Lp-Lq multipliers on locally compact groups

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

L

-L

multipliers on locally compact groups

Rauan Akylzhanov^a, Michael Ruzhansky^b,a,∗

aSchoolofMathematicalSciences,QueenMaryUniversityofLondon, United Kingdom

bDepartmentofMathematics:Analysis,LogicandDiscreteMathematics, Ghent University,Belgium

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received14June2018 Accepted19September2019 Availableonline25September2019 CommunicatedbyStefaanVaes

MSC:

primary43A85,43A15 secondary35S05

Keywords:

Locallycompactgroups Fouriermultipliers Spectralmultipliers Hörmandertheorem

In this paper we discuss the L^p-L^q boundedness of both spectralandFourier multipliers on general locally compact separableunimodulargroups Gfor therange 1< p ≤2 ≤ q <∞.AsaconsequenceoftheestablishedFouriermultiplier theorem we also derive a spectral multiplier theorem on general locally compact separable unimodular groups. We thenapplyittoobtainembeddingtheoremsaswellastime- asymptotics for the L^p-L^q norms of the heat kernels for general positive unbounded invariant operators on G. We illustratetheobtainedresultsforsub-Laplaciansoncompact Lie groups and on the Heisenberg group, as well as for higherorderoperators.Weshow that ourresults implythe known results for L^p-L^q multipliers such as Hörmander’s Fourier multiplier theorem on Rⁿ or known results for FouriermultipliersoncompactLiegroups.Thenewapproach developedin thispaper relies on advancing theanalysis in thegroup vonNeumann algebra anditsapplication to the derivationofthedesiredmultipliertheorems.

©2019TheAuthors.PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

✩ The second author was supported by the FWO Odysseus grant and by the Leverhulme Grant RPG-2017-151.TheauthorswerealsosupportedbyEPSRCGrantEP/R003025/1.

* Correspondingauthor.

E-mailaddresses:r.akylzhanov@qmul.ac.uk(R. Akylzhanov),Michael.Ruzhansky@ugent.be (M. Ruzhansky).

https://doi.org/10.1016/j.jfa.2019.108324

0022-1236/©2019TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

Contents

1. Introduction . . . . 2

1.1. Hörmander’stheoremonlocallycompactgroups . . . . 3

1.2. Spectralmultipliersonlocallycompactgroups . . . . 6

2. Notationandpreliminaries . . . . 8

2.1. Fouriermultipliersonlocallycompactgroups . . . 16

3. PaleyandHausdorff-Young-Paleyinequalities . . . 17

4. Nikolskiiinequalityonlocallycompactgroups . . . 22

5. Hörmander’smultipliertheoremonlocallycompactgroups . . . 25

5.1. Thecaseoflocallycompactabeliangroups . . . 28

5.2. ThecaseofcompactLiegroups . . . 29

5.3. Thecaseofq=∞ . . . 31

5.4. Thecaseofnon-invariantoperators . . . 32

6. Spectralmultipliersonlocallycompactgroups . . . 35

7. Heatkernelsandembeddingtheorems . . . 37

7.1. Sub-RiemannianstructuresoncompactLiegroups . . . 40

7.2. Sub-LaplacianontheHeisenberggroup . . . 41

7.3. RocklandoperatorsontheHeisenberggroup . . . 43

7.4. RocklandoperatorongradedLiegroups . . . 44

References . . . 46

1. Introduction

The aim of this paper is to give sufficient conditions for the L^p-L^q boundedness of Fourier and spectral multipliers onlocally compact separable unimodulargroups. It is known thatin this casewe must havep≤q and two classicalresults are available on Rⁿ,namely,Hörmander’smultipliertheorem[37] for1< p≤2≤q <∞,andLizorkin’s multiplier theorem[41] for1< p≤q <∞.There is aphilosophical difference between these results:Hörmander’s theorem doesnot requireany regularity ofthe symbol and appliestopandqseparatedby2,whileLizorkintheoremappliesalsofor1< p≤q≤2 and 2≤p≤q <∞butimposes certainregularityconditionsonthesymbol.

InthispaperweaimatprovingtheHörmandertypetheoremexpressingconditionsin termsofthesharpdecaypropertyofthespectralinformationassociatedtotheoperator, on general locally compact separable unimodular groups based on developing a new approach relying on the analysis in the noncommutative Lorentz spaces on the group vonNeumannalgebra.Thissuggestedapproachseemsveryeffective,implyingasspecial casesknownresultsexpressedintermsofsymbols,insettingswhenthesymboliccalculus is available. The obtainedresults are for generalFourier multipliers, inparticular also implying new results for spectral multipliers. Lizorkin type theorem in the setting of locally compactgroupsrequiresarathersubstantialmodificationoftechniquesandwill appearelsewhere.

The class of groups covered by our analysis is very wide. In particular, it contains abelian,compact,nilpotentgroups,exponential,realalgebraicorsemi-simpleLiegroups, solvable groups(notallofwhicharetypeI,butwedonotneedtoassumethegroupto be oftypeIorII),andmanyothers.Asfarasweareawareourresultsarenewinallof these non-Euclideansettings.

Inthispaper wefocus ontheL^p-L^q multipliersasopposed tothe L^p-multipliersfor which theorems of Mihlin-Hörmander or Marcinkiewicz types provide resultsfor both Fourierandspectralmultipliersinsomesettings,basedontheregularityofthemultiplier.

L^p-multipliershavebeenintensivelystudiedondifferentkindsofgroups.However,most oftheresultshavebeenobtainedforL^pspectralmultipliers,forwhichawealthofresults isavailable: e.g.[44,45] onHeisenberg typegroups,[13] on solvable Lie groups,[43] on nilpotent and stratified groups, to mention only very very few. L^p Fourier multiplies havealsobeen studiedbutto alesser extentdue tolack ofsymbolic calculusthatwas notavailable until recently,e.g.Coifman andWeiss [14,15] on SU(2),[58,59] and then [26] oncompact Liegroups,or[12,27] ongradedLie groups.A characteristicfeatureof theL^p-L^q multipliersisthatlessregularity ofthesymbolisrequired.Therefore,inthis paper we concentrate on the L^p-L^q multiplier theorems aiming at obtaining unifying resultsforgenerallocally compactgroups.Wegiveseveralapplications oftheobtained resultstoquestionssuchas embeddingtheorems anddispersiveestimates forevolution PDEs.

Theapproachto the L^p-Fouriermultipliersis different from thetechniqueproposed in this paper allowing us to avoid making the assumption that the group is compact or nilpotent. In this paper we are interested in both Fourier multipliers and spectral multipliers,forthelattersomeL^p-L^qresultsbeingavailableinsomespecialsettings,see e.g.[17],andalso[18],aswellas[1] forthecaseofSU(2),andforthediscussionofsome relationsbetweenthoseinthegroupsettingwecanreferto [59] andreferencestherein.

In the context of harmonic analysis, group von Neumann algebras are used in two differentways.Inthefirstapproach[32],thegroupGcanbeunderstoodasthe‘frequency domain’and itsgroup von Neumannalgebra VNR(G) can be thought of as the ‘space domain’.Inthispaper,weviewGas‘spacedomain’anditsgroupvonNeumannalgebra VNR(G) asthe‘frequencydomain’.

Bythe combinatorialmethod it is possible to establish theL^p-L^q estimates for the Poisson-type semigroup Pt on discrete groups G [38]. Finally we note that multiplier estimates on noncommutative groups are in general considerably more delicate than those in the commutative case, recall e.g. the asymmetry problem and its resolution in[21]. A linkbetween Fourier multipliersand Lorentzspaces on groupvon Neumann algebrashasbeenoutlined in[4].

Wecan alsomention afew related works:[19] for generalC^∗-algebras presentation, vonNeumann’soriginalworks[68,69],and[23] forrecentworkondirectintegraldecom- positionsforvonNeumannalgebras.

Wenowproceedto makingamorespecificdescriptionoftheconsideredproblems.

1.1. Hörmander’s theoremon locallycompactgroups

Toputthisincontext,werecallthatin[37, Theorem1.11],LarsHörmandershowed thatfor1< p≤2≤q <∞,ifthesymbolσ_A:Rⁿ→C ofaFouriermultiplierAacting ontheSchwartzspaceS(Rⁿ) satisfiesthecondition

sup

s>0

s

⎛

⎜⎝

ξ∈Rⁿ:|σA(ξ)|≥s

dξ

⎞

⎟⎠

1 p−¹q

<+∞, (1.1)

then AisaboundedoperatorfromL^p(Rⁿ) toL^q(Rⁿ).Here,asusual, theFouriermul- tiplier AonRⁿ actsbymultiplicationontheFouriertransformside,i.e.

Af(ξ) =σA(ξ)f(ξ), ξ∈Rⁿ, f ∈ S(Rⁿ). (1.2) Moreover, itthenfollowsthat

AL^p(Rⁿ)→L^q(Rⁿ)sup

s>0

s

⎛

⎜⎜

⎜⎝

ξ∈Rⁿ

|σA(ξ)|≥s

dξ

⎞

⎟⎟

⎟⎠

1 p−¹_q

, 1< p≤2≤q <+∞. (1.3)

TheL^p-L^q boundednessofFouriermultipliershasbeenalsorecentlyinvestigatedinthe contextofcompactLiegroups,andwenowbrieflyrecalltheresult.LetGbeacompact Lie group and G its unitary dual. For π ∈ G, we write dπ for the dimension of the (unitaryirreducible)representation π.In[2] theauthorshaveshownthat,foraFourier multiplierA actingvia

Af(π) =σA(π)f(π) byitsglobalsymbol σA(π)∈C^dπ×dπ wehave

AL^p(G)→L^q(G)sup

s>0

s

⎛

⎜⎜

⎝

π∈G σA(π)op≥s

d²_π

⎞

⎟⎟

⎠

1 p−¹_q

, 1< p≤2≤q≤ ∞. (1.4)

Here forπ∈G, theFouriercoefficientsaredefinedas f(π) =

G

f(x)π(x)^∗dx,

and σA(π)op is the operatornorm of σA(π) as the linear transformation of the rep- resentation space of π ∈ G identified with C^dπ. For a general development of global symbols and the corresponding global quantization of pseudo-differential operators on compact Liegroupswecanreferto [56,57].

One ofthe resultsofthis paper generalises both multipliertheorems (1.3) and (1.4) to thesettingofgenerallocallycompactseparableunimodulargroupsG.

ByaleftFouriermultiplierinthesettingofgenerallocallycompactunimodulargroups wewill meanleft invariantoperatorsthat aremeasurablewith respecttotheright group vonNeumannalgebra VNR(G),seeSection2.1foradiscussion.

Thus,inTheorem 5.1weprovethefollowing inequality

AfL^p(G)→L^q(G)sup

s>0

s

⎡

⎢⎣

t∈R+: μt(A)≥s

dt

⎤

⎥⎦

1 p−¹_q

fL^p(G), 1< p≤2≤q <+∞, (1.5) whereμt(A) arethet-thgeneralisedsingularvaluesofA,seeDefinition2.8fortheprecise definitionandproperties(following[67]).Anextensionof(1.5) to q=∞will beshown inTheorem 5.9.

Remark 1.1.The measurability assumption on A implies that the domain Dom(A) ⊂ L²(G) isdenseinL²(G).WithoutlossofgeneralityweassumethatDom(A) isdensein L^p(G) forevery1< p≤ ∞.

The proof of inequality (1.5) is based on a version of the Hausdorff-Young-Paley inequalityonlocally compactseparablegroupsthatweestablishforthispurpose.

The key idea behind the extension (1.5) is that Hörmander’s theorem (1.3) can be reformulatedas

AL^p(Rⁿ)→L^q(Rⁿ)sup

s>0

s

⎛

⎜⎜

⎜⎝

ξ∈Rⁿ

|σA(ξ)|≥s

dξ

⎞

⎟⎟

⎟⎠

1 p−¹q

σAL^r,∞(Rⁿ) AL^r,∞(VN(Rⁿ)),

(1.6) where ¹_r = ¹_p − ¹_q, σ_AL^r,∞(Rⁿ) is the Lorentz space norm of the symbol σ_A, and AL^r,∞(VN(Rⁿ)) is thenorm of the operatorA inthe Lorentzspace on thegroup von NeumannalgebraVN(Rⁿ) ofRⁿ.Inturn,ourestimate(1.5) isequivalenttotheestimate

AL^p(G)→L^q(G)AL^r,∞(VNR(G)) sup

s>0

s

⎡

⎢⎣

t∈R+: μt(A)≥s

dt

⎤

⎥⎦

1 r

, (1.7)

whereA isactingontheSchwartz-BruhatspaceS(G) andAL^r,∞(VNR(G)) isthenon- commutative Lorentz space norm on the right group von Neumann algebra VN_R(G) of G. Thus, the Lorentz spaces become a key point for the extension of Hörmander’s theoremto thesetting ofgenerallocally compact(unimodular)groups.

The assumption ofunimodularity ofGensures theexistenceof thePlancherel trace onthegroupvonNeumannalgebra VNR(G) andthuscanbeviewedasnaturalallowing onetousebasicresultsofFourieranalysisovervonNeumannalgebras,suchas,forexam- ple, Plancherel formula(see Segal [61]).Otherwise, thenoncommutative Lorenzspaces cannot be constructedas subsetsofτ-measurable operators.Nevertheless,theunimod- ularity assumption maybe inprinciple avoided,see e.g.[22], butthe expositionwould involvetheTomita-TakesakimodulartheoryandtheHaagerupreductiontechnique.For amoredetaileddiscussionofpseudo-differentialoperatorsinthegeneralsettingoflocally compact groups(possiblynon-unimodular)wereferto[42].

1.2. Spectral multiplierson locallycompactgroups

LetusillustratetheuseoftheFouriermultipliertheorem(1.5) intheimportantcase of spectral multipliers on locally compact groups. Later, in Theorem 6.1 we will give a spectral multiplier result on general semifinite von Neumann algebras, however, we nowformulateitsspecialcaseforthecaseofgroupvonNeumannalgebrasassociatedto locally compactgroups.

Interestingly, this resultasserts thatthe L^p-L^q norms of spectral multipliersϕ(|L|) depend essentially only on the rate of growth of traces of spectral projections of the operator|L|:

Theorem 1.2.Let G be a locally compact separable unimodular group and let L be a left Fourier multiplier on G. Assume that ϕ is a monotonically decreasing continuous function on[0,+∞)suchthat

ϕ(0) = 1,

u→+∞lim ϕ(u) = 0.

Then wehavetheinequality ϕ(|L|)L^p(G)→L^q(G)sup

u>0

ϕ(u)

τ(E_(0,u)(|L|))¹_p−¹q, 1< p≤2≤q <∞. (1.8) Here E_(0,u)(|L|) are the spectral projections associated to the operator |L| to the interval (0,u), seeSection 2forprecise definitions, andτ is thecanonical traceon the right groupvon NeumannalgebraVN_R(G),seealsoSection2foradiscussion.

Also wenote thatmoregeneralstatements,weakening theaboveassumptionson ϕ, are possible,seeCorollary 6.3.

The estimate (1.8) says that if the supremum on the right hand side is finite then the operator ϕ(|L|) is bounded from L^p(G) to L^q(G). Moreover, the estimate for the operator norm can be used for deriving asymptotics for propagators for equations on G. For example, we get the following consequences for the L^p-L^q norm for the heat

kernelofL,applyingTheorem1.2withϕ(u)=e^−tu,orembeddingtheoremsforLwith ϕ(u)= _(1+u)¹ γ.

We note that estimates of the type (1.10) are exactly those leading to subsequent Strichartzestimates.Here,ourmethodisverydifferentfromtheusualonesaswedonot getitbyinterpolationfromtheend-pointcase.

Corollary 1.3. Let G be a locally compact unimodular separable group and let L be a positiveleft Fouriermultiplier suchthat forsomeαwehave

τ(E_(0,s)(L))s^α, s→ ∞. (1.9)

Thenforany 1< p≤2≤q <∞ thereisaconstant C=Cα,p,q>0suchthat wehave e^−tLL^p(G)→L^q(G)≤Ct^−α

1 p−¹_q

, t >0. (1.10)

Wealso havetheembeddings

fL^q(G)≤C(1 +L)^γfL^p(G), (1.11) providedthat

γ≥α 1

p−1 q

, 1< p≤2≤q <∞. (1.12) The number α in (1.9) is determined based on the spectral properties of L and is oftencomputable.Forexample,wehave

(a) if L is the sub-Laplacian ona compact Lie groupG then α= ^Q₂, where Q is the HausdorffdimensionofGwithrespectto thecontroldistanceassociated toL; (b) if L is the sub-Laplacian on the Heisenberg group G = Hⁿ then α = ^Q₂, where

Q= 2n+ 2 isthehomogeneousdimensionofHⁿ; (c) More generally, if L = (−1)^N

_n

k=1

X_k^2N+ n k=1

Y_k^2N

, where the vector fields X1,. . . ,Xn,Y1,. . . ,Yn,T isthe(usual)basisintheLiealgebrahⁿoftheHeisenberg groupHⁿ suchthat[X_k,Y_k]=T andallothercommutatorsarezero,thenα= _2N^Q . Consequently,inbothofthesub-Laplaciancases(a)and(b),Corollary1.3impliesthat forany1< p≤2≤q <∞thereisaconstantC=Cp,q>0 such thatwehave

e^−tLL^p(G)→L^q(G)≤Ct^−Q2

1 p−¹q

, t >0. (1.13) The embeddings (1.11) under conditions (1.12) show that the statement of The- orem 1.2 is in general sharp. Taking ϕ(s) = _(1+s)¹a/2 and applying (1.8) to the

sub-Laplacian Δsub in either of examples (a) or (b) above, we get that the operator ϕ(−Δsub)= (I−Δsub)^−a/2 isL^p(G)-L^q(G) boundedandtheinequality

fL^q(G)≤C(1−Δsub)^a/2fL^p(G) (1.14) holds trueprovidedthat

a≥Q 1

p−1 q

, 1< p≤2≤q <∞. (1.15) However, this yields the Sobolev embedding theorem which is well-known to be sharp at least inthe case(b) ofthe Heisenberg group([29]), showingthe sharpness of Theo- rem1.2andhencealsooftheFouriermultipliertheorem(1.7).Moreexamplesaregiven inSection7.

Throughout thepaper wewill usethenotation ofthetypefX fY ifwe have fX ≤CfY withtheconstantC thatmaydependonthespacesX,Y butnotonf. 2. Notationandpreliminaries

In thissection wefix thenotation andbriefly recall somepreliminarieson vonNeu- mannalgebrastobeusedfordevelopingsubsequentharmonicanalysisonlocallycompact groups. For exposition purposes it seems beneficial to recall several generalnotions in thecontextofgeneralvonNeumannalgebrasM.However,forourapplicationtomulti- pliersonlocallycompactgroupsGwewillbelatersettingM to betherightgroupvon NeumannalgebraVN_R(G).Inparticular,wewillbe abletoreadilyapplythenotionof noncommutative Lorentz spaces onM as developed in[51], one of thekey ingredients forouranalysis.

LetM⊂ L(H) beasemifinitevonNeumannalgebraactinginaHilbertspaceHwith a traceτ. The semifiniteassumption simplifies the formulationsand is satisfied inour main exampleM= VNR(G).

Definition 2.1(Affiliated operators).A linearclosedoperatorA (possiblyunboundedin H)issaidtobe affiliatedwith M,symbolicallyAνM,ifitcommuteswiththeelements of thecommutant M^! ofM,i.e.

AU=U A, for all U ∈M^!. (2.1)

This relation ν is anaturalrelaxation ofthe relation ∈: ifA is abounded operator affiliated withM, then bythedouble commutant theorem A∈M.Oneof theoriginal motivations [46,47] of Johnvon Neumannwas to build amathematical foundation for quantummechanics.Inthisframework,theobservableswithunboundedspectrumcorre- spondtoclosed denselydefinedunboundedoperators.AlthoughthealgebraM consists

of bounded operators, the technique of projections makes it possible to approximate unboundedoperators.

Somepropertiesof tracesshall be usedintheproofs ofour theorems.Therefore, we giveabriefbackgroundontracessummarisingtheresultsthatwillbeusedinthesequel.

For some descriptionof measurable fields of operators and links to the representation theoryandgeneralvonNeumannandC^∗-algebraswereferto[28,AppendicesBandC].

Thefollowingdefinitionistakenfrom[20,Definition I.6.1,p. 93]:

Definition 2.2.Let M be avon Neumannalgebra. A traceon the positivepartM+ = {A ∈ M:A^∗ = A > 0} of M is a functional τ defined on M+, taking non-negative, possiblyinfinite,real values,possessingthefollowingproperties:

• IfA∈M+ andB∈M+,wehaveτ(A+B)=τ(A)+τ(B);

• IfA∈M+andλ∈R+,wehaveτ(λA)=λτ(A) (withtheconventionthat0·+∞= 0);

• IfA∈M+ andifU isaunitaryoperatorofM,then τ(U AU⁻¹)=τ(A).

Wesaythatτisfaithful(orexact)iftheconditionA∈M₊, τ(A)= 0,implythatA= 0.

We say thatτ is finite if τ(A)< +∞ for all A ∈ M+. We say that τ is semifinite if, foreachA∈M+, τ(A) isthesupremum ofthenumbersτ(B) overthoseB ∈M+ such thatB≤Aandτ(B)<+∞.Wesaythatτ isnormalif,foreachincreasingfilteringset S ⊂M+withsupremumS∈M+,τ(S) isthesupremumof{τ(B)}B∈S.AvonNeumann algebraM issaidto besemifinite ifthere existsasemifinitefaithfulnormaltraceτ on M₊.

Definition 2.3 (τ-measurable operators S(M)).A closeable operator A (possibly un- bounded) affiliated with M is said to be τ-measurable if for each ε > 0 there exists aprojectionpinM suchthatpH ⊂D(A) andτ(I−p)≤ε. Here D(A) isthedomain ofAinH. Wedenote byS(M) theset ofallτ-measurableoperators.

Werecallthefollowing resultwhichwillbepartially used.

Theorem 2.4 ([62, Theorem 4, p. 412]).If operators A and B are τ-measurable with respecttoavonNeumannalgebra M,thensoare A^∗,A+B andAB,i.e.themaps

+ :M×M(A, B)→A+B∈M, (2.2)

·:M×M(A, B)→AB∈M, (2.3)

∗:MA→A^∗∈M (2.4)

arewell-defined.

Wenote thatthenotionofτ-measurability doesnotappearintheclassicaltheoryof SchattenclassessinceforM =L(H) wehaveS(L(H))=L(H).

It canbe seenthatA∈M⁺ ifandonlyifA= (A^1/2)^∗A^1/2.

Example 2.5.Let G be a locally compact unimodular group with VNR(G) the group von Neumannalgebragenerated bythe rightregular representationπ_Rof G.LetA be alinearboundedoperatorcommutingwiththeleft regularrepresentation. Thenbythe double commutanttheoremA∈VNR(G) anditsactionisgivenby

L²(G)h→Ah=h∗KA∈L²(G),

where K_Aisitsconvolution kernel.Wecandefineatraceτ onVN⁺_R(G) by

τ(A) :=

K

A¹²²L²(G), ifK_A1/2∈L²(G),

∞, otherwise. (2.5)

The trace τ on M₊ can also be extended to the space S(M) of all τ-measurable positiveoperators.

Proposition2.6. Let(M,τ)beavonNeumannalgebraandletAbeaτ-measurablelinear operator. Assumethat ϕisaBorelfunctionon sp(|A|)⊂[0,+∞).Thenwehave

τ(ϕ(|A|)) =

+∞

0

ϕ(t)dμ(t), (2.6)

where μ_t=τ(E_t)and

|A|=

+∞

0

tdE_t(|A|).

Takingϕ(t)=t,wecanalternativelydefineatraceτ as follows τ(A) =

tdμ(t).

Proof of Proposition 2.6. For the spectral measure we can take the family {E_[0,t)}t≥0

of spectral projectionsE_(0,t)corresponding to theintervals [0,t). Thereadercancheck thatthespectralmeasure axiomshold true.

The traceτ is continuous with respect to the τ-measure. In view of the monotone convergence theorem(see [67,Theorem 3.5])wecanassume,withoutloss ofgenerality, thatAisaboundedτ-measurableoperator.Indeed,foreveryτ-measurableoperator|A| there existsasequence{An}ofτ-measurablebounded operators

An= n 0

tdEt(|A|)≤A

convergingtoAintheτ-measuretopology.Then,takingthelimit

nlim→∞τ(An) =τ(A),

we justify the claim. We notice that every Borel function can be uniformly approxi- matedbyboundedBorelfunctions.Thus,weconcentratetoestablish(2.6) forbounded measurableAandboundedBorelfunctionsϕon[0,AB(H)].Bythespectralmapping theoremwehave

sp(ϕ(|A|)) =ϕ([0,AB(H)]).

Let 0 ≤ λ₁ ≤ λ₂ ≤ . . . λ_N be a partition of the interval ϕ([0,AB(H)]). Then the Riemann-likesums

RN = N k=1

λkE_ϕ−1(λk−1,λk)(|A|)

convergeto ϕ(|A|) intheτ-measure topology.Thetraceτ onR_N isgiven by

τ(RN) = N k=1

λkτ(E_ϕ−1(λ_k−1,λk)(|A|)). (2.7) Onecannoticethatthesumin(2.7) isaLebesgueintegralsum

N k=1

λ_kμ_{(λk−1,λk)}

fortheintegral

A

0

ϕ(t)dμ(t),

whereweset themeasureμ((a,b))=τ(E_(a,b)), (a,b)⊂[0,AB(H)]. 2

Example 2.7.Let M = {Mϕ: L²(X,μ) f → Mϕf = ϕf ∈ L²(X,μ)}ϕ∈L^∞(X,μ)

and take τ(Mϕ) :=

X

ϕdμ, where (X,μ) is a measure space. Then an operator Mϕ

isτ-measurable ifand onlyifϕisaμ-almosteverywherefinitefunction.

The∗-algebraS(M) isabasicconstruction forthenoncommutativeintegration.Let A=U|A|be thepolardecomposition.Thespectraltheorem yieldsthat

|A|=

Sp(|A|)

λdE_λ(|A|), (2.8)

where {Eλ(|A|)}λ∈Sp(|A|) are thespectral projectionsassociated with the operator|A|. Here dEλ(|A|) shouldbeunderstoodastherelativedimensionfunctionfirstconstructed in[46].Since A isaffiliated withM, theprojections satisfyE_λ(|A|)∈M. Now, weare ready to‘measurethespeedofdecay’oftheoperatorA.

Definition 2.8 (Generalised t-th singular numbers).For an operator A ∈ S(M), define thedistributionfunctiond_λ(A) by

dλ(A) :=τ(E_(λ,+∞)(|A|)), λ≥0, (2.9) where E_(λ,+∞)(|A|) is the spectral projection of |A| corresponding to the interval (λ,+∞).Forany t>0,wedefinethegeneralisedt-thsingularnumbersby

μt(A) := inf{λ≥0 : dλ(A)≤t}. (2.10) Forthesakeoftheexpositionclaritywenowformulatesomepropertiesofthedistri- butionfunctiondAwhichwewill beusingintheproofs.

Proposition 2.9. LetA∈S(M).Thenwehave

dA(μA(t))≤t; (2.11)

μA(t)> s if and only if t < dA(s); (2.12) sup

t>0

t^αμA(t) = sup

s>0

s[dA(s)]^α for 0< α <∞. (2.13) Theproofofthispropositionisalmostverbatimtotheproofof[33,Proposition1.4.5 on page46]. Theword ‘almost’standsfor theright-continuity ofthe non-commutative distribution function dA(s) which is discussed after [67, Definition 1.3 on page 272].

Therefore, inthefollowingproof weshallusetheright-continuity ofdA(s) withoutany justification.

Proof of Proposition 2.9. Let sn ∈ {s>0 :dA(s)≤t}be suchthatsn μA(t).Then dA(sn)≤tandtheright-continuityofdAimpliesthatdA(μA(t))≤t.Thisproves(2.11).

Now,weapplythispropertytoderive(2.12).Ifs< μ_A(t)= inf{s>0 :d_A(s)≤t},then sdoesnotbelongtotheset{s>0 : d_A(s)≤t} =⇒ d_A(s)> t.Conversely,ifforsomet ands,wehadμA(t)< s,thentheapplicationofdAandproperty(2.12) wouldyieldthe contradiction dA(s) ≤ dA(μA(t)) ≤t. Property (2.12) is established. Finally, we show

(2.13).Givens>0,pickεsatisfying0< ε< s.Property(2.12) yieldsμA(dA(s)−ε)> s whichimpliesthat

sup

t>0

t^αμ_A(t)≥(d_A(s)−ε)^αμ_A(d_A(s)−ε)>(d_A(s)−ε)^αs. (2.14) Wefirst letε→0 and thentakethe supremumover alls>0 to obtainonedirection.

Conversely,givent>0,pick0< ε< μ_A(t).Property(2.12) yieldsthatd_A(μ_A(t)−ε)> t.

Thisimplies thatsup_s>0s(d_A(s))^α ≥(μ_A(t)−ε)(d_A(μ_A(t)−ε))^α >(μ_A(t)−ε)t^α. We firstletε→0 andthentakethesupremumoverallt>0 toobtaintheoppositedirection of(2.13). 2

Hereweformulatesomepropertiesofμ_tthatweuseintheproofof Theorem5.1.

Lemma 2.10 ([67,Lemma 2.5, p. 275]).Let A,B be τ-measurable operators. Then the followingpropertiesholdtrue.

(1) The map (0,+∞) t → μ_t(A) is non-increasing and continuous from the right.

Moreover,

tlim→0μt(A) =A ∈[0,+∞]. (2.15)

(2) μ_t(A) =μ_t(A^∗). (2.16)

(3)

μt+s(AB)≤μt(A)μs(B). (2.17) (4)

μt(ACB)≤ ABμt(C), for any τ-measurable operatorC. (2.18) (5) Forany continuous increasingfunction f on[0,+∞) wehave

μ_t(f(|A|)) =f(μ_t(|A|)). (2.19) InLemma2.10,weformulateonlythepropertiesweuse,whereasin[67,Lemma 2.5, p. 275] thereadercanfindmoredetails.

Example2.11.FortheoperatorM_ϕinExample2.7,fromDefinition2.8wecanshowits generalisedt-thsingularnumberstobe

μt(Mϕ) =ϕ^∗(t),

whereϕ^∗(t) istheclassicaldecreasingrearrangement(seee.g.[6]).

Asanoncommutativeextension[51] oftheclassicalLorentzspaces,wedefineLorentz spacesL^p,q(M) associatedwithasemifinitevonNeumannalgebraM asfollows:

Definition 2.12 (Noncommutative Lorentz spaces).For 1≤p<∞, 1≤q <∞, denote byL^p,q(M) thesetofalloperatorsA∈S(M) satisfying

AL^p,q(M):=

⎛

⎝ +∞

0

t^1pμt(A)qdt t

⎞

⎠

1 q

<+∞. (2.20)

Forq=∞,wedefineL^p,∞(M) asthespaceofalloperatorsA∈S(M) satisfying AL^p,∞(M):= sup

t>0

t^p1μt(A). (2.21)

Withthis, for1≤p<∞,wecanalso defineL^p-spacesonM by

AL^p(M):=AL^p,p(M)=

⎛

⎝

+∞

0

μt(A)^p dt

⎞

⎠

1 p

.

The classical Lorentz spaces L^p,q(X,μ) correspond to the case of commutative von Neumannalgebra.Modulustechnicaldetails[20,p. 132,Theorem 1],anarbitraryabelian von Neumann algebra ina Hilbert space H is isometrically isomorphic to the algebra {Mϕ}ϕ∈L^∞(X,μ) fromExample2.7.ThennoncommutativeLorentzspacescoincidewith theclassicalones:

Example 2.13 (Classical Lorentz spaces).Let M be the algebra {Mϕ}ϕ∈L^∞(X,μ) from Example 2.7consisting ofallthemultiplicationoperators Mϕ: L²(X,μ)f →Mϕf = ϕf ∈L²(X,μ).ByExample 2.11, wehave

μt(Mϕ) =ϕ^∗(t).

Thus, theLorentzspaceL^p,q(M) consistsofalloperatorsMϕ suchthat

+∞

0

[t^1pϕ^∗(t)]^qdt

t <+∞, whichgivestheclassicalLorentzspace.

Concerning the structure of semifinite von Neumann algebras, given an arbitrary semifinitevonNeumannalgebraM withatraceτ,thereisanisomorphismofM ontoa certain HilbertalgebraU ([20, p. 99,Theorem 2]).Thus,we constructthetraceonthe

Hilbertalgebra yielding thetrace onM due to isomorphism. We referto [20], [48] for moredetailsonthis.

Letnow G be alocally compact unimodularseparable group.Denote by πL(g) and π_R(g) theleft andtherightactionofGonL²(G),respectively:

πL(g)f(x) :=f(g⁻¹x), π_R(g)f(x) :=f(xg),

andbyVNL(G) thegroupvonNeumannalgebrageneratedbyalltheπL(g) withg∈G, i.e.

VNL(G) :={πL(g)}^!!_g∈G, andsimilarly

VNR(G) :={πR(g)}^!!g∈G,

where !! is thebicommutant ofthe self-adjoint subalgebras {πL(g)}g∈G,{πR(g)}g∈G ⊂ L(L²(G)).It hasbeen shownin[60] that

VNL(G)^!= VNR(G), (2.22)

VNR(G)^!= VNL(G). (2.23)

We do notmake assumption that Gis either of type I or type II. The decomposition theory forunitary representationsof locally compact separableunimodulargroupshas beenestablishedin[24,25].

FromnowonwetakeM= VNR(G).

Forf ∈L¹(G)∩L²(G), wesay thatf on Ghas a Fouriertransform whenever the convolutionoperator

Rfh(x) := (h∗f)(x) =

G

h(g)f(g⁻¹x)dg (2.24)

is a τ-measurable operator with respect to VN_R(G), i.e. R_f ∈ S(VN_R(G)). The Plancherelidentitytakes ([61,Theorem3onpage282])theform

RfL²(VNR(G)) =fL²(G). (2.25) Inthissetting,theHausdorff-Younginequalityhasbeenestablishedin[40] intheform

RfL^p(VNR(G)) fL^p(G), 1< p≤2. (2.26)

The constantin (2.26) has been computed in [55] for simply connected real nilpotent Lie groups with explicitly computable Plancherel measures. It canbe shown that the constantislessthan1 forlocallycompactgroupswithnocompactopensubgroups[31].

In [51], as an application of the technique of the t-th generalised singular values, the Hardy-Littlewood theorem ([34]) has been generalised to an arbitrary locally compact separableunimodulargroupG:

Theorem 2.14([51]). Let1< p≤2andf ∈L^p(G).Thenwehave

R_fL^p,p(VNR(G))≤ fL^p(G). (2.27) Remark 2.15.ThePlancherel equality (2.25) by Segal [61] and Kosaki’s version[51] of Hardy-Littlewood inequality(2.27) have been originally established forthe left convo- lution L_fh=f ∗h. However, the sameline of reasoningyields inequalities (2.27) and (2.25) withtherightconvolution Rf.Weworkwiththerightconvolution operatorsRf

heresinceitnaturallycorrespondstoleft-invariantoperatorswhenanalysingtheFourier multipliersongroups.

Using thetechniqueof thet-thgeneralisedsingular valuesdevelopedin[67], wecan formulateboth theHausdorff-Young(2.26) andHardy-Littlewood(2.27) inequalitiesin thefollowing forms(for1< p≤2):

⎛

⎝

+∞

0

μt(Rf)^pdt

⎞

⎠

p1

≡ RfL^p(VNR(G)) ≤ fL^p(G), (2.28)

⎛

⎝

+∞

0

t^p−2μ_t(R_f)^pdt

⎞

⎠

1 p

≡ R_fL^p,p(VNR(G))≤ fL^p(G). (2.29)

Inthesequel,whenweprovePaleyinequalityinTheorem3.1,theHardy-Littlewood inequalities(2.27) and(2.29) (fortherightconvolutionR_f)willalsofollowasitsspecial cases.

2.1. Fouriermultiplierson locallycompactgroups

LetGbealocally compactseparableunimodulargroup.Thefirstquestionishowto understandthenotionofFouriermultipliers.Inthefirstinstanceweadoptthefollowing definition:

Definition 2.16.A linearoperatorA is saidto be aleft Fouriermultiplier on G ifA∈ S(VNR(G)).

Ifwe now recallDefinition 2.1we cansee thatA isa leftFourier multiplieron Gif and only if A is affiliated with the right group von Neumann algebra VNR(G) and is τ-measurable.WecanthenclarifyDefinition2.16further:

Remark2.17.For M = VN_R(G) the operators affiliated with M areprecisely those A thatareleft-invariant onG,namely,

Ais affiliated with VNR(G) ⇐⇒ AπL(g) =πL(g)A, for allg∈G. (2.30) Summarising this observation with Definition 2.16, left Fourier multipliers on G are preciselytheleft-invariantoperatorsthatare measurable(inthesenseofDefinition2.3).

Forclarityand inviewofitsimportance,wegiveashort justificationofthis.

Proof of Remark2.17. =⇒.ByDefinition2.16,wehave

AU=U A, for all U ∈VN_R(G)^!. (2.31) Thenby(2.23),and bytaking U =πL(g),g∈G, weseethatA mustbeleft-invariant.

⇐=.Wehave

Aπ_L(g) =π_L(g)A, for all g∈G.

Bydefinition,thealgebraVNL(G) istheclosure oftheinvolutivesubalgebra {π_L(g)}g∈G⊂ L(L²(G))

inthestrongoperatortopology.Therefore,we obtain

AU=U A, for all U ∈VN_R(G)^!, (2.32) whereweused (2.23).Thiscompletes theproof ofRemark2.17. 2

3. PaleyandHausdorff-Young-Paleyinequalities

OuranalysisofL^p-L^q multiplierswillbe basedonaversionoftheHausdorff-Young- Paleyinequalitythatweestablishinthissectioninthecontextoflocallycompactgroups.

Itwill be obtainedbyinterpolation betweentheHausdorff-Younginequalityand Paley inequalitythatwediscussfirst.

Westartfirstwithaninequalitythatcanbe regardedas aPaleytypeinequality.

Theorem3.1(Paleyinequality). LetGbealocallycompactunimodularseparablegroup.

Let1< p≤2.Supposethat apositivefunctionϕ(t)satisfies thecondition

Mϕ:= sup

s>0

s

t∈R+

ϕ(t)≥s

dt <+∞. (3.1)

Then forallf ∈L^p(G) wehave

⎛

⎝

+∞

0

μ_t(R_f)^pϕ(t)^2−pdt

⎞

⎠

1 p

≤M

2−p

ϕp fL^p(G). (3.2)

As usual,theintegraloveranemptyset in(3.1) isassumedto bezero.

Wenotethattakingϕ(t)=¹_t werecoverKosaki’sHardy-Littlewoodinequality(2.29).

Inthissense,thePaleyinequalitycanbeviewedasanextensionof(one of)theHardy- Littlewoodinequalities.AsasmallbyproductofourproofofTheorem3.1wethusgeta simpleproofof Theorem2.14.

Proof of Theorem3.1. Let measure ν be absolute continuous with respect to the Lebesguemeasure onRⁿ₊,i.e.i.e.

ν(t)

dt :=ϕ²(t), t∈R+ (3.3)

where ^ν(t)_dt is theRadon-Nikodym derivativeat the point t∈R+. We definethe corre- spondingspaceL^p(R+,ν),1≤p<∞,asthespaceofcomplex(orreal)valuedfunctions f =f(t) suchthat

fL^p(R+,ν):=

⎛

⎜⎝

R+

|f(t)|^pϕ²(t)dt

⎞

⎟⎠

1 p

<∞. (3.4)

Wewill showthatthesub-linearoperator

T:L^p(G)f →T f :=μt(Rf)/ϕ(t)∈L^p(R+, ν)

is well-definedandboundedfromL^p(G) toL^p(R+,ν) for1< p≤2.Inotherwords,we claimthatwehavetheestimate

T fL^p(R+,ν)=

⎛

⎜⎝

R+

μt(Rf) ϕ(t)

p

ϕ²(t)dt

⎞

⎟⎠

1 p

M

2−p

ϕp fL^p(G), (3.5)

whichwouldgive(3.2),andwhere wesetM_ϕ:= sup_t>0t

t∈R+

ϕ(t)≥s

dt.Wewill showthatT is ofweak-type(2,2)andofweak-type(1,1).Moreprecisely,weshow that

ν{t∈R+: |T f(t)| ≥y} ≤

M2fL²(G)

y

2

with norm M2= 1, (3.6) ν{t∈R+: |T f(t)| ≥y} ≤ M₁fL¹(G)

y with normM1=Mϕ, (3.7) whereν isdefinedin(3.3).Then(3.5) would followfrom(3.6) and(3.7) bytheMarcin- kiewiczinterpolationtheorem.Now,toshow(3.6),usingPlancherel’sidentity(2.25),we get

y²

t∈R+ μt(Rf)

ϕ(t) ≥y

ϕ²(t)dt≤ T f²L²(R+,ν)=

R+

μ_t(R_f) ϕ(t)

2

ϕ²(t)dt

=

R+

μ²_t(Rf)dt=Rf²L²(V NR(G))=f²L²(G).

Thus,Tisofweak-type(2,2)withnormM₂≤1.Further,weshowthatT isofweak-type (1,1)withnormM₁=M_ϕ;moreprecisely, weshowthat

t∈R+ μt(Rf)

ϕ(t) ≥y

ϕ²(t)dtM_ϕfL¹(G)

y . (3.8)

FromthedefinitionoftheFouriertransformitfollowsthat

μt(Rf)≤ fL¹(G). (3.9)

Indeed,fromtheDefinition2.8,wehave

μt(Rf)≤ RfL²(G)→L²(G).

TheYounginequalityforconvolution(e.g. [30,p.52,Proposition2.39])yields R_fgL²(G)≤ fL¹(G)gL²(G).

Thus

RfL²(G)→L²(G)≤ fL¹(G).

Thisproves(3.9).Therefore, wehave y < μt(Rf)

ϕ(t) ≤fL¹(G)

ϕ(t) .