Sur un opérateur bilinéaire discret singulier

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Harmonic analysis

On a discrete bilinear singular operator

Sur un opérateur bilinéaire discret singulier

Dong Dong

DepartmentofMathematics,UniversityofIllinoisatUrbana–Champaign,1409W.GreenStreet,Urbana,IL61801,USA

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received27September2016 Acceptedafterrevision15March2017 Availableonline29March2017 PresentedbyYvesMeyer

We prove that for a large class of functions P and Q, the discrete bilinear operator TP,Q(f,g)(n)=

m∈Z\{⁰}f(n−^P(m))g(n−^Q(m))_m¹ isboundedfroml²×^{l2intol1+,∞} forany

∈(0,1]^.

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

ré s u m é

Nousmontrons, quepour unegrande classe defonctions P et Q,l’opérateur bilinéaire discret TP,Q(f,g)(n)=

m∈Z\{0}f(n−^P(m))g(n−^Q(m))_m¹ est borné de l²×^{l2 dans} l^1+,∞,pourtout

_∈(0,1]^.

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

1. Introduction

TheHilberttransform(HTforshort)isdefinedby

H

(

f

)(

x

) =

f

(

x

−

^t

)

^dt

t

,

f

∈

S

(R),

whereS(Rⁿ

)

,n∈N^{,istheSchwartzspaceon}R^{n.Itwasprovedin1928([21])thatHTisboundedonLp forp}∈

(

1

,

∞

)

.An interestinggeneralizationofHTistheso-calledHTalongcurves:

HC

(

f

)(

x

) =

f

(

x

− γ (

t

))

^dt

t

,

f

∈

S

( R

ⁿ

).

Here

γ

:R→R^{n isa}well-behavedcurve.The L^p boundednessofHC hasbeenobtainedforvarious curves.See[22]fora comprehensivesurvey and[2]forageneralization ofHC tothenon-translation-invariantsetting.When

γ

isapolynomial withintegercoefficients,thereisadiscreteversionofHC definedby

E-mailaddress:ddong3@illinois.edu.

http://dx.doi.org/10.1016/j.crma.2017.03.010

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

H^dis_C

(

f

)(

n

) =

m∈Z\{0}

f

(

n

− γ (

m

))

¹

m

,

f

∈

^D

( Z

ⁿ

),

whereD

(Z

ⁿ

)

isthespaceofcompactlysupportedcomplex-valuedfunctionsdefinedonZ^{n.Ontheonehand,Hdis}_C ^hasmany applicationsinergodictheory([6,16–18,20]),butontheotherhandthisdiscreteoperatorismoresubtletohandlethanits continuous counterpart HC,asmanynumbertheoreticaltoolsare involved. H^dis_C wasatfirstproved tobe boundedonl^p forp∈

(

³₂

,

3

)

([23]).Thisrestrictedrangewasextendedtothefullrange

(

1

,

∞)alongtimelater([7,15]).

AnotherdirectionofgeneralizingHTistoconsideritsbilinear analogue,whichissignificantlymoredifficulttoanalyze sincePlancherelTheoremisunavailableinthebilinearsetting.ThebilinearHilberttransform(BHTforshort)canbedefined as

B

(

f

,

g

)(

x

) =

f

(

x

−

^t

)

g

(

x

+

^t

)

^dt

t

,

f

,

g

∈

S

( R ).

Itwasabout70yearsafterthefirstproofoftheboundednessofHTthatLaceyandThiele([9,10])obtainedtheL^p estimates forBHT.Veryrecently,L^p estimatesforBHTalongcurves

B_C

(

f

,

g

)(

x

) =

f

(

x

−

^t

)

g

(

x

− γ (

t

))

^dt

t

,

f

,

g

∈

S

( R ),

werealsoestablishedwhen

γ

isapolynomial([14]).Notethat BC isanaturalbilinearversionofHC. Followingthedevelopmentofthelinearcase,inthispaperweconsiderthediscreteversionofB_C,thatis,

B^dis_C

(

f

,

g

)(

n

) =

m∈Z\{⁰}

f

(

n

−

^m

)

g

(

n

−

^P

(

m

))

¹

m

,

f

,

g

∈

^D

(Z),

where P isapolynomialwithintegercoefficients. Thisoperatorcanalsobe viewedasabilinearanalogueofH^dis_C .As H^dis_C ishardertohandlethan HC,itisreasonabletoexpectthatprovingtheboundednessofB^dis_C shouldbemoredifficultthan that of BC. As astarting pointof thelong journey ofinvestigation on B^dis_C ,in thisarticlewe show thel²×^l2→^l1+,∞

boundednessof B^dis_C (Theorem 1.1).

Wewillstudyanoperatorthatismoregeneralthan B^dis_C (see (1.1)).Giventwofunctions P and Q thatmap Z^intoZ^, define

A^P,Q

:=

(

m₁

,

m₂

) ∈ ( Z \ {

⁰

} )

²

:

^P

(

m₁

) −

^Q

(

m₁

) =

^P

(

m₂

) −

^Q

(

m₂

)

.

Wesaythatthepairoffunctions

(

P

,

Q

)

satisfiescondition

()

ifthereareconstants D1 andD2suchthat ^|_|^m_m^1|

2|≤^D1forall

(

m1

,

m2

)

∈^{AP,Q andforeachm}1∈Z^{,thereareatmostD}2 pairs

(

m1

,

m2

)

intheset A^P,Q.

Theorem1.1.GiventwofunctionsP andQ thatmapZ^intoZ^,let

T_P,Q

(

f

,

g

)(

n

) :=

m∈Z\{⁰}

f

(

n

−

^P

(

m

))

g

(

n

−

^Q

(

m

))

¹

m

,

f

,

g

∈

^D

(

Z

).

(1.1)

Assumethat

(

P

,

Q

)

satisfiescondition

()

.Thenforany ∈

(

0

,

1]^{,thereisaconstantC}dependingonlyon ,D1andD2suchthat

^TP,Q

(

f

,

g

)

l¹⁺_,∞

≤

^C

^f

l²

^g

l²

.

(1.2)

Remarks.(1).Condition

()

ismild. Apairofpolynomials withintegercoefficients

(

P

,

Q

)

satisfiescondition

()

aslong as P−^{Q isnotconstant.Notethat D}1 dependsonthecoefficientsof P and Q,sodoesC inthe theorem.It isnatural toexpectthatthisdependencecanberemoved,asuniformestimatesexistforrelatedoperators([3,4,11,12,14,23,24]).We shallnotpursuitthishere.

(2).WeconjecturethatatleastforsomespecialpairsofP andQ (forexample,P

(

t

)

=^tandQ

(

t

)

=^t2),TP,Q isbounded froml^p×^{lqintolr,wherep}

,

q∈

(

1

,

∞)^{, 1}_p+¹_q=¹_r^{.Thisproblemisverydifficultandcurrentlyoutofreach.}

(3). A useful operator related with TP,Q is the corresponding maximal operator T^∗_P,Q

(

f

,

g

)(

n

)

=^supM∈[¹,∞)|_M¹ ×

M

m=1 f

(

n−^P

(

m

))

g

(

n−^Q

(

m

))

|^{,whichisatfirstprovedtobeboundedfroml2}×^{l2 tolr whenr}

>

1 ([5]).ByusingHölder inequalityandboundednessofthecorrespondingdiscretelinearmaximalfunction f→^supM∈[¹,∞)]M¹

M

m=¹|^f

(

n−^P

(

m

))

| (see,forexample,[1,8,19,26]),we canprovethat T^∗_P,Q isboundedfroml^p×^{lq intolr,wheneverp}

,

q∈

(

1

,

∞)^{, 1}_p+¹_q=¹_r^, andr

>

1 (seep. 75in[25]forasimilartrick).Whethertherestrictionr

>

1 canbedroppedisstillunknown.

(4).See[13]foradiscussionaboutanergodicanalogueofT_P,Q.

TherestofthepaperisdevotedtotheproofofTheorem 1.1.

2. ProofofTheorem 1.1

We will use A^{B todenote thestatement that A}≤^{C B forsome positive constant C.When theimplied constant C} dependsonr,wewrite ArB.AlltheconstantsmaydependonD1 andD2 (appearedinthedefinitionofcondition

()

), but thisdependence will be suppressed since D1 and D2 are oftenfixed inapplications. A^{B is short for AB and} B^{A.Foranysetofintegers E,}|^E|^and

χ

_{E willbe usedtodenotethecountingmeasureandtheindicatorfunctionofE,}

receptively.

Let P andQ beapairoffunctionssatisfyingcondition

()

.Fornotationalconvenience,wewillsimplywrite T forTP,Q andr:=¹+

.Forany

λ >

0 and f

,

g∈^D

(

Z

)

,define thelevelset Eλ= {ⁿ∈Z: |^T

(

f

,

g

)(

n

)

|

> λ

}^{.Ourgoalistoprovethe} followinglevelsetestimate

|

Eλ

|

r

1

λ

^r

,

whenever

f

_l²

=

g

_l²

=

1

.

(2.3)

Wefirstwrite T=

m∈Z\{⁰}f

(

n−^P

(

m

))

g

(

n−^Q

(

m

))

_m¹ asabilinearmultiplieroperator.RecalltheFouriertransformfor any f∈^D

(

Z

)

isdefinedby ˆf

(ξ )

:=

m∈Zf

(

m

)

e^−2πiξm.Hence

T

(

f

,

g

)(

n

) =

T

T

ˆ

_f

(ξ )

g

ˆ ( η )

e^2πi(ξ+η)n

σ (ξ, η )

d

ξ

d

η ,

whereTistheunitcircleand

σ

istheperiodicmultiplier(a.k.a. symbol)givenby

σ (ξ, η ) =

m∈Z\{0} 1

me^{−2πi(P(m)ξ+Q(m)η)}

.

Then we decomposedyadicallythesymbol

σ

asfollows.Pickan oddfunction

ρ

_∈S

(

R

)

supportedintheset {^x: |^x|∈

(

¹₂

,

2

)

}^{withthepropertythat}

1 x

=

∞ j=⁰

1 2^j

ρ

^x

2^j

for anyx

∈ R

^with

|

^x

| ≥

¹

.

Sothesymbol

σ

canbewrittenas

σ (ξ, η )

=_∞

j=0

σ

j

(ξ, η )

,where

σ

j

(ξ, η ) :=

¹ 2^j

m∈Z

ρ

^m

2^j

e^{−2πi(P(m)ξ+Q(m)η)}

.

Correspondingly T=_∞

j=0T_j,where

T_j

(

f

,

g

)(

n

) =

T

T

ˆ

f

(ξ )

^g

ˆ ( η )

e^2πi(ξ+η)n

σ

j

(ξ, η )

d

ξ

d

η

=

¹ 2^j

m∈Z

ρ

^m

2^j

f

(

n

−

^P

(

m

))

g

(

n

−

^Q

(

m

)).

By thesupportof

ρ

andHölderinequality,itis easyto see^Tj

(

f

,

g

)

l^1fl^2gl².Sowe havethefollowing levelset estimateforeach Tj.

Lemma2.1.Foranyf

,

g∈^D

(Z)

^{withl2-norm1, j}∈N^,and

λ >

0,wehave

|{

ⁿ

∈ Z : |

^Tj

(

f

,

g

)(

n

)| > λ}|

¹

λ .

This lemma says that each single T_j is under good (and uniform) control. The difficulty is how to get the desired estimatesforthesumofTj’s.Inthefollowing,wewillapplytheideaoftheT T^∗ method.

Defineanauxiliaryfunctionh

(

n

)

=^T_T⁽₍^f_f^,_,^g_g⁾⁽₎₍ⁿ_n⁾₎

χ

_E

λ

(

n

)

.Itiseasytoverifythat

λ

²

|

^Eλ

|

²

≤

n∈Z

T

(

f

,

g

)(

n

)

h

(

n

)

2

.

(2.4)

ByFubinitheoremandthedefinitionofFouriertransform,

n∈Z

T

(

f

,

g

)(

n

)

h

(

n

) =

T

T

ˆ

f

(ξ )

^g

ˆ ( η ) σ (ξ, η )

h

ˆ (−(ξ + η ))

d

ξ

d

η

=

T

T

ˆ

_f

(ξ − η )

g

ˆ ( η ) σ (ξ − η , η )

_h

ˆ ( − ξ )

d

ξ

d

η .

ApplytheCauchy–SchwarzinequalityandthePlancherelTheorem,andweget

n∈Z

T

(

f

,

g

)(

n

)

h

(

n

)

₂

≤

B

|

Eλ

|,

(2.5)

where

B

:=

^sup

ξ∈^T

T

| σ (ξ − η , η )|

^2d

η .

Combining(2.4)and(2.5),weseethat|^Eλ|≤_λ^B2.Hence,toprove(2.3),itsufficestoobtaintheestimate

B

r

λ

^2−r

.

(2.6)

Tocontrol B,wemakeuseofthedyadicdecompositionof

σ

,aimingforsomecancellations.Forany

ξ

∈^T,

T

| σ (ξ − η , η )|

^2d

η =

T

∞ j=⁰

σ

j

(ξ − η , η )

2

d

η

≤

∞ j₁,j₂=⁰

1 2^j1

1 2^j2

m1,m2∈Z

ρ

^m1

2^j1

ρ

^m2

2^j2

χ

A^P,Q

(

m₁

,

m₂

).

(2.7)

Bycondition

()

, ^|_|^m_m^1|

2|≤^D1 forall

(

m₁

,

m₂

)

∈^{AP,Q.The supportof}

ρ

forces|^m1|^{2j1 and} |^m2|^{2j2.Thesefacts show} that|^j1−^j2|^{1.Alsonotethatforeachm}1,thereareonlyboundednumberofm2’ssuchthat

(

m1

,

m2

)

∈^{AP,Q.Thus(2.7)} implies

B

=

^sup

ξ∈T

T

| σ (ξ − η , η ) |

^2d

η

∞

j=0

1 2^j

.

When

λ

≥^{1, asr}∈

(

1

,

2]^{, trivially B}

λ

^2−r andwe are done.Let M= [(²−^r

)

log_2λ¹]+^1,where[^x] ^{denotestheinteger} partof x.Inthe case

λ <

1,since _∞

j=^M+¹2^1j r

λ

^2−r,the abovemethod stillgives thedesiredestimate for_∞

j=^M+¹Tj, theoperatorassociatedwiththesymbol _∞

j=^M+¹

σ

j.ItremainstocontrolthelevelsetoftheoperatorM

j=⁰T_j for

λ <

1.

ApplyingLemma 2.1,wehave

⎧ ⎨

⎩

ⁿ

∈ Z :

M j=0

Tj

(

f

,

g

)(

n

) > λ

⎫ ⎬

⎭

^M2

λ

r

1

λ

^r

,

whereweusedthefactsr

>

1 and

λ <

1 inthelastinequality.ThisfinishestheproofofTheorem 1.1.

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