Holomorphic extension on product Lipschitz surfaces in two complex variables

0.36 MB·

by  Jarod Hart

HOLOMORPHIC EXTENSION ON PRODUCT LIPSCHITZ SURFACES IN TWO COMPLEX
VARIABLES
JARODHARTANDALESSANDROMONGUZZI
5
1
0
ABSTRACT. InthisworkweproveanewLpholomorphicextensionresultforfunctionsdefinedonproductLipschitzsurfaces
2
withsmallLipschitzconstantsintwocomplexvariables. WedefinebiparameterandpartialCauchyintegraloperatorsthat

r playtheroleofboundaryvalues forholomorphic functions onproductLipschitzdomain. Inthespiritoftheapplication
p
ofDavid-Journe´-Semmes and Christ’s Tbtheorem to theCauchy integral operator, weprove abiparameter Tbtheorem
A andapplyittoproveLpspaceboundsforthebiparameterCauchyintegraloperator. Wealsoprovesomenewbiparameter
Littlewood-Paley-SteinestimatesandusethemtoprovethebiparameterTbtheorem.
1

]
A
C
h. 1. INTRODUCTION
at Inthiswork,wesolveaholomorphicextensionproblemforcertainproductsurfacesinC2 andprovesomeresults
m in harmonic analysis pertaining to biparameter singular integral operators and Littlewood-Paley-Stein theory. To
[ motivate our results, we start with a brief history of holomorphic extension and boundary values of holomorphic

functionsresultsrelatedtoourproblem.
3
Thefirst situationwe describe isone onthe upperhalf planeH={x+it :x∈R, t >0}in C. Givena function
v
1 f ∈Lp(R)for1<p<¥ ,onecanextend f toaholomorphicfunction
6 1 f(y)
3 F(x+it)= dy; forx∈R, t6=0.
2 2p iZRy−(x+it)
. ThisfunctionF isaholomorphicextensionof f inthethesensethatF isholomorphiconC\Rand f(x)= f (x)−
1 +
0 f−(x)forx∈R,where
4
f (x)= lim F(x+it) and f (x)= lim F(x−it).
1 + t→0+ − t→0+
:
v TheselimitsholdalmosteverywhereinRandinLp(R). Sometimesthissortofholomorphicextensionresultisknown
i asaHilbert-Riemanntypeproblem. Italsofollowsthat f = 1(±I+iH)f whereI istheidentityoperatorandH is
X ± 2
theHilberttransform
r
a 1 x−y
Hf(x)= lim f(y)dy.
t→0+p ZR(x−y)2+t2
There is a rich history involving the Hilbert transform and boundary behavior of holomorphic functions, which is
intrinsically related to the study of Hardy spaces. The Lp(R) extension results mentioned here were solved by the
combinedworkofmanypeopleintheearly1900’s,includingclassicalworksofHilbertandRiesz,amongothers.
ThenextsituationwediscussisaLipschitzperturbedupperhalfspaceoftheformHG ={g (x)+it:x∈R, t>0}
where g :R→C is a Lipschitz graph. Problemsrelated to holomorphicfunctionson HG can oftenbe solvedusing
the correspondingsolution on H and the Riemann mappingtheorem, but that is not the case in generalwith the Lp
boundarybehaviorofholomorphicfunctionsonHG . Theholomorphicextensionresultcorrespondingtotheoneinthe
Date:April2,2015.
1991MathematicsSubjectClassification. Primary42B02;Secondary44A02.
Keywordsandphrases. SquareFunction,Littlewood-Paley,Biparameter,Caldero´n-ZygmundOperators,HolomorphicExtension,SeveralComplex
Variables.
ThesecondauthorpartiallysupportedbythegrantPRIN2010-11RealandComplexManifolds:Geometry,TopologyandHarmonicAnalysis.
1
2 JARODHARTANDALESSANDROMONGUZZI
lastparagraphisthefollowing:givenafunctiong∈Lp(G )for1<p<¥ ,onecanextendgtoaholomorphicfunction
1 g(x )
G(z+it)= dx ; forz∈G , t6=0,
2p iZG x −(z+it)
whichisa holomorphicextensionof gin thethe sensethatG isholomorphiconC\G andg(z)=g (z)−g (z) for
+ −
z∈G ,where
g (z)= lim G(z+it) and g (x)= lim G(z−it)
+ −
t→0+ t→0+
andtheselimitsexistinpointwisealmosteverywhereonG andinLp(G ). TheboundaryvaluesofGcanberealizedin
thissettingaswellbyg±(z)= 12(±I+iCG )g(z),whereCG istheCauchyintegraltransform
1 z−x
CG g(z)=t→lim0+p ZG (z−x )2+t2g(x )dx .
Progressing from the extension problem on H to the one on HG was not an easy feat. It took more than 40 years
fromtheproofofLp boundsfortheHilberttransformtoprovetheLp boundsfortheCauchyintegraltransformalong
Lipschitzcurveswithsmallconstants,whichwasduetoCaldero´n[Cal77]. TheproofforageneralLipschitzconstant
appearedsomeyearslaterinworksofCoifman,McIntoshandMeyer[CMM82b,CMM82a]. Later,newproofsand
generalizationsappearedin the work of David–Journe`–Semmes[DJS85], Jones[Jon89], and Chist [Chr90], among
others.
These results were extendedto upperhalf spaces of type Rn+1 =Rn×(0,¥ ) in place of H by Stein in terms of
+
systemsofconjugateharmonicfunctions,seee.g.[Ste67].Inthissituation,theroleoftheHilberttransformisreplaced
bytheRiesz transformsR onRn, andconvergenceresultsholdin Lp(Rn)for1< p<¥ andappropriatefunctions
j
f :Rn→CwithitsharmonicconjugatesR f(x)for j=1,...,n. Then-dimensionalLipschitzupperhalfspaceswere
j
also addressed in a series of papers, Fabes-Kenig-Neri [FKN81], Jerison-Kenig [JK82], and Kenig-Pipher [KP87].
TheysolvedproblemsrelatedtoharmonicfunctionsonupperhalfdomainsoftheformRn+1=Rn×{L(x)+t:x∈
L+
Rn,t>0},amongothers,whereL:Rn→RisaLipschitzfunction.In[FKN81,JK82,KP87],doublelayerpotentials
replacetheRiesztransformsinStein’swork,andtheirassociatedHardyspacesaredefined.
Another setting where this type of problem has been solved is on the product upper half plane H×H in C2.
The correspondingHilbert-Riemannpropertyfor the productupperhalfplane is stated as follows: givena function
f ∈Lp(R2)for1<p<¥ ,onecanextend f toaholomorphicfunction
1 f(y)
F(x+it)= dy; forx=(x ,x )∈R2, t=(t ,t )
(2p i)2ZR2 (y1−(x1+it1))(y2−(x2+it2)) 1 2 1 2
witht ,t 6=0. ThisfunctionF isa holomorphicextensionof f inthe thesense thatF isholomorphicon(C\R)×
1 2
(C\R)and f(x)= f (x)−f (x)−f (x)+f (x)forx∈R2,where
++ +− −+ −−
f (x)= lim F(x +it ,x +it ), f (x)= lim F(x +it ,x −it ),
++ 1 1 2 2 +− 1 1 2 2
t1,t2→0+ t1,t2→0+
f (x)= lim F(x −it ,x +it ), and f (x)= lim F(x −it ,x −it ).
−+ 1 1 2 2 −− 1 1 2 2
t1,t2→0+ t1,t2→0+
TheselimitsholdalmosteverywhereinR2 andinLp(R2). Inthissituation,itfollowsthat f = 1(±I+iH )(±I+
±,± 4 1
iH )f(x) where H f and H f are the Hilbert transforms applied to the first and second variable of f respectively.
2 1 2
These operators H , H , and H H are sometimes called the partial and biparameter Hilbert transforms, which are
1 2 1 2
boundedon Lp(R2), see e.g. [Fef81, FS82]. These boundednessresultsare related to the biparameterHardy space
theory that is addressed in [MM77, GS79, Gun80, CF80, Fef81, FS82, Fef86, Fef87], among many others. Many
of these articles workon the polydiskinstead of productsof upperhalf planes, butworkingin these two settings is
essentiallyequivalent;look,forexample,in[GS79].
Inthiswork,weaddressaholomorphicextensionresultsimilartotheonesaboveforproductLipschitzupperhalf
spaces,whichisstatedasfollows.GivenanappropriateLipschitzboundarysurfaceG =G ×G ⊂C2 andafunction
1 2
g:G →C,thereisafunctionGthatisholomorphicon(C\G )×(C\G )satisfying
1 2
(1.1) g(z)=g (z)−g (z)−g (z)+g (z),
++ +− −+ −−
HOLOMORPHICEXTENSIONONPRODUCTLIPSCHITZSURFACESINTWOCOMPLEXVARIABLES 3
forz=(z ,z )∈G ,where
1 2
(1.2) g (z)= lim G(z +it ,z +it ), g (z)= lim G(z +it ,z −it ),
++ 1 1 2 2 +− 1 1 2 2
t1,t2→0+ t1,t2→0+
g (z)= lim G(z −it ,z +it ), and g (z)= lim G(z −it ,z −it ).
−+ 1 1 2 2 −− 1 1 2 2
t1,t2→0+ t1,t2→0+
For now we leave the sense in which (1.1) holds, the sense that the limits in (1.2) hold, and the conditions on G
unspecified,butthesethingswillbedefinedlaterinthissection.
Beforewestateourholomorphicextensionresult,wewillsetafewdefinitions.WesaythatG(w ,w )isholomor-
1 2
phicat(w ,w )∈C2ifGhasanabsolutelyconvergentpowerseriesrepresentationonaneighborhoodof(w ,w ).We
1 2 1 2
willcalltheLipschitzsurfacesthatweworkwithproductLipschitzsurfaceswithsmallLipschitzconstants,andthey
aredefinedasfollows.LetL ,L :R→RbeLipschitzfunctions.Defineg (x )=x +iL (x ),g (x )=x +iL (x ),
1 2 1 1 1 1 1 2 2 2 2 2
andg (x)=(g (x ),g (x ))∈C2forx=(x ,x )∈R2. ThenwecallG =G ×G =g (R)×g (R)aproductLipschitz
1 1 2 2 1 2 1 2 1 2
surfaceinC2. WesaythatG isaproductLipschitzsurfacewithsmallLipschitzconstantsiftheLipschitzconstantsl
1
andl 2 ofL1 andL2 respectivelyarebothsmallerthan1. TheupperhalfspaceassociatedtoG isdefinedHG 1×HG 2,
whereHG j ={g j(xj)+itj:xj∈R, tj>0}. WealsodefineLp(G )foraproductLipschitzsurfaceG asfollows.Given
a productLipschitz surface G =g (R)×g (R), let Lp(G ) be the collectionof measurablefunctionsg:G →C such
1 2
that
||g||p = |g(g (x))|p|g ′(x )g ′(x )|dx dx <¥ .
Lp(G ) ZR2 1 1 2 2 1 2
Nowwestateourholomorphicextensionresult.
Theorem1. LetG beaproductLipschitzsurfacewithsmallLipschitzconstantsinC2definedbyg =(g ,g ):R2→C2.
1 2
Assumethat
g (x ) g (x )
1 1 2 2
lim =c and lim =c
1 2
|x1|→¥ x1 |x2|→¥ x2
for some c ,c ∈C. If g∈Lp(G ) for some 1< p<¥ , then there exists a function G:(C\G )×(C\G )→C that
1 2 1 2
isaholomorphicextensionofg,where(1.1)holdsalmosteverywhereonG andthelimitsin(1.2)holdinLp(G )and
pointwisealmosteverywhereonG .
In addition to the problems mentioned above, some other boundary value problems related to Theorem 1 can
be found in the work of Bochner [Boc44], Weinstock [Wei69], Stein [Ste70, Ste73], Jacewicz [Jac73], and Krantz
[Kra80,Kra07]. Theseworksproveanumberresultsaboutthebehaviorofholomorphicfunctionsondomainswith
smoothboundariesinCn,butthepointofviewtakenin[Boc44,Wei69,Ste70,Ste73,Jac73,Kra80,Kra07]isdifferent
thantheonetakeninthiswork.TheystartwithaholomorphicfunctionGdefinedonadomainDandmakeconclusions
about the G near or on the boundary ¶ D. Whereas we are given a boundary G with initial data g and construct a
holomorphicfunctionGonthedomainHG ×HG whosebehaviorattheboundaryisdeterminedbyg. Themeaning
1 2
ofboundarybehaviorforusisdescribedin(1.1)and(1.2).
Wetakethis“extensionfromtheboundary”pointofviewbecausewewantthisworktoemphasizetheboundedness
ofboundaryvaluesingularintegraloperatorsthattaketheplaceofthepartialandbiparameterHilberttransformsfrom
the extension problems above; we call these operators the biparameter and partial Cauchy integral transforms, and
theywillbedefinedlaterinthissection.
ItisnaturaltoeventuallydefineHardyspacesofholomorphicfunctionsassociatedtoourproductupperhalfspace
in the same way thatHardyspaces are definedon H, HG , Rn++1, RnL++1, and H×H. These Hardyspaces are related
to the holomorphic extension problems briefly described in the beginning of the Introduction. It is also natural to
expectthateveryholomorphicfunctionin these new Hardyspaceswouldbe realizedas oneofourextensionsfrom
theboundaryG . However,wedonotwanttodealwiththeextratechnicalitiesinvolvedwithdevelopingthesespaces
inthiswork.InsteadwefocusontheholomorphicextensionproblemforG asstatedinTheorem1.
ThesituationinTheorem1ismoregeneralthanholomorphicextensionresultsfrom[Boc44,Wei69,Ste70,Ste73,
Jac73, Kra80, Kra07] in terms of the regularity required for the boundary. In all of these works, the domain D is
assumedtohavesmoothboundary,atleastC2.WhereasTheorem1canbeviewedasaboundaryresultforholomorphic
functionsonHG ×HG ,whichrequiresonlyLipschitztypesmoothnessfortheboundaryG .
1 2
4 JARODHARTANDALESSANDROMONGUZZI
ToproveTheorem1,wetakeanapproachrelatedtotheonesin[MM77,Cha79,Fef79,GS79,Ste79,CF80],which
aremoregeometricinnatureandusestheboundednessofbiparameterandpartialHilberttransforms. Inplaceofthe
Hilberttransforms,wedefinebiparameterandpartialCauchyintegraltransformsforz=(z ,z )∈G andappropriate
1 2
g:G →C,
1 z −x z −x
CG g(z)=t1,tl2i→m0+Ctg(z); Ctg(z)= (2p i)2ZG (z1−1x 1)21+t12(z2−2x 2)22+t22g(x )dx ,
1 z −x t
Cp1g(z)= lim Cp1g(z); Cp1g(z)= 1 1 2 g(x )dx ,
G t1,t2→0+ t t (2p i)2ZG (z1−x 1)2+t12(z2−x 2)2+t22
1 t z −x
Cp2g(z)= lim Cp2g(z); Cp2g(z)= 1 2 2 g(x )dx .
G t1,t2→0+ t t (2p i)2ZG (z1−x 1)2+t12(z2−x 2)2+t22
ThelimitsdefiningCG ,CGp1,andCGp2aretakeninthefollowingpointwisesense:givenc∈Candct∈Cfort=(t1,t2)∈
(0,¥ )2, we say c →c ast ,t →0+ if for all e >0, there exists d >0 such that 0<t ,t <d implies |c −c|<e .
t 1 2 1 2 t
Wealsodefineconvergenceinnormedspacesast ,t →0+: givenanormedfunctionspaceX,F ∈X,andF ∈X for
1 2 t
t=(t1,t2)∈(0,¥ )2,wesayFt→F ast1,t2→0+if||Ft−F||X →0ast1,t2→0+. TheoperatorsCG g,CGp1g,andCGp2g
aredefinedinitiallyaspointwiselimitsontestfunctions,andwewillprovelaterthattheselimitsholdinLp(G )aswell
for1< p<¥ andappropriateg. TheseconvergenceresultswillbeprovedinSections5and6. Acrucialpartofthe
proofoftheseconvergenceresultsistheLp(G )boundednessofCG ,CGp1,andCGp2,whichwestatenowinTheorem2.
Theorem2. LetG beaproductLipschitzsurfacewithsmallLipschitzconstantinC2definedbyg =(g ,g ):R2→C2.
1 2
Assumethat
g (x ) g (x )
1 1 2 2
lim =c and lim =c
1 2
|x1|→¥ x1 |x2|→¥ x2
for some c1,c2 ∈C. Then the operators CG , CGp1, and CGp2 can be continuously extended to bounded operators on
Lp(G )andforg∈Lp(G )
lim Ctg=CG g, lim Ctp1g=CGp1g, and lim Ctp2g=CGp2g
t1,t2→0+ t1,t2→0+ t1,t2→0+
inLp(G )when1<p<¥ andpointwisealmosteverywhereonG .
We take a moment now to discuss why Theorem 2 cannot be proved with techniques currently available in the
literature;inparticularwhytheanalysisofholomorphicfunctionsrelatedtoH×HandtheHilberttransformsisnot
applicabletoourproblem.MuchofthemachineryusedintheanalysisofholomorphicfunctionsrelatedtoH×Hand
theHilberttransformsisnotavailablewhenwemovetothesettingofG =G ×G andtheCauchyintegraltransforms.
1 2
IfonedefinesthebiparameterHilberttransformHbp asCG isdefinedabove(butwithg j(xj)=xj for j=1,2),then
withtheaidoftheFouriertransformitiseasytoshowthatHbp=H H . SinceHbpcanberealizedasthiscomposition
1 2
ofH andH inthisway,theLp(R2)boundsforHbp triviallyfollowfromthoseofH andH . Furthermore,thefact
1 2 1 2
thatHbp=H H saysthatthetwodimensionallimitt ,t →0+ definingHbpcanactuallyberealizedasiteratedone
1 2 1 2
dimensionallimitst1→0+ andt2→0+. ThereisnosuchformulatowriteCG asacompositionCGp1 andCGp2 thatwe
know of since the Fouriertransformis nota viable tool in this setting; that is, it is notknownin generalif the two
dimensionallimitdefiningCG canbe realizedasan iteratedonedimensionallimit. Thisprecludes,atleast with the
toolscurrentlyavailable,anyrelativelysimpleproofofLp(G )boundsforCG ,andhencemotivatesthedevelopmentof
theharmonicanalysistheoryinthisarticle.
To obtain the pointwise convergenceresult stated in Theorem2, we need a little more that then boundednessof
thepartialandbiparameterCauchyintegraltransforms. Forz=(z ,z )∈G andappropriatefunctionsg:G →C,we
1 2
definethemaximalbiparameterCauchyintegraltransform
(1.3) CG∗g(z)= sup |Ctg(z)|,
t1,t2>0
wheret=(t ,t ). Then,weprovethefollowingboundednessresult.
1 2
HOLOMORPHICEXTENSIONONPRODUCTLIPSCHITZSURFACESINTWOCOMPLEXVARIABLES 5
Theorem3. LetG beasinTheorem2. ThemaximaloperatorCG∗extendstoaboundedoperatorCG∗:Lp(G )→Lp(G )
for1<p<¥ . Moreover,forallginLp(G ),CtgconvergestoCG galmosteverywhereonG .
We proveTheorem2 using theapproachthatDavid-Journe´-Semmesused to applytheir Tb theoremto proveLp
boundsforCauchyintegraltransformin[DJS85]. Forthis,weprovethefollowingreducedbiparameterTbtheorem.
Theorem4. Letb1,b˜1∈L¥ (Rn1)andb2,b˜2∈L¥ (Rn2)bepara-accretivefunctions,anddefineb(x)=b1(x1)b2(x2)
and b˜(x) =b˜1(x1)b˜2(x2) for x =(x1,x2) ∈Rn1+n2. Also let T be a biparameter operator of Caldero´n-Zygmund
typeassociatedtobandb˜. IfT satisfiestheweakboundednessproperty,mixedweakboundednessproperties,andthe
Tb=T∗b˜=0conditions,thenT canbecontinuouslyextendedtoaboundedlinearoperatoronLp(Rn)for1<p<¥ .
TherehavebeenanumberofresultsforbiparametersingularintegraloperatorsofCaldero´n-Zygmundtype,going
backto R. Fefferman, Stein, and Journe´, amongothers. There were differentversionsof T1 theoremsprovedin R.
Fefferman-Stein [FS82], Journe´ [Jou85], Pott-Villaroya [PV11], Han-Lin-Lee [HLL13], Ou [Ou13], and Hart-Lu-
Torres[HLT]. Infact,therecentarticles[HLL13,Ou13]includebiparameterTbtheoremsaswell. Theformulation
ofTheorem4isdifferentthantheonesin[HLL13,Ou13],andeventhedefinitionsofbiparameterCaldero´n-Zygmund
operatorsaredifferent.InSection4,wedefinebiparametersingularintegraloperatorsrelyingonlyoncontinuityintest
functionspaces,afullkernelrepresentation,andtestingconditionsonnormalizedbumps,whereasin[HLL13,Ou13]
the singularintegraloperatorsaddressedare requiredto havefullandpartialkernelrepresentationsas wellas some
aprioripartialL2 bounds. Inourformulation,wedonotusepartialkernelrepresentationsorpartialL2 boundedness
hypotheses. Instead, we introducea mixedweak boundedness,which is a testing conditionsimilar to the fullweak
boundednesspropertyused in many of the aforementionedworks. The formulationof Theorem4 in this work is a
natural extension of the single parameter theory, and the sufficient conditions seem to be easy to verify, as will be
demonstratedinSection5. Unfortunately,Theorem4isstillnotafullcharacterizationofLp boundsforbiparameter
Caldero´n-ZygmundoperatorssincedifficultiesofworkingwithproductBMOpersist,butthisreducedTb=T∗b=0
Theorem4issufficienttoprovethetheboundednessresultsinTheorem2andhencetheholomorphicextensionresult
ofinTheorem1. Theformulationofthebiparametersingularintegraloperatorsinthisworkisessentiallythesameas
theonebyHart-Lu-Torresin[HLT],butwerepeattheconstructionstofittheaccretivefunctionsettinginTheorem4.
EventhoughwewillonlyapplyTheorem4whenn =n =1,weproveitforgeneraldimensionsn ,n ∈N. Our
1 2 1 2
strategytoproveTheorem4istodecomposetheoperatorT,
hTf,gi= (cid:229) Q f,g ,
~k
~k∈Z2
(cid:10) (cid:11)
whereQ aresmoothtruncationsofT. ThesetruncationsQ arebiparameterLittlewood-Paley-Steinoperators,which
~k ~k
have been studied extensively in the single parametersetting, see e.g. [DJ84, DJS85, Sem90, Han94]. There are a
few results for biparameter Littlewood-Paley-Steinoperators due to R. Fefferman, Stein, and Journe´ [Fef81, FS82,
Fef86, Jou85], amongothers. All of these results are foroperatorsof convolutiontype. We proveestimates for the
square function associated to a larger class of operators including non-convolution operators, which we call bipa-
rameterLittlewood-Paley-Steinoperators. Inparticular,weproveboundsforsquarefunctionoperatorsassociatedto
biparameterLittlewood-Paley-Steinoperators,definedby
(1.4) Sf(x)2= (cid:229) |Q f(x)|2
~k
~k∈Z
forx∈Rnandappropriate f :Rn→C.
Theorem 5. Let b1 ∈L¥ (Rn1) and b2 ∈L¥ (Rn2) be para-accretive functions, and define b(x)=b1(x1)b2(x2) for
x=(x1,x2)∈Rn1+n2. Also let Q ~k for~k∈Z2 be a collection of biparameterLittlewood-Paley-Steinoperators with
kernelsq . If
~k
q (x,y)b (y )dy = q (x,y)b (y )dy =0
ZRn1 ~k 1 1 1 ZRn2 ~k 2 2 2
for all~k∈Z2 and x,y∈Rn, then ||Sf||Lp(Rn) .||f||Lp(Rn) for all f ∈Lp(Rn) when 1< p<¥ . Note that S is the
squarefunctionoperatordefinedin(1.4)
6 JARODHARTANDALESSANDROMONGUZZI
Infact,wewillproveTheorem5foraslightlylargerclassofoperatorsthanthebiparameterLittlewood-Paley-Stein
operators. Theseclassesofoperatorswillbedefinedinthecomingsections,anditwillbespecifiedhowtheycanbe
generalizedtoaslightlylargerclassbyweakeningtheregularitypropertiesofq .
~k
TheformulationsandproofsofTheorems4and5wereintroducedbyHart-Lu-Torres[HLT]inaslightlydifferent
setting,whereb=b˜=1.InSections3and4,wereproducetheproofsfrom[HLT],andaddresstheadditionaltechnical
difficultiesthatarisewhenaccretivefunctionsbandb˜ areusedinplaceof1.
Thisarticleisorganizedinthefollowingway.InSection2,weprovetheholomorphicextensionresultinTheorem
1assumingthatTheorem2holds.InSection3,wedevelopsomebiparameterLittlewood-Paley-Steintheoryandprove
Theorem5. InSection4,weprovethebiparameterTbTheorem4usingresultsfromSection3. FinallyinSection5,
we provepartofTheorem2byapplyingTheorem4toa parameterizedversionofCG , andweproveTheorem3. In
Section6weprovetherestofTheorem4byapplyingtheoneparameterTbtheoremfrom[DJS85]toparameterized
versionsofCp1andCp2.
G G
2. HOLOMORPHICEXTENSION FROM PRODUCT LIPSCHITZ DOMAINS
FixLipschitzfunctionsL ,L :R→RwithLipschitzconstantsl <1andl <1. Defineg (x )=x +iL (x ),
1 2 1 2 1 1 1 1 1
g (x )=x +iL (x ),andg (x)=(g (x ),g (x ))forx=(x ,x )∈R2.ThenG =G ×G isaproductLipschitzsurface
2 2 2 2 2 1 1 2 2 1 2 1 2
withsmallLipschitzconstantsinC2,whereG =g (R)andG =g (R). Itfollowsthat
1 1 2 2
0<1−l 2≤ (xj−yj)2−(Lj(xj)−Lj(yj))2 = |Re (g j(xj)−g j(yj))2 | ≤2.
j (x −y )2 (x −y )2
j j (cid:2) j j (cid:3)
Throughoutthiswork,wewillusethefactthatRe (g (x )−g (y ))2 and(x −y )2 arecomparablewithconstants
j j j j j j
only dependingon the Lipschitz constantsof g , noton x and y . We also remarkthat the normsof g and g◦g are
j j
comparableinthefollowingsense: foranyg∈Lp(G(cid:2)), (cid:3)
||g◦g ||Lpp(R2)≤||(g ′1)−1||L¥ (R)||(g ′2)−1||L¥ (R)||g||Lpp(G )≤||g||Lpp(G )
(2.1) ≤||g ′1||L¥ (R)||g ′2||L¥ (R)||g◦g ||Lpp(R2)≤2||g◦g ||Lpp(R2).
NotethatsinceRe[g ′(x )]=1forallx ∈R,wehave|g ′(x )|≥Re[g ′(x )]=1forallx ∈R. Nowgivenafunction
j j j j j j j j
g:G →C,wedefineforw =(w ,w )=(z +it ,z +it )∈(C\G )×(C\G )where(z ,z )∈G andt ,t 6=0,
t1 t2 1 1 2 2 1 2 1 2 1 2
1 g(x )dx
(2.2) G(w ,w )= .
t1 t2 (2p i)2ZG (x 1−w t1)(x 2−w t2)
Itfollowsthat
1
G(w ,w )= p (z −x )p (z −x )−q (z −x )q (z −x )
t1 t2 4ZG t1 1 1 t2 2 2 t1 1 1 t2 2 2
(cid:18)
+iq (z −x )p (z −x )+ip (z −x )q (z −x ) g(x )dx ,
t1 1 1 t2 2 2 t1 1 1 t2 2 2
(cid:19)
where
1 t 1 w
p (w )= j and q (w )= j forw ∈C.
tj j p w 2+t2 tj j p w 2+t2 j
j j j j
Alsodefinefort=(t ,t )∈(0,¥ )2,g :G →C,g :G →C,g:G →C,andz=(z ,z )∈G ,theoperators
1 2 1 1 2 2 1 2
P g (z )= p (z −x )g (x )dx , P g (z )= p (z −x )g (x )dx ,
t1 1 1 ZG t1 1 1 1 1 1 t2 2 2 ZG t2 2 2 2 2 2
1 2
and Pg(z)= p (z −x )p (z −x )g(x )dx .
t ZG t1 1 1 t2 2 2
WeusetheindicesofP ,P ,andP toidentifytheoperators.NotethatPg=P P gforg:G →C,whereweusethe
t1 t2 t t t1 t2
notation
P g(z)= p (z −x )g(x ,z )dx and P g(z)= p (z −x )g(z ,x )dx
t1 ZG t1 1 1 1 2 1 t2 ZG t2 2 2 1 2 2
1 2
HOLOMORPHICEXTENSIONONPRODUCTLIPSCHITZSURFACESINTWOCOMPLEXVARIABLES 7
Thisisanabuseofnotation,butitisclearincontextwhichoperatorisbeingused. Westartwithalemmaaboutthe
convergenceoftheoperatorsP g,P g,andPgforg∈Lp(G ).
t1 t2 t
Lemma 2.1. Let G be a product Lipschitz surface with small Lipschitz constants in C2 and g ∈Lp(G ) for some
1<p<¥ . Then
lim P g=g, lim P g=g, and lim Pg=g,
t1→0+ t1 t2→0+ t2 t1,t2→0+ t
whereeachlimitholdsinthetopologyofLp(G )andpointwisealmosteverywhereonG .
Proof. WefirstverifythatP 1=1foreach j=1,2. LetR>0and
tj
E ={z ∈G :|z |≤R}∪{z ∈C:|z |=R, Im(z )>L (Re(z ))}.
R j j j j j j j j
E is aclosed, andforR sufficientlylarge, itdefinestheboundaryofan open,simplyconnectedregionU ={z ∈
R R j
C:|z |<R, Im(z )>L (Re(z ))}. For z ∈G ,t >0, and R sufficiently large, it follows that z +it ∈U and
j j j j j j j j j R
z −it ∈/U . Then
j j R
t
j
x −(z −it )
j j j
isholomorphicinx onU forsuchz ,t ,andR. Usingthedecayof p andaresiduetheorem,itfollowsthat
j R j j tj
1 t
p (z −x )dx = lim j dx
ZG j tj j j j R→¥ p ZER (x j−(zj+itj))(x j−(zj−itj)) j
1 2p it
j
= lim =1.
R→¥ p (zj+itj)−(zj−itj)
ConsiderthefollowingparameterizedversionsofP,P ,andP : for f :R2→Candx∈R2
t t1 t2
P f(x)= p (g (x )−g (y ))g ′(y )f(y ,x )dy ,
t1 ZR t1 1 1 1 1 1 1 1 2 1
Pet2f(x)=ZRpt2(g 2(x2)−g 2(y2))g ′2(y2)f(x1,y2)dy2, and
Petf(x)=Pt1Pt2f(x)=ZR2pt1(g 1(x1)−g 1(y1))pt2(g 2(x2)−g 1(y2))g ′1(y1)g ′2(y2)f(y)dy.
ThekernelsofPet1,Pt2,andePet are
p (x ,y )=p (g (x )−g (y ))g ′(y ), p (x ,y )=p (g (x )−g (y ))g ′(y ),
t1e 1 e1 te1 1 1 1 1 1 1 t2 2 2 t2 2 2 2 2 2 2
and p(x,y)=p (x ,y )p (x ,y ), respectively.
t t1 1 1 t2 2 2
e e
NotethatP 1(x )=P 1(g (x ))=1forallx ∈R. AlsosincetheLipschitzconstantofL andL aresmall,itfollows
tj j e tj j ej e j 1 2
that
e 1 tj|g ′j(yj)| tj t−j 1
|p (x ,y )|= ≤ . .
tj j j p (cid:12)(cid:12)t2j +(g j(xj)−g j(yj))2(cid:12)(cid:12) t2j +(1−l 2j)(xj−yj)2 (1+t−j 1|xj−yj|)2
Tfohlelonw{sptthja:ttgj>◦eg0∈}Lfopr(mRs2)a,naanp(cid:12)(cid:12)(cid:12)dphreonxcimeathtiaotngt◦ogi(d·e,nxti)ty∈(cid:12)(cid:12)(cid:12)oLnpR(Rf)ofroeraaclhmjo=ste1v,e2r.yFxix∈gR∈.LNp(oGw)fifoxrxso∈meR1o<utspid<eo¥ f.aInt
2 2 2
appropreiateexceptionalset,sothat||g◦g (·,x2)||Lp(R)<¥ . Itfollowsthatg◦g (·,x2)∈Lp(R)andhencethat
t1l→im0+||Pt1(g◦g )(·,x2)−g◦g (·,x2)||Lp(R)=0.
Bydominatedconvergence,italsofollowsthat
e
lim ||P (g◦g )−g◦g ||p = lim ||P (g◦g )(·,x )−g◦g (·,x )||p dx =0.
t1→0+ t1 Lp(R2) ZRt1→0+ t1 2 2 Lp(R) 2
ThereforePt1(g◦g )→ge◦g in Lp(R2), and in light of (2.1)eit easily followsthat Pt1g→g in Lp(G ). By symmetry,
it follows that P g→g in Lp(G ) as well. Now for g∈Lp(G ), we verify that Pg→g in Lp(G ) as t ,t →0+ for
t2 t 1 2
e
8 JARODHARTANDALESSANDROMONGUZZI
1<p<¥ ,asdefinedintheintroduction.First,defineM tobetheHardy-Littlewoodmaximalfunctionactingonthe
1
firstvariableofafunction f :R2→C,i.e.
1
M f(x)=sup |f(y ,x )|dy ,
1 I∋x1|I|ZI 1 2 1
where the supremum is taken over all intervals I ⊂R that contain x . It is not hard to verify that M is bounded
1 1
on Lp(R2) for 1< p≤¥ and that |P h(g (x))|.M (h◦g )(x) uniformly in t >0 for any h∈Lp(G ). The Lp(G )
t1 1 1
convergenceofPgfollows:
t
t1l,ti2m→0||Ptg−g||Lp(G )≤t1l,ti2m→0||Pt1(Pt2g−g)||Lp(G )+||Pt1g−g||Lp(G )
.t1l,ti2m→0||M1(Pt2(g◦g )−g◦g )||Lp(R2)+||Pt1g−g||Lp(G )
.tl2i→m0||Pt2(g◦eg )−g◦g ||Lp(R2)+tl1i→m0||Pt1g−g||Lp(G )=0.
Inthelastline,weusethatPt2(g◦g )→g◦g inLep(R2)andthatPt1(g◦g )→g◦g inLp(R2). Thiscompletestheproof
oftheLp(G )convergencepropertiesinLemma2.1. Nowweprovethepointwiseconvergenceresults. Forg∈Lp(G ),
itfollowsthatg◦g (·,x2)∈Lep(R)foralmosteveryx2∈R. Forafixedx2∈Routsideofanappropriatemeasurezero
set,bytheLebesguedifferentiationtheoremitfollowsthat
lim P (g◦g )(x ,x )=g(g (x ,x ))
t1→0+ t1 1 2 1 2
for almost every x1 ∈R. Hence Pt1(g◦g )→eg◦g as t1 →0+ pointwise almost everywhere in R2 and hence that
P g→gast →0+ pointwisealmosteverywhereinG . Bysymmetry,P (g◦g )→g◦g ast →0+ pointwisealmost
t1 1 t2 2
everywhereinR2 andhencethatPeg→gast →0+ pointwisealmosteverywhereinG .
t2 2
NowweverifythepointwiseconvergenceforPgonG . Fixx∈R2suechthatP (g◦g )(x)→g◦g (x)ast →0+and
t t1 1
||g◦g (·,x2)||Lp(R)<¥ ,whichistrueforalmosteveryx∈R2. Nowwebound
e
|P(g◦g )(x)−g◦g (x)|≤|P (P (g◦g )−(g◦g ))(x)|+|P (g◦g )(x)−(g◦g )(x)|
t t1 t2 t1
(2.3) . p (g (x )−g (y ))|P (g◦g )(y ,x )−(g◦g )(y ,x )|dy
e ZeR te1 1 1 1 1 t2 e 1 2 1 2 1
+|P (g◦g )(x)−(g◦g )(x)|.
e t1
We verifythatthe first term of (2.3) tendsto zero ast ,t →0+: let e >0. Since P (g◦g )(y ,x )→(g◦g )(y ,x )
1 2 e t2 1 2 1 2
pointwise as t →0+ for almost every y ∈R, there exists d >0 such that 0<t <d implies |P (g◦g )(y ,x )−
2 1 2 t2 1 2
g◦g (y ,x )|<e for almost every y ∈R such that |x −y |≤1 (recall we have fiexed x and x ). The selection of
1 2 1 1 1 1 2
d does not dependon y1 as long as it is within the compactset defined by |x1−y1|≤1. Now wee take 0<t1,t2 <
min(d ,e )/(1+||g◦g (·,x2)||Lp(R)),whichispossiblesincex∈R2wasselectedsothat||g◦g (·,x2)||Lp(R)isfinite.Then
p (g (x )−g (y ))|P (g◦g )(y ,x )−g◦g (y ,x )|dy
ZR t1 1 1 1 1 t2 1 2 1 2 1
.e Z pt1(g 1(xe1)−g 1(y1))dy1
|x1−y1|≤1
t (|P (g◦g )(y ,x )|+|g◦g (y ,x )|)
+ 1 t2 1 2 1 2 dy
Z|x1−y1|>1 (g 1(x1)−g 1(y1))2+t12 1
e
(|P (g◦g )(y ,x )|+|g◦g (y ,x )|)
.e +t t2 1 2 1 2 dy
1Z|x1−y1|>1 (x1−y1)2 1
e 1
..ee ++tt11|(cid:16)|g||◦Petg2((·g,x◦2g))||(L·,px(R2))|.|Lpe(.R)+||g◦g (·,x2)||Lp(R)(cid:17)(cid:18)Z|x1−y1|>1(x1−dyy11)2p′(cid:19)p′
HOLOMORPHICEXTENSIONONPRODUCTLIPSCHITZSURFACESINTWOCOMPLEXVARIABLES 9
Itfollowsthatthefirsttermof(2.3)tendstozeroast ,t →0+foralmosteveryx∈R2. Thesecondtermin(2.3)also
1 2
tendstozeroast ,t →0+ sincexwaschosensothatP f(x)→ f(x)ast →0+. Againusing(2.1),iteasilyfollow
1 2 t1 1
thatPg→gast ,t →0+pointwisealmosteverywhereonG . (cid:3)
t 1 2
e
NowweproveTheorem1assumingTheorem2;wewillproveTheorem2inSection5.
Proof. Let1<p<¥ ,g∈Lp(G ),anddefineGasin(2.2).Notethatp (z −x )=−p (z −x )andq (z −x )=
−tj j j tj j j −tj j j
q (z −x )fort 6=0,z ∈G ,and j=1,2. Thenitfollowsthatfor(z ,z )∈G andt ,t >0,wehave
tj j j j j j 1 2 1 2
1
G(z +it ,z +it )= Pg(z)−Cg(z)+iCp1g(z)+iCp2g(z) ,
1 1 2 2 4 t t t t
1(cid:16) (cid:17)
G(z +it ,z −it )= −Pg(z)−Cg(z)−iCp1g(z)+iCp2g(z) ,
1 1 2 2 4 t t t t
1(cid:16) (cid:17)
G(z −it ,z +it )= −Pg(z)−Cg(z)+iCp1g(z)−iCp2g(z) ,
1 1 2 2 4 t t t t
1(cid:16) (cid:17)
G(z −it ,z −it )= Pg(z)−Cg(z)−iCp1g(z)−iCp2g(z) .
1 1 2 2 4 t t t t
ByTheorem2,itfollowsthatCG g,CGp1g,CGp2g∈L(cid:16)p(G )andCtg→CG g,Ctp1g→CGp1g,an(cid:17)dCtp2g→CGp2gast1,t2→0+
inLp(G )andpointwisealmosteverywhereonG . Thenforz=(z ,z )∈G
1 2
1
g++(z)= g(z)−CG g(z)+iCGp1g(z)+iCGp2g(z) ,
4
1(cid:16) (cid:17)
g+−(z)= −g(z)−CG g(z)−iCGp1g(z)+iCGp2g(z) ,
4
1(cid:16) (cid:17)
g−+(z)= −g(z)−CG g(z)+iCGp1g(z)−iCGp2g(z) , and
4
1(cid:16) (cid:17)
g−−(z)= g(z)−CG g(z)−iCGp1g(z)−iCGp2g(z) .
4
Thenitalso followsthat(1.1) holds, i. e(cid:16). g=g −g −g +g , as Lp(cid:17)(G )functionsand almosteverywhere
++ +− −+ −−
inG . ItisalsonothardtoverifythatG(w ,w )isholomorphicfor(w ,w )∈(C\G )×(C\G ): forz =(z ,z )∈
1 2 1 2 1 2 1 2
(C\G )×(C\G ),wehavethefollowingpowerseriesrepresentation
1 2
G(w ,w )= 1 (cid:229)¥ g(x )dx (w −z )k1(w −z )k2,
1 2 (2p i)2k1,k2=0(cid:18)ZG (x 1−z 1)k1+1(x 2−z 2)k2+1(cid:19) 1 1 2 2
when|w −z |<dist(z ,G )/2and|w −z |<dist(z ,G )/2. ThereforeGisaholomorphicextensionofg. (cid:3)
1 1 1 1 2 2 2 2
3. LITTLEWOOD-PALEYSQUARE FUNCTION THEORY
Inthissection,wedevelopsomebiparameterLittlewood-Paley-Steintheory. WeworkinarbitrarydimensionRn,
wheren=n +n . WestartbyfixingsomenotationanddefiningbiparameterLittlewood-Paley-Steinoperatorsand
1 2
squarefunction.Fork ∈Z,N >0,andx ∈R
j j j
F Nj(x )= 2njkj
kj j (1+2kj|xj|)Nj
for j=1,2. Againwe willusethe subscriptsof kj, Nj, andxj todistinguishbetweenfunctionsonRn1 andRn2. A
collectionoffunctionsq :R2n→Cfor~k∈Z2 isacollectionofbiparameterLittlewood-Paley-Steinkernelsifforall
~k
10 JARODHARTANDALESSANDROMONGUZZI
x1,y1,x′1,y′1∈Rn1 andx2,y2,x′2,y′2∈Rn2
(3.1) |q (x,y)|.F N1+g (x −y )F N2+g (x −y )
~k k1 1 1 k2 2 2
|q ~k(x,y)−q ~k(x′1,x2,y)|.(2k1|x1−x′1|)g
(3.2) × F N1+g (x −y )+F N1+g (x′ −y ) F N2(x −y )
k1 1 1 k1 1 1 k2 2 2
|q (x,y)−q (x ,x′,y)|.(2k2|x(cid:16)−x′|)g (cid:17)
~k ~k 1 2 2 2
(3.3) ×F N1(x −y ) F N2+g (x −y )+F N2+g (x′ −y )
k1 1 1 k2 2 2 k2 2 2
|q ~k(x,y)−q ~k(x,y′1,y2)|.(2k1|y1−y′1|)g (cid:16) (cid:17)
(3.4) × F N1+g (x −y )+F N1+g (x −y′) F N2(x −y )
k1 1 1 k1 1 1 k2 2 2
(cid:16) (cid:17)
|q ~k(x,y)−q ~k(x,y1,y′2)|.(2k2|y2−y′2|)g
(3.5) ×F N1(x −y ) F N2+g (x −y )+F N2+g (x −y′)
k1 1 1 k2 2 2 k2 2 2
for some N >n , N >n , and 0<g ≤1. We say that a co(cid:16)llection of operators Q for~k∈Z(cid:17)2 is a collection of
1 1 2 2 ~k
biparameterLittlewood-Paley-Steinoperatorsif
(3.6) Q f(x)= q (x,y)f(y)dy.
~k ZRn ~k
forsomecollectionofbiparameterLittlewood-Paley-Steinkernelsq satisfying(3.1)-(3.5).
~k
Remark3.1. Properties(3.1)-(3.5)holdifandonlyifq satisfiesthealternateconditionset:
~k
|q (x,y)|.F N1′(x −y )F N2′(x −y ),
~k k1 1 1 k2 2 2
|q ~k(x,y)−q ~k(x′1,x2,y)|.2n1k12n2k2(2k1|x1−x′1|)g′,
|q ~k(x,y)−q ~k(x1,x′2,y)|.2n1k12n2k2(2k2|x2−x′2|)g′,
|q (x,y)−q (x,y′,y )|.2n1k12n2k2(2k1|y −y′|)g′,
~k ~k 1 2 1 1
|q (x,y)−q (x,y ,y′)|.2n1k12n2k2(2k2|y −y′|)g′
~k ~k 1 2 2 2
forsomeN′ >n ,N′ >n ,and0<g ′≤1.
1 1 2 2
Proof. Itisobviousthat(3.1)-(3.5)implytheaboveconditionsetsinceF Nkjj(xj)≤2kjnj. AssumethereexistN1′ >n1,
N′ >n , and0<g ′ ≤1suchthatthealternateconditionsetholdsandchooseh ∈(0,1)smallenoughso thatN =
2 2 1
(1−h )N′−gh ′>n andN =(1−h )N′−gh ′>n ,whichispossiblesinceN′ >n andN′ >n .Alsodefineg =gh ′,
1 1 2 2 2 1 1 2 2
anditfollowsthat
h
|q (x,y)−q (x′,x ,y)|. 2k1n12k2n2(2k1|x −x′|)g′
~k ~k 1 2 1 1
(cid:16) × F N1′(x −y )+F(cid:17)N1′(x′ −y ) 1−h F N2′(x −y )1−h
k1 1 1 k1 1 1 k2 2 2
.(2k1|x1−x′1|)(cid:16)g F Nk11+g (x1−y1)+F Nk11+g (x′1(cid:17)−y1) F Nk22+g (x2−y2).
Theotherconditionsfollowbysymmetry,andhenc(cid:16)etheconditionsetsareequivalent(cid:17). (cid:3)
Weusethedefinitionofpara-accretivegivenbyHanin[Han94].
Definition3.1. A functionb∈L¥ (Rn) is para-accretiveif b−1∈L¥ (Rn) and there exists a c >0 such thatfor all
0
cubesQ⊂Rn thereexistsacubeR⊂Qsuchthat
1
b(x)dx ≥c .
|Q| ZR 0
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)