Also, an iteration of the inequality in theorem 3.5 and lemma 3.1 give re-spectively that for |x− z| ≤

√t− τ − θr2, k ≥ 1 and y ∈ Rn

∫B2kr

(z)

G(X, y, τ + θr2) dy ≤ Nk+1

∫Br(z)

G(X, y, τ + θr2) dy

u(y, τ + θr2) ≤ Ne−|y−z|2

Nr2

and (3.6) follows from (3.7) and the exponential decay of u.

Now, if we set θ = 1 in (3.5), (3.6) and integrate the inequalities withrespect to the time variable the following follows.

19

Lemma 3.3. (Space-time doubling property) There is a constant N suchthat for X, Z ∈ Rn+1 the inequalities

∫B2r(z)

∫ τ+4r2

τ

G(X,Y ) dY ≤ N

∫Br(z)

∫ τ+4r2

τ

G(X,Y ) dY∫B2r(z)

∫ τ

τ−4r2G(X,Y ) dY ≤ N

∫Br(z)

∫ τ

τ−r2G(X,Y ) dY

hold respectively when when |x− z| ≤√t− τ − 3r2, t ≥ τ + 4r2, and when

|x− z| ≤√t− τ .

Lemma 3.4. Let W be a nonnegative adjoint solution for P on C4r(Z),then (

—∫

Cr(Z)

Wn+1

n dY

) nn+1

≤ N—∫

C2r(Z)

W dY

Proof. The proof of this result in the elliptic case is done in [22, Theorem 2.1].There the argument is based in the Pucci-Aleksandrov inequality (Theorem2.1) and integration by parts. In the parabolic setting, a proof of the aboveestimate can be obtained with similar arguments but replacing the Pucci-Aleksandrov estimate by its parabolic analog (inequality (3.1)).

Proof of theorem 1.3. To prove the existence of the global nonnegative adjointsolution define for T > 0

WT (Y ) =G(0, T, Y )α(T )

where α(T ) = —∫

B1×(0,1)

G(0, T, y, s) dY .

Then, WT (B1 × (0, 1)) = |B1| and lemmas 3.2, 3.3 and 3.4 give

WT (B2r(z), τ + θr2) ≤ N WT (Br(z), τ)

WT (B2r(z), τ) ≤ N WT (Br(z), τ + θr2)(—∫

Qr(Z)

WT

n+1n dY

) nn+1

≤ N—∫

Qr(Z)

WT dY

for |z| ≤√T/2, τ ≤ T/4, 0 < r ≤

√T/8 and 0 ≤ θ ≤ 1. Then, proceeding

as in section 2 and from the analog properties of parabolic Muckenhouoptweights, compactness and (1.12) one obtains the existence of a nonnegativeadjoint solutionW defined on Rn+1 verifying the properties stated in theorem1.3.

20

Definition. A function w is a normalized adjoint solution for P relative toW in an open set Ω ⊂ Rn+1 if wW ∈ L1

loc(Ω) is an adjoint solution in Ω.

Theorem 3.7. (Harnack inequality for normalized adjoint solutions) LetZ ∈ Rn+1 and w be a nonnegative normalized adjoint relative to W inC2r(Z). Then,

supBr(z)×(τ−2r2,τ−r2)

w ≤ N infBr(z)×(τ−4r2,τ−3r2)

w

Proof. By translation and scaling we may assume Z = (0, 4), r = 1, andthat w is a nonnegative normalized adjoint solution relative to W in C2(Z).Defining the normalized Green’s function for the cylinder Q2 as g2(X,Y ) =g2(X,Y )/W (Y ) and denoting the surface measure on the lateral boundaryof the cylinder as dσ, the following representation formula holds for w andY ∈ C2(Z)

w(Y ) =∫

B2

g2(x, 4, Y )w(x, 4)W (x, 4)dx+∫

∂B2

∫ 4

s

∂g2∂ν

(X,Y )w(X)W (X)dσ

In particular, for Y ∈ B × (2, 3), Y ∈ B × (0, 1)

w(Y ) ≤∫

B2

g2(x, 4, Y )w(x, 4)W (x, 4)dx+∫

∂B2

∫ 4

2

∂g2∂ν

(X,Y )w(X)W (X)dσ

w(Y ) ≥∫

B2

g2(x, 4, Y )w(x, 4)W (x, 4)dx+∫

∂B2

∫ 4

2

∂g2∂ν

(X,Y )w(X)W (X)dσ

From theorems 3.1, 3.2 and 3.3

g2(x, 4, Y ) ≤ N g2(0, 5, Y )(4− |x|2) ∂g2∂ν

(X,Y ) ≤ N g2(0, 5, Y )

N g2(x, 4, Y ) ≥ g2(0, 5, Y )(4− |x|2) N∂g2∂ν

(X,Y ) ≥ g2(0, 5, Y )

for all x ∈ B2, X ∈ ∂B2 × [2, 4], and from these inequalities

w(Y )w(Y )

≤ Ng2(0, 5, Y )g2(0, 5, Y )

.

Since the constant function one is a normalized adjoint solution relative toW , for Y ∈ C2(Z)

1 =∫

B2

g2(x, 4, Y )W (x, 4) dx+∫

∂B2

∫ 4

s

∂g2∂ν

(X,Y )W (X) dσ

21

and then

1 ≥∫

B2

g2(x, 4, Y )W (x, 4) dx+∫

∂B2

∫ 4

318

∂g2∂ν

(X,Y )W (X) dσ

1 ≤∫

B2

g2(x, 4, Y )W (x, 4) dx+∫

∂B2

∫ 4

0

∂g2∂ν

(X,Y )W (X) dσ .

Using again theorems 3.1, 3.2 and 3.3 it follows that for x ∈ B2, X ∈∂B2 × [ 318 , 4]

N g2(x, 4, Y ) ≥ (4− |x|2)g2(0, 5, Y ) N∂g2∂ν

(X,Y ) ≥ g2(0, 5, Y )

and for x ∈ B2, X ∈ ∂B2 × [0, 4]

g2(x, 4, Y ) ≤ N(4− |x|2)g2(0, 5, Y )∂g2∂ν

(X,Y ) ≤ Ng2(0, 5, Y ) ,

and these inequalities imply

g2(0, 5, Y )g2(0, 5, Y )

≤ N

∫B2

(4− |x|2

)W (x, 4) dx+

∫∂B2

∫ 4

0W (X) dσ∫

B2(4− |x|2) W (x, 4) dx+

∫∂B2

∫ 4318W (X) dσ

.

Thus, to finish the proof suffices to show that the above fraction is boundedby a constant depending on n and ν. Now, integration by parts and theidentity L(4 − |x|2) = −2trace a(X), where a(X) denotes the coefficientsmatrix of P give∫

B2

(4− |x|2) W (x, τ) dx+∫

B2

∫ 4

τ

2trace a(X) W (X) dX

=∫

B2

(4− |x|2) W (x, 4) dx+∫

∂B2

∫ 4

τ

a(X)x · x W (X) dσ

for τ ≤ 4, and from (1.1) we obtain that the above quotient is bounded by

N

∫B2

(4− |x|2

)W (x, 0) dx+

∫B2

∫ 4

0W (X) dX∫

B2(4− |x|2) W (x, 31

8 ) dx+∫

B2

∫ 4318W (X) dX

and this last ratio is bounded by a universal constant due to the doublingproperties of W , in particular (1.11).

A standard consequence of Harnack’s inequality is the following theorem

22

Theorem 3.8 (Holder continuity). Let w be a normalized adjoint solutionrelative to W in Qr(Z). Then, for some α = α(n, ν) ∈ (0, 1]

|w(Y )− w(Z)| ≤ N

(|y − z|2 + |s− τ |

r2

)α/2

supQr(Z)

|w|

This result implies a Liouville-type theorem for bounded normalized ad-joint solutions defined in Rn+1 and gives again as in the elliptic case theuniqueness of a nonnegative and global adjoint solution W for P verifying(1.11) and the normalization condition W (B1 × (0, 1)) = |B1|.

Defining G(X,Y ) = G(X,Y )/W (Y ), theorem 3.7 with Z = (y, t) andr =

√t− s implies

G(X,Y ) ≤ N G(X, z, 4s− 3t) for z ∈ B√t−s(y) , s < t .

Multiplying this inequality by W (z, 4s − 3t) and integrating the result overB√t−s(y) with respect to dz one obtains

W (B√t−s(y), 4s− 3t) G(X,Y ) ≤ N

∫B√t−s(y)

G(X, z, 4s− 3t) dz ,

and from lemma 3.1

(3.8) W (B√t−s(y), 4s− 3t) G(X,Y ) ≤ Ne−|x−y|2N(t−s) .

Then, the upper bound in theorem 1.3 follows from (3.8) and the inequalities

W (B√t−s(y), s) ≤ N W (B√t−s(y), 4s− 3t)

W (B√t−s(x), s) ≤ N

(1 +

|x− y|√t− s

)N

W (B√t−s(y), s)

which are standard consequences of (1.11).To obtain the lower bound we apply theorem 3.7 with Z = (y, t), r =√t− s/2 obtaining

N G(X,Y ) ≥ G(X, z, t+s2 ) for z ∈ B√t−s/4(y) , s < t ,

and multiplying this inequality by W (z, t+s2 ) and integrating the result over

B√t−s/4(y) with respect to dz

N W (B√t−s/4(y),t+s2 ) G(X,Y ) ≥

∫B√t−s/4(y)

G(X, z, t+s2 ) dz .

23

Also, from (1.11)

N W (B√t−s(y), s) ≥ W (B√t−s/4(y),t+s2 )

N

(1 +

|x− y|√t− s

)N

W (B√t−s(x), s) ≥W (B√t−s(y), s) .

The function u(.) =∫

B√t−s/4(y)G(., z, t+s

2 ) dz is nonnegative and verifies

Pu = 0 on Rn × ( t+s2 ,+∞), and with an standard iteration of theorem 3.1

(See the proof of the lower bound in [21, Theorem 2.7]) we have

u(X) ≥ N−1e−N|x−y|2

t−s u(y, t+s2 + (

√t−s8 )2)

and from theorem 3.6, Nu(y, t+s2 + (

√t−s8 )2) ≥ 1, obtaining∫

B√t−s/4(y)

G(X, z, t+s2 ) dz ≥ N−1e−N

|x−y|2t−s ,

and the lower bound follows from the previous inequalities. Finally, fromthe bounds we proved there is a constant N = N(n, ν,K, ε) such thatN−1W (Y ) ≤ G(X, .) ≤ N W (Y ) for Y ∈ Ω(X,K, ε), and this implies thelast claim.

Remark 4.1. A standard consequence of the Harnack’s inequality for nor-malized adjoint solutions is that when the coefficients of P are smooth andQ = Ω×(−∞, T ), where Ω is a bounded Lipschitz domain in Rn, ϕ ∈ C(∂pQ)and ∂pQ = Ω×T∪∂Ω×(−∞, T ), there exists a unique normalized adjointsolution wϕ in Q verifying wϕ ∈ C(Q) , wϕ = ϕ on ∂pQ. Also, the maximumprinciple holds in this setting and there is α0 = α0(n, ν,Ω) ∈ (0, 1) such thatif ϕ ∈ Cα,α/2(∂pQ) and α ≤ α0, then wϕ ∈ Cα,α/2(Q).

From the arguments in [19],[20] and [41], once Harnack’s inequality fornormalized adjoint solutions and the bounds in theorem 1.3 for the funda-mental solution are established, the results proved in the above works (theso called: Carleson-type inequality, Interior elliptic-type Harnack inequal-ity, Comparison principle, Backward Harnack inequality near the boundary,Holder continuity for quotients and the Doubling property of parabolic mea-sure) can be carried out again in the context of normalized adjoint solutions,and in particular, once those results are established the analog of theorem1.4 can be proved again in the parabolic setting with similar arguments.

ACKNOWLEDGMENTS

Part of this work was done at the time the author was attending the half-year program in Harmonic Analysis held at M.S.R.I. from July to December

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1997. He would like to thank the members of the Institute and the organizersof the program for their hospitality. The author is supported by Universidaddel Paıs Vasco grant UPV 127.310 EA-210/96 and by the Basque Governmentgrant P-I997/22.

REFERENCES

[1] Aleksandrov A.D., Uniqueness conditions and estimates for the solutions of theDirichlet problem, Vestnik Lennin-grad Univ. 18 (1996), 5–29; II, English tranl. Amer.

Math. Soc. Trans. 68 (1968), 89–119.

[2] Aronson D.G., Bounds for the fundamental solution of a parabolic equations, Bull. ofthe AMS 73 (1967), 890–896.

[3] , Non-negative solutions of linear parabolic equations, Ann. Sci. Norm. Sup.

Pisa 22 (1968), 607–694.

[4] Bakelman I.Ya, Theory of quasilinear elliptic equations, Siberian Math. J. 2 (1961),

179–186.

[5] Bauman P., Positive solutions of elliptic equations in nondivergence form and their

adjoints, Ark. Mat. 22 (1984), 153–173.

[6] , Equivalence of the Green’s function for diffusion operators in Rn: a coun-

terexample, Proc. Amer. Math. Soc. 91 (1984), 64–68.

[7] , A Wiener test for nondivergence structure second order elliptic equations,Indiana U. Math. J. (1985), 825–844.

[8] Cabre X., On the Alexandroff-Bakelman-Pucci estimate and the reversed Holder in-

equality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math.11 (1994), 1–33.

[9] Cerutti M.C., Escauriaza L., Fabes E.B., Uniqueness in the Dirichlet problem for

some elliptic operators with discontinuous coefficients, Annali di Mat. Pura Appl.

CLXIII (1993), 161–180.

[10] , Uniquennes for some diffusions with discontinuous coefficients, Annals ofProbability 19 (1991), 525–537.

[11] Chanillo S., Li Yanyan, Continuity of solutions of uniformly elliptic equations, Manus-

cripta Math. 77 (1992), 415–433.

[12] De Giorgi E., Sulla differentiabilita e analiticita delle extremali degli integrali multipleregolari, Mem. Accad. Sci. Torino S.III, Parte I (1957), 25–43.

[13] Escauriaza L., Weak type-(1,1) inequalities and regularity properties of adjoint and

normalized adjoint solutions to linear nondivergence form operators with VMO coef-ficients, Duke Math. J. 74 (1994), 177–201.

[14] Escauriaza E., Kenig C.E., Area integral estimates for solutions and normalized ad-

joint solutions to nondivergence form elliptic equations, Ark. Mat. 31 (1993), 275–

296.

[15] Fabes E.B., Gaussian upper bounds on fundamental solutions of parabolic equa-tions:the method of Nash (to appear).

[16] Fabes E.B., Garofalo N., S.Salsa, A control on the set where a Green’s function

vanishes, Coll. Mat. LX/LXI (1990), 637–647.

[17] Fabes E.B., Garofalo N., Marın Malave S., Salsa S., Fatou theorems for some non-linear elliptic equations, Rev. Mat. Ib. 4 (1988), 227–242.

[18] Fabes E.B., Kenig C.E., Examples of singular parabolic measures and singular tran-

sition probability densities, Duke Math. J. 48 (1981), 845–856.

[19] Fabes E.B., Safonov M.V., Behavior near the boundary of positive solutions of secondorder parabolic equations, Proceedings of El Escorial 96. CRC Press.

25

[20] Fabes E.B., Safonov M.V., Yu Yuan Λ, Behavior near the boundary of positive solu-

tions of second order parabolic equations II, Trans. Amer. Math. Soc. (to appear).[21] Fabes E.B., Stroock D.W., A new proof of Moser’s parabolic Harnack inequality via

the old ideas of Nash, Ark. Rat. Mech. Anal. 96 (1986), 327–338.[22] , The Lp-integrability of Green’s functions and fundamental solutions for el-

liptic and parabolic equations, Duke Math J. 51 (1984), 977–1016.

[23] Garofalo N., Second order parabolic equations in nonvariational form: boundary Har-nack principle and comparison theorems for nonnegative solutions, Ann. Mat. Pura

Appl. (4) 138 (1984), 267–296.

[24] Gilbarg D., Serrin J., On isolated singularities of solutions of second order ellipticdifferential equations, J. Analyse Math. 4 (1955/56), 309–340.

[25] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order,

2nd ed., Springer-Verlag, Berlin-Heildeberg-New York-Tokyo, 1983.[26] Gruter M., Widman K.O., The Green function for uniformly elliptic equations, Manus

Math. 37 (1982), 303–342.

[27] Kenig C.E., Wei-Ming Ni, On the elliptic equation Lu−k+Kexp[2u] = 0, Ann. Scuo.Norm. Sup. Pisa (Series IV) 12 (1985), 191–224.

[28] Krylov N.V., Sequences of convex functions and estimates of the maximum of thesolution of a parabolic equation, Siberian Math. J. 17 (1976), 226–236.

[29] , On one-point uniqueness for elliptic equations, Comm. Parial Differential

Equations 17, n. 11/12 (1992), 1759–1784.[30] Krylov N.V., Safonov M.V., A certain property of solutions of parabolic equations

with measurable coefficients, Math. USSR Izv. 16 (1981), 151–164.

[31] , An estimate of the probability that a diffusion process hits a set of positivemeasure, Soviet Math. Dokl. 20 (1979), 253–255.

[32] Littman W., Stampachia G., Weinberger H., Regular points for elliptic equations with

discontinuous coefficients, Ann. Sc. Norm. Sup. Pisa 17 (1963), 45–79.[33] Moser J., On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl.

Math. 14 (1961), 577–591.[34] , A Harnack inequality for parabolic differential equations, Comm. Pure Appl.

Math. 17 (1964), 101–134.

[35] Muckenhoupt B., Weighted norm inequalities for the Hardy maximal functions, Trans.Amer. Math. Soc. 165 (1972), 207–226.

[36] Nadirashvili N.S., Nonuniqueness in the martingale problem and the Dirichlet problem

for uniformly elliptic equations (1995) (to appear).[37] Nash J., Continuity of solutions of parabolic and elliptic equations, Amer. J. Math.

80 (1958), 931–954.

[38] Pucci C., Limitazioni per soluzioni di equaziones ellittiche, Ann. Mat. Pura Appl. 74(1966), 15–30.

[39] Safonov M.V., On one-point weak uniqueness for elliptic equations, Comm. Partial

Differential Equations 19 (1994), 943–957.[40] , Nonuniqueness for second order elliptic equations with measurable coeffi-

cients, SIAM J. Math. Anal. (Jan. 1998) (to appear).[41] Safonov M.V., Yu Yuan, Doubling properties for second order parabolic equations,

Annals of Math. (to appear).

[42] Tso K., On an Aleksandrov-Bakelman type maximum principle for second order par-abolic equations, Comm. Partial Differential Equations 10 (1985), 543–553.