[Borichev an Tomilov] Optimal Polynomial Decay of Functions and Operator

8/16/2019 [Borichev an Tomilov] Optimal Polynomial Decay of Functions and Operator

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Math. Ann. (2010) 347:455–478DOI 10.1007/s00208-009-0439-0 Mathematische Annalen

Optimal polynomial decay of functions and operator

semigroups

Alexander Borichev · Yuri Tomilov

Received: 4 June 2009 / Revised: 7 September 2009 / Published online: 30 October 2009© Springer-Verlag 2009

Abstract We characterize the polynomial decay of orbits of Hilbert space C 0-semi-groups in resolvent terms. We also show that results of the same type for generalBanach space semigroups and functions obtained recently in Batty and Duyckaerts(J Evol Eq 8:765–780, 2008) are sharp. This settles a conjecture posed in Batty andDuyckaerts (2008).

Mathematics Subject Classification (2000) Primary 47D06; Secondary 34D05 ·46B20

1 Introduction

One of the main issues in the theory of partial differential equations is to determinewhether the solutions to these equations approach an equilibrium and if yes then

The authors were partially supported by the Marie Curie “Transfer of Knowledge” programme, project“TODEQ”. The first author was also partially supported by the ANR project DYNOP. The second authorwas also partially supported by a MNiSzW grant Nr. N201384834.

A. BorichevCentre de Mathématiques et Informatique, Université d’Aix-Marseille I,39 rue Frédéric Joliot-Curie, 13453 Marseille, Francee-mail: [email protected]

Y. Tomilov (B)

Faculty of Mathematics and Computer Science, Nicolas Copernicus University,ul. Chopina 12/18, 87-100 Toruń, Polande-mail: [email protected]

Y. TomilovInstitute of Mathematics, Polish Academy of Sciences,Śniadeckich str. 8, 00-956 Warsaw, Poland

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456 A. Borichev Y. Tomilov

how fast do the solutions approach it. Recently, an essential progress was achievedin treating such asymptotic problems by operator-theoretical methods involvingC 0-semigroups. For different accounts of developments, highlights, and techniquesof asymptotic theory of C 0-semigroups see [2,5,10,26].

In particular, the following result was proved in [4, p. 803], see also [5, p. 41–42].(The result is implicitly contained already in [1].)

Theorem 1.1 Let (T (t ))t ≥0 be a bounded C 0-semigroup on a Banach space X withgenerator A. Suppose that iR is contained in the resolvent set ( A) of A. Then

T (t ) A−1 → 0, t → ∞. (1.1)

In other words, all classical solutions to the abstract Cauchy problem

˙ x (t ) = Ax (t ), t ≥ 0, x (0) = x 0, x 0 ∈ X , (1.2)

given by x (t ) = T (t ) x 0, t ≥ 0, x 0 ∈ D( A), converge uniformly (on the unit ball of D( A) with the graph norm) to zero at infinity if A satisfies the conditions of Theo-rem 1.1.

In general, without any additional assumptions, the decay in (1.1) can be arbi-trarily slow. However, in a number of special situations involving concrete PDE’s, e.g.damped wave equations, the rate of decay in (1.1) corresponds to the rate of decay

of the energy of the system described by (T (t ))t ≥0, and it is of interest to determinewhether this rate of decay can be achieved.

By rewriting equations in the abstract form (1.2), the rates of the decay of suffi-ciently smooth orbits for the corresponding semigroup (T (t ))t ≥0 (and equivalently of solutionsto(1.2))canbeassociatedwiththesizeoftheresolvent R(λ, A) = (λ− A)−1of A on the imaginary axis. This approach was initiated in [19] and later pursued, inparticular, in [7,8,13,20]. However, with a few exceptions, the issue of optimality (ornon-optimality) of the rates of growth has not been studied so far.

The above applications (mainly in the abstract set-up) motivated a thorough study

of the decay rates on T (t ) A−1 for bounded C 0-semigroups (T (t ))t ≥0 on Banachspaces in [3,21], and, most recently, in [6]. In the latter paper, the authors developeda unified and simplified approach for estimating the decay rates for T (t ) A−1 interms of the growth of R(i s, A), s ∈ R, using the contour integrals technique byNewman–Korevaar. In particular, the following theorem is proved there.

For A as in Theorem 1.1 we define a continuous non-decreasing function

M (η) = maxt ∈[−η,η]

R(i t , A), η ≥ 0, (1.3)

and the associated function

M log(η) := M (η) (log(1 + M (η)) + log(1 + η)) , η ≥ 0. (1.4)

Let M −1log be the inverse of M log defined on [ M log(0), +∞).

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Optimal polynomial decay 457

Theorem 1.2 (Batty, Duyckaerts) Let (T (t ))t ≥0 be a bounded C 0-semigroup on a Banach space X with generator A, such that iR ⊂ ( A). Let the functions M and M log be defined by (1.3) and (1.4). Then there exist C , B > 0 such that

T (t ) A−1 ≤ C M −1log (t /C )

, t ≥ B. (1.5)

Note that in the case α > 0, M (η) ≤ C (1 + ηα), η ≥ 0, the Batty–Duyckaertsresult gives

T (t ) A−1 ≤ C

log t

t

1α

, t ≥ B. (1.6)

It was conjectured in [6] that Theorem 1.2 can be improved by removing the loga-rithmic factor in (1.6) in the case when X is a Hilbert space, and that this factor isnecessary if X is merely a Banach space, see [6, Remark 9]. (See also the introductionin [3] and the comments after Theorem 3.5 therein.)

In our paper, we address the problem of optimality of the rate of decay in (1.6) andconfirm the conjecture from [6] for the case of polynomially growing M . We show thatthe logarithmic factor can be dropped in (1.6) if X is a Hilbert space. Thus, variousresults on polynomial decay of solutions of PDE’s, e.g. in [3,8,21,22]canbeimprovedto sharp formulations or shown to be sharp. See also [6] and references therein.

On the other hand, we show that Theorem 1.2 is essentially sharp (see Theorems 3.1and 4.1 below). This is done by a function-theoretical construction which may be inter-esting for its own sake and may be useful in other instances related to C 0-semigroupsas well. We prove, in particular, that given α > 0 there exists a Banach space X α anda bounded C 0-semigroup (T (t ))t ≥0 in X α with generator A such that

R(i s, A) = O(|s|α), |s| → ∞,

and

lim supt →∞

t

log t

1/αT (t ) A−1 > 0.

The classical problem of estimating the local energy for solutions to wave equationsleads to the study of decay rates for functions of the form T 1T (t )T 2, where T 1, T 2are bounded operators on X , so that the assumptions are imposed on the cut-off resol-vent F (λ) = T 1 R(λ, A)T 2 rather than on the resolvent itself, see e.g [6,7,13,27]. Dueto the lack of the Neumann series expansions for F we have to assume that F extendsanalytically to the region of known shape and satisfies certain growth restrictionsthere. The domain and the growth of F in are not in general related to each other.Moreover, the operator-theoretical approach can hardly be used to deal with F sincethe resolvent identity is not available as well. Thus, it is natural to put the problem intothe framework of decay estimates for L∞(R+, X ) functions given that their Laplace

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transforms extend to the specific with, in our case, polynomial estimates. In thisdirection, we obtain a result on the polynomial rates of decay for bounded functionswhich partially generalizes [6, Theorem 10]. The result has its version for the rates of decay of

T 1T (t )T 2

thus improving [6, Corollary 11].

We use standard notations. Given a closed linear operator A we denote by σ ( A),ρ( A), D( A), and Im ( A) the spectrum of A, the resolvent set of A, the domain of Aand the image of A respectively. By C , C 1 etc. we denote generic constants whichmay change from line to line.

The plan of the paper is as follows. In the Sect. 2 we characterize the rate of poly-nomial decay of the semigroup orbits in the Hilbert space via the growth rate of theresolvent on the imaginary axis. Examples of functions constructed in the Sect. 3 showthat the version of Theorem 1.2 for functions given in [6] is sharp. These examplesare used in the Sect. 4 to show that Theorem 1.2 is itself sharp.

2 Decay of Hilbert space semigroups

We start with recalling two simple and essentially known statements on C 0-semi-groups. The first one is a version of the well-known criterion on the generators of bounded Hilbert space C 0-semigroups, [14,24]. In our case, the proof is particularlyeasy and, to make our presentation self-contained, we give a short argument. LetC+ := { z ∈ C : Re z > 0}.

Lemma 2.1 Let (T (t ))t ≥0 be a C 0-semigroup on a Hilbert space H with generator A. Then (T (t ))t ≥0 is bounded if and only if

C+ ⊂ ( A), and supξ >0

ξ

R

R(ξ + i η, A) x 2 + R(ξ + i η, A∗) x 2

d η 0

ξ

R

R(ξ + i η, A) x 2 + R(ξ + i η, A∗) x 2

d η ≤ C x 2.

Proof of Lemma 2.1 The necessity is a direct consequence of the Plancherel theoremapplied to the families {e−ξ t T (t ) x : x ∈ H , ξ > 0}, {e−ξ t T ∗(t ) x : x ∈ H , ξ > 0}.The sufficiency follows from the representation

T (t ) x , x ∗ = 12π i t

1t +i∞

1t −i∞

eλt R2(λ, A) x , x ∗ d λ, t > 0,

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where the integral converges absolutely by the Hölder inequality and our assumptions,see e.g. [14,24,9].

The second statement allows us to cancel the growth of the resolvent by restricting

it to sufficiently smooth elements of H .

Lemma 2.3 Let (T (t ))t ≥0 be a bounded C 0-semigroup on a Hilbert space H withgenerator A, such that iR ⊂ ( A). Then for a fixed α > 0

R(λ, A)(− A)−α ≤ C , Re λ > 0,

if and only if

R(i s, A) = O(|s|α ), s → ∞. (2.1)Proof The lemma is proved in [18, Lemma 3.2], [15, Lemma 1.1] in a version saying,in particular, that the condition

R(λ, A) ≤ C 1 + |λ|α , 0


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(i)

R(i s, A) = O(|s|α), s → ∞. (2.2)

(ii)

T (t )(− A)−α = O(t −1), t → ∞. (2.3)

(iii)

T (t )(− A)−α x = o(t −1), t → ∞, x ∈ H .

(iv)

T (t ) A−1 = O(t −1/α ), t → ∞. (2.4)

(v)

T (t ) A−1 x = o(t −1/α), t → ∞, x ∈ H . (2.5)

Proof The implication (ii) ⇒ (i) was proved in [6]; the implication (iii) ⇒ (ii)is a consequence of the uniform boundedness principle. Moreover, the equivalence

(iv) ⇐⇒ (ii) was obtained in [3, Proposition 3.1] as a consequence of the momentinequalities for A. Its ‘o’-counterpart (iii) ⇐⇒ (v) can be obtained by the sameargument. Thus, it remains to prove that (i) ⇒ (iii).

(i) ⇒ (iii): Let H = H ⊕ H be the direct sum of two copies of H . Consider theoperator A on H given by the operator matrix

A =

A (− A)−αO A

with the diagonal domain D(A

) = D( A) ⊕ D( A). Then σ ( A) = σ (A

), and theresolvent R(λ, A) of A is of the form

R(λ, A) =

R(λ, A) R2(λ, A)(− A)−αO R(λ, A)

, λ ∈ ρ ( A).

The operator A is the generator of the C 0-semigroup (T (t ))t ≥0 on H defined by

T (t ) = T (t ) t T (t )(− A)−α

O T (t ) , (2.6)because the resolvents of A and of the generator of (T (t ))t ≥0 coincide. By (2.2) andLemma 2.3,

R(λ, A)(− A)−α ≤ C , Re λ > 0.

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Hence, for every x = ( x 1, x 2) ∈ H and λ ∈ C+,

R(λ,A)x2 ≤ C

R(λ, A) x 12 + R(λ, A) x 22

, (2.7)

and similarly

R(λ, A∗)x2 ≤ C R(λ, A∗) x 12 + R(λ, A∗) x 22

. (2.8)

By Lemma 2.1,

supξ >0

ξ

R R(ξ + i η, A) x 2 + R(ξ + i η, A∗) x 2

d η 0

ξ

R

R(ξ + i η,A)x2 + R(ξ + i η,A∗)x2

d η


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and since (T (t ))t ≥0 is bounded, the last relation holds for all x ∈ H . By the simpleintegral resolvent stability criterion from [25, Theorem 3.1], this means that (T (t ))t ≥0satisfies (2.9). Using the estimates (2.7), (2.8), we conclude that R (ξ +i η,A) x , ξ > 0,satisfies an analog of (2.10) for every x

∈ H. Since the semigroup (T (t ))t

≥0 is

bounded, by the same stability criterion, (T (t ))t ≥0 is stable. Therefore, in particular,

T (t )(− A)−α x = o( 1t

), t → ∞, x ∈ H .

Remark 2.6 Consider the elementary example of a multiplication C 0-semigroup(T (t ))t ≥0,

(T (t ) f )( z) = et z f ( z), t ≥ 0,

on L2(S , µ), where S := { z ∈ C : Re z


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3 Decay of functions

Let M be a continuous non-decreasing function, and let M log be defined by (1.4).The following result is an analog of Theorem 1.2 for the decay of functions f

∈ L∞(R+, X ) (seealso[6, Theorem 10]). Given f ∈ L∞(R+, X ), itsLaplace transformis defined by

f ( z) = ∞ 0

e− zt f (t ) dt .

Theorem 3.1 (Batty, Duyckaerts) Let f ∈ L∞(R+, X ) be such that

a) f extends analytically to the domain := { z ∈ C : Re z > −1/ M (|Im z|)} , and b)

f ( z) ≤ M (|Im z|), z ∈ .Then there exist C , C 1 > 0, t 0 (depending on f and M ) such that

f (0) −t

0

f (s) ds ≤ C 1 M −1log (t /C ) , t ≥ t 0.If we are interested only in polynomial rate of growth of f and in (possibly differ-

ent) polynomial rate of narrowing of \C+, we can formulate the following variantof [6, Theorem 10].

Proposition 3.2 Let α, β, c1, c2 > 0 , := { z ∈ C : Re z > −c1(1 + |Im z|)−α}.Suppose that f ∈ L∞(R+, X ) is such that

f admits an analytic extension to and

f ( z) ≤ c2(1 + |Im z|)β , z ∈ .Then f (0) −

t 0

f (s) ds

≤ C

log t

t

1/α, t ≥ 2,

with C depending only on c1

, c2

,

f ∞

, α , β.

Remark 3.3 Thus, given polynomial growth of f in , the rate of polynomial decayof ( f (0) − t 0 f (s) ds) is determined only by the shape of . This is in sharp contrastto the semigroup case where the growth of R(λ, A) in and the size of are relatedto each other due to the Neumann series expansions of the resolvent.

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Without loss of generality we assume that f is continuous up to ∂ .Lemma 3.4 Under the conditions of Proposition 3.2 , for any ε > 0 and for someC 1, C 2 depending only on c1, c2, f ∞, α , β , ε we have

f ( z) ≤ C 2(1 + |Im z|)α+ε, z ∈ ,where := { z ∈ C : Re z > −C 1(1 + |Im z|)−α}.

Proof Suppose that β > α + ε (otherwise, there is nothing to prove). We use the factthat the function log f is subharmonic in . Fix A > 2β/ε and y > 1 (we dealwith the case y


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where Q = { z : −c1( y + 1)−α 1 wehave

f (0) − t 0

f (s) ds = 12π i

γ

1 + z

2

R2

f ( z) − t 0

e− zs f (s) ds

e zt dz z

. (3.2)

Let γ 1 =

RT∩C

+, γ

2 = RT

∩(

\C

+), γ

3 = RT

\C

+, γ

4 = ∂

∩ RD, where D is

the unit disc of C, and T = ∂D. Then

2π

f (0) −t

0

f (s) ds

≤ γ 1

1 + z

2

R2

f ( z) − t 0

e− zs f (s) ds

e zt d z z

+ γ 2

1 + z

2

R2

f ( z)e zt

d z

z

+ γ 3

1 + z

2

R2

t 0

e− zs f (s) ds e zt d z

z

+ γ 4

1 + z

2

R2

f ( z)e zt d z z

= I 1 + I 2 + I 3 + I 4.Now

I 1 ≤ cπ/2

−π/2

∞ 0

e−s R cos θ f (s + t ) ds cos θ d θ ≤ c · f L∞(R+, X )

R,

I 2 ≤ cc1( R+1)−α

0

c2( R + 1)β y d y R2

≤ cc21c2( R + 1)β−2α−2,

I 3 ≤ c

π/2

−π/2

t

0

e−s R cos θ

f (t − s) ds cos θ d θ ≤ c · f L∞(R+, X ) R , I 4 ≤ c

R 0

c2(s + 1)β e−c1t (s+1)−α ds

s + 1 .

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Setting R = c(c1,α,β)(t / log t )1/α, we obtain

I 1 + I 2 + I 3 + I 4 ≤ C (c1, c2, f ∞, α , β )log t

t 1/α

, t ≥ 2.

Remark 3.5 We can avoid using Lemma 3.4 by replacing the term (1 + z2 R2

) in the

right hand side of (3.2) by (1 + z2 R2

) N for a suitable N . However, Lemma 3.4 provides

some additional information on the growth of f close to the imaginary axis.The statement of Proposition 3.2 can be sharpened if f belongs to the space of

bounded uniformly continuous X -valued functions BUC(R

+, X ).Proposition 3.6 Suppose that f ∈ BUC(R+, X ) satisfies the assumptions of Propo-sition 3.2. Then f (0) −

t 0

f (s) ds

α

= o

log t

t

, t → ∞. (3.3)

Proof By Arendt et al. [2, Corollary 4.4.6], our hypothesis on f imply that

f (t ) = o(1), t → ∞. (3.4)

Arguing as in the proof of Proposition 3.2, and using (3.4) we obtain

I 1 ≤c

Rsups≥0

f (s + t ) = o

1

R

, t → ∞,

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Then there exist c = c( M , M 1, α, β) > 0 such that

T 1T (t )( I − A)−1T 2 ≤ clog t

t 1/α

, t ≥ 2. (3.5)

Proof As in the proof of [6, Corollary 11] we consider the function

f (t ) = d dt

T 1T (t )( I − A)−1T 2

= T 1T (t ) A( I − A)−1T 2

= −T 1T (t )T 2 + T 1T (t )( I − A)−1T 2.

Its Laplace transform

f extends analytically to with the same estimates as F (up to

a constant multiple). Since

f (0) − t 0

f (s) ds = −T 1T (t )( I − A)−1T 2,

Proposition 3.2 yields the claim.

Next result shows that Proposition 3.6 is sharp, at least for β > α/2. It suffices to

consider just scalar-valued functions.Theorem 3.8 Given α > 0 , β > α /2 , and a positive function γ ∈ C 0(R+) , thereexists a function f ∈ C 0(R+) such that

a) f admits an analytic extension to the region := { z ∈ C : Re z > −1/(1 +|Im z|)α} and

f ( z)(1 + |Im z|)−β → 0, | z| → ∞, z ∈ ,

and

b)

lim supt →∞

t

γ (t ) log t

f (0) −t

0

f (s) ds

α

> 0.

To prove Theorem 3.8 we imitate the multiplication semigroup example describedin Remark 2.6. However, we choose an infinite charge µ so that while formal inte-gral expressions for the resolvent, the semigroup and its orbits remain the same as inRemark 2.6, the corresponding size estimates behave in a different way due to the lackof absolute convergence of the integrals.

The proof of Theorem 3.8 is based on the following lemma. Denote by χ A thecharacteristic function of a set A.

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Lemma 3.9 Given Q > 0 and ε > 0 , there exist an integer k > Q and a complex measure µ with compact support in C\ such that for some B = B(α,β) , we have

C\

d µ(ζ) z − ζ

≤ B(1 + |Im z|)β · χ{ξ :|ξ |>Q}( z) + ε, z ∈ , (3.6) C\

et ζ d µ(ζ)

≤ Bχ{r :|r |>Q}(t ) + ε, t ≥ 0, (3.7)1/ B

≤

C\ e

k ζ d µ(ζ)

≤ B, (3.8)

C\

et ζ d µ(ζ)

ζ

α

≤ B log t t

· χ(Q,2k )(t ) +ε

t + 1 , t ≥ 0, (3.9)

and

C\

ek ζ d µ(ζ)

ζ

α

≥ log k Bk

. (3.10)

Proof Choose H > Q large enough, and for an integer k , k ≥ 2, such that

H α ≤ k ≤ H 3α/2,

define

A := 2k log k ,τ := Ak −1/

√ k ,

q := e2π i / k ,w := i H − 1.

We define also a finite measure

µ := τ

1≤s≤k qs (1 + qs /( Aw))δw+qs / A,

where δ x is the unit mass at x .

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Observe that

1≤s≤k qs

x

−qs

= k x k

−1

,

1≤s≤k q2s

x

−qs

= k x x k

−1

. (3.11)

Indeed, to prove the first identity we use that

1≤s≤k

qs

x − qs =P( x )

x k − 1 ,

forsomepolynomial P with deg P 1, then

| z − w| ≥ 1 − 1 H α

.

Indeed, if Im z < H − 1, then | z − w| > 1, and otherwise, Re z ≥ − H −α.Now if H 2 ≤ Im z ≤ | z| ≤ 2 H , z ∈ , then we use that A| z − w| > 2,

| Ak ( z − w)k − 1| > Ak e−k / H α /2, to obtain that

zw k Aτ Ak ( z − w)k − 1 ≤ cτ k A1−k ek / H α = c√ kek / H α .From now on we assume that k satisfies the condition

√ kek / H

α ≤ H β . (3.12)

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Then

|C µ( z)| ≤ c H β , H 2 ≤ Im z ≤ | z| ≤ 2 H , z ∈ . (3.13)

If z ∈ and |Im z − H | + || z| − H | > H /2, then | z − w| > c max(| z|, H ), andunder condition (3.12) we have for large H :

|C µ( z)| ≤ c1| z|√

k

H (c max(| z|, H ))k ≤c1| z|

√ k

H (c max(| z|, H ))α+1 ≤ ε.

This, together with (3.13), proves (3.6).Next,

Lµ(t ) def =

C\

et ζ d µ(ζ) = τ

1≤s≤k qs (1 + qs /( Aw))et (w+qs / A),

and

|Lµ(t )| = τ e−t 1≤s≤k qs (1 + qs /( Aw))eqs t / A

. (3.14)Furthermore, we have

1≤s≤k

qs (1 + qs /( Aw))eqs t / A

= 1≤s≤k

n≥0

(qs + q2s /( Aw))(qs t / A)n 1n!

= k m≥1

t km−1

Akm−1 · 1

( km − 1)! +t km−2

Akm−2 · 1

( km − 2)! ·1

Aw

= kt k −1

Ak −1(k − 1)!m≥1

t k

Ak

m−1(k − 1)!

( km − 1)!

1 + km − 1

t w

,

and

|Lµ(t )| = k 3/2t k −1e−t

k !m≥1

t k

Ak

m−1(k − 1)!

( km − 1)!

1 + km − 1

t w

.

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Thus, for some constants c, c1, c2, c3 we have

|Lµ(t )

| ≤ c

k 3/2t k −2e−t

k ! m≥1 k

H Ak (m−1)

(k − 1)!( km − 1)!

km

H

≤ c1e−t k 5/2t k −2/(k ! H ), 0 ≤ t ≤ k / H ,

and

c2e−t k 3/2t k −1/ k ! ≤ |Lµ(t )| ≤ c3e−t k 3/2t k −1/k !, k / H ≤ t ≤ A.

If t ≥ A, then by (3.14),

|Lµ(t )| ≤ 2τ e−t ke t / A = 2√

k Ak −1e−t (1−1/ A).

The function t → e−t t k −2 attains its maximum on [0, k / H ] at t = k / H ; thefunction t → e−t t k −1 attains its maximum on [k / H , A] at t = k − 1; the functiont → e−t (1−1/ A) attains its maximum on [ A, +∞) at t = A. Using that A = 2k log k ,by Stirling’s formula, we obtain that

0


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Here we use the formula

1≤s≤k qs eq

s t / A =

1≤s≤k n≥0qs (qs t / A)n

1

n

!= k

m≥1

t km−1

Akm−1 · 1

( km − 1)!

= kt k −1

Ak −1(k − 1)!m≥1

t k

Ak

m−1(k − 1)!

( km − 1)!

= (1 + o(1)) kt k −1

Ak −1(k − 1)! , 0 ≤ t ≤ A, H → ∞.

Therefore, by Stirling’s formula, we have for large H :

| N µ(k )| ≥ c1τ e−k

H · k

k

Ak −1(k − 1)! ≥c2

H . (3.15)

Moreover,

t 1/α

| N µ(t )

| ≤ c

k

t H · t k k

ek −t t 1/α

≤ ε + c1(k 1/α/ H ) · χ(k /2,2k )(t ), 0 ≤ t ≤ A, (3.16)

and

t 1/α| N µ(t )| ≤ cτ e−t

H · kt 1/αet / A ≤ c1

Ak −1+1/α√

ke− A

H ≤ ε, t ≥ A. (3.17)

Now we fix 0 < ψ < β −

α

2

and k =

ψ H α log H in such a way that k ∈N. Then

(3.12) is satisfied for large H , (3.15) implies (3.10), and (3.16), (3.17) imply (3.9).

Proof of Theorem 3.8 Without loss of generality, we can assume that γ is non-increas-ing.

Our function f will be defined by an inductive construction. Set Q1 = 1. On stepn ≥ 1 weuse Lemma 3.9 with Q = Qn , ε = 2−n tofind µn , k n satisfying (3.6)–(3.10).Since

lim| z|→∞, z∈C µn ( z) = 0,

limt →∞Lµn (t ) = 0,

we can find Qn+1 > k n such that

1 3


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|C µn ( z)| ≤ ε, | z| > Qn+1, z ∈ , (3.18)|Lµn (t )| ≤ ε, t > Qn+1, (3.19)

which completes the induction step.Finally, for some numbers φn ∈ C, |φn | = 1, to be chosen iteratively later on, we

set

f =n≥1

φn γ (k n )1/αLµn .

The series above converges absolutely by the choice of µn , and therefore f ∈ C 0(R+).Similarly, the function

f = n≥1

φnγ (k n)1/αC µn

extends analytically to , and

f ( z)(1 + |Im z|)−β → 0, | z| → ∞, z ∈ .Finally, for m ≥ 1 we have

k m

γ (k m ) log k m

f (0) −k m

0

f (s) ds

α

= k mγ (k m ) log k m

n≥1

φn γ (k n)1/α N µn (k m )

α

≥ck m

γ (k m ) log k m 1≤n≤m φnγ (k n )1/α N µn(k m )α

−n>m

φnγ (k n )1/α N µn (k m )

α .

On step m ≥ 1 we choose φm in such a way that

1≤n≤m φn γ (k n )1/α N µn(k m )=γ (k m )1/α N µm (k m )+

1≤n


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Then

k m

γ (k m ) log k m 1≤n≤m φnγ (k n)1/α N µn (k m )

α

− n>m φn γ (k n )1/α N µn(k m )

α

≥ k m

γ (k m ) log k m

γ (k m )| N µm (k m )|α −

n>m

φnγ (k n)1/α N µn (k m )

α

≥ c1 −c2

log k m

n>m

2−n/αα

≥ c3 > 0.

Remark 3.10 A variant of our construction works with γ = 1 in Theorem 3.8, if onelooks just for f ∈ L∞(R+).

4 Decay of Banach space semigroups

Using the construction of Theorem 3.8, we show next that the analogue of Theorem 1.2for C 0-semigroups on Banach spaces, Theorem 3.1, is also sharp. Estimates for thelocal resolvents of group generators similar to the ones used below has been also

employed, in particular, in [2] and [11, Sect. II.4.6].

Theorem 4.1 Given α > 0 , there exist a Banach space X α and a bounded C 0-semi-group (T (t ))t ≥0 on X α with generator A such that

(a) R(i s, A) = O(|s|α ), |s| → ∞,

and

(b) lim supt →∞

t log t

1α T (t ) A−1 > 0.Proof Let (S (t ))t ≥0 be the left shift semigroup on BUC(R+), let := {λ ∈ C :Re λ > −1/(1 + |Im λ|)α}, and let 0 := ∩ {λ ∈ C : |Re λ|


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is a Banach space. Moreover, S (t ) X α ⊂ X α , t ≥ 0, and the restriction (T (t ))t ≥0 of (S (t ))t ≥0 to X α is also a C 0 -semigroup. To prove this assertion it suffices to observethat

f (λ) − T (t ) f (λ) = ∞ 0

e−λs f (s) ds −∞

0

e−λs f (t + s) ds

= (1 − eλt ) f (λ) + eλt t 0

e−λs f (s) ds, Re λ > 0,

and the same equality holds on 0. By the definition of X α ,

T (t ) f − f α → 0, t → 0+,

and then

T (t ) f − f X α → 0, t → 0 + .

Let A stand for the generator of (T (t ))t ≥0.Next we prove that for every f

∈ X α the local resolvent R(λ, A) f satisfies the

estimate

R(λ, A) f X α ≤ C (1 + |Im λ|)α f X α , 0


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Fix λ with Re λ ∈ (0, 1). To estimate R(λ, A) f α note that

( R(λ, A) f )(µ)

=

∞

0 e−µt

∞

0 e−λs f (t

+s) ds dt

=∞

0

e(λ−µ)t ∞

t

e−λs f (s) ds dt

= − 1λ − µ

∞ 0

e−λs f (s) ds + 1λ − µ

∞ 0

e−µt f (t ) dt

= − f (λ) − f (µ)λ − µ , Re µ > 1.Therefore, R(λ, A) f extends analytically to 0, and

( R(λ, A) f )(µ) =−

f (λ)− f (µ)λ−µ , λ = µ, µ ∈ 0,

−

f

(µ), λ = µ.

Now, if |λ − µ| ≥ 1, µ ∈ 0, then| ( R(λ, A) f )(µ)| ≤ | f (λ)| + f (µ)| ≤ c(1 + |Im µ|)α (1 + |Im λ|)α f α.

Furthermore, if

1 > |λ − µ| ≥ 12(1 + |Im λ|)α , µ ∈ 0,

then we have

| ( R(λ, A) f )(µ)| ≤ 2(1 + |Im λ|)α | f (λ)| + | f (µ)|≤ c(1 + |Im λ|)α(1 + |Im µ|)α f α.

Finally, if

|λ

−µ

| ≤

1

2(1 + |Im λ|)α

,

then, applying Cauchy’s formula on the circle

C λ := { z ∈ C : | z − λ| =2

3(1 + |Im λ|)−α},

1 3


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we obtain that

| ( R(λ, A) f )(µ)| ≤ c(1 + |Im µ|)α (1 + |Im λ|)α f α.

Thus,

R(λ, A) f α ≤ C (1 + |λ|)α f α , 0


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[Borichev an Tomilov] Optimal Polynomial Decay of Functions and Operator

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