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408 N.V. Krylov

In particular, since



|χ

(i)

(x)|



|χ

(i)

(x − x

we have

−Nκ

(t−t

)

= N

∗

−Nκ

(t−t

)



|χ

(i)

(x − y)|

≤ N

∗



|χ

(i)

(x)|

≤ N

∗

Nκ

(t−t

)



|χ

(i)

(x − y)|

dy = e

Nκ

(t−t

)

where N

∗

= |B

−1

−d

and |B

| is the volume of B

. It follows that



|χ

(1)

(x)|

≤ (N

∗

)

−1

Nκ

(t−t

)

∗

)

−1

−Nκ

(t−t

)

≤



|χ

(2)

(x)|

Furthermore, since u

=0ift ≤ t

and T



≤ t

+ κ

−1

, in evaluating the norms

in (5.20) we need not integrate with respect to t such that κ

(t − t

) ≥ 1or

(t − t

) ≤ 0, so that for all t really involved we have



|χ

(1)

(x)|

≤ (N

∗

)

−1

, (N

∗

)

−1

−N

≤



|χ

(2)

(x)|

After this observation it only remains to integrate (5.20) through with respect to

and use the fact that N

∗

−d

∗

. The lemma is proved. 

Proof of Theorem 3.1. Obviously we may assume that S = −∞.Thenﬁrstwe

show how to choose an appropriate γ = γ(d, δ, p) ∈ (0, 1]. For one, we take

it smaller than the one from Lemma 5.7. Then call N

the constant factor of

p/q

I

Du

in (5.19). We know that N

= N

(d, δ, p)andwechooseγ ∈ (0, 1]

so that N

p/q

≤ 1/2. Then under the conditions of Lemma 5.9 we have

p/2

I

u

+ I

Du

≤ N

i=1

I



+ Nλ

−p/2

I



+ N

∗

−p/2

I

Du

+ N

∗

I

u

+ N

∗

−p/2

i=1

I



. (5.21)

After γ has been ﬁxed we recall that κ

= κ

(d, p, ρ

,K)andtakeaζ ∈ C

∞

(R)

with support in (0,κ

−1

) such that



∞

−∞

(t) dt =1. (5.22)

For s ∈ R deﬁne ζ

= ζ(t − s), u

(x)=u

(x)ζ

. Obviously u

=0ift ≤ s ∧ T .

Therefore, we can apply (5.21) to u

with t

= s ∧ T observing that

∂

= D

+ b

)+b

− (c

+ λ)u

+ D

(ζ

)+ζ

+(ζ

)



Divergence Equations with Growing Coeﬃcients 409

Then from (5.21) for λ ≥ λ

,whereλ

= λ

(d, δ, p, ρ

) is taken from Lemma 5.7,

we obtain

p/2

I

s∧T

u

+ I

s∧T

Du

(5.23)

≤ N

i=1

I

s∧T



+ Nλ

−p/2

I

s∧T



+ NI

s∧T

(ζ

)



u

+ N

∗

−p/2

I

s∧T

Du

+ N

∗

I

s∧T

u

+ N

∗

−p/2

i=1

I

s∧T



We integrate through (5.23) with respect to s ∈ R,observethat

s∧T<t<[(s∧T )+κ

−1

]∧T

= I

t<T

s∧T<t<(s∧T )+κ

−1

= I

t<T

s<t<s+κ

−1

and that (5.22) yields



∞

−∞

s∧T

(t)(ζ

)

ds =



∞

−∞

s∧T<t<[(s∧T )+κ

−1

]∧T

(t − s) ds

= I

t<T



t−κ

−1

(t − s) ds = I

t<T

We also notice that, since κ

depends only on d, p, ρ

,K,wehave



∞

−∞

|ζ



(s)|

ds = N

∗

Then we conclude

p/2

u

(T )

+ Du

(T )

≤ N

i=1

f



(T )

+ N

−p/2

f



(T )

+ N

∗

−p/2

Du

(T )

+ N

∗

u

(T )

+ N

∗

−p/2

i=1

f



(T )

Without losing generality we assume that N

≥ 1 and we show how to choose

= λ

(d, δ, p, ρ

,ρ

,K) ≥ 1. Above we assumed that λ ≥ λ

(d, δ, p, ρ

), where

(d, δ, p, ρ

) is taken from Lemma 5.7. Therefore, we take

= λ

(d, δ, p, ρ

,ρ

,K) ≥ λ

(d, δ, p, ρ

)

such that λ

p/2

≥ 2N

∗

. Then we obviously come to (3.3) (with S = −∞). The

theorem is proved. 

410 N.V. Krylov

6. Proof of Theorems 3.3 and 3.4

We need two auxiliary results.

Lemma 6.1. For any τ, R ∈ (0, ∞), we have



−τ



(|b

(x)|



+ |b

(x)|



+ c



(x)) dxds < ∞. (6.1)

Proof. Obviously it suﬃces to prove (6.1) with B

)inplaceofB

for any x

In that case, for instance, (notice that q ≥ p



, see (3.1))



)

(x)|



dx ≤ N





)

(x) −





+ N|



Accordingto(5.6)



)

(x)|



dx ≤ N + N |



and in what concerns b it only remains to use Assumption 3.1 (iii). Similarly, b

and c

are treated. The lemma is proved. 

The solution of our equation will be obtained as the weak limit of the solutions

of equations with cut-oﬀ coeﬃcients. Therefore, the following result is appropriate.

By the way, observe that usual way of proving the existence of solutions based on

a priori estimates and the method of continuity cannot work in our setting mainly

because of what is said after Theorem 3.6.

Lemma 6.2. Let φ ∈ C

∞

, τ ∈ (0, ∞).Letu

, u ∈ W

, m =1, 2,...,besuch

that u

→ u weakly in W

.Form =1, 2,... deﬁne χ

(t)=(−m) ∨ t ∧ m,

= χ

), b

= χ

),andc

= χ

). Then the functions



,φ) ds,



φ) ds,



,φ) ds (6.2)

converge weakly in the space L

([−τ,τ]) as m →∞to



,φ) ds,



φ) ds,



,φ) ds, (6.3)

respectively.

Proof. By Corollary 5.6 and by the fact that (strongly) continuous operators are

weakly continuous we obtain that



,φ) ds →



,φ) ds

as m →∞weakly in the space L

([−τ,τ]). Therefore, in what concerns the ﬁrst

function in (6.2), it suﬃces to show that



, (b

− b

)φ) ds → 0

Divergence Equations with Growing Coeﬃcients 411

weakly in L

([−τ,τ]). In other words, it suﬃces to show that for any ξ ∈L



([−τ,τ])



−τ





, (b

− b

)φ) ds



dt → 0.

This relation is rewritten as



−τ

,η

− b

)φ) ds → 0, (6.4)

where



τ sgn s

is bounded on [−τ,τ]. However, by the dominated convergence theorem and Lemma

6.1, we have η

−b

)φ → 0asm →∞strongly in L



(−τ,τ) and by assump-

tion Du

→ Du weakly in L

(−τ,τ). This implies (6.4). Similarly, one proves our

assertion about the remaining functions in (6.2). The lemma is proved. 

Proof of Theorem 3.3. OwingtoTheorem3.1implyingthatthesolutionon

(−∞,T] ∩ R is unique, without loss of generality we may assume that T = ∞.

Deﬁne b

, b

,andc

as in Lemma 6.2 and consider equation (2.1) with b

,andc

in place of b

, b

,andc

, respectively. Obviously, b

, b

,andc

satisfy Assumption 3.2 with the same γ and K as b

, b

,andc

do. By Theorem

3.1 and the method of continuity for λ ≥ λ

(d, δ, p, ρ

,ρ

,K) there exists a unique

solution u

of the modiﬁed equation on R.

By Theorem 3.1 we also have

u



+ Du



≤ N,

where N is independent of m. Hence the sequence of functions u

is bounded in

the space W

and consequently has a weak limit point u ∈ W

. For simplicity of

presentation we assume that the whole sequence u

converges weakly to u.Take

a φ ∈ C

∞

. Then by Lemma 6.2 the functions (6.2) converge to (6.3) weakly in

([−τ,τ]) as m →∞for any τ. Obviously, the same is true for (u

,φ) → (u

,φ)

and the remaining terms entering the equation for u

. Hence, by passing to the

weak limit in the equation for u

we see that for any φ ∈ C

∞

equation (2.3) holds

for almost any s, t ∈ R.

Now notice that, for each t ∈ R, owing to Corollary 5.5 the equation

(ˆu

,φ):=



,φ) ds +







−(c

+ λ)u

+ f

,φ)

−(a

+ b

+ f

φ)



ds (6.5)

deﬁnes a distribution. Furthermore, by the above for any φ ∈ C

∞

we have (u

,φ)=

(ˆu

,φ) (a.e.). A standard argument shows that for almost all t ∈ R,(u

,φ)=(ˆu

,φ)

for any φ ∈ C

∞

,thatisu

=ˆu

(a.e.) and ˆu

∈ W

.Inparticular,weseethatwe

412 N.V. Krylov

can replace u

in (6.5) with ˆu

. Finally, for any t

∈ R

(ˆu

,φ) − (ˆu

,φ)







ˆu

−(c

+ λ)ˆu

+ f

,φ) − (a

ˆu

+ b

ˆu

+ f

φ)





ˆu

− (c

+ λ)ˆu

+ f

,φ) − (a

ˆu

+ b

ˆu

+ f

φ)

and the theorem is proved. 

Proof of Theorem 3.4. (i) One reduces the general case to the one that u

=0as

in the proof of Theorem 3.2. Also, obviously, one can assume that λ is as large

as we like, say satisfying (3.4), since S and T are ﬁnite. By continuing u

(x)as

zero for t ≤ S we see that we may assume that S = ∞.Ifwesetf

=0for

t ≥ T and use Theorem 3.3 about the existence of solutions on (−∞, ∞)along

with Theorem 3.1, which guarantees uniqueness of solutions on (−∞,T], then we

see that we only need to prove assertion (ii) of the theorem.

(ii) In the above proof of Theorem 3.3 we have constructed the unique so-

lutions of our equations as the weak limits of the solutions of equations with

cut-oﬀ coeﬃcients. Therefore, if we knew that the result is true for equations with

bounded coeﬃcients, then we would obtain it in our general case as well.

Thus it only remains to concentrate on equations with bounded coeﬃcients.

Existence an uniqueness theorems also show that it suﬃces to prove that, if u is

the solution corresponding to p = p

,thenu ∈ W

Take a ζ ∈ C

∞

d+1

) such that ζ(0) = 1, set ζ

(x)=ζ(t/n, x/n), and notice

that u

:= u

satisﬁes

∂

= L

− λu

+ D

+ f

where

= f

− u

,i≥ 1,

= f

− f

−(a

+ a

−b

+ u

∂

Since u

has compact support and p

≤ p

, it holds that u

∈ W

for any p ∈ [1,p

]

and by Theorem 3.1 for p ∈ [p

]wehave

u



≤ N

i=0

f



. (6.6)

One knows that

f



≤ N(f



+ f



so that by H¨older’s inequality

f



≤ N + N uDζ



≤ N + u

Dζ



Divergence Equations with Growing Coeﬃcients 413

with constants N independent of n,where

q =

−p

Similar estimates are available for other terms in the right-hand side of (6.6). Since

∂



+ Dζ



= Nn

−1+(p

−p)(d+1)/(p

→ 0

as n →∞if

−

d +1

, (6.7)

estimate (6.6) implies that u ∈ W

Thus knowing that u ∈ W

allowed us to conclude that u ∈ W

as long as

p ∈ [p

] and (6.7) holds. We can now replace p

with a smaller p and keep going

in the same way each time increasing 1/p by the same amount until p reaches p

Then we get that u ∈ W

. The theorem is proved 

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N.V. Krylov

University of Minnesota

127 Vincent Hall

Minneapolis

MN, 55455, USA

e-mail: krylov@math.umn.edu

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 415–432

 2011 Springer Basel AG

Global Properties of Transition Kernels

Associated with

Second-order Elliptic Operators

Karima Laidoune, Giorgio Metafune, Diego Pallara

and Abdelaziz Rhandi

Dedicated to Prof. H. Amann on the occasion of his 70th birthday

Abstract. We study global regularity properties of transitions kernels asso-

ciated with second-order diﬀerential operators in R

with unbounded drift

and potential terms. Under suitable conditions, we prove Sobolev regularity of

transition kernels and pointwise upper bounds. As an application, we obtain

suﬃcient conditions implying the diﬀerentiability of the associated semigroup

on the space of bounded and continuous functions on R

Mathematics Subject Classiﬁcation (2000). Primary 35K65; Secondary 47D07,

60J35.

Keywords. Semigroups, transition kernels, parabolic regularity, Lyapunov

functions.

1. Introduction

Given a second-order elliptic partial diﬀerential operator with real coeﬃcients

A =

i,j=1

i=1

−V = A

+ F · D −V, (1.1)

where A

i,j=1

), we consider the parabolic problem



(x, t)=Au(x, t),x∈ R

,t>0,

u(x, t)=f(x),x∈ R

(1.2)

where f ∈ C

416 K. Laidoune, G. Metafune, D. Pallara and A. Rhandi

We assume the following conditions on the coeﬃcients of A which will be

kept in the whole paper without further mentioning.

(H) a

= a

: R

→ R, V : R

→ [0, +∞), with a

∈ C

1+α

V, F

∈ C

loc

)forsome0<α<1and

λ|ξ|

≤

i,j=1

(x)ξ

≤ Λ|ξ|

for every x, ξ ∈ R

and suitable 0 <λ≤ Λ.

Notice that neither the drift F =(F

,...,F

)orthepotentialV are assumed to

be bounded in R

Problem (1.2) has always a bounded solution but, in general, there is no

uniqueness. However, if f is nonnegative, it is not diﬃcult to show that (1.2) has

a minimal solution u among all non negative solutions. Taking such a solution u

one constructs a semigroup of positive contractions T (·)onC

) such that

u(x, t)=T (t)f(x),t>0,x∈ R

solves (1.2). Furthermore, the semigroup can be represented in the form

T (t)f(x)=



p(x, y, t)f(y) dy, t > 0,x∈ R

for f ∈ C

). Here p is a positive function and for almost every y ∈ R

,it

belongs to C

2+α,1+α/2

loc

×(0, ∞)) as a function of (x, t) and solves the equation

∂

p = Ap, t > 0. We refer to Section 2, [8, Chapter 1] and [11] (in the case V =0)

for a review of these results as well as for conditions ensuring uniqueness for (1.2).

Now, we ﬁx x ∈ R

and consider p as a function of (y,t). Then p satisﬁes

∂

p = A

∗

p, t > 0, (1.3)

in the following sense (see [10, Lemma 2.1]): Let 0 ≤ t

and ϕ ∈ C

2,1

(Q(t

))

(see below for the notation) be such that ϕ(·,t) has compact support for every

t ∈ [t

]. Then



Q(t

)

(∂

ϕ(y,t)+Aϕ(y, t)) p(x, y, t) dy dt (1.4)



(p(x, y, t

)ϕ(y,t

) − p(x, y, t

)ϕ(y,t

)) dy.

The aim of this paper is to study global regularity properties of the kernel p

as a function of (y, t) ∈ R

× (a, T )for0<a<T.

We prove that p(x, ·, ·) belongs to W

1,0

× (a, T )) (see below for the no-

tation) provided that





V (y)

+ |F (y)|



p(x, y, t) dy dt < ∞, ∀k>1

Estimates on Transition Kernels 417

for ﬁxed x ∈ R

and 0 <a

<a. This generalizes [10, Corollary 3.1 and

Lemma 3.1] and in some sense Theorem 4.1 in [2]. Assuming that certain Lyapunov

functions (exponentials or powers) are integrable with respect to p(x, y, t) dy for

(x, t) ∈ R

× (a, T ), pointwise upper bounds for p are obtained. If in addition

V ∈ W

1,∞

loc

),F ∈ W

1,∞

loc

, R

)aresuchthatDV, DF are dominated by

some exponential functions, then p ∈ W

2,1

× (a, T )) for all k>1. As a con-

sequence, we obtain also upper bounds for |D

p|. In the case where F and V and

their corresponding derivatives up to the second order satisfy growth conditions

of exponential type, upper bounds are also obtained for |D

p| and |∂

p|.Asa

consequence, we deduce that the semigroup T (·) is diﬀerentiable on C

)for

t>0.

In the case where V = 0, regularity and pointwise estimates for p can be

found in [10], [14] and for the solution of (1.3) with a L

-function as the initial

datum we refer to [3], [4].

Other bounds for the transition densities p are obtained in [1], using time

dependent Lyapunov functions techniques.

Notation. B

(x) denotes the open ball of R

of radius R and center x.Ifx =0we

simply write B

.For0≤ a<b,weuseQ(a, b)forR

×(a, b)andQ

for Q(0,T)

(here the intervals can be either open or closed). We write C = C(a

,...,a

)to

point out that the constant C depends on the quantities a

,...,a

. To simplify

the notation, we understand the dependence on the dimension N andonquanti-

ties determined by the matrix (a

) as the ellipticity constant or the modulus of

continuity of the coeﬃcients.

If u : R

× J → R,whereJ ⊂ [0, ∞[ is an interval, we use the following

notation:

∂

u =

∂u

∂t

u =

∂u

∂x

u = D

Du =(D

u,...,D

u),D

u =(D

and

|Du|

i=1

, |D

i,j=1

Let us come to notation for function spaces. C

)isthespaceofj times diﬀer-

entiable functions in R

, with bounded derivatives up to the order j. C

∞

)is

the space of test functions. C

) denotes the space of all bounded and α-H¨older

continuous functions on R

For 1 ≤ k ≤∞,j∈ N,W

) denotes the classical Sobolev space of all

-functions having weak derivatives in L

) up to the order j. Its usual norm

is denoted by ·

j,k

and by ·

when j =0.

Let us now deﬁne some spaces of functions of two variables (following basically

the notation of [7]). C

(Q(a, b)) is the Banach space of continuous functions u

deﬁned in Q(a, b) such that lim

|x|→∞

u(x, t) = 0 uniformly with respect to t ∈