Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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

[a, b]. C

2,1

(Q(a, b)) is the space of all bounded functions u such that ∂

u, Du

and D

u are bounded and continuous in Q(a, b). For 0 <α≤ 1wedenoteby

2+α,1+α/2

(Q(a, b)) the space of all bounded functions u such that ∂

u, Du and

u are bounded and α-H¨older continuous in Q(a, b) with respect to the parabolic

distance d((x, t), (y,s)) := |x − y| + |t − s|

.LocalH¨older spaces are deﬁned, as

usual, requiring that the H¨older condition holds in every compact subset.

We shall also use parabolic Sobolev spaces. We denote by W

r,s

(Q(a, b))

the space of functions u ∈ L

(Q(a, b)) having weak space derivatives D

u ∈

(Q(a, b)) for |α|≤r and weak time derivatives ∂

u ∈ L

(Q(a, b)) for β ≤ s,

equipped with the norm

u

r,s

(Q(a,b))

:= u

(Q(a,b))

|α|≤r

D

u

(Q(a,b))

|β|≤s

∂

u

(Q(a,b))

k,1

) denotes the space of all functions u ∈ W

1,0

)with∂

u ∈ (W

1,0



))



the dual space of W

1,0



), endowed with the norm

u

k,1

)

:= ∂

u

1,0



))



+ u

1,0

)

where



= 1. Finally, for k>2, V

) is the space of all functions u ∈

1,0

) such that there exists C>0forwhich



u∂

φdxdt

≤ C



φ

k−2

)

+ Dφ

k−1

)



for every φ ∈ C

2,1

(Q(a, b)). Notice that

k−1

= k



k−2







. V

)isaBanach

space when endowed with the norm

u

)

= u

1,0

)

+ ∂

uk

,k;Q

where ∂

uk

,k;Q

is the best constant C such that the above estimate holds.

The space H

k,1

) was introduced and studied by Krylov [6]. All properties

of the spaces H

k,1

)andV

) needed here, can be found in [10, Appendix].

In the whole paper the transition density p will be considered as a function

of (y,t) for arbitrary but ﬁxed x ∈ R

. The writing p therefore stands for any

norm of p as function of (y,t), for a ﬁxed x.

2. Local regularity and integrability of transition densities

As a ﬁrst step we recall some local regularity results for the kernel p associated

with the minimal semigroup

T (t)f(x)=



p(x, y, t)f(y) dy,

i.e., the semigroup which deﬁnes the minimal bounded positive solutions of equa-

tion (1.2) when f ≥ 0.

Estimates on Transition Kernels 419

Regularity properties of the kernels p with respect to the variables (y,t)are

known even under weaker conditions than our hypothesis (H), see [2]. We combine

the results of [2] with the Schauder estimates to obtain regularity of p with respect

to all the variables (x, y, t). The proof is similar to the one of Proposition 2.1 in [10].

Proposition 2.1. Under assumption (H) the kernel p = p(x, y, t) is a positive con-

tinuous function in R

×R

×(0, ∞) which enjoys the following properties.

(i) For every x ∈ R

, 1 <s<∞, the function p(x, ·, ·) belongs to H

s,1

loc

(0, ∞)).Inparticularp, D

p ∈ L

loc

×(0, ∞)) and p(x, ·, ·) is continuous.

(ii) For every y ∈ R

the function p(·,y,·) belongs to C

2+α,1+α/2

loc

×(0, ∞))

and solves the equation ∂

p = Ap, t > 0. Moreover

sup

|y|≤R

p(·,y,·)

2+α,1+α/2

×[ε,T ])

< ∞

for every 0 <ε<T and R>0.

(iii) If, in addition, F ∈ C

),thenp(x, ·, ·) ∈ W

2,1

s,loc

) for every x ∈ R

1 <s<∞, and satisﬁes the equation ∂

p = A

∗

p,where

∗

= A

−F · D − (V +divF )

is the formal adjoint of A.

The uniqueness of the bounded solution of (1.2) does not hold in general, but

it is ensured by the existence of a Lyapunov function (cf. [10, Proposition 2.2]),

that is a C

2+α

loc

-function W : R

→ [0, ∞) such that lim

|x|→∞

W (x)=+∞ and

AW ≤ λW for some λ>0. Lyapunov functions are easily found imposing suitable

conditions on the coeﬃcients of A. For instance, W (x)=|x|

is a Lyapunov

function for A provided that

(x)+F (x) · x −|x|

V (x) ≤ C|x|

for some

C>0. The following result can be proved as in [10, Proposition 2.2].

Proposition 2.2. Let W be a Lyapunov function for A and let u, v ∈ C

[0,T]) ∩ C

2,1

× (0,T]) solve (1.2).Thenu = v.

Now we turn our attention to integrability properties of p and show how they

can be deduced from the existence of suitable Lyapunov functions. In the proof

of Proposition 2.4 below we need to approximate the semigroup (T (t))

t≥0

with

semigroups generated by uniformly elliptic operators. This is done in the next

lemma.

Lemma 2.3. Assume that A has a Lyapunov function W .Takeη ∈ C

∞

(R) with

η(s)=1for |s|≤1,η(s)=0for |s|≥2, and deﬁne η

(x)=η





, F

= η

F ,

:= η

V and A

= A

·D −V

. Consider the analytic semigroup (T

(t))

t≥0

generated by A

in C

). Then, for every f ∈ C

2+α

) there exists a sequence

) such that T

(·)f(·) → T (·)f(·) in C

2,1

× [0,T]).

Proof. Let u

(x, t)=T

(t)f(x), u(x, t)=T (t)f(x) and ﬁx a radius >0. If

n>+ 1 the Schauder estimates for the operator A (see, e.g., [5, Theorem 8.1.1])

yield

u



2+α,1+α/2



×[0,T ])

≤ C



f

2+α

)

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

By a standard diagonal argument we ﬁnd a subsequence (n

) such that u

con-

verges to a function u in C

2,1

×(0, ∞)). Since ∂

−Au

=0inB



×[0,T]

for n

>we have ∂

u − Au =0inR

× [0,T]. Moreover, u(x, 0) = f(x)

and |u(x, t)|≤f 

∞

, since this is true for u

. By Proposition 2.2 we infer that

u(x, t)=T (t)f(x). 

The integrability of Lyapunov functions with respect to the measures p(x, y, t) dy

is given by the following result, which is an extension of [12, Lemma 3.9], where

the case V = 0 is considered.

Proposition 2.4. A Lyapunov function W is integrable with respect to the measures

p(x, y, t)dy. Setting

ζ(x, t)=



p(x, y, t)W (y) dy, (2.1)

the inequality ζ(t, x) ≤ e

λt

W (x) holds. Moreover, |AW | is integrable with re-

spect to p(x, ·,t), ζ ∈ C

2,1

× (0, ∞)) ∩ C(R

× [0, ∞)) and D

ζ(x, t) ≤



p(x, y, t)AW (y) dy.

Proof. For α ≥ 0, set W

:= W ∧ α and ζ

(x, t):=



p(x, y, t)W

(y) dy.

Let us consider, for every 0 <ε<1,ψ

∈ C

∞

(R) such that ψ

(t)=t for

t ≤ α, ψ

constant in [α + ε, ∞),ψ



≥ 0, and ψ



≤ 0. Since ψ



≤ 0 one deduces

that

tψ



(t) ≤ ψ

(t), ∀t ≥ 0. (2.2)

Now we approximate A with A

:= A

+ F

·∇−V

and (T (t))

t≥0

with (T

(t))

t≥0

as in Lemma 2.3. Denoting by p

(x, y, t)thekernelof(T

(t))

t≥0

,sinceψ

◦ W ∈

2+α

)wehave

∂

(t)(ψ

◦ W )(x)=



(x, y, t)A

(ψ

◦ W )(y) dy.

On the other hand, by (2.2), we obtain

(ψ

◦ W )(x)=ψ



(W (x))A

W (x)+V

(x)[ψ



(W (x))W (x) − ψ

(W (x))]

+ ψ



(W (x))

i,j=1

(x)D

W (x)D

W (x)

≤ ψ



(W (x))A

W (x).

Thus,

∂

(t)(ψ

◦ W )(x) ≤



(x, y, t)ψ



(W (y))A

W (y) dy

and also

∂

(t)(ψ

◦ W )(x) ≤



(x, y, t)ψ



(W (y))AW (y) dy

Estimates on Transition Kernels 421

if n is suﬃciently large since, for ﬁxed ε, the function ψ



◦ W has compact support.

Letting n →∞and using Lemma 2.3 (possibly passing to a subsequence) we

deduce

∂

T (t)(ψ

◦ W )(x) ≤



p(x, y, t)ψ



(W (y))AW (y) dy. (2.3)

Next we observe that ψ

◦ W ≤ α +1,ψ



(t) → χ

(−∞,α]

(t), and ψ

◦ W → W

pointwise as ε → 0. From [8, Proposition 2.2.9] we deduce that T (t)(ψ

◦ W ) →

T (t)W

in C

2,1

× (0, ∞)). So, letting ε → 0 in (2.3) and using dominated

convergence in the right-hand side (all the integrals can be taken on the compact

set {W ≤ α +1},whereAW is bounded) we get

(x, t) ≤



{W ≤α}

p(x, y, t)AW (y) dy. (2.4)

To conclude we proceed as in the proof of [12, Lemma 3.9]. From (2.4) we obtain

(x, t) ≤ λζ

(x, t) (2.5)

and hence, by Gronwall’s lemma, ζ

(x, t) ≤ e

λt

(x). Letting α →∞we obtain

ζ(x, t) ≤ e

λt

W (x)andthenW is summable with respect to the measure p(x, ·,t).

The inequality 0 ≤ ζ

≤ ζ and the interior Schauder estimates show that the

family (ζ

) is relatively compact in C

2,1

× (0, ∞)). Since ζ

→ ζ pointwise

as α → +∞, it follows that ζ ∈ C

2,1

× (0, ∞)). Moreover, the inequality

(x, t) ≤ ζ(x, t) ≤ e

λt

W (x) implies that ζ(·,t) → W (·)ast → 0

, uniformly on

compact sets. Set E = {x ∈ R

: AW (x) ≥ 0}. Clearly



p(x, y, t)AW (y) dy ≤ λ



p(x, y, t)W (y) dy ≤ λζ(x, t) < ∞. (2.6)

Moreover, letting α → +∞ in (2.3), we obtain that

ζ(x, t) ≤ lim inf

α→+∞



{W ≤α}

p(x, y, t)AW (y) dy.

This fact and (2.6) imply that |AW | is summable with respect to p(x, ·,t)andthat

the above lim inf is a limit, so that the proof is complete. 

Assuming that AW tends to −∞ faster than −W one obtains, by Proposition

2.4, that the function ζ in (2.1) is bounded with respect to the space variables, see

[12, Theorem 3.10] for the case V =0.

Proposition 2.5. Assume that the Lyapunov function W satisﬁes the inequality

AW ≤−g(W ) where g :[0, ∞) → R is a diﬀerentiable convex function such that

g(0) ≤ 0, lim

s→+∞

g(s)=+∞ and 1/g is integrable in a neighbourhood of +∞.

Then for every a>0 the function ζ deﬁned in (2.1) is bounded in R

× [a, ∞).

Moreover, the semigroup (T (t))

t≥0

is compact in C

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

Proof. Observe that g(s) ≤ sg



(s), since g is convex with g(0) ≤ 0. Let us prove

that



p(x, y, t)g(W (y)) dy ≥ g(ζ(x, t)). (2.7)

For, ﬁx x and t and set s

= ζ(x, t). Then, for all y ∈ R

we have

g(W (y)) ≥ g(s

)+g



)(W (y) −s

)

and therefore, multiplying by p(x, y, t) and integrating



p(x, y, t)g(W (y)) dy

≥ g(s

)



p(x, y, t) dy + g



1 −



p(x, y, t) dy



≥ g(s

From Proposition 2.4 and (2.7) we deduce

ζ(x, t) ≤



p(x, y, t)AW (y) dy ≤−



p(x, y, t)g(W (y)) dy ≤−g(ζ(x, t))

and therefore ζ(x, t) ≤ z(x, t), where z is the solution of the ordinary Cauchy

problem





= −g(z)

z(x, 0) = W (x).

Let  denote the greatest zero of g.Thenz(x, t) ≤  if W (x) ≤ . On the other

hand, if W (x) >,thenz is decreasing and satisﬁes

t =



W (x)

z(x,t)

g(s)

≤



∞

z(x,t)

g(s)

. (2.8)

This inequality easily yields, for every a>0, a constant C(a) such that z(x, t) ≤

C(a) for every t ≥ a and x ∈ R

The compactness of (T (t))

t≥0

in C

) can be proved as in [12, Theorem

3.10]. 

Remark 2.6. If



p(x, y, t) dy =1(asisthecaseifV = 0) then (2.7) follows

from Jensen’s inequality and the condition g(0) ≤ 0 is not needed.

Let us state a condition under which certain exponentials or polynomials are

Lyapunov functions. Using the same procedure as for the case V =0(see[10,

Proposition 2.5 and 2.6]) we obtain the following results.

Proposition 2.7. Let Λ be the maximum eigenvalue of (a

) as in (H). Assume that

lim sup

|x|→∞

|x|

1−β



F (x) ·

|x|

−

V (x)

δβ|x|

β−1



< −c, (2.9)

Estimates on Transition Kernels 423

0 <c<∞,forsomec, δ > 0,β > 1 such that δ<(βΛ)

−1

c.ThenW (x)=

exp{δ|x|

} is a Lyapunov function. Moreover, if β>2, there exist positive con-

stants c

such that

ζ(x, t) ≤ c

exp



−β/(β−2)



(2.10)

for x ∈ R

,t>0.

Proposition 2.8. Assume that

lim sup

|x|→∞

|x|

1−β



F (x) ·

|x|

−

|x|

2α

V (x)



< 0, (2.11)

for some α>0, β>2.ThenW (x)=



1+|x|



is a Lyapunov function and

there exists a positive constant c such that

ζ(x, t) ≤ ct

−(2α)/(β−2)

(2.12)

for x ∈ R

, 0 <t≤ 1.

Remark 2.9. Proposition 2.7 will be used to check the integrability of |F |

and V

with respect to p, assuming that |F|,V grow at inﬁnity not faster than exp{|x|

}

for some γ<β.

3. Uniform and pointwise bounds on transition densities

In this section we ﬁx T>0 and consider p as a function of (y, t) ∈ R

×(0,T)for

arbitrary, but ﬁxed, x ∈ R

.Further,ﬁx0<a

<a<b<b

≤ T and assume

for deﬁniteness b

−b ≥ a − a

. Setting

Γ(k, x, a

):=



Q(a

)

(1 + |F (y)|

+ V (y)

)p(x, y, t) dy dt

, (3.1)

the proofs of Proposition 3.1, Lemma 3.1 and Proposition 3.2 in [10] remain valid

for the case V = 0. So, we obtain that

p ∈H

s,1

(Q(a, b)) for all s ∈ (1,k),

provided that Γ(k, x, a

) < ∞ for some k>N+ 2. Hence, by the embedding

theorem for H

s,1

,s>N+2, (see [10, Theorem 7.1]), we have

Theorem 3.1. If Γ(k, x,a

) < ∞ for some k>N+2,thenp belongs to

∞

(Q(a, b)).

To obtain uniform and pointwise bounds on p we introduce the functions

(k, x, a



Q(a

)

(1 + |F (y)|

)p(x, y, t) dy dt

, (3.2)

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

(k, x, a



Q(a

)

(y)p(x, y, t) dy dt

. (3.3)

Clearly Γ

(k, x,a

)+Γ

(k, x, a

) ≤ CΓ(k, x,a

). The following re-

sult shows that only the assumption Γ

(k, x, a

), Γ

(k, x, a

) < ∞ for some

k>N+ 2 is needed to obtain the boundedness of p.

Theorem 3.2. If Γ

(k, x, a

), Γ

(k, x,a

) < ∞ for some k>N+2 then

p

∞

(Q(a,b))

≤ C

(k, x,a

)+Γ

(k, x, a

− a

(a − a

)

. (3.4)

Proof. Step 1. Assume ﬁrst that Γ(k, x,a

) < ∞ so that p ∈ L

∞

(Q(a, b)) for

every a

<a<b<b

by Theorem 3.1 and consider q = η

p ∈ L

∞

)whereη is

a smooth function with compact support in (a

) such that 0 ≤ η ≤ 1, η(t)=1

for a ≤ t ≤ b. Clearly q ∈ L

∞

Let ϕ ∈ C

2,1

) be such that ϕ(·,t) has compact support for every t.From

(1.4) we obtain



q(∂

ϕ + A

ϕ) dy dt



(qF ·Dϕ − Vqϕ+

pϕη

k−2

∂

η) dy dt

Next we note that

pη

k−2



)

≤q

k−2

∞

)

− a

)

and that

Fq

)

≤q

k−1

∞

)

(k, x, a

)

Vq

)

≤q

k−2

∞

)

(k, x, a

Since also

q

)

≤q

k−1

∞

)

−a

)

q

)

≤q

k−2

∞

)

−a

)

Theorem 7.3 in [10] now implies that

q

∞

)

≤ C



q

k−1

∞

)

(k, x, a

)

+ q

k−2

∞

)



(k, x, a

− a

)

a −a



and hence, after a simple calculation,

q

∞

)

≤ C

(k, x, a

)+Γ

(k, x,a

− a

(a −a

)

and (3.4) follows.

Estimates on Transition Kernels 425

Step 2. Let us now consider the general case. Fix a smooth function θ ∈ C

∞

(R)

such that θ(s)=1for|s|≤1, θ(s)=0for|s|≥2 and deﬁne θ

(x)=θ



|x|



= Vθ

. We consider the minimal semigroup (U

(t))

t≥0

generated in C

)by

the operator A

= A

+F ·D−V

.SinceV

≤ V the procedure for constructing the

minimal semigroup recalled in Section 2 and the maximum principle yield U

(t)f ≤

T (t)f for every f ∈ C

). If p

denotes the kernel of U

the above inequality

is equivalent to p

(x, y, t) ≤ p

(

x, y, t). To show that p

converges pointwise to p

we consider the analytic semigroup (T

(t))

t≥0

generated by A on C

), under

Dirichlet boundary conditions (B

is the ball of centre 0 and radius n). Since

= V in B

, the maximum principle gives T

(t)f ≤ U

(t)f ≤ T (t)f in B

for

every f ∈ C

), f ≥ 0. Then r

(x, y, t) ≤ p

(x, y, t) ≤ p(x, y, t)forx, y ∈ B

with ρ<n,wherer

is the kernel of T

in B

. Letting n →∞we see that p

→ p

pointwise, since this is true for r

, see [11, Theorem 4.4].

The proof now easily follows by approximation from Step 1. Let Γ

(k, x,a

)be

the functions deﬁned in (3.1), (3.2), (3.3) relative to p

.Sincep

≤ p and V

≤ V ,

it follows that Γ

(k, x, a

) ≤ Γ

(k, x, a

). Moreover, Γ

(k, x, a

) < ∞ for

every n,sinceV

is bounded. Then we obtain from Step 1

p



∞

(Q(a,b))

≤ C

(k, x, a

)+Γ

(k, x, a

− a

(a −a

)

and the statement follows letting n →∞. 

Now we apply similar techniques to obtain pointwise bounds.

We consider the following assumption depending on the weight function ω

which, in our examples, will be a polynomial or an exponential.

(H1) W

are Lyapunov functions for A, W

≤ W

and there exists 1 ≤ ω ∈

) such that

(i) ω ≤ cW

, |Dω|≤cω

k−1

, |D

ω|≤cω

k−2

(ii) ωV

≤ cW

and ω|F |

≤ cW

for some k>N+2anda constantc>0.

We denote by ζ

,ζ

the functions deﬁned by (2.1) and associated with W

respectively.

By Proposition 2.4 we know that (H1) implies Γ

(k, x,a

) < ∞ for i =1, 2.

In particular, since k>N+ 2, Theorem 3.2 shows that p(x, ·, ·) ∈ L

∞

(Q(a, b)) for

every x ∈ R

The use of diﬀerent Lyapunov functions allows us to obtain more precise

estimates in the theorem below and its corollaries.

The proof of the following result is similar to the one of [10, Theorem 4.1].

For reader’s convenience we give the details.

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

Theorem 3.3. Assume (H1). Then, there exists a constant C>0 such that

0 <ω(y)p(x, y, t) ≤ C



(x, t) dt +

(a − a

)



(x, t) dt

(3.5)

for all x, y ∈ R

,a≤ t ≤ b.

Proof. Step 1. Assume ﬁrst that ω is bounded. As in the proof of Theorem 3.2 we

choose a smooth function η(t) such that η(t)=1fora ≤ t ≤ b and η(t)=0for

t ≤ a

and t ≥ b

, 0 ≤ η



≤

a−a

.Weconsiderψ ∈ C

2,1

) such that ψ(·,T)=0

andsuchthatψ(·,t) has compact support for all t. Setting q = η

p and taking

ϕ(y,t)=η

(t)ω(y)ψ(y, t), from (1.4) we obtain



ωq (−∂

ψ − A

ψ) dydt =





ψA

ω +2

i,j=1

ωD

ψ+ (3.6)

ωF ·Dψ + ψF ·Dω − Vωψ



pωψη

k−2

∂

dy dt.

Since ω is bounded, then ωq ∈ L

) ∩ L

∞

),byTheorem3.2andthen[10,

Theorem 7.3] yields

ωq

∞

)

≤ C



ωq

)

+ ωq

)

+ qD

ω

)

+ qDω

)

+ ωqF

)

+ qFDω

)

+ qV ω

)

a − a

pωη

k−2

)



Next observe that

ωq

)

≤ωq

k−1

∞

)

ωq

)

≤ωq

k−1

∞

)



ωq

)

≤ωq

k−2

∞

)

ωq

)

≤ωq

k−2

∞

)



and that, by (H1)(ii),

ωqF

)

≤ωq

k−1

∞

)

ωqF



)

≤ωq

k−1

∞

)



ωqV 

)

≤ωq

k−2

∞

)

ωqV



)

≤ωq

k−2

∞

)



Moreover, as in the proof of Theorem 3.2 one has

ωpη

k−2



)

≤ωq

k−2

∞

)

ωp

(Q(a

))

≤ωq

k−2

∞

)



Estimates on Transition Kernels 427

Next we combine (H1)(i) and (H1)(ii) to estimate the remaining terms

DωqF 

)

≤





k−2



≤ωq

k−2

∞

)



and, similarly,

D

ωq

)

≤ωq

k−2

∞

)



Dωq

)

≤ωq

k−1

∞

)



Collecting similar terms and recalling that W

≤ W

we obtain

ωq

∞

)

≤ Cωq

k−1

∞

)



+ Cωq

k−2

∞

)

⎛

⎝



a −a



⎞

⎠

hence, after simple computations,

ωq

∞

)

≤ C



dt +

(a − a

)



and (3.5) follows for a bounded ω.

Step 2. If ω is not bounded, we consider ω

1+εω

. A straightforward computation

shows that ω

satisﬁes (H1) with a constant C independent of ε. Therefore, from

Step 1 we obtain

0 <ω

(y)p(x, y, t) ≤ C



(x, t) dt +

(a − a

)



(x, t) dt

, (3.7)

with C independent of ε and, letting ε → 0, the statement is proved. 

Corollary 3.4. Assume that

lim sup

|x|→∞

|x|

1−β



F (x) ·

|x|

−

V (x)

δβ|x|

β−1



< −c, 0 <c<∞ (3.8)

for some δ>0,c > 0,β > 2 such that δ<(βΛ)

−1

c,whereΛ is the maximum

eigenvalue of (a

), and that V (x)+|F (x)|≤c

|x|

β−ε

for some ε, c

> 0.

Then

0 <p(x, y, t) ≤ c

exp



−

β−2



exp



−δ|y|



for x, y ∈ R

, 0 <t≤ T , for suitable c

> 0.