linear parabolic equations in locally uniform spacesjarrieta/papers/2004_linear_parabolic.pdf ·...

February 6, 2004 10:40 WSPC/103-M3AS 00323

Mathematical Models and Methods in Applied SciencesVol. 14, No. 2 (2004) 253–293c© World Scientific Publishing Company

LINEAR PARABOLIC EQUATIONS

IN LOCALLY UNIFORM SPACES

JOSE M. ARRIETA∗ and ANIBAL RODRIGUEZ-BERNAL

Departamento de Matematica Aplicada,

Universidad Complutense de Madrid, 28040 Madrid, Spain∗[email protected]

JAN W. CHOLEWA and TOMASZ DLOTKO

Institute of Mathematics, Silesian University,

40-007 Katowice, Poland

Received 28 January 2003Revised 27 June 2003

Communicated by E. Zuazua

We analyze the linear theory of parabolic equations in uniform spaces. We obtain sharpL

p − Lq-type estimates in uniform spaces for heat and Schrodinger semigroups and

analyze the regularizing effect and the exponential type of these semigroups. We alsodeal with general second-order elliptic operators and study the generation of analyticsemigoups in uniform spaces.

Keywords: Uniform spaces; heat equation; Schrodinger semigroup; analytic semigroups;L

p − Lq estimates; regularizing effect; exponential type.

AMS Subject Classification: 35K

1. Introduction

In this paper we study the linear theory associated to parabolic equations in RN ,

of the form

ut −N

∑

k,l=1

akl(x)∂k∂lu +

N∑

j=1

bj(x)∂ju + c(x)u = 0, x ∈ RN , t > 0

u(0) = u0

(1.1)

with complex valued coefficients and initial data in suitable uniform spaces as ex-

plained below.

As a motivation for considering these sort of spaces, observe that, contrary to the

case of a bounded domain, in an unbounded domain typical Sobolev embeddings

are not compact and the family of spaces Lp are not nested; that is, for p 6= q

no space Lp or Lq contains the other. These two drawbacks and the fact that

253

254 J. M. Arrieta et al.

constant functions are not in Lp(RN ), 1 ≤ p < ∞, makes the analysis of linear and

nonlinear evolutionary equations much more involved. For instance, an equation like

ut = ∆u + u− u3 is much more difficult to treat in RN than in a bounded domain.

Notice that, for this equation, the solutions u = 1, u = −1 and the traveling wave

connections between u = 0 and u = ±1, are no longer included if we pose the

problem in a space of functions like Lp(RN ), 1 ≤ p < ∞. Moreover, the traveling

waves do no connect 0 and ±1 in L∞(RN ).

To overcome this difficulty several approaches have been proposed. First,

weighted spaces have been used among others in Refs. 1, 10, 13, 17 and 20. Some

basic results on the underlying linear theory can be found in Ref. 34. Also, a set-

ting based on the space of bounded and uniformly continuous functions have been

carried over in Refs. 26 and 30 and references therein. Recently a setting in non-

weighted, standard Lebesgue spaces has been presented in Ref. 9, where some basic

fact on the underlying linear theory were taken from Refs. 5 and 34.

On the other hand, Refs. 5, 17, 19, 27, 29, 31, 32 and 34 make use of spaces

of functions which have the property that their elements have some uniform size

when measured in balls of fixed radius but arbitrary center. That is, for instance,

LpU (RN ) = u ∈ Lp

loc(RN ) : ∃C > 0 , ‖u‖Lp(B(x,1)) ≤ C for all x ∈ R

N ,

with norm ‖u‖LpU(RN ) = supx∈RN ‖u‖Lp(B(x,1)). These are the so-called locally uni-

form spaces. It turns out that these spaces are very natural and useful for equations

in unbounded domains since, as in the case of bounded ones, they enjoy suitable

nesting propertie (i.e. if p ≤ q, then LqU (RN ) → Lp

U (RN )) they have locally com-

pact embeddings and constant functions lie within them. In particular, when trying

to analyze parabolic equations in unbounded domains, these spaces will allow to

consider large classes of initial data with no prescribed behavior at infinity and

even allowing for local singularities.

To the best of our knowledge, these spaces were first introduced by Kato in

Ref. 27. He studied hyperbolic equations in standard Sobolev spaces in RN although

the coefficients of the equations belong to this class of uniform spaces. Similarly,

in Ref. 34 the author studies Schrodinger operators ∆ + V in standard Sobolev

spaces with potentials V ∈ LpU (RN ), for some appropriate p ≥ 1. As a matter of

fact the theory is constructed for potentials in a larger class that contains LpU (RN ),

see Ref. 34. In Ref. 5 the authors study the generation of analytic semigroups

in standard Sobolev spaces by rather general second-order elliptic operators. The

coefficients of these operators live in certain classes of uniform spaces. It is important

to note that none of these three papers,5,27,34 use the spaces LpU (RN ) as the space

where solutions live, but rather, where the coefficients of the differential operators

involved are.

On the other hand, Refs. 17, 19, 29, 31 and 32 study different nonlinear equations

and the properties of their nonlinear dynamics in uniform spaces. In Refs. 31 and

32 they study complex Ginzburg–Landau and Swift–Hohenberg equations in uni-

form spaces. They analyze the existence and different properties of the attractors.


Linear Parabolic Equations in Locally Uniform Spaces 255

Actually, the study is done in the subspace of LpU (RN ) consisting of the functions

which are translation continuous. In our paper we denote these spaces as LpU (RN ).

In Refs. 19 the author studies a damped hyperbolic equations in uniform spaces

and studies their asymptotic dynamics and in Ref. 17 they study nonlinear dynam-

ics for reaction-diffusion systems in weighted and uniform spaces in R3. Finally in

Ref. 29 they study the instantaneous smoothing property of the Hermite semigroup

in uniform spaces.

Although the results obtained in these papers are nice and relevant, there are

still many important questions, not treated in the literature yet, regarding just

the linear theory of evolutionary operators in the class of uniform spaces. Among

these aspects we would like to mention the following: sharp LpU −Lq

U estimates for

heat and Schrodinger semigroups; characterizations of the exponential type (de-

cay or growth) of Schrodinger semigroups and its relation with the exponential

type in regular Lp(RN ) spaces; the generation of analytic semigroups by general

second-order elliptic operators (including the Laplace and Schrodinger operators)

with the characterization of the domain and fractional power spaces or interpolation

spaces; other properties of the linear semigroups like the regularizing effect and or-

der preserving properties. All of these properties are extremely important and they

constitute a necessarily first step for a systematic study of nonlinear evolutionary

equations. For instance, the LpU − Lq

U estimates and the regularizing effect of the

linear semigroup will determine the class of nonlinearities for which the associated

nonlinear problem is well posed and in turn it will also determine the smoothness

and compactness properties of invariant sets of the nonlinear equation.

Hence, in this work we present a rather complete linear theory of parabolic

equations with initial data in uniform spaces and in particular we give an answer

to the questions posed in the previous paragraph.

We have organized the results of this paper starting from the less general op-

erator, the Laplace operator and therefore obtaining the strongest results, to the

more general operators for which less strong results are obtained.

In Sec. 2 we study the behavior of the heat equation ut = ∆u in these uniform

spaces. The results are obtained through the integral expression of the solution of

the heat equation. We will show that the semigroups obtained are analytic for all

1 ≤ p ≤ ∞, characterize the domain and will prove sharp LpU −Lq

U estimates. These

estimates are obtained as follows. Notice that trying to estimate the LpU (RN )-norm

of the solution of the heat equation T (t)u0, amounts to obtain estimates of the

Lp(B(x, 1))-norm, independent of x ∈ RN . This can be accomplished by separating

the initial condition u0 into two parts: u0 = u0XB(x,2)+u0XRN\B(x,2), which implies

that T (t)u0 = T (t)(u0XB(x,2)) + T (t)(u0XRN\B(x,2)). Since u0XB(x,2) ∈ Lp(RN ),

the first term can be estimated by regular Lp − Lq estimates. Moreover, since

u0XRN\B(x,2) is null in B(x, 2), the kernel in the integral expression is not singular

anymore when estimated in B(x, 1). This allows us to obtain estimates of the type

‖T (t)(u0XRN\B(x,2))‖L∞(B(x,1)) ≤ C(t)‖u0XRN\B(x,2)‖L1U (RN ), with C(t) bounded.

Combining both estimates we get the desired LpU−Lq

U estimates, see Proposition 2.1

and Corollaries 2.1, 2.2 below.

In Sec. 3 we analyze in detail the semigroups generated by Schrodinger operators

of the form ∆ + V with V ∈ LσU (RN ) for some σ > N/2. These semigroups will

also be analytic in any LpU (RN ) for 1 ≤ p ≤ ∞. We also obtain sharp Lp

U − LqU

estimates and study the decay properties of the semigroups in different uniform

spaces. We will see that the decay rate of the semigroup generated by ∆ + V in

LpU (R) is independent of p and it actually coincides with the decay rate in Lq(RN )

for any 1 ≤ q ≤ ∞. The basic ingredient when analyzing the semigroup generated

by Schrodinger operators is the variation of constants formula. Note that once we

have obtained LpU −Lq

U -estimates for the heat semigroup in Sec. 2 the variation of

constants formula gives us the right tool to derive these estimates for Schrodinger

semigroups.

In Sec. 4 we study the locally uniform spaces and their relation with weighted

and regular Sobolev spaces. We prove sharp embedding results, compactness prop-

erties of the embeddings, density theorems and interpolation inequalities like

Gagliardo–Nirenberg type inequalities. As a difference with other results appear-

ing in the literature, see for instance Ref. 27, we obtain results for fractional order

uniform Sobolev spaces.

In Sec. 5 we give results on the generation of analytic semigroups by general

elliptic operators −aij∂ij + bi∂i + c, with coefficients in suitable uniform spaces,

and describe the associated fractional power spaces. We study the smoothing effect

of the semigroup and will characterize when the semigroup is order preserving.

In particular, we extend to uniform spaces the results in Ref. 5. As a matter of

fact, the results from Ref. 5 are the starting point in the analysis of this section.

They proved, among other things, the generation of analytic semigroups by general

second-order elliptic operators in regular Lp(RN ) spaces. Through an appropriate

isomorphism, see Lemma 5.1, we will translate this property to weighted Sobolev

spaces for a suitable class of weights. The fact now that all the estimates will be

valid for a weight ρ and all its translations ρ(· − y), y ∈ RN and that the estimates

are independent of y will allow one to conclude the results for uniform spaces.

2. The Heat Equation in Locally Uniform Spaces

We define, for 1 ≤ p < ∞, the uniform space LpU (RN ) as the set of functions

φ ∈ Lploc(R

N ) such that

supx∈RN

∫

B(x,1)

|φ(y)|p dy < ∞ (2.1)

with norm

‖φ‖LpU (RN ) = sup

x∈RN

‖φ‖Lp(B(x,1)).

Observe that for p = ∞, using the analogous definition, we have L∞U (RN ) =

L∞(RN ) with norm ‖φ‖L∞U (RN ) = supx∈RN ‖φ‖L∞(B(x,1)) = ‖φ‖L∞(RN ).

Observe that LpU (RN ) contains L∞(RN ), Lr(RN ) and Lr

U (RN ) for any r ≥ p.

Also, in the above definition, the fact that we take the ball of radius one, B(x, 1),

is purely instrumental and changing the radius to any other positive number does

not change the space and gives an equivalent norm in LpU (RN ) and the equivalence

constants depend only on N but not on p.

Also denote by LpU (RN ) a subspace of Lp

U (RN ) consisting of all elements which

are translation continuous with respect to ‖ · ‖LpU (RN ), i.e.

‖τyφ − φ‖LpU (RN ) → 0 as |y| → 0 ,

where τy, y ∈ RN denotes the group of translations. Note that Lp(RN ) ⊂ LpU (RN )

for 1 ≤ p < ∞ and for p = ∞ we get L∞U (RN ) = BUC(RN ).

Finally we also define the uniform Sobolev space W k,pU (RN ) as the set of functions

φ ∈ W k,ploc (RN ) such that

‖φ‖W k,pU (RN ) = sup

x∈RN

‖φ‖W k,p(B(x,1)) < ∞ (2.2)

for k ∈ N, see Sec. 4 below.

Note that an analogous definition allows one to define LpU (Ω) and W k,p

U (Ω) for

an arbitrary unbounded domain Ω. For this just take x ∈ Ω and B(x, 1) ∩ Ω in

(2.1).

Our goal in this section is to give some results on the solutions of the heat

equation in the spaces LpU (RN ), that is, on the solutions of

ut − ∆u = 0 in RN , t > 0

u(0) = u0 ∈ LpU (RN ) .

(2.3)

Note that, formally, the solution of (2.3) is given by the convolution with the heat

kernel

u(t, x) = T (t)u0 = (4πt)−N/2

∫

RN

e−|x−y|2

4t u0(y)dy . (2.4)

The following results show some smoothing properties of (2.4):

Proposition 2.1. There exist a constant M0 depending only on the dimension N,

such that for every u0 ∈ LpU (RN ), the following estimates hold

‖T (t)u0‖LqU (RN ) ≤ M0(t

−N2 ( 1

p− 1q ) + 1)‖u0‖Lp

U (RN ) , 1 ≤ p ≤ q ≤ ∞ , t > 0 .

(2.5)

Proof. Let us consider a cube decomposition of RN as follows. For any index

i ∈ ZN , denote by Qi the open cube in RN of center i with all edges of length 1

and parallel to the axes. Then Qi ∩ Qj = ∅ for i 6= j and RN = ∪i∈ZN Qi. For a

given i ∈ ZN let us denote by N(i) the set of indices near i, that is, j ∈ N(i) if and

only if Qi ∩ Qj 6= ∅. Obviously

dij := infdist(x, y), x ∈ Qi, y ∈ Qj



satisfies dij = 0, if j ∈ N(i), dij ≥ 1, if j 6∈ N(i), and as a matter of fact it is not

difficult to see that dij ≥ ‖i − j‖∞ − 1. Let us denote by Qneari = ∪j∈N(i)Qi and

Qfari = RN \ Qnear

i .

Assume that u0 ∈ LqU (RN ) and, for a fixed i, decompose

T (t)u0 = (T (t)u0)neari + (T (t)u0)

fari

with

(T (t)u0)neari = T (t)(u0XQnear

i) , (T (t)u0)

fari = T (t)(u0XQfar

i)

where X denotes the characteristic function of a set.

The proposition will follow from the following estimates of the two terms of the

decomposition. First,

‖(T (t)u0)neari ‖Lq(Qi) ≤ (4πt)−

N2 ( 1

p− 1q )‖u0‖Lp(Qnear

i ) , t > 0 , (2.6)

and, second,

‖(T (t)u0)fari ‖L∞(Qi) ≤ c(t)‖u0‖L1

U (Qfari ), t ≥ 0 , (2.7)

for some bounded monotonic function c(t) such that c(0) = 0 and 0 ≤ c(t) ≤Ct−N/2e−α/t as t → 0, where C and α > 0 depend only on N .

In fact since the constants for the embedding L∞(Qi) → Lq(Qi) and the re-

strictions LpU (RN ) → Lp(Qnear

i ), LpU (RN ) → L1

U (Qfari ) depend on N but can be

chosen independent of q, p and i, (2.7),(2.6) imply

‖T (t)u0‖Lq(Qi) ≤ C(t−N2 ( 1

p− 1q ) + c(t))‖u0‖Lp

U (RN ) .

Since the LqU (RN ) can be bounded by a constant, only depending on N , times the

supremum of the Lq(Qi) norms, this shows the result.

Now observe that (2.6) follows from standard Lp(RN ) − Lq(RN ) estimates for

the heat equation, since in fact, for t > 0,

‖(T (t)u0)neari ‖Lq(Qi) ≤ ‖T (t)(u0XQnear

i)‖Lq(RN ) ≤ (4πt)−

N2 ( 1

p− 1q )‖u0‖Lp(Qnear

i ) .

We show now (2.7). Observe that we have u0XQfari

=∑

j∈ZN\N(i) uj0 where

uj0 = u0XQj . Hence for all j, we have

T (t)uj0(x) = (4πt)−N/2

∫

RN

e−|x−y|2

4t uj0(y)dy = (4πt)−N/2

∫

Qj

e−|x−y|2

4t uj0(y)dy

which implies that for j 6∈ N(i)

‖T (t)uj0‖L∞(Qi) ≤ (4πt)−N/2e

−d2ij

4t ‖uj0‖L1(Qj) ≤ (4πt)−N/2e

−d2ij

4t ‖u0‖L1U (Qfar

i ) ,

where dij is defined above. Hence,

‖(T (t)u0)fari ‖L∞(Qi) ≤

∑

j∈ZN\N(i)

‖T (t)uj0‖L∞(Qi)

≤ (4πt)−N/2‖u0‖L1U (Qfar

i )

∑

j∈ZN\N(i)

e−d2

ij4t .

But, using that #j ∈ Z, dij = k ≤ CkN−1

∑

j∈ZN\N(i)

e−d2

ij4t ≤ C

∞∑

k=1

kN−1e−k2

4t = tN/2c(t)

with c(t) as in (2.7).

A careful examination of the proof above allows us to conclude that

Corollary 2.1. If u0 ∈ LpU (RN ), fixing x0 ∈ RN , and r < R, then setting

T (t)u0 = (T (t)u0)nearx0

+ (T (t)u0)farx0

with

(T (t)u0)nearx0

= T (t)(u0XB(x0,R)), (T (t)u0)farx0

= T (t)(u0XRN\B(x0,R)) ,

where X denotes the characteristic function of a set, we have for 1 ≤ p ≤ q ≤ ∞

‖(T (t)u0)nearx0

‖Lq(B(x0,r)) ≤ (4πt)−N2 ( 1

p− 1q )‖u0‖Lp(B(x0,R)) , t > 0 (2.8)

and

‖(T (t)u0)farx0

‖L∞(B(x0,r)) ≤ c(t)‖u0‖L1U (RN\B(x0,R)) , t ≥ 0 (2.9)

for some bounded monotonic function c(t) such that c(0) = 0 and 0 ≤ c(t) ≤Ct−N/2e−α/t as t → 0, where C and α > 0 depend only on R, r and N but not in t,

x0 or u0. Moreover, for fixed N, t and r, we have c(t) → 0 and α → ∞ if R → ∞.

Concerning higher regularity of (2.4) we indeed get

Corollary 2.2. With the notations above, if u0 ∈ LpU (RN ), we have for any k ∈ N

and 1 ≤ p ≤ q ≤ ∞

‖(T (t)u0)nearx0

‖W k,q(B(x0,r)) ≤ Ct−12 (k− N

q + Np )‖u0‖Lp(B(x0,R)) , t > 0 , (2.10)

and

‖Dβ(T (t)u0)farx0

‖L∞(B(x0,r)) ≤ t−k2 c(t)‖u0‖L1

U (RN\B(x0,R)), t ≥ 0 (2.11)

for any 1 ≤ |β| = k, where c(t) is as in (2.9). Note that t−k2 c(t) → 0 as t → 0.

In particular

‖DβT (t)u0‖LqU(RN ) ≤ Ct−

12 (k−N

q + Np )‖u0‖Lp

U (RN ) t > 0 ,

for any 1 ≤ |β| = k, and (2.4) satisfies the heat equation (2.3) pointwise for t > 0.

Proof. Note that (2.10) again follows from the known smoothing properties of the

heat kernel in Lp(RN ).

As for (2.11), observe that, using for simplicity the notations of the proof of

Proposition 2.1, we have that for any multi-index β

DβT (t)uj0(x) = (4πt)−N/2

∫

Qj

Dβxe−

|x−y|2

4t uj0(y)dy

= (4πt)−N/2

∫

Qj

pβ(|x − y|, t)e− |x−y|2

4t uj0(y)dy ,

where pβ(|x − y|, t) is the sum of polynomials in |x − y| over powers of t and

that the leading term is a polynomial of degree |β| over t|β|. Therefore |pβ(|x −y|, t)|e− |x−y|2

4t ≤ Ct|β|/2 e−c |x−y|2

t , for some positive constants C, c, and arguing as in

Proposition 2.1, we get the result.

As for the attainment of the initial data in (2.4), we have

Corollary 2.3. For any bounded set B ⊂ RN and for any u0 ∈ LpU (RN ), 1 ≤ p <

∞, we have

‖T (t)u0 − u0‖Lp(B) → 0, as t → 0 .

If moreover u0 ∈ LpU (RN ), 1 ≤ p ≤ ∞, then T (t)u0 ∈ Lq

U (RN ) for every t > 0 and

any 1 ≤ q ≤ ∞ and

‖T (t)u0 − u0‖LpU (RN ) → 0 , as t → 0 .

Proof. Clearly it is enough to take B = B(x0, r) with x0 ∈ RN and r > 0. Then

decomposing T (t)u0 as in Corollary 2.1, we have that, by the continuity at t = 0 of

the heat semigroup Lp(RN ), as t → 0,

(T (t)u0)nearx0

= T (t)(u0XB(x0,R)) → u0 in Lp(B(x0, R)) .

On the other hand, from (2.9), as t → 0,

‖(T (t)u0)farx0

‖L∞(B(x0,r)) → 0

and we obtain the result.

Assume now u0 ∈ LpU (RN ), for 1 ≤ p ≤ ∞. Then from (2.4) is immediate to

get that for any y ∈ RN and t > 0, τyT (t)u0 = T (t)τyu0, which together with

(2.5), implies that T (t)u0 ∈ LqU (RN ) for every t > 0 and any p ≤ q ≤ ∞. Since the

uniform spaces are nested, we get the result for 1 ≤ q < p as well.

For the continuity at t = 0, denote K(t, x) the heat kernel, i.e. K(t, x) =

(4πt)−N/2e−|x|2

4t . Hence, using Holder’s inequality and that the integral of K(t, ·)is 1, we get, for 1 ≤ p < ∞

|T (t)u0(x) − u0(x)|p =

∣

∣

∣

∣

∫

RN

K(t, z)(u0(x − z) − u0(x))dz

∣

∣

∣

∣

p

≤∫

RN

K(t, z)|u0(x − z) − u0(x)|pdz .

Integrating on B(x0, 1) and using Fubini’s theorem, we then get

‖T (t)u0 − u0‖pLp(B(x0,1)) ≤

∫

RN

K(t, z)‖u0(· − z) − u0‖pLp(B(x0,1))dz .

We split the integral over the sets |z| ≤ τ and |z| > τ. Then, since u0 ∈Lp

U (RN ), for sufficiently small τ > 0, the first term can be made arbitrarily small

uniformly in x0 ∈ RN , while the second term can be bounded by

C(‖u0‖LpU (RN ))

∫

|z|>τK(t, z)dz

which tends to 0 as t → 0. Hence, we get the continuity at t = 0. On the other

hand, if p = ∞, the result follows similarly as above.

From the results above we get

Theorem 2.1. The expression (2.4) defines an order preserving analytic semigroup

in LpU (RN ), for 1 ≤ p ≤ ∞, which is continuous even at t = 0 if u0 ∈ Lp

U (RN ). In

particular, (2.4) is a strong solution of the heat equation (2.3) in LpU (RN ).

Moreover the generator of the analytic semigroup is the operator ∆ in LpU (RN )

with domain

D(∆)

⊂ W 1,qU for any q <

N

N − 1if p = 1

= W 2,pU for 1 0,

T (t)u0 ∈ W 1,pU (RN ) ⊂ Lp

U (RN ), see Remark 4.2 below. Therefore, Corollary 2.3

implies that (2.4) defines a semigroup in LpU (RN ) which is continuous for t > 0

and even up to t = 0 if u0 ∈ LpU (RN ). Since the heat kernel is non-negative, the

semigroup is order preserving.

Finally, Proposition 2.1 with q = p and Corollary 2.2 with k = 2 and q = p

imply that there exists a constant C > 0 such that for t ∈ (0, 1],

‖T (t)‖L(LpU(RN )) ≤ C

and

‖∆T (t)‖L(LpU (RN )) ≤

C

t.

Since the semigroup is clearly one to one it suffices to show that it is in fact

differentiable in LpU (RN ) since then Proposition 2.1.9 in p. 41 of Ref. 28 implies

analyticity of the semigroup.



Note that since we know that u(t) = T (t)u0 given in (2.4) satisfies (2.3) point-

wise and ut = ∆u(t) ∈ LpU (RN ), for each t > 0 we must show that

δ(h) =T (t + h)u0 − T (t)u0

h− ∆T (t)u0

goes to zero in LpU (RN ), as h → 0.

For this note that from (2.4) we get, for any t > 0 and x ∈ RN

δ(h)(x) =

∫

RN

(

K(t + h, x − y) − K(t, x − y)

h− Kt(t, x − y)

)

u0(y)dy ,

where K(t, x) = (4πt)−N/2e−|x|2

4t denotes the heat kernel. Note then that we have

δ(h)(x) =

∫

RN

(Kt(t + θh, x − y) − Kt(t, x − y))u0(y)dy

=

∫

RN

θhKtt(t + θh, x − y)u0(y) dy

for some θ, θ ∈ [0, 1] and the kernel H(t, x − y) = θhKtt(t + θh, x − y) can be

bounded in modulus by a term of the form hR(t)e−c |x−y|2

t with a rational function

of time R(t) and c > 0.

Proceeding as in Proposition 2.1, we fix the cube Qi and split u0, and hence

the integral for δ(h), into the near-i and far-i components. Then the near-i part

converges to zero as h → 0 in Lp(RN ) because of the known properties of the heat

equation in this space. In particular this component goes to zero in Lp(Qi) and in

fact we get

‖δ(h)neari ‖Lq(Qi) ≤ ‖δ(h)near

i ‖Lp(RN ) ≤ Ch‖u0‖Lp(Qneari )

with C independent of i.

For the far-i part, we split the integral again in the cubes Qj with j 6∈ N(i) and

we get that the far-i part satisfies

‖δ(h)fari ‖L∞(Qi) ≤ Ch‖u0‖L1U (Qfar

i )

with C independent of i. Hence δ(h) goes to zero in LpU (RN ) as h → 0 and we get

the differentiability.

To conclude the theorem it remains to prove the statement about the domain of

the generator. For this in what follows we replace −∆ by −∆ + I which introduces

an exponential factor e−t in (2.4).

By Lemma 2.1.6 on p. 40 in Ref. 28, we have that for any f ∈ LpU (Ω) we have,

for A = ∆ − I ,

u0 = A−1f =

∫ ∞

0

e−tT (t)f dt.

We start first with the case 1 < p < ∞. Plugging (2.4) into this expression we

get, for x ∈ RN ,

u0(x) =

∫

RN

G(x − y)f(y)dy

with G(x) the Green’s function for −∆ + I in RN . Hence, it remains to show that

u0 ∈ W 2,pU (RN ) for every f ∈ Lp

U (Ω). Note that formally we have that −∆u0 +

u0 = f and to obtain the equivalent to the Calderon–Zigmud estimates now in the

uniform spaces.

For this we again fix the cube Qi and split f into the near-i and far-i components.

Then the near-i part of u0 is in W 2,p(RN ) because of the known properties of the

Laplace equation in this space. In particular this component is in W 2,p(Qi) and in

fact

‖(u0

)near

i‖W 2,p(Qi) ≤ ‖(u0

)near

i‖W 2,p(RN ) ≤ C‖f‖Lp(Qnear

i )

with C independent of i.

For the far-i part, we split the integral again in the cubes Qj with j 6∈ N(i) and,

since G is of class C∞ and decays exponentially as at infinity together with all its

derivatives, we get that the far-i part belongs to W k,∞(Qi) for any k ∈ N and in

fact

‖(u0)fari ‖W k,∞(Qi) ≤ C‖f‖L1

U (Qfari )

with C independent of i. Thus

‖u0‖W k,pU (RN ) ≤ C‖f‖Lp

U(RN )

and the proof is complete.

Now, for the case p = 1 notice that, from (2.5) and Corollary 2.2 with k = 1 we

get

‖u0‖W 1,qU (RN ) ≤ ‖f‖L1

U(RN )

∫ ∞

0

e−tg(t)dt

with g(t) ≈ t−12 (1+N−N

q ) for t → 0, hence the integral is convergent for q <

N/(N − 1).

Finally for p = ∞, since the uniform spaces are nested it is clear that the domain

of the generator must be included in W 2,qU (RN ) for any q < ∞.

Note that a basic ingredient in the proof above is the Gaussian structure of the

heat kernel. Hence, the same results apply to any parabolic operator with a similar

Gaussian bound for the kernel, see Ref. 14. Hence, we get

Theorem 2.2. Assume that the differential operator L is given by

L(u) = −N

∑

i=1

∂i(ai,j(x)∂ju + ai(x)u) + bi(x)∂iu + c0(x)u (2.12)

with real coefficients ai,j , ai, bi, c0 ∈ L∞(RN ) and satisfies the ellipticity condition

N∑

i,j=1

ai,j(x)ξiξj ≥ α0|ξ|2 (2.13)

for some α0 > 0 and for every ξ ∈ RN .

Then the fundamental solution of the parabolic problem ut + Lu = 0 in RN

satisfies a Gaussian bound

0 ≤ k(x, y, t, s) ≤ C(t − s)−N/2eω(t−s)e−c |x−y|2

(t−s)

for t > s and x, y ∈ RN where C, c, ω depend on the L∞ norm of the coefficients.

Therefore

u(t, x) = TL(t)u0 =

∫

RN

k(x, y, t, 0)u0(y)dy (2.14)

defines an order preserving semigroup in LpU (RN ), for 1 ≤ p ≤ ∞, which is contin-

uous even at t = 0 if u0 ∈ LpU (RN ). Moreover, Proposition 2.1 and Corollaries 2.1

(with an extra factor eωt in the estimates) and 2.3 hold true.

Proof. The Gaussian bounds are obtained from Ref. 14 while the positivity of the

kernel comes from the maximum principle, see Ref. 22, Chap. 8.

On the other hand, for slightly more regular coefficients we have, from Ref. 21

(see also the remark on p. 255).

Theorem 2.3. Assume that the differential operator A is given by

A(u) = −N

∑

i,j=1

ai,j(x)∂2i,ju +

N∑

i=1

bi(x)∂iu + c0(x)u (2.15)

with Holder continuous real coefficients ai,j , bi, c0 and satisfies the ellipticity condi-

tion (2.13).

Then, for any given T > 0, the fundamental solution of the parabolic problem

ut + Au = 0 in RN satisfies the Gaussian bounds

|Dβxk(x, y, t, s)| ≤ C(t − s)−(N+|β|)/2e−c

|x−y|2

(t−s)

for |β| ≤ 2, 0 ≤ s < t ≤ T and x, y ∈ RN where C, c depend on the Holder norm of

the coefficients and T .

Therefore

u(t, x) = TA(t)u0 =

∫

RN

k(x, y, t, 0)u0(y) dy (2.16)

defines an order preserving analytic semigroup in LpU (RN ), for 1 ≤ p ≤ ∞, which is

continuous even at t = 0 if u0 ∈ LpU (RN ). Moreover, Proposition 2.1 and Corollaries

2.1, 2.2, up to k = 2, and 2.3 hold true for 0 ≤ t ≤ T .

See also Ref. 15 for similar bounds on the fundamental solution but independent

of T > 0.

3. Schrodinger Operators in Uniform Spaces

Because of their importance in applications we discuss in this section the behavior

of Schrodinger operators in uniform spaces. Although most of the results below hold

true for more general elliptic operators replacing the Laplace operator, we will focus

below in the operator −∆ − V where V is a potential in LσU (RN ) with σ > N/2,

σ ≥ 1. Hence, we state our results for the solutions of the linear parabolic equation

ut − ∆u = V (x)u , x ∈ RN t > 0 ,

u(0) = u0 ∈ LpU (RN )

(3.1)

see Ref. 34 for properties of this equation in Lp(RN ) spaces.

Our first result shows that, for small p, (3.1) defines an analytic semigroup

in LpU (RN ). Note that in the proof below we treat the Schrodinger operator as a

perturbation of the Laplace operator.

Proposition 3.1. If V ∈ LσU (RN ) with σ > N/2 then ∆ + V, with domain

W 2,pU (RN ), generates an order preserving analytic semigroup in Lp

U (RN ) for any

1 ≤ p < σ.

Proof. From the previous section we know that ∆, with domain W 2,pU (RN ), gen-

erates an analytic semigroup in LpU (RN ) for 1 < p < ∞.

If 1 < p < σ, then the multiplication operator denoted by V is a bounded

linear operator from V : LrU (RN ) → Lp

U (RN ), with 1p = 1

σ + 1r and ‖V u‖Lp

U (RN ) ≤‖V ‖Lσ

U (RN )‖u‖LrU(RN ). Notice that the condition p < σ guarantees that 1 < r < ∞.

But we have that W 2,pU (RN ) → Ls

U (RN ) with 2N − 1

p = − 1s and in particular

2N − 1

σ = 1r − 1

s . Since σ > N/2, we have that s > r. By standard interpolation

inequalities, see Lemma 4.1 (i) below, we have that for all ε > 0 there exists a Cε

satisfying

‖V u‖LpU (RN ) ≤ ε‖u‖W 2,p

U (RN ) + Cε‖u‖LpU(RN ) .

Applying now perturbation results for sectorial operators, we get that ∆ + V gen-

erates an analytic semigroup in LpU (RN ), 1 < p < σ.

The order preserving property will be obtained below in Proposition 5.3, using

the order preserving property in Lp(RN ) from Ref. 34.

Note that the case p = 1 follows as above, since the LrU (RN ) norm above can

be controlled using the description of the domain of ∆ in L1U (RN ) obtained in

Theorem 2.1.

Note that by taking p = 1, we have a semigroup well defined in a very large

space. Hence, since the spaces are nested, if u0 ∈ LpU (RN ) for p > σ, the semigroup

e(∆+V )tu0 is well defined in a larger space LrU (RN ) for 1 ≤ r < σ.

Now we prove LpU (RN ) − Lq

U (RN ) estimates for the Schrodinger semigroup as

the ones in Sec. 2 for the heat equation.

Proposition 3.2. There exist constants a and M1 depending only on N, σ and

‖V ‖LσU(RN ) such that

‖e(∆+V )tu0‖LqU (RN ) ≤ M1e

att−N2 ( 1

p− 1q )‖u0‖Lp

U (RN ) , 1 ≤ p ≤ q ≤ ∞ . (3.2)

Proof. We will show that estimate (3.2) is valid for a small time interval 0 < t ≤ τ0,

for some τ0 > 0, depending only on N , σ and ‖V ‖LσU (RN ). Observe that once this

is proved, we will obtain the estimate for an arbitrary t, may be with different

constants M1 and a. Indeed if t ≥ τ0 we may decompose t = nτ0 + s for some

0 ≤ s < τ0. Iterating n times (3.2) with p = q and denoting by u(t) = e(∆+V )tu0,

we obtain

‖u(t)‖LqU(RN ) ≤ M1e

as‖u(nτ0)‖LqU (RN ) ≤ M1e

as(M1eaτ0)n−1‖u(τ0)‖Lq

U (RN )

≤ (M1eaτ0)n‖u(τ0)‖Lq

U (RN ) .

Therefore,

‖u(t)‖LqU(RN ) ≤ (M1e

aτ0)n+1τ−N

2 ( 1p− 1

q )

0 ‖u0‖LpU (RN ) ≤ M1e

att−N2 ( 1

p− 1q )‖u0‖Lp

U (RN )

for some constants M and a depending only on M1, a and τ0, that is, depending

only on N , σ and ‖V ‖LσU (RN ). Putting together the estimate for 0 < t ≤ τ0 and

t ≥ τ0 we get (3.2) for arbitrary t > 0.

We show now that estimate (3.2) is valid for 0 < t ≤ τ0 for some small τ0 > 0.

Notice that, if we denote by u(t) = e(∆+V )tu0, the semigroup e(∆+V )t can be

expressed in terms of the variation of constants formula as

u(t) = e(∆+V )tu0 = e∆tu0 +

∫ t

0

e∆(t−s)V (x)u(s)ds .

We define γ = maxσ′, q, where σ′ is the conjugate exponent for σ. Then, for

1 < p ≤ q ≤ ∞ and u0 ∈ LpU (RN ) we have by Proposition 2.1 (using (2.5) for a

small time interval)

‖u(t)‖LqU (RN ) ≤ ‖e∆t‖p,q‖u0‖Lp

U (RN ) +

∫ t

0

‖e∆(t−s)‖r,q‖V u(s)‖LrU (RN )ds

≤ M0t−N

2 ( 1p− 1

q )‖u0‖LpU(RN ) +

∫ t

0

M0(t − s)−N2 ( 1

r − 1q )‖V ‖Lσ

U (RN )‖u(s)‖LγU(RN )ds ,

with 1r = 1

σ + 1γ and where we denote by ‖ · ‖r,p the norm of the operator as a

mapping from LrU (RN ) → Lp

U (RN ). Notice that by the choice of γ, if 1 ≤ q ≤ σ′,then γ = σ′ and r = 1. If q ≥ σ′, then γ = q ≥ σ′ which implies that r ≥ 1. In

particular we have

N

2

(

1

r− 1

q

)

≤

N

2

(

1 − 1

q

)

≤ N

2

1

σ< 1 , if 1 ≤ q ≤ σ′ ,

N

2

1

σ< 1 , if q ≥ σ′ .

(3.3)

We will divide the rest of the proof into several steps.

Step 1. Assume that q ≥ σ′ and 0 ≤ N2 ( 1

p − 1q ) ≤ α, where α = N

2σ < 1 if σ < ∞or α = 1

2 if σ = ∞.

We have,

‖u(t)‖LqU(RN ) ≤ M0t

−N2 ( 1

p− 1q )‖u0‖Lp

U (RN )

+ M0‖V ‖LσU (RN )

∫ t

0

(t − s)−N2

1σ ‖u(s)‖Lq

U (RN )ds .

Denoting by h(t) = tN2 ( 1

p− 1q )‖u(t)‖Lq

U (RN ), we have the inequality

h(t) ≤ M0‖u0‖LpU(RN ) + t

N2 ( 1

p− 1q )M0‖V ‖Lσ

U (RN )

∫ t

0

(t − s)−N2

1σ s−

N2 ( 1

p− 1q )h(s)ds

≤ M0‖u0‖LpU (RN ) + t1−

N2

1σ M0‖V ‖Lσ

U (RN ) sup0≤s≤t

h(s)

∫ 1

0

(1 − z)−N2

1σ z−

N2 ( 1

p− 1q )dz ,

where we have performed the change of variables s = zt in the integral.

But the integral in the above expression is finite with a bound independent of

p and q. Therefore, we can choose a τ0 > 0 depending only on M0, σ, ‖V ‖LσU(RN ),

such that

τ1− N

21σ

0 M0‖V ‖LσU (RN )

∫ 1

0

(1 − z)−N2

1σ z−

N2 ( 1

p− 1q ) dz ≤ 1

2.

Hence for 0 ≤ t ≤ τ0 we get

h(t) ≤ 2M0‖u0‖LpU (RN ) .

This implies that

‖u(t)‖LqU(RN ) ≤ 2M0t

−N2 ( 1

p− 1q )‖u0‖Lp

U (RN ), 0 < t ≤ τ0 . (3.4)

Step 2. Assume 1 ≤ q < σ′, 1 ≤ p ≤ q.

Notice first that if σ = ∞ this case is empty. Hence, we are implicitly assuming

that σ < ∞.

Considering q = σ′ from Step 1, we have that for all p with 0 ≤ 1p − 1

q ≤ 1σ ,

estimates (3.4) are valid. But this last restriction is equivalent to 1 ≤ p ≤ q.

If q < σ′, we have γ = σ′. Therefore, we get for any 1 ≤ p ≤ q,

‖u(t)‖LqU (RN ) ≤ M0t

−N2 ( 1

p− 1q )‖u0‖Lp

U (RN )

+ M0‖V ‖LσU (RN )

∫ t

0

(t − s)− N

2q′ ‖u(s)‖Lσ′

U (RN )ds .

But using expression (3.4) for q = σ′ and 1 ≤ p ≤ q, we get for 0 ≤ t ≤ τ0,

‖u(t)‖LqU(RN ) ≤ M0t

−N2 ( 1

p− 1q )‖u0‖Lp

U (RN )

+ M0‖V ‖LσU (RN )2M0‖u0‖Lp

U (RN )

∫ t

0

(t − s)− N

2q′ s−N2 ( 1

p− 1σ′ )ds .

Using the change of variables s = zt in the integral above and noting that∫ 1

0

(1 − z)− N

2q′ z−N2 ( 1

p− 1σ′ )dz ≤ C

for some constant C depending only on N and σ, we have

‖u(t)‖LqU (RN ) ≤ M0t

−N2 ( 1

p− 1q )‖u0‖Lp

U (RN )[1 + 2M0C‖V ‖LσU (RN )t

1− N2σ ] .

Hence, for 1 ≤ p ≤ q ≤ σ′, we have

‖u(t)‖LqU (RN ) ≤ Mt−

N2 ( 1

p− 1q )‖u0‖Lp

U (RN ), 0 < t ≤ τ0 ,

where M depends on N , and σ.

Step 3. Observe that summarizing the results from Steps 1 and 2, we have found

constants M , τ0 depending only on N , σ and ‖V ‖LσU (RN ) such that for any 1 ≤ p ≤

q ≤ ∞ satisfying N2 ( 1

p − 1q ) ≤ α, we have

‖u(t)‖LqU(RN ) ≤ Mt−

N2 ( 1

p− 1q )‖u0‖Lp

U (RN ) , 0 < t ≤ τ0 . (3.5)

Let us define now a positive integer K such that we have a partition of the

interval [1,∞], of the form 1 = r0 < r1 < · · · < rK = ∞, with the property

that for any p, q with rk ≤ p ≤ q ≤ rk+1 we have N2 ( 1

p − 1q ) ≤ α. Hence, for any

1 ≤ p ≤ q ≤ ∞, we will have k and h such that p ∈ [rk , rk+1] and q ∈ [rk+h, rk+h+1].

This allows us to iterate the inequality above jumping from p → rk+1 → rk+2 →· · · → rk+h → q. The number of jumps is h + 1 which is always bounded by K + 1.

Hence, for any 0 < t ≤ τ0, decomposing the interval [0, t] in h+1 intervals of length

t/(h + 1) we will have

‖u(t)‖LqU(RN ) ≤ M

(

t

h + 1

)−N2 ( 1

rk+h− 1

q ) ∥

∥

∥

∥

u

(

ht

h + 1

)∥

∥

∥

∥

Lrk+hU (RN )

∥

∥

∥

∥

u

(

ht

h + 1

)∥

∥

∥

∥

Lrk+hU (RN )

≤ M

(

t

h + 1

)−N2 ( 1

rk+h−1− 1

rk+h) ∥

∥

∥

∥

u

(

(h − 1)t

h + 1

)∥

∥

∥

∥

Lrk+h−1U (RN )

...∥

∥

∥

∥

u

(

t

h + 1

)∥

∥

∥

∥

Lrk+1U (RN )

≤ M

(

t

h + 1

)−N2 ( 1

p− 1rk+1

)

‖u(0)‖LpU(RN ) .

Multiplying all the expressions above, we obtain

‖u(t)‖LqU(RN ) ≤ Mh+1

(

t

h + 1

)−N2 ( 1

p− 1q )

‖u0‖LpU (RN )

≤ Mh+1(h + 1)N2 ( 1

p− 1q )t−

N2 ( 1

p− 1q )‖u0‖Lp

U (RN )

≤ Mt−N2 ( 1

p− 1q )‖u0‖Lp

U (RN ), 0 < t ≤ τ0 ,

where M = MK+1(K + 1)N/2, which depends only on N and ‖V ‖LσU (RN ). This

concludes the proof of the proposition.

Note that we cannot expect that e(∆+V )tu0 behaves continuously at t = 0 in

LpU (RN ) since the domain is not dense, see Ref. 32 and Sec. 5 below. Neverthe-

less, as the following result shows, continuity in Lploc(R

N ) can be recovered, see

Corollary 2.3.

Corollary 3.1. If V ∈ LσU (RN ) for some σ > N/2, σ ≥ 1, then for any bounded

set B ⊂ RN and for any u0 ∈ LpU (RN ), 1 ≤ p < ∞, we have

‖e(∆+V )tu0 − u0‖Lp(B) → 0, as t → 0 .

If moreover u0 ∈ LpU (RN ), 1 ≤ 4p ≤ ∞, then

‖e(∆+V )tu0 − u0‖LpU (RN ) → 0, as t → 0 .

Proof. If we denote by u(t) = e(∆+V )tu0, we have u(t) − u0 = e∆tu0 − u0 +∫ t

0e∆(t−s)V u(s)ds.

The integral term can be treated as in the proposition above to show that in

fact it converges to zero in LpU (RN ). For instance, if p ≥ σ′, then

∫ t

0

‖e∆(t−s)V u(s)‖LpU(RN )ds ≤ M

∫ t

0

(t − s)−N2σ ‖V ‖Lσ

U (RN )‖u(s)‖LpU (RN )ds

≤ M‖V ‖LσU (RN )‖u0‖Lp

U (RN )

∫ t

0

(t − s)−N2σ ds → 0, as t → 0 .

If 1 ≤ p ≤ σ′, then∫ t

0

‖e∆(t−s)V u(s)‖LpU (RN )ds ≤ M

∫ t

0

(t − s)− N

2p′ ‖V ‖LσU (RN )‖u(s)‖Lσ′

U (RN )ds

≤ M‖V ‖LσU(RN )‖u0‖Lp

U (RN )

∫ t

0

(t − s)− N

2p′ s−N2 ( 1

p− 1σ′ ) ds → 0, as t → 0 .

The rest follows from Corollary 2.3.

Now we can prove that in fact (3.1) defines an analytic semigroup in LpU (RN ),

for 1 ≤ p ≤ ∞. For this we will make use of the following useful lemma. Observe

that we do not require the semigroup below to be continuous at t = 0.

Lemma 3.1. Assume S(t)t≥0 is an analytic semigroup in a Banach space X.

Assume that for some Banach space Y and for t > 0,

S(t) : X → Y

is continuous.



Then for each u0 ∈ X, the curve of the semigroup (0,∞) 3 t 7→ S(t)u0 is ana-

lytic in Y . Moreover for each t0, the Taylor series in Y has a radius of convergence

not smaller than the one in X.

In particular if Y ⊂ X, with continuous injection, then S(t)t≥0 defines an

analytic semigroup in Y .

Proof. Denote u(t) = S(t)u0 for u0 ∈ X . Since the semigroup is analytic in X we

have that, for any fixed t0 > 0 the Taylor series

u(t) = S(t)u0 =

∞∑

m=0

1

m!um)(t0)(t − t0)

m, with um)(t0) = Sm)(t0)u0

has a positive radius of convergence in X . Also, for any ε > 0, um)(t0) =

AmS(t0)u0 = S(ε)AmS(t0 − ε)u0, where −A is the infinitesimal generator of S(t)

in X . Hence, we get

‖um)(t0)‖Y ≤ C(ε)‖AmS(t0 − ε)u0‖X = C(ε)‖um)(t0 − ε)‖X

and so the Taylor series has a positive radius of convergence in Y not smaller than

in X .

In particular if Y ⊂ X , with continuous injection, and u0 ∈ Y , the proof above

gives that

‖Sm)(t0)u0‖Y ≤ C(ε)‖um)(t0 − ε)‖X ≤ C(ε)‖S(m)(t0 − ε)‖L(X)‖u0‖X

≤ C(ε)‖S(m)(t0 − ε)‖L(X)‖u0‖Y .

Since the semigroup is analytic in X , the Taylor series converges in L(X) and

from the estimate above the Taylor series also converges in L(Y ).

Using the lemma and (3.2) we get at once

Theorem 3.1. If V ∈ LσU (RN ) for some σ > N/2, then ∆ + V generates an order

preserving analytic semigroup in LpU (RN ), for 1 ≤ p ≤ ∞ which is continuous even

at t = 0 if u0 ∈ LpU (RN ).

Moreover, for each 1 ≤ p ≤ ∞ and u0 ∈ LpU (RN )

(0,∞) 3 t 7−→ e(∆+V )tu0 ∈ L∞U (RN ) = BUC(RN )

is analytic.

Proof. First take X = L1U (RN ) and Y = Lp

U (RN ) and apply Lemma 3.1 and

Corollary 3.1. Then we do the same by taking Y = L∞U (RN ).

Finally we take σ > r > N/2 and first Y = LrU (RN ), use the semigroup in this

space and finally we apply Lemma 3.1 with X = LrU (RN ) and Y = W 2,r

U (RN ) ⊂L∞

U (RN ).

Now we turn to the question of the exponential type of the Schrodinger

semigroups.

By Proposition 3.2, with q = p, the semigroup e(∆+V )t has at most an expo-

nential growth in the spaces LpU (RN ). We are interested in relating the optimal

exponential growth of this operator in the different uniform spaces and also to

see the relationship of this growth with the exponential growth in regular Lp(RN )

spaces, 1 ≤ p ≤ ∞. It is then natural to define,

Definition 3.1. The exponential type of a semigroup S(t) in a space X is the

number ν(X) ∈ R such that for all b > ν(X) there exists a constant M = M(b)

such that the following estimate holds

‖S(t)‖L(X) ≤ Mebt, t ≥ 0. (3.6)

As a consequence of the previous result, we have

Proposition 3.3. The exponential type of ∆ + V is the same in all the spaces

LpU (RN ) for all 1 ≤ p ≤ ∞.

Moreover, if we denote by ν this exponential type, then for all η > ν there exists

M = M(η, N, σ, ‖V ‖LσU (RN )) such that the Lp

U −LqU estimates (3.2) hold with a = η

and M1 = M, for all 1 ≤ p ≤ q ≤ ∞.

Proof. Notice first that we can replace t > 0 by t ≥ 1 in Definition 3.1. Observe

also that the spaces LpU (RN ) are nested, i.e. if p ≤ q then Lq

U (RN ) → LpU (RN ) with

embedding constant equal to |B(0, 1)| 1p− 1

q ≤ |B(0, 1)|, which is independent of p,

q. In particular, if 1 ≤ p < q ≤ ∞, we have that for all η > ν(LpU (RN ))

‖e(∆+V )tu0‖LpU (RN ) ≤ Meηt‖u0‖Lp

U (RN ) ≤ Meηt‖u0‖LqU (RN ).

Using the LpU − Lq

U estimate (3.2) at time t = 1, we have that

‖e(∆+V )(t+1)u0‖LqU (RN ) ≤ M1e

a‖e(∆+V )tu0‖LpU (RN ) ,

where M1 and a are given by Proposition 3.2. Hence, we obtain,

‖e(∆+V )(t+1)u0‖LqU (RN ) ≤ M1e

aMeηt‖u0‖LqU (RN )

which implies that for all t ≥ 1, we have

‖e(∆+V )tu0‖LqU(RN ) ≤ M1e

ae−ηMeηt‖u0‖LqU (RN )

and therefore ν(LqU (RN )) ≤ η. Since this is done for arbitrary η > ν(Lp

U (RN )), we

have ν(LqU (RN )) ≤ ν(Lp

U (RN )).

Now, if η > ν(LqU (RN )), then, similarly as we have done above, for t ≥ 1, we

will have

‖e(∆+V )tu0‖LpU (RN ) ≤ ‖e(∆+V )tu0‖Lq

U (RN )

≤ Meη(t−1)‖e(∆+V )1u0‖LqU (RN ) ≤ Me−ηM1e

aeηt‖u0‖LpU (RN )

(3.7)

which implies that ν(LpU (RN )) ≤ η. Again, since this is done for arbitrary η >

ν(LqU (RN )) we have that ν(Lp

U (RN )) ≤ ν(LqU (RN )). This shows that ν(Lp

U (RN )) =

ν(LqU (RN )) ≡ ν for all 1 ≤ p ≤ q ≤ ∞.

We prove now the second statement of the proposition. Let us consider η > ν.

For 0 < t ≤ 1, we have from (3.2)

‖e(∆+V )tu‖LqU(RN ) ≤ M1e

att−N2 ( 1

p− 1q )‖u0‖Lp

U (RN )

≤ M1ea+|η|eηtt−

N2 ( 1

p− 1q )‖u0‖Lp

U (RN ) .

And for t > 1 we get from (3.7) that

‖e(∆+V )tu0‖LqU (RN ) ≤ M1e

ae−ηMeηt‖u0‖LpU (RN ).

Now, for any ε > 0, denote by K(ε) = infeεtt−N/2, t ≥ 1 ≤infeεtt−

N2 ( 1

p− 1q ), t ≥ 1. Then, we have

‖e(∆+V )tu0‖LqU (RN ) ≤

M1eae−ηM

K(ε)e(η+ε)tt−

N2 ( 1

p− 1q )‖u0‖Lp

U (RN ) .

Since this is obtained for arbitrary η > ν and ε > 0, we prove the proposition.

Notice that by Ref. 34 the exponential type ν0 of the semigroup generated by

∆ + V in Lp(RN ) coincides for all 1 ≤ p ≤ ∞ and is given by the formula

ν0 = − inf

∫

RN

|∇φ|2 − V (x)|φ|2; φ ∈ C∞0 (RN )

. (3.8)

It is not difficult to prove now the following result.

Corollary 3.2. The exponential type of ∆+V is the same in all the spaces LpU (RN )

and Lq(RN ) for any 1 ≤ p, q ≤ ∞. In particular, the semigroup generated by ∆+V

in LpU (RN ) decays exponentially if and only if ν0 < 0 in (3.8).

Proof. We just need to note that ν(LpU (RN )) = ν(L∞

U (RN )), and since L∞U (RN ) =

L∞(RN ) we get ν(LpU (RN )) = ν(L∞(RN )) = ν(Lq(RN )) for any 1 ≤ p, q ≤ ∞.

Remark 3.1. Observe that for potentials V ≤ 0 the characterization given in

Refs. 7 and 8 states that the semigroup generated by ∆ + V decays exponentially

in L2(RN ) iff∫

G

V = −∞

on all sets G that contain arbitrary large balls. Hence, by Corollary 3.2, the same

condition is necessary and sufficient for the semigroup generated by ∆+V to decay

exponentially in LpU (RN ), 1 ≤ p ≤ ∞.

4. Alternative Characterization of Locally Uniform Spaces

In order to deal with more general differential operators than in the two previous

sections, we need to give an alternative characterization of the locally uniform

spaces introduced before, and of some associated Sobolev-like spaces.

Consider now a continuous strictly positive weight function ρ : RN → (0, +∞).

Let us denote for 1 ≤ p < ∞, the weighted space

Lpρ(R

N ) =

φ ∈ Lploc(R

N ),

∫

RN

|φ(x)|pρ(x)dx < ∞

,

with norm

‖φ‖Lpρ(RN ) =

(∫

RN

|φ(x)|pρ(x)dx

)1/p

.

We consider the translated weights

τyρ(x) = ρ(x − y) , x ∈ RN , (4.1)

and the corresponding weighted spaces, Lpτyρ(R

N ). Then we, define as in Ref. 30

and 32 the locally uniform space

Lplu(RN ) =

φ ∈ Lploc(R

N ), supy∈RN

‖φ‖Lpτyρ(RN ) < ∞

(4.2)

with norm

‖φ‖Lplu(RN ) = sup

y∈RN

‖φ‖Lpτyρ(RN ).

Observe that if ρ ∈ L1(RN ) ∩ L∞(RN ), then Lp(RN ), L∞(RN ) ⊂ Lplu(RN ) ⊂

Lpρ(R

N ).

We also consider the subspace of Lplu(RN ) consisting of all elements which are

translation continuous with respect to ‖ · ‖Lplu

(RN ), i.e.

Lplu(RN ) = φ ∈ Lp

lu(RN ) : ‖τyφ − φ‖Lplu(RN ) → 0 as |y| → 0 . (4.3)

Note that if the weight ρ is bounded above and bounded away from zero, then all

spaces we have defined above coincide with Lp(RN ). That is Lplu(RN ) = Lp

lu(RN ) =

Lpρ(R

N ) = Lp(RN ). Below we will consider some weights decreasing at infinity in a

certain way, which will define larger spaces than Lp(RN ).

Definition 4.1. Consider the class I of continuous and strictly positive weight

functions ρ such that

(i) ρ ∈ L1(RN ),

(ii) there exist positive constants λ, c > 0 and r ≥ 0 such that for each ξ ∈ RN

with |ξ| ≥ r

ρ(ξ) ≤ c min|x−ξ|≤λ

ρ(x) . (4.4)



Remark 4.1. (i) If ρ ∈ I, then ρ ∈ L∞(RN ) and

supx∈RN

ρ(x) ≤ c

V (λ)

∫

RN

ρ(x)dx , (4.5)

where V (λ) = V (1)λN denotes volume of N -dimensional ball of radius λ.

(ii) If ρ ∈ I, then ρα ∈ I whenever α ≥ 1.

(iii) Condition (ii) in Definition 4.1 is equivalent to the following: for any λ > 0

there exists c = c(λ) > 0 such that ρ(ξ) ≤ c min|x−ξ|≤λ ρ(x) for any ξ ∈ RN .

In particular if ρ ∈ I, then for any ε > 0 the weight ρε(x) = ρ(εx) ∈ I and

the spaces Lpρε

(RN ) and Lpρ(R

N ) are exactly the same with equivalent norms.

(iv) Observe that for weights that decay faster at infinity than those in the class

I defined above one can perform the construction of the Lplu(RN ) and the

associated Sobolev spaces, e.g. Ref. 29. In fact, fast decaying weights have

been used for parabolic problems in many references, see for example Refs. 18,

25 and 29 and references therein. As it can be seen from these references,

the Sobolev spaces and the semigroups on these scales have very different

properties than in our construction. For instance Sobolev embeddings may

become compact,18 or the smoothing of the semigroups may take place only

after a strictly positive threshold time.25

It is important to note that different continuous and strictly positive weights ρ

in the class I lead to the same locally uniform space which in fact coincide with

the uniform spaces of Sec. 2. This observation will follow from

Proposition 4.1. Assume ρ : RN → (0, +∞) is a continuous strictly positive

weight function.

(i) Then

Lplu(RN ) ⊂ Lp

U (RN )

with continuous inclusion. The same holds for Lplu(RN ) and Lp

U (RN ).

(ii) Assume further that ρ ∈ I. Then the spaces LpU (RN ) and Lp

lu(RN ) coincide

algebraically and topologically. The same holds for LpU (RN ) and Lp

lu(RN ).

Proof. (i) Note that for any y ∈ RN ,∫

B(y,1)

|φ(x)|pdx ≤∫

B(y,1)

|φ(x)|p ρ(x − y)

mdx ≤ 1

m

∫

RN

|φ(x)|pρ(x − y)dx , (4.6)

where m = minz∈B(0,1) ρ(z) > 0. Hence, we get the estimate ‖φ‖LpU (RN ) ≤

m−1/p‖φ‖Lplu(RN ).

(ii) Now we prove LpU (RN ) ⊂ Lp

lu(RN ). If Q ⊂ RN is a closed cube centered

at zero with all edges of length 1 and parallel to the axes, then the family

z + Q, z ∈ ZN covers RN and the interiors of these cubes have disjoint

intersections. Therefore we have, for any y ∈ RN ,

‖φ‖pLp

τyρ(RN )=

∫

RN

|φ(x)|pρ(x − y)dx =

∫

RN

|φ(x + y)|pρ(x)dx

=∑

z∈ZN

∫

x∈z+Q|φ(x + y)|pρ(x)dx

=∑

z∈ZN

ρ(xz)

∫

x∈z+Q|φ(x + y)|pdx

≤ C‖φ‖pLp

U (RN )

∑

z∈ZN

ρ(xz) , (4.7)

where C > 0 and xz ∈ z + Q are chosen using the integral mean value

theorem.

Since ρ is in the class I, for z ∈ ZN and |z| ≥ r we obtain that

ρ(xz) ≤ c min|x−xz|≤λ

ρ(x) ≤ c min|x−xz|≤λ∩z+Q

ρ(x)

≤ c

V (λ, z)

∫

|x−xz|≤λ∩z+Qρ(x)dx ≤ c

V

∫

x∈z+Qρ(x)dx ,

where V (λ, z) denotes the volume of the set |x − xz| ≤ λ ∩ z + Q, and

V := infz∈ZN , |z|>r

V (λ, z) > 0.

Consequently,

∑

z∈ZN ,|z|>r

ρ(xz) ≤c

V

∑

z∈ZN ,|z|>r

∫

x∈z+Qρ(x)dx ≤ c

V

∫

RN

ρ(x)dx

and hence the series∑

z∈ZN ρ(xz) converges and we get ‖φ‖Lplu(RN ) ≤ C‖φ‖Lp

U (RN ).

Now we can define the corresponding locally uniform Sobolev spaces.

Definition 4.2. For k ∈ N we denote by W k,ploc (RN ) the space of all φ ∈ Lp

loc(RN )

having distributional derivatives Dσφ ∈ L1loc(R

N ) for all |σ| ≤ k. We define further,

for 1 ≤ p ≤ ∞,

(i) W k,pρ (RN ), the Banach space consisting of all φ ∈ W k,p

loc (RN ) such that

‖φ‖W k,pρ (RN ) =

∑

|σ|≤k

‖Dσφ‖Lpρ(RN ) < ∞.

(ii) W k,plu (RN ), the Banach space consisting of all φ ∈ W k,1

loc (RN ) such that

‖φ‖W k,plu (RN ) =

∑

|σ|≤k

‖Dσφ‖Lplu(RN ) < ∞.

Also, consider W k,plu (RN ), the Banach subspace of W k,p

lu (RN ) consisting of the

elements which are translation continuous in the ‖ · ‖W k,plu (RN ) norm.

(iii) W k,pU (RN ), the Banach space consisting of all φ ∈ W k,p

loc (RN ) such that

‖φ‖W k,pU (RN ) =

∑

|σ|≤k

‖Dσφ‖LpU (RN ) < ∞,

or equivalently,

‖φ‖W k,pU (RN ) = sup

x∈RN

‖φ‖W k,p(B(x,1)) < ∞ .

Also, consider W k,pU (RN ), the Banach subspace of W k,p

U (RN ) consisting of the ele-

ments which are translation continuous in the ‖ · ‖W k,pU (RN ) norm.

Remark 4.2. Note that from standard characterizations of Sobolev spaces we get

that for any x, h ∈ RN such that |h| ≤ 1 we have that for u ∈ W 1,p(B(x, 2))

‖u(· − h) − u‖Lp(B(x,1)) ≤ C|h| ,

where C = ‖∇u‖Lp(B(x,2)).

This implies that W k+1,pU (RN ) ⊂ W k,p

U (RN ).

For the applications to PDEs it is useful to consider intermediate spaces to the

ones defined above. For this, following Refs. 3, 4, 16, 24, 28 and 35 we consider any

interpolation functor that we denote (( , ))θ, for θ ∈ (0, 1).

Then according to the definition above we have

Definition 4.3. For 1 ≤ p < ∞, k ∈ N∪0 and s ∈ (k, k +1) we define θ ∈ (0, 1)

such that s = θ(1+k)+(1−θ)k, that is θ = s−k. Then we define the intermediate

spaces

(i) For Ω = B(y, 1), for y ∈ RN , or more generally for any smooth domain in RN

we define

W s,p(Ω) = ((W k+1,p(Ω), W k,p(Ω)))θ ,

W s,p(RN ) = ((W k+1,p(RN ), W k,p(RN )))θ

and

W s,pρ (RN ) = ((W k+1,p

ρ (RN ), W k,pρ (RN )))θ.

(ii)

W s,plu (RN ) = ((W k+1,p

lu (RN ), W k,plu (RN )))θ

and

W s,plu (RN ) = ((W k+1,p

lu (RN ), W k,plu (RN )))θ .

(iii) Alternatively, we can define AW s,plu (RN ), the space of functions such that

supy∈RN

‖φ‖W s,pτyρ(RN ) < ∞

with norm

‖φ‖AW s,plu (RN ) = sup

y∈RN

‖φ‖W s,pτyρ(RN )

and consider the subset of elements which are translational continuous,AW s,p

lu (RN ).

(iv)

W s,pU (RN ) = ((W k+1,p

U (RN ), W k,pU (RN )))θ ,

and

W s,pU (RN ) = ((W k+1,p

U (RN ), W k,pU (RN )))θ .

(v) Alternatively, we can define AW s,pU (RN ), the space of functions such that

supy∈RN

‖φ‖W s,p(B(y,1)) < ∞

with norm

‖φ‖AW s,pU (RN ) = sup

y∈RN

‖φ‖W s,p(B(y,1))

and consider the subset of elements which are translational continuous,AW s,p

U (RN ).

Observe that the families of spaces defined above depend on the interpolation

functor and are decreasing with increasing s and/or p. Also note that from elemen-

tary properties of interpolation, translation operators are continuous in W s,plu (RN )

and in W s,pU (RN ).

From the results above, we have, again for any given choice of the interpolation

functor (( , ))θ , θ ∈ (0, 1)

Proposition 4.2. Assume ρ : RN → (0, +∞) is a continuous strictly positive

weight function.

Then, for 1 ≤ p ≤ ∞, k ∈ N ∪ 0, s = k + θ with θ ∈ [0, 1),

(i) For each y ∈ RN

W s,plu (RN ) ⊂ AW s,p

lu (RN ) ⊂ W s,pρ (RN )

∩ ∩ ∩W s,p

U (RN ) ⊂ AW s,pU (RN ) ⊂ W s,p(B(y, 1))

with continuous inclusion. The same holds for the dotted spaces.

(ii) Assume further that ρ ∈ I. Then the spaces W s,pU (RN ) and W s,p

lu (RN ) coincide

algebraically and topologically. The same holds for W s,pU (RN ) and W s,p

lu (RN ).

Proof. The claim follows easily by elementary properties of interpolation functors,

since from Proposition 4.1 and estimate (4.6) the result is true for spaces of integer

order.

We will extend now the Gagliardo–Nirenberg inequality and Sobolev embed-

dings to the case of locally uniform spaces.

For this recall that for k ∈ N ∪ 0 we denote by Ckbd(RN ) the Banach space

of functions having bounded and continuous partial derivatives up to the order k.

We also write C∞bd (RN ) for the intersection

⋂

k∈NCk

bd(RN ) and BUCk(RN ) for a

subspace of Ckbd(R

N ), consisting of functions bounded and uniformly continuous

together with the derivatives up to the order k. Further, Ck+µ(RN ), k ∈ N, µ ∈(0, 1), denotes the Banach space, which consists of functions φ ∈ BUCk(RN ) being

uniformly Holder continuous in RN together with derivatives up to the order k,

endowed with the norm

‖φ‖Ck+µ(RN ) =∑

|σ|≤k

supx∈RN

|Dσφ(x)| +∑

|σ|=k

sup0<|x−y|<1

|Dσφ(x) − Dσφ(y)||x − y|µ .

Obviously, Ck+µ(RN ) ⊂ BUCk(RN ) ⊂ Ckbd(RN ) ⊂ BUCk−1(RN ).

In order to get optimal results below we will consider henceforth spaces obtained

as in Definition 4.3 by choosing as (( , ))θ, for θ ∈ (0, 1) the complex interpolation

functor which is usually denoted by [ , ]θ, for θ ∈ (0, 1).12,35 Note that, as proved

in Ref. 35, the complex interpolation between Sobolev spaces of integer order give

the so-called Bessel potential spaces.

Lemma 4.1. (i) If s1 ≥ s2 ≥ 0, 1 < p1 ≤ p2 < ∞ and s1 − Np1

≥ s2 − Np2

, then

AW s1,p1

U (RN ) ⊂ AW s2,p2

U (RN ) (4.8)

and if s1 > s2 and p1 ≤ p2, the inclusions are locally compact, i.e. for any bounded

smooth set Ω ⊂ RN the inclusions

AW s1,p1

U (RN ) ⊂ W s2,p2(Ω)

are compact. Moreover,

‖φ‖AW s,pU (RN ) ≤ C‖φ‖θ

AWs1,p1U (RN )

‖φ‖1−θAW

s2,p2U (RN )

, (4.9)

where θ ∈ [0, 1], 1p ≤ θ

p1+ 1−θ

p2, 1 < p, p1, p2 < ∞ and

s − N

p≤ θ

(

s1 −N

p1

)

+ (1 − θ)

(

s2 −N

p2

)

.

(ii) If 1 ≤ p < ∞ and s ≥ 0 let k = [s − N/p], the integer part of s − N/p, and

0 < µ < s − N/p− k < 1. Then the inclusion

AW s,pU (RN ) ⊂ Ck+µ(RN ) (4.10)

is continuous and locally compact. Moreover,

‖φ‖Ck+µ(RN ) ≤ c‖φ‖θAW s,p

U (RN )‖φ‖1−θLq

U (RN )(4.11)

if 0 < µ < 1, θ ∈ [0, 1], and 1 < p, q < ∞

k + µ ≤ θ

(

s − N

p

)

− (1 − θ)N

q.

(iii) If ρ is in the class I, then the inclusion AW s1,p1

U (RN ) ⊂ W s2,p2ρ (RN ) is compact

provided that s2 ∈ N, s1 > s2, 1 < p1 ≤ p2 < ∞ and s1 − Np1

> s2 − Np2

.

Proof. The proof of (i) and (ii) is obvious, since the result holds for the spaces

W k,p(B(y, 1)), for each y ∈ RN and k ∈ N ∪ 0, with embedding constants inde-

pendent of y ∈ RN .

For the proof of (iii), consider |σ| ≤ s2, and a bounded sequence φn inAW s1,p1

U (RN ). Then, from (4.8) and (4.7), for any ε > 0 there exists Nε ∈ N,

such that

‖Dσφn − Dσφm‖p2

Lp2ρ (RN )

=∑

z∈ZN

ρ(xz)

∫

z+Q|Dσφn(x) − Dσφm(x)|p2dx

≤ Mε +∑

|z|≤Nε

ρ(xz)

∫

z+Q|Dσφn(x) − Dσφm(x)|p2dx ,

(4.12)

since∑

z∈ZN ρ(xz) is a convergent series.

From the compactness of the embedding W s1,p1(B(z, 1)) ⊂ W s2,p2(B(z, 1)), for

|z| ≤ Nε, we can extract a subsequence, that we still denote Dσφn, which is

convergent in Lp2(z + Q) for |z| ≤ Nε. Hence, from (4.12), Dσφn is a Cauchy

sequence in Lp2ρ (RN ). Since the above holds for any |σ| ≤ s2, we can extract a

subsequence which is convergent in W s2,p2ρ (RN ).

Note that the lemma above holds true for spaces W s,pU (RN ) for integer s ∈ N;

hence it also holds true for some intermediate spaces by interpolation.

Concerning density results, we have, see Ref. 13.

Lemma 4.2. If ρ is in class I, then C∞bd (RN ) ⊂ ⋂

l∈NW l,p

U (RN ) is a dense subset

of W s,pU (RN ) for each s ≥ 0 and 1 ≤ p < ∞.

Proof. Taking a mollifier Jε ∈ C∞0 (B(0, ε)) and using Holder’s inequality we get

for any x, y ∈ RN ,

|(Jε ∗ φ)(x)τyρ1p (x)| =

∣

∣

∣

∣

∫

RN

Jp−1

pε (z)J

1pε (z)φ(x − z)(τyρ)

1p (x) dz

∣

∣

∣

∣

≤(

∫

RN

Jε(z)|φ(x − z)|pτyρ(x)dz

)1p

.

From here, by Fubini’s theorem and using Proposition 4.1 (ii), we obtain∫

RN

|(Jε ∗ φ)(x)|pτyρ(x)dx ≤ C‖φ‖pLp

U (RN )

which gives ‖Jε ∗ φ‖LpU (RN ) ≤ C‖φ‖Lp

U (RN ) and moreover

‖Jε ∗ φ − τy(Jε ∗ φ)‖LpU (RN ) = ‖Jε ∗ (φ − τyφ)‖Lp

U (RN ) ≤ C‖φ − τyφ‖LpU (RN ) .

In particular Jε ∗ φ ∈ LpU (RN ) when φ ∈ Lp

U (RN ).

Furthermore, applying Fubini’s theorem leads to the estimate

‖Jε ∗ φ − φ‖LpU (RN ) ≤ C sup

y∈RN

[∫

RN

Jε(z)

∫

RN

|φ(x − z) − φ(x)|pτyρ(x)dxdz

]1p

≤ C sup|z|<ε

‖τzφ − φ‖LpU (RN ) ,

which shows that ‖Jε ∗ φ − φ‖LpU (RN ) → 0 as ε → 0+ whenever φ ∈ Lp

U (RN ).

Since Jε ∈ C∞0 (B(0, ε)) and φ ∈ Lp

U (RN ) we get that Jε ∗ φ is in C∞bd (RN ). If

moreover φ ∈ W k,pU (RN ), with k ∈ N, the argument above shows that Jε ∗ φ → φ

in W k,pU (RN ), since Dα(Jε ∗ φ) = Jε ∗ (Dαφ) for |α| ≤ k.

5. Elliptic Operators and Semigroups in Uniform Spaces

We study in this section the generation of analytic semigroups, in the locally uni-

form spaces constructed in the previous section, by second-order linear differential

operators of the form

A := −N

∑

k,l=1

akl(x)∂k∂l +

N∑

j=1

bj(x)∂j + c(x) , (5.1)

with complex valued coefficients satisfying weak regularity assumptions.

It will be useful below to consider the partial differentiations Dj := −i∂j and,

relabeling coefficients, (5.1) can be rewritten as

A =

N∑

k,l=1

akl(x)DkDl +

N∑

j=1

bj(x)Dj + c(x) . (5.2)

We will first recall, in Sec. 5.1, several known results on elliptic operators and the

generation of analytic semigrops in Lp(RN ). Later, in Sec. 5.2 we will show how, for

operators (5.2), −A generates analytic semigroups in weighted and uniform spaces.

5.1. Elliptic operators and semigroups in Lp(RN)

Recalling the definition from Ref. 5, we say that (5.2) is elliptic, if it satisfies the

following:

Definition 5.1. (Uniform (M, θ0)-ellipticity condition) Denoting by A0(x, ξ) =∑N

k,l=1 akl(x)ξkξl, with x, ξ ∈ RN , the principal symbol of A, we assume that the

coefficients ak,l are bounded and, for some constants M > 0 and θ0 ∈ (0, π/2), the

following holds:

A0(x, ξ) ≥ 1

M> 0, |arg(A0(x, ξ))| ≤ θ0, for all x, ξ ∈ R

N with |ξ| = 1 .

Remark 5.1. Observe that as in Ref. 5 we could write (5.2) and the ellipticity

conditions above for the case of elliptic systems; that is for the case where the

coeffients are matrix-valued functions. In such a case all the results that follow

remain valid. For simplicity we stick to the case of scalar equations.

Two important examples of operators satisfying Definition 5.1 are the Laplace

operator −∆ = D21 + · · · + D2

n and the complex Ginzburg–Landau operator −(1 +

iα)∆, with α ∈ R.

Given 1 < p < ∞, the complex-valued coefficients akl, bj , c are assumed to

satisfy the following, regularity requirements, see Ref. 5.

Definition 5.2. (Regularity requirements) (i) akl : RN → C, k, l = 1, . . . , N ,

belong to the set of bounded uniformly continuous functions, BUC(RN ),

(ii) bj : RN → C, j = 1, . . . , N , belongs to Lpj

U (RN ) where

pj = p if p > N,

pj > N otherwise,(5.3)

(iii) c : RN → C is an element of Lp0

U (RN ) where

p0 = p if p > N/2,

p0 > N2 otherwise.

(5.4)

Remark 5.2. Recall that a modulus of continuity ω : R+ → R+ is an increasing

function continuous at zero, positive in (0,∞) and satisfying ω(2s) ≤ c0ω(s) for

certain c0 > 0, see Ref. 5. Bounded uniformly continuous complex valued functions

possessing a modulus of continuity ω forms a Banach space BUC(RN , ω) having

the norm

‖φ‖BUC(RN ,ω) = supx∈RN

|φ(x)| + supx,y∈RN

x6=y

|φ(x) − φ(y)|ω(|x − y|) .

If φ ∈ BUC(RN ), then there exists certain modulus of continuity ω for which φ ∈BUC(RN , ω). In particular, if akl, k, l = 1, . . . , N ⊂ BUC(RN ), then akl, k, l =

1, . . . , N ⊂ BUC(RN , ω) for some modulus of continuity ω.

We next define a class of differential operators E(M, θ0, p).

Definition 5.3. An operator A given by (5.2) is of class E(M, θ0, p) if and only if

A is uniformly (M, θ0)-elliptic as in Definition 5.1, the coefficients of A satisfy the

regularity in Definition 5.2 and the following estimate holds:

N∑

k,l=1

‖akl‖BUC(RN ,ω) +

N∑

j=1

‖bj‖LpjU (RN )

+ ‖c‖Lp0U (RN ) ≤ M . (5.5)

We recall now some definitions and results on sectorial operators and analytic

semigroups.

Definition 5.4. Let K ≥ 1, a ∈ R and θ ∈ (0, π/2). We say that a linear operator

Λ : D(Λ) ⊂ Y → Y is (K, θ, a)-sectorial in a Banach space Y if the resolvent set of

Λ contains the sector Sa,θ = θ ≤ |arg(z − a)| ≤ π ∪ a and

(1 + |λ − a|)‖(λI − Λ)−1‖L(Y ) ≤ K for all λ ∈ Sa,θ. (5.6)

Note that the definition above does not require that the domain D(Λ) is dense

in Y ; however the operator is necessarily closed.

Hence if Λ is sectorial, then −Λ generates an analytic semigroup which is more-

over continuous up to t = 0 if the domain of Λ is dense.

One of the main results in Ref. 5 states that the Lp(RN )–realization of elliptic

differential operators (5.2) in the class E(M, θ0, p) generate analytic semigroups in

Lp(RN ). In fact, from Ref. 5, we have

Proposition 5.1. Given 1 0, θ0 ∈ (0, π/2) and θ ∈ (θ0, π/2), there

exist constants c, K ≥ 1 and µ > 0 such that any A ∈ E(M, θ0, p) is a (K, θ,−µ)-

sectorial operator in Lp(RN ) with domain W 2,p(RN ). Furthermore, µI + A is a

linear isomorphism from W 2,p(RN ) onto Lp(RN ) and

‖µI + A‖L(W 2,p(RN ),Lp(RN )) + ‖(µI + A)−1‖L(Lp(RN ),W 2,p(RN )) ≤ c . (5.7)

Therefore −A, with domain W 2,p(RN ), generates a C0 analytic semigroup

S(t) in Lp(RN ). Hence, the linear equation

ut +

N∑

k,l=1

akl(x)DkDlu +

N∑

j=1

bj(x)Dju + c(x)u = 0

u(0) = u0 ∈ Lp(RN )

has a unique solution u(t) = S(t)u0 for t ≥ 0.

Using Lemma 3.1 and Sobolev embeddings allows us to do a bootstrap argument

to show the following result:

Corollary 5.1. Assume that A is a differential operator as in (5.2) with coefficients

akl ∈ BUC(RN ), bj ∈ Lpj

U (RN ) and c ∈ Lp0

U (RN ) where pj > N and p0 > N/2.

Then for any 1 < r ≤ minpj , p0 and for any u0 ∈ Lr(RN ) we have that there

exists a unique solution of the linear equation

ut +N

∑

k,l=1

akl(x)DkDlu +N

∑

j=1


u(0) = u0 ∈ Lr(RN ) ,

u(t) = S(t)u0, for t ≥ 0, which satisfies that

(0,∞) 3 t 7→ S(t)u0 ∈ BUC(RN )

is analytic.

As it is well known,3,4,24,28,33 sectorial operators have a naturally associated

scale of fractional power spaces, Xα, for α ≥ 0, which are the domains of the

fractional powers of A (provided Reσ(A) > 0). This space allows one to measure

the degree of smoothing of the solution of the evolution equation, since

‖S(t)u0‖Xα ≤ M0eδt

tα−β‖u0‖Xβ for t > 0 (5.8)

for any α ≥ β ≥ 0 and some δ ∈ R.

Therefore for a given sectorial operator, it is very useful to have a good descrip-

tion of its fractional power spaces. An important property of the class of operators

defined above is that their fractional power spaces can be characterized in terms

of Bessel potential spaces. This property relies on the fact that if an operator has

bounded imaginary powers, then its fractional power spaces are given by complex

interpolation.35 This last property has been established for the Lp(RN )-realization

of elliptic differential operators (5.2) in Ref. 5. Another ingredient for the result is

that Bessel potential spaces are the complex interpolation spaces between Lp(RN )

and W 2,p(RN ), see Ref. 35.

Hence, following Ref. 4, we define

Definition 5.5. Assume Λ is a densely defined sectorial operator in a Banach space

Y such that Reσ(Λ) > 0. For N ≥ 1 and θ ≥ 0 let us define

Λ ∈ BIP (Y,N , θ) iff ‖Λit‖L(Y ) ≤ N eθ|t|, t ∈ R. (5.9)

Then, from Ref. 5 and the results above, we have

Proposition 5.2. Let 1 0, θ0 ∈ (0, π/2), θ ∈ (θ0, π/2) and

A ∈ E(M, θ0, p) be given such that∫ 1

0

ω1/3(t)

tdt < ∞ , (5.10)

where ω is the modulus of continuity appearing in (5.5).

Then there exists µ > 0 (the same one as in Proposition 5.1), and N =

N (M, θ0, θ, p, ω) ≥ 1 such that

µI + A ∈ BIP(

Lp(RN ),N , θ)

. (5.11)



In particular, if Xαp are the domains of fractional powers of A above Lp(RN ), then

for any α ∈ (0, 1)

Xαp = [Lp(RN ), W 2,p(RN )]α = H2α

p (RN ) ,

1

c(N , θ)‖ · ‖H2α

p (RN ) ≤ ‖ · ‖Xαp≤ c(N , θ)‖ · ‖H2α

p (RN ),(5.12)

where H2αp (RN ) denote the Bessel potential spaces.

Remark 5.3. Another important property of these semigroups in Lp(RN ) concerns

the maximum principle or, in other words, the order preserving property. Some

important classes of operators that satisfy such a property are the Schrodinger

operators

A = −∆ + V (x) , (5.13)

where V is a real potential in LσU (RN ) with σ > N/2, see Ref. 34. Also, operators

of the form (2.12) with bounded coefficients or operators of the form

A = −N

∑

k,l=1

akl(x)∂k∂l +

N∑

j=1

bj(x)∂j + c(x) , (5.14)

with real bounded coefficients and akl continuous.6

5.2. Elliptic operators and semigroups in weighted

and uniform spaces

Our main concern in this section will be to prove that linear second-order elliptic

operators of the class defined before generate analytic semigroups in both weighted

and locally uniform spaces. Therefore we will generalize Propositions 5.1, and 5.2

to these spaces. For this it will be useful to consider further properties of the weight

functions that can be used for the alternative definition of the uniform spaces of

the previous section. Hence, we have the following definition; compare with Refs. 1,

10, 16, 17, 20, 30 and 32.

Definition 5.6. We say that a weight function ρ : RN → (0,∞) is in the class

R = Rρ0,ρ1 if

(i) ρ ∈ C2(RN ),

(ii) | ∂ρ∂xj

(x)| ≤ ρ0ρ(x), for x ∈ RN , j = 1, . . . , N ,

(iii) | ∂2ρ∂xj∂xk

(x)| ≤ ρ1ρ(x), for x ∈ RN , j, k = 1, . . . , N , where ρ0, ρ1 are positive

constants.

Remark 5.4. If ρ is in the class R, then |∇ρ(x−yt)|ρ(x−yt) ≤

√Nρ0 which leads to the

estimate

ρ(x) ≤ ρ(x − y)e√

Nρ0|y| , x, y ∈ RN , (5.15)

see Refs. 32 and 31.



In particular if ρ ∈ R = Rρ0,ρ1 , then (4.4) is satisfied and so if moreover

ρ ∈ L1(RN ) then ρ ∈ I.

The main idea that we will follow below is as follows. Take X = Lpρ(R

N ) and

observe that the mapping Φ(φ) = ρ1p φ is an isomorphism between X and Y :=

Lp(RN ), which is moreover an isometry. Furthermore, if ρ ∈ R, then it is not hard

to check that Φ(W 2,pρ (RN )) = W 2,p(RN ). The following result will then be very

useful below,

Lemma 5.1. Let Y be a Banach space and Λ : D(Λ) ⊂ Y → Y be a (K, θ, a)-

sectorial operator. Suppose that X is another Banach space such that there exists a

linear isomorphism Φ : X → Y . Then

(i) The operator ΛΦ := Φ−1ΛΦ : D(ΛΦ) ⊂ X → X, with domain D(ΛΦ) =

Φ−1(D(Λ)), is (K, θ, a)-sectorial operator in X with K = ‖Φ‖‖Φ−1‖K.

Moreover, the resolvent sets of ΛΦ and Λ coincide and the semigroups are

related by the formula e−ΛΦt = Φ−1e−ΛtΦ.

(ii) The operator −aI + ΛΦ is one-to-one from D(ΛΦ) onto X whenever −aI + Λ

is one-to-one from D(Λ) onto Y .

(iii) In addition,

‖Φ‖L(D(ΛΦ),D(Λ)) + ‖Φ−1‖L(D(Λ),D(ΛΦ)) ≤ c ,

and

‖ − aI + Λ‖L(D(Λ),Y ) + ‖(−aI + Λ)−1‖L(Y,D(Λ)) ≤ c

imply

‖ − aI + ΛΦ‖L(D(ΛΦ),X) + ‖(−aI + ΛΦ)−1‖L(X,D(ΛΦ)) ≤ c c2 ,

(iv) Furthermore, if Λ is densely defined then −aI + ΛΦ ∈ BIP (X, ‖Φ‖‖Φ−1‖KN , θ) provided that −aI + Λ ∈ BIP (Y,N , θ),

(v) Finally if Λ is densely defined then, for each α ≥ 0, Φ is an isomorphism from

Xα onto Y α, where the latter represent the fractional power spaces associated

to Λ and ΛΦ. If moreover Φ : X → Y is an isometry, then it is also an

isometry, between Xα and Y α.

Proof. For the proof of (i) it is sufficient to note that

(λI − ΛΦ) = Φ−1(λI − Λ)Φ . (5.16)

From here it is clear that the resolvent sets coincide. Also, for λ ∈ Sa,θ we then

have (λI − ΛΦ)−1 = Φ−1(λI − Λ)−1Φ and (5.6) ensures that

(1 + |λ − a|)‖(λI − ΛΦ)−1φ‖X ≤ ‖Φ−1‖(1 + |λ − a|)‖(λI − Λ)−1Φφ‖Y

≤ ‖Φ−1‖K‖Φφ‖Y ≤ ‖Φ−1‖K‖Φ‖‖φ‖X ,(5.17)

for φ ∈ X .

Clearly items (ii) and (iii) follows directly from (5.16). Also the proof of (iv)

follows from Ref. 4.

Considering (v), suppose that the operator Λ (and so ΛΦ) is densely defined.

Note that it may be assumed, without loss of generality, that both Re σ(Λ) > 0

and Re σ(ΛΦ) > 0. Let SΛ(t), SΛΦ(t) be, respectively, the strongly continuous

analytic semigroups generated by −Λ, −ΛΦ. Then the exponential formula gives

SΛΦ(t)φ = limn→∞

(

I +t

nΦ−1ΛΦ

)−n

φ = Φ−1SΛ(t)Φφ, φ ∈ X . (5.18)

Using the integral expression for Λ−α, see Ref. 24, and (5.18) we obtain for

φ ∈ X that

(Φ−1Λ−αΦ)φ =1

Γ(α)Φ−1

∫ ∞

0

tα−1SΛ(t)(Φφ)dt

=1

Γ(α)

∫ ∞

0

tα−1SΛΦ(t)φdt = Λ−αΦ φ .

(5.19)

Hence, since Xα is the image of X under Λ−αΦ , we have

Xα = Λ−αΦ (X) = Φ−1Λ−αΦ(X) = Φ−1Λ−α(Y ) = Φ−1(Y α) . (5.20)

If moreover Φ is an isometric isomorphism from X onto Y , we have

‖φ‖Xα = ‖ΛαΦφ‖X = ‖Φ−1ΛαΦφ‖X = ‖ΛαΦφ‖Y = ‖Φφ‖Y α , φ ∈ Xα . (5.21)

Hence, considering an elliptic operator A from (5.2) in X = Lpρ(R

N ) with domain

W 2,pρ (RN ), we can then define the operator AΦ = Φ−1AΦ in Y = Lp(RN ) with

domain W 2,p(RN ). After writing down explicitly the expression for AΦ, see (5.24)

below, we will apply the results in the previous section to get the corresponding

result for A. Repeating the above argument with the shifted weights (4.1) we get

the results for uniform spaces, provided we can control, uniformly with respect to

the shift, the constants appearing in all the estimates.

In particular, we have

Theorem 5.1. Let 1 0, θ0 ∈ (0, π/2), θ ∈ (θ0, π/2) and ρ0, ρ1 > 0

be given.

Then there exist constants c,K ≥ 1 and µ > 0 such that given any elliptic

operator A ∈ E(M, θ0, p) and any weight ρ ∈ Rρ0,ρ1 , then A defines (K, θ,−µ)-

sectorial operator in the space X = Lpρ(R

N ) with domain DX(A) = W 2,pρ (RN ).

Moreover Aµ := µI + A is a linear isomorphism from its domain DX(A) onto X

and

‖Aµ‖L(DX(A),X) + ‖A−1µ ‖L(X,DX(A)) ≤ c . (5.22)

If in addition (5.10) holds, then

Aµ ∈ BIP (Lpρ(R

N ), N , θ) (5.23)

for certain N ≥ 1 which depends on p, M, θ0, θ, ω, ρ0, ρ1.

In particular, for α ∈ (0, 1), the fractional power spaces coincide with the spaces

W 2α,pρ (RN ) defined in Definition 4.3, which are given by

W 2α,pρ (RN ) = [Lp

ρ(RN ), W 2,p

ρ (RN )]α = ρ−1p H2α

p (RN )

and normed with

‖φ‖W 2α,pρ

= ‖ρ 1p φ‖H2α

p (RN ) .

Hence, −A generates a C0 analytic semigroup S(t) in Lpρ(R

N ) and the linear

equation

ut +

N∑

k,l=1

akl(x)DkDlu +

N∑

j=1


u(0) = u0 ∈ Lpρ(R

N )

has a unique solution u(t) = S(t)u0 for t ≥ 0 and the smoothing estimate (5.8)

holds true for the fractional power spaces described above.

Proof. As mentioned above for X = Lpρ(R

N ) we use Lemma 5.1 with Y := Lp(RN )

and Φ : X → Y such that Φ(φ) = ρ1p φ which is an isometry. Then we get an operator

Λ : D(Λ) = W 2,p(RN ) → Lp(RN ), defined by Λw = ρ1p Aρ−

1p w, which is explicitly

given as

Λw = ρ1p Aρ−

1p w

=N

∑

k,l=1

aklDkDlw +N

∑

j=1

bj −1

p

[

N∑

k=1

akjDkρ

ρ+

N∑

l=1

ajlDlρ

ρ

]

Djw

+

c +1

p

N∑

k,l=1

akl

[(

1 +1

p

)

DkρDlρ

ρ2− DkDlρ

ρ

]

− 1

p

N∑

j=1

bjDjρ

ρ

w , (5.24)

so that A = ΛΦ.

Since A ∈ E(M, θ0, p) and ρ ∈ Rρ0,ρ1 , it is easy to check that Λ belongs to

E(M, θ0, p) for some M = M(M, ρ0, ρ1, p). Thus, Proposition 5.1 and Lemma 5.1

give the results. Note here that N = KN where N is such that Λµ = ρ1p Aµρ−

1p ∈

BIP (Lp(RN ),N , θ) via Proposition 5.2, and K is as above.

Now, for the locally uniform spaces we get

Theorem 5.2. Let 1 0, θ0 ∈ (0, π/2), θ ∈ (θ0, π/2) and ρ0, ρ1 > 0

be given.

Then there exist constants c,K ≥ 1 and µ > 0 such that any elliptic operator

A ∈ E(M, θ0, p) defines a (K, θ,−µ)-sectorial operator in the space X = LpU (RN )

with domain DX(A) = W 2,pU (RN ). Moreover, Aµ := µI +A is a linear isomorphism

from its domain DX(A) onto X and (5.22) holds true.

Furthermore, for λ ∈ S−µ,θ ∪ −µ,‖λI − A‖L(W 2,p

U (RN ),LpU (RN )) + ‖(λI − A)−1‖L(Lp

U (RN ),W 2,pU (RN )) ≤ cλ,µ,K . (5.25)

Hence, −A generates an analytic semigroup S(t) in LpU (RN ) and the linear

equation

ut +

N∑

k,l=1

akl(x)DkDlu +

N∑

j=1


u(0) = u0 ∈ LpU (RN )

has a unique solution u(t) = S(t)u0 for t ≥ 0.

Proof. Take any weight ρ ∈ I ∩ Rρ0,ρ1 and observe that, all the constants in the

estimates derived in Theorem 5.1 above depend on the weight only in terms of the

values of ρ0 and ρ1. Also note that if ρ ∈ I ∩Rρ0,ρ1 then all shifted weights remain

in the same class. Hence for any y ∈ RN , A is a (K, θ,−µ)-sectorial operator in

Lpτyρ(R

N ), with domain W 2,pτyρ(R

N ). Therefore, if h ∈ LpU (RN ) and λ ∈ S−µ,θ∪−µ,

then the equation (λI − A)v = h has a unique solution v which belongs to each

W 2,pτyρ(R

N ) and, in addition,

(1 + |λ + µ|)‖v‖Lpτyρ(RN ) ≤ K‖h‖Lp

τyρ(RN ) . (5.26)

Furthermore, from (5.22) with X = Lpτyρ(R

N ) we have

1

c‖(λI − A)v‖Lp

τyρ(RN ) ≤(

1 +|λ + µ|

c

)

‖v‖W 2,pτyρ(RN )

≤ (c + |λ + µ|)(‖(λI − A)v‖Lpτyρ(RN ) + |µ + λ|‖v‖Lp

τyρ(RN )) . (5.27)

Therefore v ∈ W 2,pU (RN ) and λ is in the resolvent set of A. In particular, A is

a (K, θ,−µ)-sectorial operator in LpU (RN ) and (5.22) holds with X = Lp

U (RN ) and

as well (5.25) follows from (5.26) and (5.27).

Remark 5.5. Note that the theorem above can be obtained only assuming ρ ∈ Rso we can allow weights that increase at infinity, see Ref. 10. In such a case the

spaces W s,pU must be replaced by W s,p

lu .

As a consequence of (5.8) and the Sobolev embedings for uniform spaces, we get

the following:

Corollary 5.2. Assume A is a differential operator as in (5.2) with coefficients

akl ∈ BUC(RN ), bj ∈ Lpj


U (RN ) where pj > N and p0 > N/2.

Then for any 1 < r ≤ ∞ and for any u0 ∈ LrU (RN ), there exists a unique

solution of the linear equation

ut +

N∑

k,l=1

akl(x)DkDlu +

N∑

j=1

bj(x)Dju + c(x)u = 0 ,

u(0) = u0 ∈ LrU (RN ) ,

u(t) = S(t)u0, for t ≥ 0, which satisfies that

(0,∞) 3 t 7→ S(t)u0 ∈ BUC(RN )

is analytic.

Moreover, there exist constants µ, K and α ≥ 0 such that the following estimate

holds

‖S(t)u0‖BUC(RN ) ≤ Keµt

tα‖u0‖Lr

U(RN ) for t > 0 . (5.28)

Proof. We divide the proof in two cases, r > N/2 and 1 < r ≤ N/2.

If r > N/2, we can always choose p with N/2 N/2, W 2,pU (RN ) →

BUC(RN ) and since p < r we also have the embedding LrU (RN ) → Lp

U (RN ).

Moreover, by the regularization of the semigroup LpU (RN ) → W 2,p

U (RN ) obtained

in Theorem 5.2 and Lemma 3.1 we get the first part. Also, from Theorem 5.2 we

get that

‖S(t)u0‖BUC(RN ) ≤ C‖S(t)u0‖W 2,pU (RN ) ≤ C

eµt

t‖u0‖Lp

U (RN ) ≤ Ceµt

t‖u0‖Lr

U (RN )

which shows the corollary for r > N/2. In particular it shows the corollary for

N = 1, 2.

For the case 1 < r ≤ N/2 (N ≥ 3) we proceed as follows. By The-

orem 5.2, the semigroup has a regularizing effect LpU (RN ) → W 2,p

U (RN ) for

any 1 < p ≤ N/2. With this regularizing effect and the Sobolev embeddings

W 2,pU (RN ) → L

Np/(N−2p)U (RN ), 1 < p < N/2 we deduce the regularizing effect

LpU (RN ) → L

Np/(N−2p)U (RN ), for any 1 0. In particular, there exists a finite integer n such

that for all 1 < r < N/2 the nth iterate of the map β will satisfy βn(r) > N/2.

Hence, we obtain

‖S(nt)u0‖Lβn(r)U (RN )

≤ Ceµnt

tn‖u0‖Lr

U (RN ) .

But since βn(r) > N/2, applying the estimate obtained for the case r > N/2

we get

‖S((n + 1)t)u0‖BUC(RN ) ≤ Ceµt

t‖S(nt)u0‖L

βn(r)U (RN )

≤ Cn+1 eµ(n+1)t

tn+1‖u0‖Lr

U (RN ) .

Rescaling this last inequality by (n + 1)t → t and using again Lemma 3.1 we get

the result.

Remark 5.6. Observe that since BUC(RN ) → LqU (RN ) we also obtain the regu-

larity LrU (RN ) → Lq

U (RN ) for any r ≤ q ≤ ∞ with the same estimate (5.28).

Note that the analytic semigroup in X = LpU (RN ) will not be strongly continu-

ous because W 2,pU (RN ) is not dense in Lp

U (RN ) in general, see Ref. 32.

However, we will prove below that −A generates a strongly continuous analytic

semigroup in LpU (RN ) provided the coefficients of A satisfy the following stronger

regularity requirements.

Theorem 5.3. Let, as in Theorem 5.2, 1 0, θ0 ∈ (0, π/2), θ ∈(θ0, π/2), ρ0, ρ1 > 0 be given and assume the stronger regularity conditions

bj ∈ Lpj


U (RN ) (5.29)

where pj , j = 0, . . . , N, are defined as in (5.3), (5.4).

Then, −A, with domain D(A) = W 2,pU (RN ), generates a strongly continuous

analytic semigroup on LpU (RN ). Furthermore, if in addition (5.10) holds, then

Aµ ∈ BIP (LpU (RN ), N , θ) (5.30)

for some N ≥ 1, which depends on p, M, θ0, θ, ω, ρ0, ρ1. In particular, for α ∈(0, 1), the fractional power spaces coincide with the spaces W 2α,p

U (RN ) defined in

Definition 4.3, which are given by

W 2α,pU (RN ) = [Lp

U (RN ), W 2,pU (RN )]α . (5.31)

Hence, −A generates a C0 analytic semigroup S(t) in LpU (RN ) and the linear

equation

ut +

N∑

k,l=1

akl(x)DkDlu +

N∑

j=1


u(0) = u0 ∈ LpU (RN )

has a unique solution u(t) = S(t)u0 for t ≥ 0 and the smoothing estimate (5.8)

holds true for the fractional power spaces described above.

Proof. Since we know that A is (K, θ,−µ)-sectorial in LpU (RN ), it suffices to prove

that the resolvent map transforms W 2,pU (RN ) onto Lp

U (RN ), i.e.

(λI − A)W 2,pU (RN ) = Lp

U (RN ) , for all λ ∈ S−µ,θ ∪ −µ . (5.32)

After this is done, the first assertion will follow from the resolvent estimate of

Theorem 5.2.

If h ∈ LpU (RN ), then we know from Theorem 5.2 that (λI − A)−1h = v ∈

W 2,pU (RN ) and we may write

(λI − A)[τzv − v] = [τzh − h] +N

∑

k,l=1

[τzakl − akl]DkDlτzv

+

N∑

j=1

[τzbj − bj ]Djτzv + [τzc − c]τzv . (5.33)



From (5.33), (5.25) and Holder inequality we next have

1

cλ,µ,K‖τzv − v‖W 2,p

U (RN )

≤ ‖(λI − A)[τzv − v]‖LpU (RN )

≤ ‖τzh − h‖LpU (RN ) +

N∑

k,l=1

supx∈RN

|τzakl − akl|‖DkDlτzv‖LpU (RN )

+

N∑

j=1

‖τzbj − bj‖LpjU (RN )

‖Djτzv‖j + ‖τzc − c‖Lp0U (RN )‖τzv‖0 , (5.34)

where

‖Djτzv‖j =

‖Djτzv‖L∞(RN ) if p > N ,

‖Djτzv‖L

ppjpj−p

U (RN )

if p ≤ N ,

and

‖τzv‖0 =

‖τzv‖L∞(RN ) if p > N/2 ,

‖τzv‖L

pp0p0−p

U (RN )if p ≤ N/2 .

Since v ∈ W 2,pU (RN ), in either case the norms ‖Djτzv‖j and ‖τzv‖0 are bounded

by a multiple of ‖v‖W 2,pU (RN ). Thanks to (5.29) the right-hand side of (5.34) tends

to zero as z → 0. Hence v ∈ W 2,pU (RN ).

To complete the proof of (5.32), it remains to show that if v ∈ W 2,pU (RN ),

then h := (λI − A)v belongs to LpU (RN ). This follows as above by estimating

‖τzh − h‖LpU (RN ) with the aid of (5.33) and (5.25).

Finally (5.30) is a consequence of (5.23). Observe that, via Proposition 5.2,

(τyρ)1p Aµ(τyρ)−

1p ∈ BIP (Lp(RN ),N , θ) for each y ∈ RN , so that Aµ ∈

BIP (Lpτyρ(R

N ),KN , θ), where K is a constant from Theorem 5.2. We thus have

the characterization of the fractional power spaces.

Remark 5.7. Note that the norm of the fractional power spaces obtained in the

theorem coincides with norm of the space AW 2α,plu (RN ), i.e.

‖φ‖W s,plu (RN ) = sup

y∈RN

‖φ‖W s,pτyρ(RN ) .

For this just note that

‖Aαµφ‖Lp

lu(RN ) = supy∈RN

‖Aαµφ‖Lp

τyρ(RN ) = supy∈RN

‖φ‖W 2α,pτyρ

= ‖φ‖AW s,plu (RN ) , (5.35)

where we have used the results of Theorem 5.1.

Concerning order preserving properties we have the following general result.

Proposition 5.3. If an operator A is in the conditions of Corollary 5.2 and it

generates a semigroup S(t) in Lr0(RN ), Lr1

U (RN ) and Lr2ρ (RN ), for some 1 <

r0, r1, r2 < ∞, then S(t) is order preserving in Lr0(RN ) if and only if it is or-

der preserving in Lr1

U (RN ), if and only if it is order preserving in Lr2ρ (RN ).

Proof. The proof uses a combination of embedding and density results to translate

the order preserving property from one space to another. Observe that for any r

we have C∞0 (RN ) ⊂ Lr(RN ) ⊂ Lr

U (RN ) ⊂ Lrρ(R

N ). In particular, any of the three

statements imply that the semigroup is order preserving in C∞0 (RN ).

Hence, if the semigroup is order preserving in C∞0 (RN ), by density it will also

be order preserving in Lr0(RN ) and in Lr2ρ (RN ). Moreover, since by Theorems 5.1

it is well defined in Lsρ(Ω) for all 1 < s small, that we may assume s < r1 and since

Lr1

U ⊂ LsU (RN ) ⊂ Ls

ρ(RN ) we get that it will also be order preserving in Lr1

U (RN ).

Recall that Remark 5.3 gives some examples of operators for which the above

applies.

On the other hand, note that the results in this section apply, in particular to the

Laplace operator as well as to the complex Ginzburg–Landau operator, see Remark

5.1. In the former case it is not difficult to show that the semigroup constructed

above coincides with the one given in Sec. 2 by (2.4).

Acknowledgment

Partially supported by Project BFM2000–0798, DGES Spain and by KBN grant

No. 2 P03A 035 18, Poland.

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