Stability theory for semigroups using (Lp,Lq) Fourier multipliers
Rozendaal, Jan; Veraar, Mark DOI
10.1016/j.jfa.2018.06.015 Publication date
2018
Document Version Final published version Published in
Journal of Functional Analysis
Citation (APA)
Rozendaal, J., & Veraar, M. (2018). Stability theory for semigroups using (L p,L q) Fourier multipliers. Journal of Functional Analysis, 275(10), 2845-2894. https://doi.org/10.1016/j.jfa.2018.06.015
Important note
To cite this publication, please use the final published version (if applicable). Please check the document version above.
Copyright
Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy
Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.
This work is downloaded from Delft University of Technology.
Green Open Access added to TU Delft Institutional Repository
‘You share, we take care!’ – Taverne project
Contents lists available atScienceDirect
Journal
of
Functional
Analysis
www.elsevier.com/locate/jfa
Stability
theory
for
semigroups
using
(L
p,
L
q)
Fourier
multipliers
✩Jan Rozendaala,b, Mark Veraarc,∗
aMathematicalSciencesInstitute,AustralianNationalUniversity,ActonACT 2601,Australia
bInstituteofMathematics,PolishAcademyof Sciences,ul.Śniadeckich8,00-656 Warsaw,Poland
cDelftInstituteofAppliedMathematics,DelftUniversityof Technology,P.O.Box 5031,2628CDDelft,theNetherlands
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received21November2017 Accepted19June2018 Availableonline3July2018 CommunicatedbyDanVoiculescu
MSC: primary47D06 secondary34D05,35B40,42B15, 46B20 Keywords: C0-semigroup
Polynomialandexponentialstability Fouriermultipliers
Typeandcotype
Westudypolynomialandexponentialstability forC0
-semi-groupsusingtherecentlydevelopedtheoryofoperator-valued (Lp,Lq) Fouriermultipliers.Wecharacterizepolynomial
de-cayoforbitsofaC0-semigroupintermsofthe(Lp,Lq) Fourier
multiplierpropertiesofitsresolvent.Usingthis characteriza-tionwederivenewpolynomialdecayrateswhichdependon thegeometryoftheunderlyingspace.Wedonotassumethat thesemigroupisuniformlybounded,ourresultsdependonly onspectralpropertiesofthegenerator.
Asacorollaryofourworkonpolynomialstabilitywereprove andunifyvariousexistingresultsonexponentialstability,and we also obtain a new theorem on exponential stability for positivesemigroups.
©2018ElsevierInc.Allrightsreserved.
✩ ThefirstauthorispartiallysupportedbygrantDP160100941oftheAustralianResearchCouncil.The
secondauthorissupportedbytheVIDIsubsidy639.032.427oftheNetherlandsOrganisationforScientific Research(NWO).
* Correspondingauthor.
E-mailaddresses:janrozendaalmath@gmail.com(J. Rozendaal),M.C.Veraar@tudelft.nl(M. Veraar).
https://doi.org/10.1016/j.jfa.2018.06.015
1. Introduction
Inthis articlewestudy theasymptoticbehavior ofsolutionsto theabstract Cauchy problem
u(t) + Au(t) = 0, t≥ 0,
u(0) = x. (1.1)
Here −A isthegeneratorofaC0-semigroup(T (t))t≥0onaBanachspaceX andx∈ X.
The unique solution of (1.1) with initial data x is given by u(t) = T (t)x for t ≥ 0.
One of the key difficulties in the asymptotic theory for solutions of (1.1) is that the classicalLyapunov stabilitycriterionis ingeneralnotvalidifX isinfinitedimensional. However, asymptotic behavior canbe deduced from theassociated resolvent operators
R(λ,A)= (λ−A)−1forλ∈ ρ(A).Forexample,onaHilbertspaceX theGearhart–Prüss theorem [3, Theorem 5.2.1] states that(T (t))t≥0 is exponentially stable if and only if
σ(A) ⊂ C+ and supRe(λ)<0R(λ,A)< ∞. A uniform bound for the resolvent is not
sufficient toensureexponential stabilityongeneralBanachspaces,butitwasshownin [26,37] (seealso[16,32,36,65])thatexponentialstabilitycanbecharacterizedintermsof
LpFouriermultiplierpropertiesoftheresolvent.OutsideofHilbertspacesthismultiplier conditionisastrictlystrongerassumptionthanuniformboundedness,andinapplications it can be difficult to verify. On the other hand, cf. [47,49,64,65], uniform bounds for the resolvent do imply exponential stability for orbitsin fractional domains, with the fractionaldomainparameterdependingonthegeometryoftheunderlyingspace.Atthe momentitisnotfullyunderstoodhowthecharacterizationofexponentialstabilityusing Fouriermultipliersisrelatedto suchconcretedecayresults.
In a separate development, over the past decade much attention has been paid to polynomial decay of semigroup orbits. The work of Lebeau [39,40] and Burq [13] on energydecayfordampedwaveequationsraisedthequestionofwhat thepreciserelation is betweengrowth ratesfor theresolventand decayratesforthesemigroup. More pre-cisely, ifonehasσ(A)⊂ C+ in(1.1) butR(iξ,A)→ ∞ as|ξ|→ ∞, then(T (t))t≥0 is notexponentiallystable andonetypically encountersotherasymptotic behavior.Since auniformrateofdecayforallsolutionsto(1.1) impliesexponentialstabilityofthe semi-group,onecanexpectuniformasymptoticbehavioronlyfororbitsinsuitablesubspaces such as fractional domains, and ingeneral the smoothnessparameter of the fractional domaininfluencesthedecaybehavior.In[4] Bátkai,Engel,PrüssandSchnaubeltproved thatforuniformlyboundedsemigroupsapolynomialgrowthrateoftheresolventimplies aspecific polynomialdecayrateforclassicalsolutionsof (1.1) andvice versa,and they showedthatthiscorrespondenceisoptimaluptoanarbitrarilysmallpolynomialloss.In [8] BattyandDuyckaertsextendedthiscorrespondencetothesettingofarbitrary resol-ventgrowthandtheyreducedthelosstoalogarithmicscale.ThenBorichevandTomilov provedin[12] thatthislogarithmiclossissharpongeneralBanachspaces,butthatitcan be removedonHilbertspaces inthecaseofpolynomialresolventgrowth.Inparticular,
on Hilbertspaces this yields a characterizationof polynomial stability interms of the growthof theresolvent. Thisresulthasbeenappliedextensivelyinthestudyofpartial differentialequations(seee.g.[1,2,9,14,24,38,42,57] andreferencestherein)andhasbeen extendedin[7,15,43,54,60,62,63] tofinerscalesofresolventgrowthandsemigroupdecay. Althoughmuchworkhasgoneintodeterminingtherelationbetweenresolventgrowth and polynomialrates of decay,it is notclear how such asymptotic behavior relates to theFourieranalyticpropertiesoftheresolventwhichcharacterizeexponential stability. Furthermore, the currently available literature on polynomial decay deals almost ex-clusively with uniformly bounded semigroups. To the best of our knowledge, the only previouslyknownresultconcerningpolynomialdecayforgeneralsemigroupsis[4, Propo-sition3.4].Therearemanynaturalclassesofexampleswherethegeneratorhasspectral propertiesasabovebutthesemigroupisnotuniformlybounded,orwhereitisunknown whetherthesemigroupisbounded. Typicalexamplesofthisphenomenoncanbe found inSection4.7andincludesemigroupswhosegeneratorisanoperatormatrixora multi-plicationoperatoronaSobolevspace. Inturn,suchoperators canbe foundindisguise in concrete partial differential equations. One example is the standard wave equation withperiodicboundaryconditions;hereuniformboundednessfails.Otherexamplescan be found in [50] for certain classes of perturbed wave equations and in [61] for delay equations.Forinfinitesystemsofequationstheuniform boundednessconditionleadsto additionalassumptionsonthecoefficientsin[51].
In this article we deal with the problems outlined above in three ways. First, we characterizepolynomialstabilityongeneralBanachspacesintermsofFouriermultiplier propertiesofpowersoftheresolvent,inTheorem4.6.IndoingsoweextendtheFourier analyticcharacterizationofexponentialstabilitytothismorerefinedsetting.Then,using thetheoryofoperator-valued(Lp,Lq) Fouriermultiplierswhichwasdevelopedin[55,56] withapplicationstostabilitytheoryinmind,wederiveconcretepolynomialdecayrates from this characterization. These results involve only growth bounds for the resolvent and areneweven on Hilbertspaces.In particular, thefollowing theorem canbe found inthemain textasCorollary 4.11.
Theorem 1.1.Let −A be the generator of aC0-semigroup (T (t))t≥0 on a Hilbert space
X such that σ(A)⊂ C+ and R(λ,A)≤ C(1+|λ|)β forsome β > 0,C ≥ 0 and all
λ∈ C withRe(λ)≤ 0.Thenforeach τ ≥ β thereexistsa Cτ ≥ 0 such that
T (t)A−τ ≤ C
τt1−τ/β (t∈ [1, ∞)). (1.2) Notethatwedonotassumethatthesemigroupisuniformlybounded.Infact,weshow that onecan derive polynomialdecay behavior for initial values insuitable fractional domainsgiven onlyspectralpropertiesofthegenerator.Inparticular, bysettingτ = β
inTheorem 1.1 oneobtains uniform boundedness of sufficiently smooth solutions. For uniformlyboundedsemigroupstheparameter1− τ/β in(1.2) canbereplacedby−τ/β, aswasshownin[12],butinExample 4.20weprovethat1− τ/β isoptimalforgeneral
semigroupsifτ = β.OurmaintheoremsallowforA tohaveasingularityatzero,oreven singularitiesat bothzeroand infinity.Wealso obtainversionsof Theorem1.1onother Banachspaces;thedecayratein(1.2) thendependsonthegeometry oftheunderlying space.
Finally, as a directcorollary of our results on polynomial stability we recover in a unifiedmannervariousresultsonexponential stabilityfrom[26,37,47,49,64,65].Wealso obtainanewstabilityresultforpositivesemigroups,Theorem5.8.
Toproveourmainresultswerelyonthetheoryofoperator-valuedFouriermultipliers fromLp(R;X) toLq(R;Y ),forX andY Banachspaces.AFouriermultiplier characteri-zationofexponentialstabilityforgeneralp∈ [1,∞) andq∈ [p,∞] wasknownfrom[37], butsofaronlythecasewherep= q hasbeen used(see[5,25,26,36,37,65]).Althoughin thissettingverypowerfulmultipliertheoremsareavailable,seeforexampleWeis’version of theMikhlinmultipliertheoremin[66] and [17,29,33], theassumptionsof these theo-remsareingeneraltoorestrictiveforapplicationstostabilitytheory.Indeed,multiplier theorems onLp(R;X) typicallyrequireboth ageometric assumptiononX,namelythe UMDconditionwhichexcludesspacesofinterestsuchasX = L1,aswellassmoothness
of the multiplier and comparatively fast decay at infinity of its derivative. The latter assumption inparticular isnotsatisfiedinmostapplications tostabilitytheory.
Inthisarticlewearguethatforthestudyofasymptoticbehavioritismorenaturalto considergeneralp∈ [1,∞) andq∈ [p,∞].Itwasobservedin[55,56] thatonecanderive boundedness ofFouriermultipliersfromLp(R;X) toLq(R;Y ) forp< q underdifferent geometricassumptionsonX andY thaninthecasewherep= q,andassumingdecayof themultiplieratinfinitybutnosmoothness.Infact,theparametersp andq depend on thegeometryofX,andtheamountofdecaywhichisrequiredatinfinityisproportional to 1p−1q.Moreover,inSection3.2weprovethatgrowthoftheresolventonX corresponds
touniformboundedness,andinfactevendecay,oftheresolventfromsuitablefractional domain and range spaces to X. Then one can determine for which fractional domain and range parameters the conditions of the (Lp,Lq) multiplier theorems are satisfied for (powers of) the resolvent, and the Fourier multiplier characterizations of stability in Theorems 4.6 and 5.3 yield the corresponding asymptotic behavior.We emphasize that, although we useFourier multiplier techniques for the proofs, our main theorems onconcrete decayratesinvolveonlygrowth boundsontheresolvent.
This articleis organized as follows. InSection 2we present some basics on Banach spacegeometry,Fouriermultipliersandsectorialoperators.InSection3wededuce mul-tiplierpropertiesoftheresolventandweproveProposition3.4andCorollary3.5.These arefundamentalinlatersectionsforrelatingresolventgrowthonX toboundednessand decayfrom fractionaldomainandrangespacestoX.InSection4westudypolynomial decayofsemigroups.WecharacterizepolynomialstabilityusingFouriermultipliers,and from thischaracterizationwe deduceconcretepolynomialdecayrateswhich dependon the geometryof theunderlyingspace. InSection5wederive from theseresultsvarious corollaries onexponential decay.Wealsoproveacharacterizationofexponential stabil-ity using multipliers on Besov spaces, which in turn is used to obtaina new stability
resultforpositivesemigroups.Anappendixcontainsestimatesforcontourintegralsand exponentialfunctions.
1.1. Notation
The set of natural numbers is N = {1,2,. . .}, and N0 := N∪ {0}. We denote by
C+={λ∈ C| Re(λ)> 0} andC−=−C+ theopencomplexrightand lefthalf-planes.
Nonzero Banach spaces over the complex numbers are denoted by X and Y . The spaceofbounded linearoperators fromX toY is L(X,Y ),and L(X):=L(X,X).The identity operatoronX isdenotedby IX,and weusually write λ forλIX whenλ∈ C. ThedomainofaclosedoperatorA onX isD(A),aBanachspacewiththenorm
xD(A):=xX+AxX (x∈ D(A)).
For aninjectiveclosed operator A weidentify therange ran(A) of A with theBanach spaceD(A−1).ThespectrumofA isσ(A) andtheresolventsetisρ(A)=C\ σ(A).We writeR(λ,A)= (λ− A)−1 fortheresolventoperatorofA atλ∈ ρ(A).
For p ∈ [1,∞] and Ω a measure space, Lp(Ω;X) is the Bochner space of equiva-lenceclassesofstronglymeasurable,p-integrable, X-valuedfunctionsonΩ.TheHölder conjugateofp∈ [1,∞] isdenotedbyp andisdefinedby1= 1p+p1.
TheclassofX-valuedSchwartzfunctionsonR isdenotedbyS(R;X),andthespace ofX-valuedtempereddistributionsbyS(R;X).TheFouriertransformoff ∈ S(R;X)
isdenotedbyFf orf . Iff ∈ L1(R;X) then
Ff(ξ) =
R
e−iξtf (t) dt (ξ ∈ R).
Weusetheconventionthat 1
0 =∞ and 0 0 =∞.
ForsetsS and Z weoccasionally denoteafunctionf : S→ Z ofavariable s simply
byf = f (s).Weusethenotationf (s) g(s) forfunctionsf,g : S→ R toindicatethat
f (s)≤ Cg(s) for all s ∈ S and a constantC ≥ 0 independent of s, and similarly for
f (s) g(s).Wewritef (s) g(s) ifg(s) f(s) g(s) holds.
2. Preliminaries
2.1. Banachspacegeometry
Here we collect some background on Banach space geometry which is used for our resultsonnon-HilbertianBanachspaces.
ABanachspace X hasFourier type p∈ [1,2] iftheFouriertransform F isbounded from Lp(R;X) to Lp(R;X).We thenset F
p,X :=FL(Lp(R;X),Lp(R;X)). Tomake our
hasFouriertypeq.Each BanachspacehasFouriertype1,and X hasFouriertype2 if and only ifX is isomorphic toa Hilbertspace. Forr∈ [1,∞] and Ω ameasure space,
Lr(Ω) has Fouriertypemin(r,r).FormoreonFouriertypesee[29,52].
A (real)Rademachervariable isarandom variabler : Ω→ {−1,1} onaprobability space(Ω,P) suchthatP(r = −1)=P(r = 1)= 12.ARademachersequence isasequence (rk)k≥1 ofindependentRademachervariablesonsomeprobabilityspace.
Let(rk)k≥1 beaRademachersequenceonaprobabilityspace(Ω,P).ABanachspace
X has type p ∈ [1,2] if there exists aconstant C ≥ 0 such that for all n ∈ N and all
x1,. . . ,xn∈ X onehas E n k=1 rkxk 21/2 ≤ C n k=1 xkp 1/p .
Also, X has cotype q ∈ [2,∞] if there exists aconstant C ≥ 0 suchthat forall n∈ N
and allx1,. . . ,xn∈ X onehas n k=1 xkq 1/q ≤ CE n k=1 rkxk 21/2 ,
with the obvious modification for q = ∞. We saythat X has nontrivial type if X has
type p ∈ (1,2], and finite cotype if X has cotype q ∈ [2,∞). Each Banach space has type p= 1 andcotype q = ∞,and X has type p= 2 and cotype q = 2 ifand only if
X is isomorphic to a Hilbert space, by Kwapień’s theorem [34]. For r ∈ [1,∞) and Ω ameasure space,Lr(Ω) hastypemin(r,2) andcotypemax(r,2).Formoreontypeand cotypesee [18,30].
LetX beaBanachlatticeandp,q∈ [1,∞].WesaythatX isp-convex ifthereexists aconstantC≥ 0 such thatforalln∈ N andallx1,. . . ,xn ∈ X onehas
n k=1 |xk|p 1/p X ≤ C n k=1 xkpX 1/p ,
with the obvious modification forp=∞. Wesay thatX is q-concave if there exists a constantC≥ 0 suchthatforalln∈ N andallx1,. . . ,xn∈ X onehas
n k=1 xkqX 1/q ≤ C n k=1 |xk|q 1/q X,
with the obvious modification for q = ∞. Each Banach lattice X is 1-convex and
∞-concave. For r ∈ [1,∞] and Ω a measure space, Lr(Ω) is r-convex and r-concave. Formoreonp-convexityandq-concavitywereferthereaderto[21,41].
Let X and Y be Banach spaces and T ⊆ L(X,Y ). We say thatT is R-bounded if
thereexistsaconstantC≥ 0 suchthatforalln∈ N,T1,. . . ,Tn∈ T andx1,. . . ,xn ∈ X onehas
E n k=1 rkTkxk 2 Y 1/2 ≤ CE n k=1 rkxk 2 X 1/2 . (2.1)
ThesmallestsuchC istheR-bound ofT andisdenotedbyR(T ).Ifwewanttospecify theunderlyingspaces X and Y thenwewrite RX,Y(T ) for the R-boundofT , and we write RX(T )= RX,Y(T ) if X = Y .Every R-bounded collectionis uniformly bounded withsupremumbound lessthanorequaltoitsR-bound, andtheconverseholdsifand onlyifX hascotype2 andY has type2.Forλ∈ C andanR-bounded collectionT ⊆
L(X,Y ),theclosedabsolutelyconvexhullaco(λT )⊆ L(X,Y ) ofλT = {λT | T ∈ T } is R-bounded,and
RX,Y(aco(λT )) ≤ 2|λ|RX,Y(T ). (2.2) Inparticular, L1-averagesof R-boundedcollectionsare againR-bounded, afactwhich
willbe usedfrequently.FormoreonR-boundednesssee[30,33,48].
ThefollowinglemmaisusedintheproofofCorollary5.5.Itcanalsobededucedfrom acorresponding statement in[31, Theorem 5.1] for theBesov space Br,11/r(R;L(X,Y )).
Here we give amoredirect proof. For r∈ [1,∞] and E a Banachspace we denote by
W1,r(R;E) the Sobolev space of weakly differentiable f : R → E such that f,f ∈
Lr(R;E),withf
W1,r(E):=fLr(R;E)+fLr(R;E).
Lemma2.1.LetX beaBanachspacewithcotypeq∈ [2,∞) andY aBanachspacewith type p∈ [1,2], and letr∈ [1,∞] be suchthat r1 = 1p−1q. Thenthere exists aconstant C ∈ [0,∞) such that forall f ∈ W1,r(R;L(X,Y )) the set {f(t)| t∈ R} ⊆ L(X,Y ) is
R-bounded,with
R({f(t) | t ∈ R}) ≤ CfW1,r(R;L(X,Y )).
Proof. Letf ∈ W1,r(R;L(X,Y )) and forj∈ Z setIj:= [j,j + 1) andTj :={f(t)| t∈
Ij}.Then [33,Example2.18] andHölder’sinequalityimply
R(Tj) fW1,1(I
j;L(X,Y )) fW1,r(Ij;L(X,Y ))
for all j ∈ Z. Now [20, Theorem 3.1] (see also [30, Proposition 9.1.10]) shows that
{f(t)| t∈ R}=j∈ZTj isR-bounded,with R({f(t) | t ∈ R}) (R Tj)jr(Z) fW1,r(Ij;L(X,Y )) jr(Z) fW1,r(R;L(X,Y )). 2
Byreplacing the Rademacher random variables in (2.1) byGaussian variables, one obtainsthe definitionof aγ-bounded collection T ⊆ L(X,Y ). Each R-bounded collec-tion is γ-bounded, and the converse holds if and only if X has finite cotype (see [35, Theorem 1.1]).WechoosetoworkwithR-boundednessinthisarticle,bothbecausethe
notionofR-boundednessismoreestablishedandbecausethosestabilitytheoremsinthis articlewhichuseR-boundednessareonlyofinterestonspaceswithfinitecotype.
2.2. Fouriermultipliertheorems
ToproperlydefineFouriermultipliersforsymbolswithasingularityatzero,webriefly introduce theclass of vector-valuedhomogeneous distributions.For moreonthese dis-tributions see[56]. ForX a Banachspacelet
˙
S(R; X) := {f ∈ S(R; X) | f(k)(0) = 0 for all k∈ N0},
endowedwiththesubspacetopology,andletS˙(R;X) bethespaceofcontinuouslinear mappingsfromS(R;˙ C) toX.ThenS(R;˙ X) isdenseinLp(R;X) forallp∈ [1,∞),and
Lp(R;X) canbenaturallyidentifiedwith asubspaceofS˙(R;X) forallp∈ [1,∞]. Let X and Y be Banach spaces. A function m : R\ {0} → L(X,Y ) is X-strongly measurable ifξ → m(ξ)x isastronglymeasurableY -valuedmapforeachx∈ X.Wesay thatm isof moderategrowth ifthere existα∈ [0,∞) andg∈ L1(R) suchthat
|ξ|α(1 +|ξ|)−2αm(ξ)
L(X,Y )≤ g(ξ) (ξ∈ R).
Letm:R\{0}→ L(X,Y ) beanX-stronglymeasurablemapofmoderategrowth.Then
Tm: ˙S(R;X)→ ˙S(R;Y ),
Tm(f ) :=F−1(m· f ) (f ∈ ˙S(R; X)), (2.3) istheFouriermultiplieroperator associatedwithm.Onecallsm thesymbol ofTm,and we identifysymbolswhichare equalalmost everywhere.Ifm(·)L(X,Y )∈ L1loc(R) then (2.3) extendstoallf ∈ S(R;X) anddefinesanoperatorTm:S(R;X)→ S(R;X).
For p ∈ [1,∞) and q ∈ [1,∞] we let Mp,q(R;L(X,Y )) be the set of all
X-strongly measurable m : R \ {0} → L(X,Y ) of moderate growth such that
Tm∈ L(Lp(R;X),Lq(R;Y )),and
mMp,q(R;L(X,Y )):=TmL(Lp(R;X),Lq(R;Y )).
We write · Mp,q =· Mp,q(R;L(X,Y )) when the spaces X and Y are clear from the
context.
We now recall several (Lp,Lq) Fourier multiplierresults from ourearlier work. The firstis [55,Proposition3.9].
Proposition 2.2. LetX be a Banachspace with Fouriertype p∈ [1,2] and Y a Banach space with Fourier cotype q ∈ [2,∞], and let r ∈ [1,∞] be such that 1
r =
1
p −
1
q. Let
m:R\{0}→ L(X,Y ) beanX-stronglymeasurablemapsuchthatm(·)L(X,Y )∈ Lr(R).
mMp,q(R;L(X,Y ))≤
1
2πFp,XFq,Ym(·)L(X,Y )Lr(R). (2.4)
Ournextresultfollowsfrom[56,Theorem4.6andRemark4.8] and[55,Theorem3.21 andRemark3.22].
Proposition 2.3.Let X be a Banach space with type p∈ [1,2] and Y a Banach space with cotype q ∈ [2,∞], and letr ∈ [1,∞] be such that 1r > 1p− 1q.Then there exists a constantC∈ [0,∞) suchthatthefollowingholds.Letm:R→ L(X,Y ) beanX-strongly measurable map such that {(1+|ξ|)rm(ξ) | ξ ∈ R} ⊆ L(X,Y ) is R-bounded. Then
m∈ Mp,q(R;L(X,Y )) and
mMp,q(R;L(X,Y ))≤ CRX,Y({(1 + |ξ|)
rm(ξ)| ξ ∈ R}). (2.5)
Moreover, if X is a complemented subspace of a p-convex Banach lattice with finite cotypeandif Y isaBanach spacecontinuouslyembeddedin aq-concave Banachlattice forq∈ [1,∞), then(2.5) also holdsif 1r= 1p−1q.
Fors∈ R andp∈ [1,∞],theinhomogeneousBesselpotentialspace Hps(R;X) consists ofallf ∈ S(R;X) suchthatTms(f )∈ L
p(R;X),wherems(ξ):= (1+|ξ|2)s/2forξ∈ R. ItisaBanachspaceendowedwiththenorm
fHs
p(R;X):=Tms(f )Lp(Rd;X) (f ∈ Hps(R; X)).
Moreover,S(R˙ d;X)⊆ Hps(Rd;X) is denselyembeddedforp<∞.
Thefollowingpropositionisprovedinthesamewayasthecorrespondinghomogeneous versionin[55, Theorem3.24].We notethatonecanoftenavoidcondition(2) byusing approximationarguments.
Proposition2.4. Letp∈ [1,∞) andq∈ [p,∞). LetX be ap-convexBanachlattice with finite cotype and let Y be a q-concave Banach lattice, and let r ∈ (1,∞] be such that
1
r =
1
p −
1
q.Then there exists a constant C ∈ [0,∞) such that the following holds.Let
m:R→ L(X,Y ) besuch that there existsa K :R→ L(X,Y ) satisfying thefollowing conditions:
(1) K(t)∈ L(X,Y ) is apositiveoperatorforallt∈ R;
(2) K(·)x∈ L1(R;Y ) forallx∈ X;
(3) F(K(·)x)(ξ)= m(ξ)x for allx∈ X andξ∈ R. ThenTm: Hp1/r(R;X)→ Lq(R;Y ) is boundedand
TmL(H1/r
p (R;X),Lq(R;Y ))≤ Cm(0)L(X,Y )≤ C sup
2.3. Sectorialoperators
ForaC0-semigroup(T (t))t≥0⊆ L(X) on aBanachspaceX welet
ω0(T ) := inf{ω ∈ R | ∃M ∈ [0, ∞) : T (t)L(X)≤ Meωt for all t∈ [0, ∞)}. Forϕ∈ (0,π) set
Sϕ:={z ∈ C \ {0} | |arg(z)| < ϕ},
andletS0:= (0,∞).RecallthatanoperatorA onaBanachspaceX issectorialofangle
ϕ∈ [0,π) ifσ(A)⊆ Sϕ andifsup{λR(λ,A)L(X)| λ∈ C\ Sθ}<∞ for allθ∈ (ϕ,π). Then we write A∈ Sect(ϕ,X) andwe letωA := min{ϕ ∈ [0,π)| A∈ Sect(ϕ,X)}.An operatorA suchthat
M (A) := sup{λ(λ + A)−1L(X)| λ ∈ (0, ∞)} < ∞ (2.6) is sectorialofangleϕ= π− arcsin(1/M(A)).
ForasectorialoperatorA onaBanachspaceX onehasN (A)∩Ran(A) = {0} and,if
X isreflexive,X = N (A)⊕ Ran(A).If−A generatesaC0-semigroup(T (t))t≥0⊆ L(X)
then T (t)x = x for all x ∈ N(A) and t ≥ 0. Moreover, the restriction of (T (t))t≥0 to Ran(A) is generated by the partof A inRan(A), which is injective. Hencefor the purposes ofstabilitytheoryitisnaturaltoassumethatA isinjective,andwewilldoso frequently.
For the definition and various properties of fractional powers of sectorial operators we refer to [23,44]. We shall use inparticular that,for ϕ∈ [0,π), A ∈ Sect(ϕ,X) and α,β,η∈ (0,∞),onehas Aα(η + A)−α−β = 1 2πi ∂Sθ zα (η + z)α+βR(z, A)dz. (2.7) Here∂SθisthepositivelyorientedboundaryofSθforθ∈ (ϕ,π).NotethatAαisinjective forA injective,and ifA is invertiblethenonemayletα = 0 in(2.7).
ForA asectorialoperatorandα,β∈ [0,∞) wesetΦα
β(A):= Aα(1+ A)−α−β∈ L(X). Wewill frequentlyusethatΦα0(A)= (A(1+ A)−1)α andthat
Φα1 β1(A)Φ α2 β2(A) = Φ α1+α2 β1+β2(A) (2.8)
forα1,α2,β1,β2∈ [0,∞),by[23,Proposition3.1.1].LetXβα:= Ran(Φαβ(A)),Xα:= X0α
and Xβ:= Xβ0.IfA isinjectivethenXβαisaBanachspace withthenorm
xXα
β :=xX+Φ
α
Itfollowsfrom[7,Proposition3.10(i)] (therestrictionα,β ∈ [0,1] isnotneededhere)that
Xα
β = ran(Aα)∩ D(Aβ) withequivalenceofnorms.Finally,notethatΦαβ(A): X→ Xβα isanisomorphism.Moreprecisely, thereexistsaconstantC≥ 0 such that
T L(Xα β,X)≤ T Φ α β(A)L(X)≤ CT L(Xα β,X) (T ∈ L(X α β, X)). (2.9) 3. Resolventestimates andmultipliers
Inthis sectionweprovesomestatementsonFouriermultipliersandresolventswhich willbe usedinlatersections.
3.1. Resolvents andFouriermultipliers
Throughout this subsection −A is the generator of a C0-semigroup (T (t))t≥0 on a
BanachspaceX.
Forthereader’sconvenienceweincludeaproofofthefollowingstandardlemma. Lemma 3.1.Let n ∈ N0, x ∈ X and ξ ∈ R. Suppose that −iξ ∈ ρ(A) and that [t →
tnT (t)x]∈ L1([0,∞);X). Then F[t → tn T (t)x](ξ) = n!(iξ + A)−n−1x, (3.1) F ∞ 0 tnT (t)g(· − t)x dt(ξ) =g(ξ)n!(iξ + A)−n−1x (g∈ L1(R)). (3.2)
Proof. It suffices to prove (3.1), as (3.2) follows from (3.1) by standard properties of convolutions.Sinceλ(λ+ A)−1x→ x asλ→ ∞,bythedominatedconvergencetheorem we mayadditionally assumethatx∈ D(A) and that [t → tnT (t)Ax]∈ L1([0,∞);X).
Also,[45,Lemma3.1.9] impliesthat[t → T (t)x]∈ C0([0,∞);X).Nowthefundamental
theoremofcalculusyields
(iξ + A) ∞ 0 e−iξtT (t)x dt = − e−iξtT (t)x ∞ 0 = x.
Hence 0∞e−iξtT (t)xdt= (iξ + A)−1x and
∞ 0 e−iξttnT (t)x dt = 1 (−i)n dn dξn ∞ 0 e−iξtT (t)x dt = n!(iξ + A)−n−1x. 2
Proposition 3.2. Let Y be a Banach space that is continuously embedded in X and let n ∈ N. Suppose that iR\ {0}⊆ ρ(A) and that there existψ ∈ L∞(R), p∈ [1,∞) and q∈ [1,∞] such that forj∈ {n− 1,n}∩ N onehas
mj1(·) := ψ(·)R(i·, A)j∈ M1,∞(R; L(Y, X)),
mj2(·) := (1 − ψ(·))R(i·, A)j ∈ Mp,q(R; L(Y, X)).
ThenTR(i·,A)n: Lp(R;Y )∩ L1(R;Y )→ L∞(R;X) isboundedandTR(i·,A)n≤ 2MCn,
where M = sup{T (t)L(X)| t∈ [0,2]}, Cn= n j=n−1 mj 1M1,∞(R;L(Y,X))+m j 2Mp,q(R;L(Y,X)) forn> 1,and
C1=m11M1,∞(R;L(Y,X))+m12Mp,q(R;L(Y,X))+IYL(Y,X).
Proof. Let K ∈ N,f1,. . . ,fK ∈ ˙S(R) andx1,. . . ,xK ∈ Y ,and set f :=Kk=1fk⊗ xk. Then Tmn 1(f )∈ Cb(R;X) and sup t∈RTm n 1(f )(t)X≤ m n 1M1,∞(R;L(Y,X))fL1(R;Y ). (3.3) Also, Tmn 2(f )Lq(R;X)≤ m n 2Mp,q(R;L(Y,X))fLp(R;Y ).
Thelatterinequalityimpliesthatforeachl∈ Z thereexists at∈ [l,l + 1] suchthat
Tmn
2(f )(t)X ≤ 2m n
2Mp,q(R;L(Y,X))fLp(R;Y ). (3.4)
Fixanl∈ Z and lett∈ [l,l + 1] besuchthat(3.4) holds.Then (3.3) implies
TR(i·,A)n(f )(t)X ≤ 2(mn1M1,∞+mn2Mp,q)fL1(R;Y )∩Lp(R;Y ). (3.5)
Letτ ∈ [0,2] andnote that
eiξτT (τ )R(iξ, A)x = R(iξ, A)x +
τ
0
eiξrT (r)x dr
T (τ )TR(i·,A)n(f )(t) =
1 2π
R
eiξ(t−τ)eiξτT (τ )R(iξ, A)nf (ξ) dξ
= 1 2π R eiξ(t−τ)R(iξ, A)nf (ξ) dξ + 1 2π R τ 0
eiξ(t−τ)eiξrT (r)R(iξ, A)n−1f (ξ) drdξ
= TR(i·,A)n(f )(t− τ) +
τ
0
T (r)TR(i·,A)n−1(f )(t− τ + r) dr.
Now(3.5) andHölder’sinequalityyield
TR(i·,A)n(f )(t− τ)X ≤ MTR(i·,A)n(f )(t)X+ τ 0 TR(i·,A)n−1(f )(t− τ + r)Xdr ≤ 2M(mn 1M1,∞+m n 2Mp,q)fL1(R;Y )∩Lp(R;Y ) + M (τTmn−1 1 (f )L∞(R;X)+ τ 1/qT mn2−1(f )L q(R;X)) ≤ 2M n j=n−1 mj 1M1,∞+m j 2Mp,q fLp(R;Y )∩L1(R;Y )
forn> 1.Forn= 1 thecomputationissimilar, butonecandirectlyestimate τ
0
f(t − τ − r)Xdr≤ IYL(Y,X)fL1(R;Y ).
Thisconcludestheproof, sinceτ ∈ [0,2] andl∈ Z arearbitraryandsinceS(R)˙ ⊗ Y ⊆
Lp(R;Y )∩ L1(R;Y ) isdense. 2
Remark3.3.Whenapplying Proposition3.2 wewill consider ψ with compactsupport. Thenonemayassumethatmj1∈ Mu,v(R;L(Y,X)) forgeneralu∈ [1,∞) andv∈ [1,∞].
Forχ∈ Cc∞(R) suchthatχ≡ 1 onsupp(ψ) onehasmj1= χmj1∈ Mu,∞(R;L(Y,X)) by Young’sinequality.The sameproof now showsthatTR(i·,A)n : Lu(R;Y )∩ Lp(R;Y )→
L∞(R;X) is bounded,with TR(i·,A)n ≤ 2M n j=n−1 mj 1Mu,v(R;L(Y,X))+m j 2Mp,q(R;L(Y,X))
for n> 1, and similarly forn = 1.However,Young’s inequalityalso showsthat mj1 =
χmj1χ∈ M1,∞(R;L(Y,X)), so thatthese assumptions are nomoregeneral thanthose
inProposition3.2.
3.2. Resolventestimates
We now present two propositions on resolvent growth. The assertions on uniform boundedness haveforthemostpartbeenobtainedbydifferentmethodsin[65,Lemma 3.3],[28,Lemma1.1],[37, Lemma3.2] and[7,Theorem 5.5].Theproofbelowallowsus to alsodeducethecorresponding statementsonR-boundednessdirectly.NotethatifA
satisfies (3.6) with α∈ (0,1) thenonemayin factletα = 0,by elementaryproperties of resolvents.
Proposition3.4. Letα∈ {0}∪[1,∞),β∈ [0,∞) andβ0∈ [0,1],andletA beaninjective
sectorial operator on aBanach spaceX. Letϕ∈ (0,π2] andΩ:=C+\ (Sϕ∪ {0}), and
suppose that −Ω⊆ ρ(A).Thenthefollowingstatements hold:
(1) The collection
{λα(λ + A)−1| λ ∈ Ω, |λ| ≤ 1} ⊆ L(X) (3.6)
isuniformly boundedifandonly if
{(λ + A)−1| λ ∈ Ω, |λ| ≤ 1} ⊆ L(Xα, X) (3.7)
isuniformlybounded.Moreover,(3.6) isR-boundedifandonlyif(3.7) isR-bounded.
(2) The collection
{λ−β(λ + A)−1| λ ∈ Ω, |λ| ≥ 1} ⊆ L(X) (3.8)
isuniformly boundedifandonly if
{λβ0(λ + A)−1| λ ∈ Ω, |λ| ≥ 1} ⊆ L(X
β+β0, X) (3.9)
isuniformlybounded.Moreover,(3.8) isR-boundedifandonlyif(3.9) isR-bounded.
(3) The collection (1− λ)β0(λ + A)−1Aα(1 + A)−α−β−β0− (−λ) α (1− λ)α+β(λ + A) −1λ ∈ Ω isR-boundedin L(X).
Proof. Fixθ∈ (max(ωA,π− ϕ),π) andletΓ:={reiθ| r ∈ [0,∞)}∪ {re−iθ | r ∈ [0,∞)} be orientedfrom∞eiθ to ∞e−iθ.
For(1) firstnotethat,bytheresolventidentity, (λ + A)−1A(1 + A)−1 = (1 + A)−1− λ(λ + A)−1(1 + A)−1 = (1 + A)−1− λ 1 + λ(λ + A) −1− λ 1 + λ(1 + A) −1 = 1 1 + λ(1 + A) −1− λ 1 + λ(λ + A) −1
forallλ∈ Ω.Now(2.2) and(2.9) yield(1) forα = 1.
Letα > 1.Then
(λ + A)−1Aα(1 + A)−α= (λ + A)−1(1 + A)Aα(1 + A)−α−1
= Aα(1 + A)−α−1+ (1− λ)(λ + A)−1Aα(1 + A)−α−1 (3.10) for allλ∈ Ω. Since the singleton {Aα(1+ A)−α−1}⊆ L(X) is R-bounded, by(2.9) it sufficestoshowthat(3.6) isuniformlybounded(or R-bounded)ifandonlyif
{(1 − λ)(λ + A)−1Aα(1 + A)−α−1| λ ∈ Ω, |λ| ≤ 1} ⊆ L(X) (3.11) isuniformlybounded (orR-bounded).Theresolventidentity and(2.7) yield
(λ + A)−1Aα(1 + A)−α−1= 1 2πi Γ zα (1 + z)α+1(λ + A)−1R(z, A) dz = 1 2πi Γ zα (1 + z)α+1(z + λ)dz(λ + A) −1 + 1 2πi Γ zα (1 + z)α+1(z + λ)R(z, A) dz forλ∈ Ω.Hence,using(A.1) ofLemma5.9,
(1− λ)(λ + A)−1Aα(1 + A)−α−1= (−λ) α (1− λ)α(λ + A) −1+ Sλ, (3.12) where Sλ:= 1 2πi Γ zα (1 + z)α+1 1− λ z + λR(z, A) dz.
Nowfixε∈ (0,min(α− 1,1)].Then z → (1+z)zε2εR(z,A) isintegrableonΓ,and sup
|z|α−ε
|1 + z|α+1−2ε|1 − λ||z + λ|λ ∈ Ω, z ∈ Γ
by (A.2) in Lemma5.9. Hence [33, Corollary 2.17] implies that{Sλ | λ ∈ Ω} ⊆ L(X) is R-bounded. Now (3.12) showsthatthe uniform boundedness(or R-boundedness) of (3.6) and (3.11) areequivalent,thereby proving(1).
For(2) wemaysupposethatβ + β0> 0.Then(2.7),appliedtotheinvertiblesectorial
operator 12+ A,andtheresolventidentity implythat
(λ + A)−1(1 + A)−β−β0= 1 2πi Γ 1 (12+ z)β+β0(λ + A) −1R(z,1 2+ A) dz = 1 2πi Γ 1 (12+ z)β+β0(z + λ−1 2) dz(λ + A)−1 + 1 2πi Γ 1 (1 2+ z)β+β0(z + λ− 1 2) R(z,1 2+ A) dz
forλ∈ Ω. Now(A.1) yields
(1− λ)β0(λ + A)−1(1 + A)−β−β0 = 1 (1− λ)β(λ + A)−1+ (1− λ) β0T λ, (3.13) where Tλ:= 1 2πi Γ 1 (12+ z)β+β0(z + λ−1 2) R(z,12+ A) dz.
Fixε∈ (0,β + β0).Thenz → (z +21)−εR(z,12+ A) isintegrableonΓ,and
sup 1 +|λ| |1 2+ z|β+β0−ε|z + λ − 1 2| λ ∈ Ω, z ∈ Γ<∞
by (A.2). Hence [33, Corollary 2.17] implies that{(1+|λ|)Tλ | λ∈ Ω} is R-bounded. Since |1− λ|β0 ≤ 1+|λ| for all λ∈ Ω,the proof of part (2) is completedusing (2.2), (3.13) and(2.9).
Finally,for(3) werestricttothecasewhere α > 1 andβ > 0.Theothercasesfollow in asimilar manner from theproofs of (1) and (2). The operatorfamilyin (3) can be writtenas Aα(1 + A)−α (1− λ)β0(λ + A)−1(1 + A)−β−β0− (1 − λ)−β(λ + A)−1 + (1− λ)−β (λ + A)−1Aα(1 + A)−α− (−λ) α (1− λ)α(λ + A) −1 =: Aα(1 + A)−αVλ1+ (1− λ)−βVλ2.
Using standardalgebraic propertiesof R-boundedness(see [30, Proposition 8.1.19]),it sufficestoprovethat{Vλi| λ∈ Ω}⊆ L(X) isR-boundedfori∈ {1,2}.Theproofof (2), andinparticular(3.13),showsthat
R({Vλ1| λ ∈ Ω}) = R({(1 − λ)β0T
λ| λ ∈ Ω}) < ∞. Forthe other termnote that,by (3.10) and (3.12), V2
λ = Aα(1+ A)−α−1+ Sλ. Hence theproofof(1) yields
R({Vλ2| λ ∈ Ω}) ≤ Aα(1 + A)−α−1L(X)+ R({Sλ| λ ∈ Ω}) < ∞. 2 Corollary3.5.Letα∈ [0,∞) andα0∈ [0,α]. LetA beaninjective sectorialoperatoron
aBanachspaceX suchthat iR\ {0}⊆ ρ(A) and
sup{λα(λ + A)−1L(X) | λ ∈ iR \ {0}, |λ| ≤ 1} < ∞.
Then
sup{λα−α0(λ + A)−1
L(Xα0,X)| λ ∈ iR \ {0}, |λ| ≤ 1} < ∞.
Proof. Firstnotethat0∈ ρ(A) forα < 1,byelementarypropertiesofresolvents.Hence, byProposition3.4(1) itsufficestoconsider α≥ 1 and α0∈ (0,α).By[23,Propositions
2.1.1.f and 3.1.9], A(1+ A)−1 is a sectorial operator and Aα0(1+ A)−α0 = (A(1+
A)−1)α0.Now themomentinequality[23,Proposition6.6.4] andanotherapplicationof [23,Proposition3.1.9] yield λα−α0(λ + A)−1Aα0(1 + A)−α0x X =|λ|α−α0(A(1 + A)−1)α0(λ + A)−1x X |λ|α−α0(λ + A)−1(A(1 + A)−1)αxα0/α X (λ + A)−1x (α−α0)/α X ≤ (λ + A)−1Aα(1 + A)−αα0/α L(X)λα(λ + A)−1(αL(X)−α0)/αxX
forallλ∈ iR\ {0} andx∈ X.Proposition3.4(1) and(2.9) concludetheproof. 2 4. Polynomialstability
InthissectionwestudypolynomialstabilityforsemigroupsusingFouriermultipliers. We first obtain some results valid on general Banach spaces. Then we establish the connection between polynomial stability and Fourier multipliers, and we use this link to deduce polynomial stability results undergeometric assumptions onthe underlying space.Wealsostudythenecessityofthespectralassumptionswhichwemake,compare ourtheoremswiththeliterature,andgiveexamplestoillustrateourresults.
Definition 4.1. Let α,β ∈ [0,∞). An operator A on a Banach space X has resolvent growth (α,β) ifthefollowing conditionshold:
(i) −A generatesaC0-semigroup(T (t))t≥0 onX;
(ii) C−\ {0}⊆ ρ(A),and λα (1 + λ)α+β(λ + A) −1 λ ∈ C +\ {0} ⊆ L(X) isuniformly bounded.
AnoperatorA hasR-resolventgrowth(α,β) ifA hasresolventgrowth (α,β) and
λ−β(λ + A)−1 λ ∈ C+,|λ| ≥ 1
⊆ L(X)
is R-bounded.
Note thatwe donotassumein(i) thatthesemigroupgenerated by−A isuniformly bounded. WewillimplicitlyusethroughoutthateachoperatorA withresolventgrowth (α,β),for α∈ [0,1) andβ ∈ [0,∞), is invertible andthus hasresolvent growth (0,β),
as followsfrom thefactthatR(λ,A)L(X) ≥ dist(λ,σ(A))−1 forallλ∈ ρ(A).
Recall thatweusetheconventionthat 00 =∞,forsimplicity ofnotation.
4.1. Resultson generalBanach spaces
The followinglemma isused to interpolatebetweendecayrates.Related resultscan be found in[4, Proposition3.1] and [7, Lemma4.2]. Recall thedefinition of thespace
Xβα,forα,β ≥ 0,fromSection2.3.
Lemma 4.2.LetA be aninjectivesectorial operatoron aBanachspaceX such that−A generates aC0-semigroup(T (t))t≥0 on X.Forj ∈ {1,2} letαj,βj ∈ [0,∞) besuchthat
α1 ≥ α2 and β1 ≥ β2, andlet fj : [0,∞)→ [0,∞) besuch that T (t)L(Xαj
βj,X)≤ fj(t)
forallt∈ [0,∞). Thenforeach θ∈ [0,1] there existsaCθ∈ [0,∞) suchthat
T (t)L(Xθα1+(1−θ)α2
θβ1+(1−θ)β2,X)≤ Cθ
(f1(t))θ(f2(t))1−θ (t∈ [0, ∞)). (4.1)
Moreover, suppose that f1(t) = Ct−μ for some C,μ ∈ [0,∞) and all t ∈ [1,∞). Then
foreach θ∈ [1,∞) there existsaCθ∈ [0,∞) such that
T (t)L(Xθα1
θβ1,X)≤ Cθ
Proof. Lett∈ [0,∞) andnote that,by(2.9) and(2.8), T (t)L(Xθα1+(1−θ)α2 θβ1+(1−θ)β2,X)≤ T (t)Φ θα1+(1−θ)α2 θβ1+(1−θ)β2(A)L(X) =T (t)Φθ(α1−α2) θ(β1−β2)(A)Φ α2 β2(A)L(X).
Letc:= α1− α2+ β1− β2. ThenΦ(β(α11−β−α22)/c)/c(A)= A(α1−α2)/c(1+ A)−1 issectorial, by
[7, Proposition3.10].Hence[23,Theorem 2.4.2] yields Φθ(α1−α2)
θ(β1−β2)(A) = A
θ(α1−α2)(1 + A)−θ(α1−α2+β1−β2)= (Φ(α1−α2)/c
(β1−β2)/c(A)) cθ. Themomentinequality[23, Proposition6.6.4] and[23,Theorem2.4.2] implythat
(Φ(α1−α2)/c (β1−β2)/c(A)) cθx X (Φ(α(β11−β−α22)/c)/c(A))cxθXx1X−θ =Φ α1−α2 β1−β2(A)x θ Xx1X−θ forx∈ D(Φα1−α2
β1−β2(A)).Combiningallthiswith(2.8) and(2.9) showsthat
T (t)L(Xθα1+(1−θ)α2 θβ1+(1−θ)β2,X)≤ T (t)Φ θ(α1−α2) θ(β1−β2)(A)Φ α2 β2(A)L(X) =(Φ(α1−α2)/c (β1−β2)/c(A)) cθT (t)Φα2 β2(A)L(X) Φα1−α2 β1−β2(A)T (t)Φ α2 β2(A) θ L(X)T (t)Φαβ22(A) 1−θ L(X) =T (t)Φα1 β1(A) θ L(X)T (t)Φαβ22(A) 1−θ L(X) T (t)θ L(Xα1 β1,X)T (t) 1−θ L(Xα2 β2,X)≤ (f1 (t))θ(f2(t))1−θ,
therebyproving(4.1).Asfor(4.2),letn∈ N.Then
T (t)L(Xnα1 nβ1,X)≤ T (t)Φ nα1 nβ1(A)L(X)≤ T ( t n)Φ α1 β1(A) n L(X) (f1(nt))n= Cnnμnt−μn,
whichimplies(4.2) for θ∈ N.Finally,applying (4.1) tointerpolate between(nα1,nβ1)
and((n+ 1)α1,(n+ 1)β1) yields(4.2) forallθ∈ [1,∞). 2
Thefollowing resultforC0-semigroupsongeneralBanachspaces extends[4,
Propo-sition3.4],where thecaseα = ρ= 0 wasconsidered.
Proposition4.3. Letα,β∈ [0,∞) andletA be aninjectivesectorialoperatorwith resol-vent growth(α,β) ona Banach spaceX. Let σ,τ ∈ [0,∞) be suchthat σ > α− 1 and τ > β + 1.Thenforeach ρ∈ [0,min(σ+1α − 1,τ−1β − 1)) thereexistsaCρ∈ [0,∞) such
that
T (t)L(Xσ
τ,X)≤ Cρt
Proof. By elementary calculationsthe proposition is equivalent to the following state-ment: foralls≥ 0 andδ,ε> 0 thereexistsaCs,δ,ε≥ 0 suchthat
T (t)L(Xμ
ν,X)≤ Cs,δ,εt
−s (t∈ [1, ∞)), (4.4) whereμ= max((s+ 1)α− 1+ δ,0) andν = (s+ 1)β + 1+ ε.Furthermore,byLemma4.2
itsufficesto prove(4.4) forn:= s∈ N0.
Letx∈ Xν+1μ andset y := Φμ
ν(A)x= A−μ(1+ A)μ+νx∈ D(A).Then
g(t) := 1 2πi −i∞ i∞ e−λt λ μ (1 + λ)μ+νR(λ, A)y dλ
is a well defined element of X for all t ≥ 0. One can check that g is continuously differentiable withg(t)=−Ag(t).Also,
g(0) = 1 2πi −i∞ i∞ λμ (1 + λ)μ+νR(λ, A)y dλ = A μ(1 + A)−μ−νy = x.
Here we havedeformed thepath of integration to the curve Γ ={reiθ | r ∈ [0,∞)}∪
{re−iθ | r ∈ [0,∞)} in(2.7),forθ∈ (ω
A,π),whichwemaydobytheassumptionsonA. Nowg(t)= T (t)x,byuniquenessoftheCauchyproblemassociatedwith−A.Integration byparts yields tnT (t)x = t n 2πi iR e−λt λ μ (1 + λ)μ+νR(λ, A)y dλ = (−1) n 2πi iR dn dλne−λt λμ (1 + λ)μ+νR(λ, A)y dλ = 1 2πi iR e−λtp(λ, A)y dλ.
Here p(λ,A) isafinite linearcombination oftermsoftheform
λμ−j
(1 + λ)μ+ν+(k−j)R(λ, A) n−k+1
for0≤ j ≤ k ≤ n,where weletj = 0 ifμ= 0.Then
tn T (t)xX ≤ 1 2π iR
withimplicitconstantsindependent oft andx.SinceXν+1μ isdenseinXνμ, theproof is concluded. 2
The following corollary of Proposition 4.3 and Lemma 4.2 takes into account the growthbehaviorof(T (t))t≥0 onX.ItalsoextendsProposition4.3byprovidingstability rates on Xσ
τ for σ ∈ [0,α− 1] andτ ∈ [0,β + 1]. The sameapproach was used in [4, Theorem3.5] for uniformlyboundedsemigroupsandα = 0.
Corollary4.4. Letα,β∈ [0,∞) andletA beaninjectivesectorialoperatorwithresolvent growth(α,β) ona BanachspaceX. Letσ,τ∈ [0,∞). Thenforeach ρ∈ [0,min(σα,τβ))
thereexistsaCρ∈ [0,∞) suchthat
T (t)L(Xσ
τ,X)≤ Cρmax(1,T (t)L(X))t
−ρ (t∈ [1, ∞)). (4.5) Proof. By elementary calculations it suffices to provethe following: for all s ≥ 0 and δ,ε> 0 thereexistsaconstantCs,δ,ε≥ 0 such that
T (t)L(Xμ
ν,X)≤ Cs,δ,εmax(1,T (t)L(X))t
−s (t∈ [1, ∞)), (4.6) where μ = sα + δ and ν = sβ + ε. Let ε > 0 and for θ ∈ (0,1) set s := s/θ, μ := max((s+ 1)α − 1+ε,0) andν := (s+ 1)β + 1+ε.Then,byLemma4.2and(4.4),
T (t)L(Xμθ νθ,X) T (t) 1−θ L(X)T (t)θL(Xμ ν,X) max(1, T (t)L(X) )t−s
forallt≥ 1.Next,notethatμθ = max(sα+θ(α−1+ε),0) andνθ = sβ+θ(β+1+ε).Now theproofisconcludedbylettingθ∈ (0,1) besuchthatμθ ≤ sα+ε andνθ ≤ sβ +ε. 2
4.2. Polynomial stabilityandFouriermultipliers
InthissubsectionwerelatepolynomialstabilityofasemigrouptoFouriermultiplier propertiesof theresolventof itsgenerator.
Inorder to state ourabstract resulton polynomialstability we introduce aclass of admissiblespaces.
Definition 4.5.Let −A be the generator of a C0-semigroup (T (t))t≥0 on a Banach space X, and let n ∈ N0. A Banach space Y which is continuously embedded in X
is(A,n)-admissible ifthefollowingconditionshold:
(i) thereexistsaconstantCT ∈ [0,∞) suchthatT (t)Y ⊆ Y and
T (t)YL(Y )≤ CTT (t)L(X) (t∈ [0, ∞));
(ii) there exists adensesubspace Y0 ⊆ Y such that[t → tnT (t)y]∈ L1([0,∞);X) for
Let α,β ∈ [0,∞) andletA be an injectivesectorial operatorwith resolvent growth (α,β).Then Y = Xτσ is(A,n)-admissible forall σ,τ ∈ [0,∞) and n∈ N0,by
Proposi-tion4.3.
The following theorem is our main result relating polynomial stability and Fourier multipliers. It follows from (4.10) and (4.12) below that one can obtain quantitative bounds ineachoftheimplicationsbetween(1) and(2).
Theorem 4.6 (Characterization of polynomial stability).Let −A be the generator of a C0-semigroup (T (t))t≥0 on aBanachspace X,and assumethat A has resolvent growth
(α,β) forsomeα,β ∈ [0,∞). Letn∈ N0 andletY be an(A,n)-admissiblespace. Then
thefollowingstatements are equivalent:
(1) sup t≥0t
nT (t)
L(Y,X)<∞;
(2) there existψ∈ Cc∞(R), p∈ [1,∞) and q∈ [p,∞] suchthat
ψ(·)R(i·, A)k∈ M1,∞(R; L(Y, X)), (1− ψ(·))R(i·, A)k∈ Mp,q(R; L(Y, X))
forallk∈ {n− 1,n,n+ 1}∩ N.
Moreover, if (1) or (2) holdsthen R(i·,A)k ∈ M
p,q(R;L(Y,X)) for: (i) n≥ 2, k∈ {1,. . . ,n− 1} and 1≤ p≤ q ≤ ∞;
(ii) k = n≥ 1 and 1≤ p< q≤ ∞;
(iii) k = n+ 1,p= 1 and q =∞.
Proof. (2)⇒ (1):Letω,Mω≥ 1 besuchthatT (t)L(X)≤ Mωet(ω−1)forallt≥ 0,and set
m(ξ) := n!(iξ + A)−n(IX+ ω(iξ + A)−1)∈ L(Y, X) (ξ∈ R \ {0}). Since (i· +A)−1 =−R(−i·,A),itfollowsfrom Proposition3.2that
Tm: Lp(R; Y ) ∩ L1(R; Y ) → L∞(R; X) is boundedwith
Tm ≤ 2Mn!(Cn+ ωCn+1). (4.7) Here M := supt∈[0,2]T (t)L(X), Ck is as in Proposition 3.2 for k ∈ N, and C0 :=
IYL(Y,X).NowletY0⊆ Y be asinDefinition4.5 andfixx∈ Y0. Lemma3.1yields
Setf (t):= e−ωtT (t)x for t≥ 0,andf ≡ 0 on(−∞,0).Then
f(t)Y ≤ e−ωtT (t)L(Y )xY ≤ CTe−ωtT (t)L(X)xY (t∈ [0, ∞)). (4.9) Hence f ∈ L1(R;Y )∩ L∞(R;Y ) and fLr(R;Y ) ≤ CTMωxY for all r ∈ [1,∞]. By
Lemma3.1,f (·)= (w + i· +A)−1x.Therefore, bytheresolventidentity,
m(ξ) f (ξ) = n!(iξ + A)−n−1x (ξ ∈ R \ {0}). Combining(4.8) and(4.9) with(4.7) yields
sup t≥0t nT (t)x X ≤ Tm fLp(R;Y )+|fL1(R;Y ) ≤ CxY, (4.10) where C = 4M n!CTMω(Cn+ ωCn+1).Therequired resultnowfollows sinceY0⊆ Y is
dense.
(1)⇒ (2):SetKn := supt≥0tnT (t)xX andletY0⊆ Y be asinDefinition4.5. Let
f ∈ ˙S(R)⊗ Y0and setSk(f )(s):= ∞
0 t
kT (t)f (s− t)dt fors∈ R andk∈ {0,1,. . . ,n}. Lemma3.1 yields
Sk(f ) = k!F−1((i· +A)−k−1f (·)) = k! T(i·+A)−k−1(f ). (4.11)
Now,forn≥ 2,k∈ {0,. . . ,n− 2} andr∈ [1,∞],
[t → tkT (t)]Lr([0,∞);L(Y,X))dt≤ M + Kn[t → t−2]Lr(1,∞)≤ M + Kn.
Similarly,forn≥ 1 andr∈ (1,∞],
[t → tn−1T (t)]Lr([0,∞);L(Y,X))≤ M +
Kn (r− 1)1/r.
By combining these estimates with (4.11) and with Young’s inequality for operator-valuedkernelsin[3,Proposition1.3.5] oneobtains,forp∈ [1,∞) andq∈ [p,∞],
R(i·, A)k Mp,q(R;L(Y,X)) ≤ M +Kn (k−1)! (n≥ 2, k ∈ {1, . . . , n − 1}), R(i·, A)n Mp,q(R;L(Y,X)) ≤ M +(r−1)−1/rK n (n−1)! (n≥ 1, p < q), R(i·, A)n+1 M1,∞(R;L(Y,X)) ≤Kn!n. (4.12)
Now(4.11) and(4.12) yieldstatements(i)-(iii) for(i·+A)−1,andbyreflectionthese state-mentshold forR(i·,A) aswell.Finally,for(2) letψ∈ Cc∞(R).ThenYoung’sinequality and (4.12) yield ψ(·)R(i·,A)k ∈ M
1,∞(R;L(Y,X)) for allk ∈ {1,. . . ,n+ 1},and one
obtains(4.12) for ψ(·)R(i·,A) with an additional multiplicativefactor F−1(ψ)L1(R). Similarly,(4.12) holdswithanadditionalmultiplicativefactorF−1(1− ψ)L1(R) upon replacingR(i·,A) by(1− ψ(·))R(i·,A). 2
The assumption inTheorem 4.6 thatA hasresolvent growth (α,β) forsome α,β ∈
[0,∞) isonly made to ensurethat TR(i·,A) is well-defined, and the specific choiceof α
andβ isirrelevanthere.InspectionoftheproofofTheorem4.6alsoshowsthatonecould assumein(2) thatforeachk∈ {n− 1,n,n+ 1}∩ N thereexistpk,qk ∈ [1,∞] suchthat
(1− ψ(·))R(i·, A)k ∈ Mpk,qk(R; L(Y, X)).
However, we willnot needthis generality inthe remainder. Aswas already mentioned inRemark3.3,theassumption
ψ(·)R(i·, A)k ∈ M1,∞(R; L(Y, X))
in(2) isthemostgeneral(Lp,Lq) Fouriermultiplierconditionforψ(·)R(i·,A)k. Remark 4.7.Thetheory of (Lp,Lp) Fourier multipliers alonecannot yield a character-ization of polynomialstability as inTheorem 4.6,and ingeneralitis necessary toalso considerthecasewherep< q incondition(2).Toseethis,considerauniformlybounded
C0-semigroup(T (t))t≥0 ⊆ L(X) withgenerator−A such thatC− ⊆ ρ(A) but A isnot
oftype(0,0).Letn= 0 andY = X.ThenR(i·,A)∈ M/ p,p(R;L(X)) foreachp∈ [1,∞)
since sup{R(iξ,A)L(X) | ξ ∈ R\ {0}}=∞. Nonetheless, (1) holdssince (T (t))t≥0 is uniformly bounded,and R(i·,A)∈ M1,∞(R;L(X)).Indeed,
F−1(R(i·, A) f (·))(t) = ∞
0
T (t− s)f(s)ds (t∈ R)
defines anelementofL∞(R;X) for eachf ∈ S(R;X).
AvariationoftheproofofTheorem4.6yieldsthefollowingresult,whichwillalsobe usedinSection5.Inparticular,itprovidesasimpleconditionforpowersoftheresolvent to beFouriermultipliers.
Proposition 4.8. Let −A be the generator of a C0-semigroup (T (t))t≥0 on a Banach
space X, and suppose that C−\ {0} ⊆ ρ(A). Let q ∈ [1,∞), n ∈ N0 and let Y be an
(A,n)-admissible space.Thenthefollowingstatements are equivalent:
(1) there exists a constant C ∈ [0,∞) such that [t → tnT (t)x]∈ Lq([0,∞);X) for all
x∈ Y ,and
[t → tnT (t)x]
Lq([0,∞);X)≤ CxY (x∈ Y );
(3) thereexistψ∈ Cc∞(R) andp∈ [1,q] such that
ψ(·)R(i·, A)k∈ M1,q(R; L(Y, X)) and (1 − ψ(·))R(i·, A)k ∈ Mp,q(R; L(Y, X))
fork∈ {n,n+ 1}∩ N.
Proof. (2)⇒ (3) istrivial.For(3)⇒ (1) oneproceedsinanalmostidenticalmannerasin theproofofTheorem4.6,exceptthatnowitisnotnecessarytoappealtoProposition3.2. (1) ⇒ (2): LetY0 ⊆ Y be as inDefinition4.5. Then [t → tkT (t)x]∈ Lq(R+;X) for
allk∈ {0,. . . ,n} andx∈ Y0.Hence,forf ∈ L1(R)⊗ Y0,Minkowski’sinequalityyields
R R (t− s)nT (t− s)f(s)ds q dt 1/q ≤ R ∞ s (t − s)nT (t− s)f(s)qdt1/qds≤ C R f(s)ds.
NowtheproofisconcludedusingLemma3.1. 2
4.3. Results underFouriertype assumptions
Here we apply Theorem 4.6 to obtain polynomial stability results under assump-tionsontheFouriertypeoftheunderlyingspace.Thefollowingtheoremcoincides with Proposition4.3 forp= 1. Inthecasewhere α = 0 it wasalready stated in[4] that an improvement of Proposition 4.3 might be possible using ideas from [46, §4.2], but no detailsaregiven there.
Theorem4.9. Letα,β∈ [0,∞) andletA beaninjectivesectorialoperatorwithresolvent growth(α,β) on aBanach space X withFourier type p∈ [1,2]. Letr∈ [1,∞] be such that 1
r =
1
p −
1
p,and letσ,τ ∈ [0,∞) be suchthat σ > α− 1 and τ > β +
1
r. Thenfor
each ρ∈ [0,min(σ+1
α − 1,τ−r
−1
β − 1)) thereexists aCρ∈ [0,∞) suchthat
T (t)L(Xσ
τ,X)≤ Cρt
−ρ (t∈ [1, ∞)). (4.13)
Ifp= 2 then(4.13) alsoholdsforτ ≥ β andρ∈ [0,∞) withρ<σ+1α − 1 andρ≤βτ− 1.
Proof. We prove the following equivalent statement: for all s ≥ 0 and δ,ε > 0 there
existsaconstantCs,δ,ε≥ 0 suchthat
T (t)L(Xμ
ν,X)≤ Cs,δ,εt
−s (t∈ [1, ∞)), (4.14) whereμ= max((s+ 1)α− 1+ δ,0),ν = (s+ 1)β +1r+ ε forp∈ [1,2),andν = (s+ 1)β forp= 2.ByLemma4.2 itsuffices to consider n:= s∈ N0,and thecasewhere p= 1
