Table Of ContentApplied and Numerical Harmonic Analysis
Sagun Chanillo, Bruno Franchi
Guozhen Lu, Carlos Perez
Eric T. Sawyer
Editors
Harmonic Analysis,
Partial Differential
Equations and
Applications
In Honor of Richard L. Wheeden
Applied and Numerical Harmonic Analysis
SeriesEditor
JohnJ.Benedetto
UniversityofMaryland
CollegePark,MD,USA
EditorialAdvisoryBoard
AkramAldroubi GittaKutyniok
VanderbiltUniversity TechnischeUniversitätBerlin
Nashville,TN,USA Berlin,Germany
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ArizonaStateUniversity DukeUniversity
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Moreinformationaboutthisseriesathttp://www.springer.com/series/4968
Sagun Chanillo (cid:129) Bruno Franchi (cid:129) Guozhen Lu (cid:129)
Carlos Perez (cid:129) Eric T. Sawyer
Editors
Harmonic Analysis, Partial
Differential Equations and
Applications
In Honor of Richard L. Wheeden
Editors
SagunChanillo BrunoFranchi
Dept.ofMath DepartmentofMathematics
RutgersUniversity UniversityofBologna
Piscataway Bologna,Italy
NewJersey,USA
GuozhenLu CarlosPerez
Dept.ofMath. DepartmentofMathematics
Univ.ofConnecticut UniversityofBilbao
StorrsCT,USA Bilbao,Spain
EricT.Sawyer
DeptofMathandStat
McMasterUniversity
Hamilton
Ontario,Canada
ISSN2296-5009 ISSN2296-5017 (electronic)
AppliedandNumericalHarmonicAnalysis
ISBN978-3-319-52741-3 ISBN978-3-319-52742-0 (eBook)
DOI10.1007/978-3-319-52742-0
LibraryofCongressControlNumber:2017933468
MathematicsSubjectClassification(2010):35R03,35J70,42B20,42B37,42B25
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Richard L. Wheeden
Preface
It is a pleasure to bring out this volume of contributed papers on the occasion of
the retirement of Richard Wheeden. Dick Wheeden as he is known to his many
friendsandcollaboratorsspentalmostallhisprofessionallifeatRutgersUniversity
since 1967, other than sabbatical periods at the Institute for Advanced Study,
Princeton, Purdue University, and the University of Buenos Aires, Argentina. He
hasmademanyoriginalcontributionstoPotentialTheory,HarmonicAnalysis,and
PartialDifferentialequations.Manyofhispapershaveprofoundlyinfluencedthese
fields and have had long lasting effects, stimulating research and shedding light.
Inadditionmanycolleaguesandespeciallyyoungpeoplehavebenefittedfromthe
generosity of his spirit, where he has shared mathematical insight and provided
encouragement. We hope this volume showcases some of the research directions
DickWheedenwasinstrumentalinpioneering.
1 Potential TheoryandWeighted Norm Inequalities
for SingularIntegrals
DickWheeden’sworkinAnalysiscanbebrokenintotwoperiods.Thefirstperiod
consistsofhisworkinPotentialTheory,thetheoryofsingularintegralswithadeep
emphasison weightednorminequalities,and a second periodfromthe late 1980s
wherehe andhis collaboratorssuccessfullyappliedweightednorminequalitiesto
thestudyofdegenerateellipticequations,subellipticoperators,andMonge-Ampère
equations.
WheedenobtainedhisPh.D.in 1965fromtheUniversityofChicagounderthe
supervisionof AntoniZygmund.One veryproductiveoutcomeof this association
withZygmundisthebeautifulgraduatetextbookMeasureandIntegral[36].
vii
viii Preface
Wheeden’sthesisdealtwithhypersingularintegrals.Thesearesingularintegrals
oftheform
ˆ
(cid:2).y/
Tf .x/D .f .xCy/(cid:2)f .x// dy; 0<˛ <2;
Rn jyjnC˛
where(cid:2).y/ishomogeneousofdegreezero,integrableonSn(cid:2)1 andsatisfies
ˆ
y(cid:2).y/d(cid:3).y/D0; 1(cid:3)i(cid:3)n:
i
Sn(cid:2)1
Sincethesingularityofthe kernel (cid:2).y/ ismorethanthatofastandardCalderón-
jyjnC˛
Zygmundkernel,oneneedssomesmoothnessonf toensureboundedness.Atypical
resultfoundin[34]is
kTfkLp.Rn/ (cid:3)CkfkW˛;p.Rn/; 1<p<1;
where W˛;p.Rn/ is the fractional Sobolev space of order ˛. These results are
developedfurtherin[35].
Another important result that Wheeden obtained at Chicago and in his early
time at Rutgers was with Richard Hunt. This work may be viewed as a deep
generalizationofaclassictheoremofFatouwhichstatesthatnonnegativeharmonic
functionsintheunitdiskinthecomplexplanehavenontangentiallimitsa.e.onthe
boundary,thatisontheunitcircle.ThetheoremofFatouwasgeneralizedtohigher
dimensionsandotherdomainsbyCalderónandCarleson.Theworks[17,18]extend
theFatoutheoremto Lipschitzdomains,wherenowoneis dealingwith harmonic
measureontheboundary.Themainresultis
Theorem1 Let (cid:2) (cid:4) Rn be a bounded domain with Lipschitz boundary. Let
!P0.Q/,Q 2 @(cid:2),denoteharmonicmeasurewithrespecttoafixedpointP0 2 (cid:2).
Then any nonnegative harmonic function u.P/in (cid:2) has nontangentiallimits a.e.
withrespecttoharmonicmeasure!P0 on@(cid:2).
Theproofreliesonconstructingcleverbarriersandinparticularonapenetrating
analysis using Harnack’sinequality on the kernelfunction K.P;Q/, P 2 (cid:2), Q 2
@(cid:2),whichistheRadon-Nikodymderivative
d!P.Q/
K.P;Q/D :
d!P0.Q/
1.1 SingularIntegralsand Weighted Inequalities
In 1967, Wheeden moved to Rutgers University and began a long and fruitful
collaborationwithhiscolleagueB.Muckenhoupt.Twoexamplesofmanyseminal
Preface ix
resultsprovedbyMuckenhouptandWheedenarethetheoremsonweightednorm
inequalities for the Hilbert transform and the fractionalintegral operator. To state
theseresultswerecalladefinition.
Definition1 Let1 < p < 1,andletw 2 L1 .Rn/ beapositivefunctiononRn.
loc
Thenw2A ifandonlyifforallcubesQ,
p
(cid:2) ˆ (cid:3)(cid:2) ˆ (cid:3)
1 1 p(cid:2)1
sup w w(cid:2)p(cid:2)11 <1:
jQj jQj
Q Q Q
The A condition had already appeared in Muckenhoupt’s pioneering work on
p
the Hardy-Littlewoodmaximal function[51]. But now Wheeden along with Hunt
and Muckenhoupt [19] carried it further. They considered the prototypical one-
dimensionalsingularintegral,theHilberttransform,
ˆ
1 f .y/
Hf .x/Dp:v: dy;
x(cid:2)y
(cid:2)1
andestablishedthefollowingtrailblazingtheorem.
Theorem2 A nonnegativew 2 L1 .R/ satisfiestheLp weightednorm inequality
loc
fortheHilberttransform,
(cid:2)ˆ (cid:3) (cid:2)ˆ (cid:3)
1 1
p p
jHfjpw (cid:3)C jfjpw ;
p
R R
ifandonlyifw2A .
p
Theirkeydifficultyinestablishingthisresultwastoproveitwhenp D 2.Then
onecanadapttheCalderón-Zygmundschemeforsingularintegralsandfinishwith
an interpolation. The case p D 2 had been studied earlier by Helson and Szegö
[47]usingacompletelydifferentfunctiontheoreticapproach,wheretheyobtained
theequivalenceoftheweightednorminequalitywithasubtledecompositionofthe
weightinvolvingtheconjugatefunction.Theorem2finallycharacterizedthesetwo
equivalentpropertiesin termsof a remarkablysimple andcheckablecriterion,the
A condition.Theorem2 wasthe forerunnerto a delugeof resultsbyWheedenin
p
thedecadessince,tomultiplieroperatorsbyKurtzandWheeden[20],totheLusin
squarefunctionby Gundyand Wheeden[16] (precededbySegoviaandWheeden
[33]), and the Littlewood-Paley g(cid:3) function by Muckenhouptand Wheeden [24],
(cid:4)
tonamejusta few.With Muckenhoupt,Wheedenalso initiateda studyofthe two
weight theory for the Hardy-Littlewood maximal operator and Hilbert transform
[25]andwithChanilloastudyofthetwoweighttheoryforthesquarefunction[6].
Thatisonenowseeksconditionsonnonnegativefunctionsv;wsothatonehas
(cid:2)ˆ (cid:3) (cid:2)ˆ (cid:3)
1 1
p p
jTfjpv (cid:3)C jfjpw ;
p
Rn Rn
x Preface
where T could be a singular integral operator, a square function, or the Hardy-
Littlewoodmaximaloperator.Thepapers[25]and[6]stimulatedmuchresearchina
searchforanappropriatetwoweighttheoryforsingularintegrals.Intheearly1990s
Wheedenreturnedto this questionandundertooka studyof two weightproblems
forthefractionalintegral.Theseresultsaredescribedlaterinthispreface.
The later “one weight” results mentioned above relied on the so-called good-
(cid:4) inequalities [37], [40], a beautiful stratagem with which Wheeden was wholly
won over. We cite two instances of results proved by Wheeden, where good-(cid:4)
inequalities play a key step in the proofs. The first example is joint work with
Chanillo [1] where he investigated a complete theory of differentiation based on
theMarcinkiewiczintegral
!
ˆ 1
jf .xCt/Cf .x(cid:2)t/(cid:2)2f.x/j2 2
Mf .x/D dt :
Rn jtjnC2
ThisworkviewedMf asaroughsquarefunctionandtheaimwastotreatitinthe
spiritoftheworkofBurkholderandGundy[37]fortheLusinsquarefunctionand
establishcontrolviaagood-(cid:4)inequalityandmaximalfunctions.
ThesecondworkwithMuckenhoupt,destinedtoplayamajorroleinWheeden’s
interestindegenerateellipticPDEinthelate1980sonward,wasthepaper[23]on
fractionalintegraloperatorsI˛.Definefor0<˛ <n,
ˆ
f .y/
I˛f .x/D Rn jx(cid:2)yjn(cid:2)˛dy:
Theorem3 Letv beapositivefunctiononRn.Thenfor 1 D 1 (cid:2) ˛, 1 < p < n
q p n ˛
and 1p C p10 D1,theweightednorminequalityforI˛,
(cid:2)ˆ (cid:3) (cid:2)ˆ (cid:3)
1 1
q p
j.I˛f/vjq (cid:3)Cp jfvjp ;
Rn Rn
holdsifandonlyifforallcubesQ,
(cid:2) ˆ (cid:3) (cid:2) ˆ (cid:3)
1 1
1 q 1 0 p0
sup vq v(cid:2)p <1:
jQj jQj
Q Q Q
Thecorrespondinginequalityforthefractionalmaximaloperator
ˆ
1
M˛f .x/DQsWux2pQjQj1(cid:2)˛n Qjf.y/jdy;
´
which is dominated by the fractional integral I˛jfj.x/ D Rn jx(cid:2)jfy.yjn/(cid:2)j˛dy, can be
established by a variety of techniques, and from this, the inequality for the larger
