Lie groups revisited Besov continuity of pseudo-differential operators on compact C.R. Acad. Sci. Paris, Ser.I

Contents lists available atScienceDirect

C. R. Acad. Sci. Paris, Ser. I

www.sciencedirect.com

Harmonic analysis/Functional analysis

Besov continuity of pseudo-differential operators on compact Lie groups revisited

Continuité de Besov des opérateurs pseudo-différentiels sur les groupes de Lie compacts revisitée

Duván Cardona

MathematicsDepartment,UniversidaddelosAndes,Carrera1No.18a10,Bogotá,Colombia

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

Articlehistory:

Received29October2016

Acceptedafterrevision28February2017 Availableonline14April2017 PresentedbytheEditorialBoard

Inthisnotewepresentsomeresultsontheactionofglobalpseudo-differentialoperators onBesovspacesoncompactLiegroups.

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

r é s um é

Danscette note,nousprésentonsquelques résultatssurl’action des opérateurspseudo- différentielsglobauxsurlesespacesdeBesovdesgroupesdeLiecompacts.

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

Versionfrançaiseabrégée

Danscetarticle,nousprésentonsplusieursrésultatssurlacontinuitédanslesespacesdeBesovdesopérateurspseudo- différentielssurlesgroupesdeLiecompacts.Plusprécisé,nousprésentonsunanaloguepoursurlesespacesdeBesovdes groupesde Lie compactsde quelques théorèmesclassiquessur lacontinuitédes multiplicateursde Fourieretdesopéra- teurs pseudo-différentiels sur les espaces L^p

(R

ⁿ

)

. Lesrésultatsque nous généralisonsici sontles suivants : le théorème de Hörmander–Mihlin[12],lacondition de Hörmander sur lacontinuité L^p–L^q desmultiplicateurs de Fourier[12] etun résultat de Fefferman [8]sur lacontinuité L^p desopérateurs pseudo-différentiels associésclasses de Hörmander sur R^n. Unedesmotivationspourcestravauxestquechacundesrésultatsmentionnésci-dessusaétéprécédemmentgénéraliséau cadredesespacesL^p surlesgroupesdeLiecompacts–voir[1],[7],[24]et[10].

1. Introduction

In thisnote, we investigatethe mapping propertiesof pseudo-differentialoperators on compact Liegroups actingon theircorrespondingBesovspaces.Inordertoillustrateourresults,werecallsomefundamentalresultsinharmonicanalysis

E-mailaddresses:[email protected],[email protected].

http://dx.doi.org/10.1016/j.crma.2017.02.012

1631-073X/©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

on themappingpropertiesofmultipliersandpseudo-differentialoperators on L^p-spaces.It wasproved byMarcinkiewicz inthefollowingmultipliertheorem[14]:if

( σ (ξ ))

_ξ∈Zisasequencesatisfying

supξ∈Z

| σ (ξ ) | < ∞ ,

sup

j∈N0

2^j≤|ξ|²<2^j+1

| σ (ξ +

¹

) − σ (ξ ) | < ∞ ,

(1) then theperiodic Fourier multiplier T definedby T f

:=

F⁻¹

[ σ (ξ )F (

f

) ]

^{extendsto a bounded operator on Lp}

(

T

)

, 1

<

p

< ∞

^{. Here} T

:= [

⁰

,

1

]

^isthe one-dimensionaltorusand F ^{is theFourier transformonthe torus. Later,Mihlin in[15]}

establishedtheFouriermultipliertheorem,whichstatesthateveryfunction

σ ( · )

ofclassCκ

(

Rⁿ

\{

⁰

} )

,

κ = [

ⁿ₂

] +

^1,satisfying

|∂

_ξ^α

σ (ξ )| ≤

^Cα

|ξ |

^−|α|

, | α | ≤ κ ,

(2)

generatesa Fouriermultiplier T definedby T f

:=

F⁻¹

[ σ (ξ )F (

f

)]

^{,whichextendstoaboundedoperatoron Lp}

(R

ⁿ

)

,1

<

p

< ∞

^{. Here} F ^{is the Euclidean Fourier transform. Hörmander improved the Mihlin theorem in [12] by imposing the} followingcondition:

sup

r>0

σ (

r

· )

H_loc^s (Rⁿ)

< ∞ ,

(3)

where H_loc^s is the standardlocal Sobolev spaceoforder s and, in thiscase, s

>

ⁿ₂ isfixed. The caseof L^p–L^q-multiplier theoremsalsoisaclassicalproblem.InR^{n,Hörmanderin[12]showsthatif}

σ

isafunctiononR^{n satisfying}

sup

r>0

r

· [ μ _{ ξ ∈ R

ⁿ

: | σ (ξ ) | >

r

}]

^1p−1q

< ∞ ,

1

<

p

≤

²

≤

^q

< ∞ ,

(4) thentheFouriermultiplierT associatedwith

σ

extendstoaboundedoperatorfromL^p

(

Rⁿ

)

intoL^q

(

Rⁿ

)

.Thegeneralcaseof pseudo-differentialoperators isvery complicated.Forafunction

σ

onR^2n,thecorrespondingpseudo-differentialoperator Tσ isdefinedontheSchwartzspaceS

(R

ⁿ

)

by

T_σf

(

x

) =

Rⁿ

e^i2πx,ξ

σ (

x

, ξ )(

Ff

)(ξ )

d

ξ, (

Ff

)(ξ ) :=

_f

(ξ ) =

Rⁿ

e^−i2πx,ξf

(

x

)

dx

,

(5)

where Ff istheFouriertransformof f

∈

S

(

Rⁿ

)

.Amultiplier is,roughlyspeaking,a specialcaseofpseudo-differential operator where the corresponding symbol dependsonly on

ξ

. If we denote by S^mρ,δ

(R

²ⁿ

)

, 0

≤ ρ , δ ≤

^{1, to the class of} functions

σ

onR^{2n satisfying}

|∂

x^β

∂

_ξ^α

σ (

x

, ξ )| ≤

^Cα,β

ξ

^{m−ρ|α|+δ|β|}

, α , β ∈ N

ⁿ

,

(6)

then we can announce a classical theorem dueto Fefferman (cf. [8]) known to be one of the more sharp theorems on the boundedness of pseudo-differential operators:for general 1

<

p

< ∞

^{and m}

≤ −

^mp

= −

ⁿ

(

1

− ρ )|

¹_p

−

¹₂

|

^{, a pseudo-} differentialoperatorwithsymbol

σ ∈

^S−_ρ,δ^mp

(R

²ⁿ

)

is L^p-bounded.Animportantandrecentproblemisthegeneralizationof thetheoremsaboveinthesettingofcompactLiegroups,whentheFouriermultipliersandthepseudo-differentialoperators are consideredindifferentfunctionspaces.ItisimportanttomentionthatthisproblemhasbeensolvedforL^p-spacesin therecentworksofAkylzhanovandRuzhansky[1],DelgadoandRuzhansky[7],RuzhanskyandWirth[24],andFischer[10].

AversionoftheFeffermantheoremfortheperiodiccaseofthetorushasbeenstudiedin[6].

TheresultsofthispaperpresentaversionoftheresultsmentionedaboveinthesettingofBesovspacesoncompactLie groups. ForpropertiesinBesovspacesofpseudo-differentialoperatorsassociatedwithHörmanderclassesonR^n,werefer toBordaud[3]andGibbons[11].

2. Besovspaces,pseudo-differentialoperatorsoncompactLiegroupsandL^p-boundedness

Inthissection,wepresentsomebasicsonthetheory ofpseudo-differentialoperatorsoncompactLiegroups. Thepre- liminaries of thetheory ofglobalpseudo-differential operatorscan be foundin [18] and[20]. Letusassumethat G is a compactLiegroupanddenotebyGitsunitarydual,i.e.thesetofequivalenceclassesofallstronglycontinuousirreducible unitary representationsofG.If A isalinearoperatorfromC^∞

(

G

)

intoC^∞

(

G

)

and

ξ :

^G

→

^U

(

H_ξ

)

denotesan irreducible unitary representation,wecanassociatewith A a matrix-valuedsymbol

σ

A

(

x

, ξ ) ∈

C^{dξ×dξ (this}correspondencegivesrise totheglobalcalculusofRuzhanskyandTurunen[18])satisfying

A f

(

x

) :=

[ξ]∈G

dξTr

[ξ(

x

) σ

A

(

x

, ξ )(

Ff

)(ξ )], (

Ff

)(ξ ) :=

_f

(ξ ) =

G

f

(

x

)ξ(

x

)

^∗dx

∈ C

^dξ×dξ

.

(7)

Here, foreach class

[ ξ ]

^{we pickjustone}representative

ξ ∈ [ ξ ]

^,dξ

=

^dim

(

Hξ

)

and

(

F^f

)(ξ )

istheFouriertransformat

ξ

. We end this section with the following formulation of Besov spaces on compact Lie groups, characterized in terms of representationsin[17]and[16].

Definition2.1. Letr

∈

R^,0

≤

^q

< ∞

^{and 0}

<

p

≤ ∞

^{.If f is ameasurable functionon G, we saythat f}

∈

^Brp,q

(

G

)

ifthe function f satisfies

f

B^r_p,q

:=

⎛

⎝

^∞

m=0

2^mrq

2^m≤ξ<2^m+1

dξTr

[ξ(

x

)

_f

(ξ )]

^q_Lp(G)

⎞

⎠

1 q

< ∞.

(8)

Ifq

= ∞

^{, Br}p,∞

(

G

)

consistsofthosefunctions f satisfying

^f

B^r_p,∞

:=

^sup

m∈N2^mr

2^m≤ξ<^2m+1

d_ξTr

[ ξ(

x

)

_f

(ξ ) ]

L^p(G)

< ∞ .

(9)

Applicationsofthematrix-valued calculus ofRuzhansky andTurunen canbe found in[19,21,23].Thework ofFischer [9]includesasummaryofessentialpropertiesontheHörmanderclassesincludingaversion oftheCalderón–Vaillancourt theorem.Theproblemofclassificationofglobaloperators inL^p-spaceshasbeentreatedrecently.Themainresultscanbe summarizedinthefollowingtheorem.

Theorem2.1.IfG isacompactLiegroupandn isitsdimension,

isthelessintegerlargerthanⁿ₂andl

:= [

ⁿ

/

p

] +

^{1,underoneofthe} followingconditions:

1.

∂

x^βD^α

σ

A

(

x

, ξ )

op

≤

^Cα

ξ

^−|α|

,

for all

| α | ≤ , |β| ≤

^{l and}

[ξ] ∈

G,

2.

D

^α_ξ

∂

x^β

σ

A

(

x

, ξ )

op

≤

^Cα,β

ξ

^{−m−ρ|α|+δ|β|}

, | α | ≤ , |β| ≤

^l

,

m

≥ (

1

− ρ )|

¹_p

−

¹₂

| + δ

l,

3.

D

^α_ξ

∂

_x^β

σ

A

(

x

, ξ )

op

≤

^Cα,β

ξ

^{−ν−ρ|α|+δ|β|}

, α ∈

Nⁿ

, | β | ≤

^l,0

≤ ν <

ⁿ₂

(

1

− ρ )

,

|

¹_p

−

¹₂

| ≤

^ν_n

(

1

− ρ )

⁻¹, 4.

σ

A

s

:=

^sup_[ξ]∈G

[ σ

A

(ξ )

op

+ σ

A

(ξ ) η (

r⁻²LG

)

H˙^s(G)

] < ∞

^,

theglobaloperatorA

≡

^TaisboundedonL^p

(

G

)

,1

<

p

< ∞

^.

ItisimportanttomentionthatthelastconditionaboveisadiscreteanalogueoftheclassicalHörmander–Mihlintheorem provedin[12].Also,thecondition

5. sup_s>0s

[ μ {[ξ] ∈

G

: ∂

x^β

σ

A

(

x

, ξ )

op

>

s

}]

^p11−p12

< ∞

^,1

<

p1

≤

²

≤

^p2

< ∞

^,

|β | ≤

^l,

impliestheL^p1–L^p2-boundednessofA.Theproofoftheassertionsabovecanbefoundinthepapers[2,7,10,22,24].Finally,itis importanttomentionthat,asaconsequenceoftheSharpGardinginequalityforglobaloperators,itwasprovedin[19]thatunderthe condition

6. A

∈

¹

(

G

)

with

σ

A

(

x

, ξ ) ≥

^0,

thelinearoperatorA extendstoaboundedoperatorfromL²

(

G

)

toL²

(

G

)

.

Inthenext section,we willusethelast resultson L^p-estimatestoproveour resultsonBesovboundednessforglobal pseudo-differentialoperators.Inthissense,thefollowingresultscanbethoughtofasanaloguesoftheL^p-resultsabovein theframeworkofBesovspaces.Themaintoolisthefollowingresult(seeCardona[4]).

Theorem2.2.LetG beacompactLiegroupandA beaFouriermultiplier.IfA isboundedfromL^p1

(

G

)

intoL^p2

(

G

)

,thenA extendsto aboundedoperatorfromB^r_p1,q

(

G

)

intoB^r_p2,q

(

G

)

,forallr

∈

R^and0

<

q

≤ ∞

^{.IfA isofweaktype}

(

1

,

1

)

,thenA extendstoabounded operatorfromB^r_1,q

(

G

)

,intoB^r_p,q

(

G

)

forallr

∈

R^,0

<

q

≤ ∞

^and0

<

p

<

1.

3. Pseudo-differentialoperatorsonBesovspaces,thecaseofcompactLiegroups

Inthissection,wepresentourmainresults.OnR^n,pseudo-differentialoperatorswithsymbolsin S⁰_1,0

(

R²ⁿ

)

areknown tobeboundedon L^p

(

Rⁿ

)

,1

<

p

< ∞

^{(see[13]).SimilarresultsonBesovandLp}-boundednesscanbeobtainedforthecase ofcompactLiegroups.Here,Dξ isthedifferenceoperatorintroducedbyRuzhanskyandTurunenin[24].

Theorem3.1.IfG isacompactLiegroupandn isitsdimension,

isthelessintegerlargerthanⁿ₂andl

:= [

ⁿ

/

p

] +

^{1,underoneofthe} followingconditions:

1.

D

^αξ

∂

_x^β

σ

A

(

x

, ξ )

op

≤

^Cα

ξ

^−|α|

,

for all

| α _{| ≤} , | β | ≤

^{l and}

[ ξ ] ∈

G,

2.

D

^α_ξ

∂

_x^β

σ

A

(

x

, ξ )

op

≤

^Cα,β

ξ

^{−ν−ρ|α|}

, α ∈

Nⁿ

, | β | ≤

^l

, |

¹_p

−

¹₂

| ≤

^ν_n

(

1

− ρ )

⁻¹

,

0

≤ ν <

ⁿ₂

(

1

− ρ )

, 3.

D

^α_ξ

∂

x^β

σ

A

(

x

, ξ )

op

≤

^Cα,β

ξ

^{−m−ρ|α|+δ|β|}

, | α | ≤ , |β | ≤

^l

,

m

≥ (

1

− ρ )|

¹_p

−

¹₂

| + δ

l,

thecorrespondingpseudo-differentialoperatorA extendstoaboundedoperatorfromB^r_p,q

(

G

)

intoB^r_p,q

(

G

)

forallr

∈

R^,1

<

p

< ∞

and0

<

q

≤ ∞

^.Inparticular,if

σ

A

(

x

, ξ ) ≥

^0andA

∈

¹

(

G

)

,thenA isboundedonB^r_2,q

(

G

)

forallr

∈

R^and0

<

q

≤ ∞

^.

Thefollowingisan L¹–L^p-boundednesstheoremofglobalFouriermultipliersfor0

<

p

<

1.

Theorem3.2.IfG isacompactLiegroupandn isitsdimension,

isthelessintegerlargerthanⁿ₂,thenunderthecondition:

4.

D

^α

σ (ξ )

op

≤

^Cα

ξ

^−|α|

,

for all

| α _{| ≤} ,

and

[ ξ ] ∈

G

thecorrespondingmultiplierA isboundedfromL¹

(

G

)

intoL^p

(

G

)

for0

<

p

<

1.Also,A isboundedfromB^r_1,q

(

G

)

intoB^r_p,q

(

G

)

forall r

∈

R^,0

<

p

<

1and0

<

q

≤ ∞

^.

Also, we have some conditionsof type Hörmander andMihlin–Hörmander(inspiredby the recentwork ofV. Fischer [8] onmultipliers on compactLie groupsandby a paper ofthe authorwithM. Ruzhansky [5] whereBesov spacesand multiplierswereconsideredinthecaseofgradedLiegroups).

Theorem3.3.LetG beacompactLiegroupandn itsdimension.Underthefollowingcondition

5. sup_s>0s

[ μ _{[ ξ ] ∈

G

: ∂

^β_x

σ

A

(

x

, ξ )

op

>

s

}]

^p11−p12

< ∞

^,1

<

p1

≤

²

≤

^p2

< ∞

^,

| β | ≤

^l

:= [

_pⁿ₁

] +

^1.

TheoperatorA isboundedfromB^r_p

1,q

(

G

)

intoB^r_p

2,q

(

G

)

forallr

∈

R^and0

<

q

< ∞

^{.Also,ifweconsiderasymbol}

σ

Asatisfying 6. max_|α|≤^l

[

^supz∈^G,[ξ]∈G

(∂

_x^αa

)(

x

, ξ ) |

x=^z

op

+ (∂

^α_xa

)(

x

, · ) |

x=^z

η (

r⁻²LG

)

H˙^s(G)

] < ∞

^,

thecorrespondingoperatorA isboundedonB^r_p,q

(

G

)

forallr

∈

R^,1

<

p

< ∞

^and0

<

q

≤ ∞

^. Acknowledgements

The author wouldlike to thankProfessor AlexanderCardona, his master thesis’ advisor, forhis support andconstant encouragementduringtherealizationofthiswork.HealsowouldliketothankProfessorMichaelRuzhanskyfordiscussions aboutsomeoftheresultspresentedinthisnoteandProfessorFlorentSchaffhauserforhisadviseinthepresentation.

References

[1]R.Akylzhanov,M.Ruzhansky,FouriermultipliersandgroupvonNeumannalgebras,C.R.Acad.Sci.Paris,Ser.I354(2016)766–770.

[2]R.Akylzhanov,E.Nursultanov,M.Ruzhansky,Hardy–LittlewoodinequalitiesandFouriermultipliersonSU(2),Stud.Math.234(2016)1–29.

[3]G.Bourdaud,L^p-estimatesforcertainnon-regularpseudo-differentialoperators,Commun.PartialDiffer.Equ.7(1982)1023–1033.

[4]D.Cardona,BesovcontinuityformultipliersdefinedoncompactLiegroups,Palest.J.Math.5 (2)(2016)35–44.

[5]D.Cardona,M.Ruzhansky,Littlewood–Paleytheorem,Nikolskiiinequality,Besovspaces,FourierandspectralmultipliersongradedLiegroups,arXiv:

1610.04701.

[6]J.Delgado,L^pboundsforpseudo-differentialoperatorsonthetorus,Oper.Theory,Adv.Appl.231(2012)103–116.

[7] J. Delgado,M. Ruzhansky, L^p-boundsfor pseudo-differential operators oncompactLiegroups, J. Inst.Math. Jussieu(2017), http://dx.doi.org/10.

1017/S1474748017000123.

[8]C.Fefferman,L^p-boundsforpseudo-differentialoperators,Isr.J.Math.14(1973)413–417.

[9]V.Fischer,Intrinsicpseudo-differentialcalculionanycompactLiegroup,J.Funct.Anal.268 (11)(2015)3404–3477.

[10]V.Fischer,HörmanderconditionforFouriermultipliersoncompactLiegroups,arXiv:1610.06348.

[11]G.Gibbons,Operateurspseudo-differentielsetespacesdeBesov,C.R.Acad.Sci.Paris,Ser.A286(1978)895–897.

[12]L.Hörmander,EstimatesfortranslationinvariantoperatorsinL^pspaces,ActaMath.104(1960)93–140.

[13]L.Hörmander,TheAnalysisoftheLinearPartialDifferentialOperators,vol.III,Springer-Verlag,1985.

[14]J.Marcinkiewicz,SurlesmultiplicateursdessériesdeFourier,Stud.Math.8(1939)78–91.

[15]S.G.Mihlin,OnthemultipliersofFourierintegrals,Dokl.Akad.NaukSSSR(N.S.)109(1956)701–703.

[16]E.Nursultanov,M.Ruzhansky,S.Tikhonov,NikolskiiinequalityandfunctionalclassesoncompactLiegroups,Funct.Anal.Appl.49(2015)226–229.

[17]E.Nursultanov,M.Ruzhansky,S.Tikhonov,NikolskiiinequalityandBesov,Triebel–Lizorkin,WienerandBeurlingspacesoncompacthomogeneous manifolds,Ann.Sc.Norm.Super.Pisa,Cl.Sci.XVI(2016)981–1017.

[18]M.Ruzhansky,V.Turunen,Pseudo-DifferentialOperatorsandSymmetries:BackgroundAnalysisandAdvancedTopics,Birkhäuser-Verlag,Basel,Switzer- land,2010.

[19]M.Ruzhansky,V.Turunen,SharpGardinginequalityoncompactLiegroups,J.Funct.Anal.260(2011)2881–2901.

[20]M.Ruzhansky,V.Turunen,Globalquantizationofpseudo-differentialoperatorsoncompactLiegroups,SU(2)and3-sphere,Int.Math.Res.Not.IMRN 2013 (11)(2013)2439–2496.

[21]M.Ruzhansky,V.Turunen,J.Wirth,Hörmanderclassofpseudo-differentialoperatorsoncompactLiegroupsandglobalhypoellipticity,J.FourierAnal.

Appl.20(2014)476–499.

[22]M.Ruzhansky,J.Wirth,OnmultipliersoncompactLiegroups,Funct.Anal.Appl.47(2013)72–75.

[23]M.Ruzhansky,J.Wirth,GlobalfunctionalcalculusforoperatorsoncompactLiegroups,J.Funct.Anal.267 (1)(2014)144–172.

[24]M.Ruzhansky,J.Wirth,L^pFouriermultipliersoncompactLiegroups,Math.Z.280 (3–4)(2015)621–642.