Table Of ContentInterpolation between Banach spaces and
continuity of Radon-like integral transforms
adiplomathesisby
Pavel Zorin-Kranich
3
1
0 Abstract
2
Wepresenttheabstractframeworkandsomeapplicationsofinterpola-
n tiontheory. ThemainnewresultconcernsinterpolationbetweenH1and Lp
a estimatesforanalyticfamiliesofoperatorsactingonSchwartzfunctions.
J
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r EberhardKarlsUniversitätTübingen
a
MathematischesInstitut
Advisors:
Prof.F.Ricci(ScuolaNormaleSuperiorediPisa,Italy)
Prof.R.Nagel(UniversitätTübingen,Germany)
PresentedinJanuary2011
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Contents
Contents iii
Introduction v
1 Realinterpolation 1
1.1 TheK-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 TheJ-methodandtheequivalencetheorem . . . . . . . . . . . . . . 2
1.3 Thereiterationtheorem,caseθ <θ . . . . . . . . . . . . . . . . . . 5
0 1
1.4 Lorentzspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Thereiterationtheorem,caseθ =θ . . . . . . . . . . . . . . . . . . 11
0 1
1.6 TheHardy-Littlewoodmaximalfunction. . . . . . . . . . . . . . . . . 13
1.7 Interpolationbetweendualspaces . . . . . . . . . . . . . . . . . . . . 15
1.8 Supplement: Maximalinequalityinlargedimension . . . . . . . . . 20
2 Complexinterpolation 25
2.1 Harmonicmajoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Thethreelineslemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Analyticityofvector-valuedfunctions . . . . . . . . . . . . . . . . . . 30
2.4 Intermediatespaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Interpolationbetween Lp spaces. . . . . . . . . . . . . . . . . . . . . . 36
3 Fractionalintegration 41
3.1 Rieszpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 CompositionofRieszpotentials . . . . . . . . . . . . . . . . . . . . . . 43
3.3 InverseofaRieszpotential . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Supplement: Localoperatorsaredifferential . . . . . . . . . . . . . . 47
4 TheRadontransform 51
4.1 Thesupporttheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Theinversionformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Lp discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Lp estimatesbycomplexinterpolation . . . . . . . . . . . . . . . . . . 57
4.5 ALorentzspaceestimateatthecriticalpoint . . . . . . . . . . . . . . 59
4.6 Radontransformasaconvolutionoperator . . . . . . . . . . . . . . . 62
5 Rearrangementinequalities 65
6 TheHardyspace H1 71
6.1 Atomicdecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
iii
6.2 Finiteatomicdecompositionandextensionofoperators . . . . . . . 78
6.3 BMO,thedualspaceofH1 . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 VMO,apredualofH1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.5 Thesharpfunctionandtheinverse Lp inequality . . . . . . . . . . . 86
6.6 Interpolationbetween Lp andH1 . . . . . . . . . . . . . . . . . . . . . 88
6.7 Whitneydecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.8 Calderón-Zygmunddecomposition . . . . . . . . . . . . . . . . . . . . 92
7 The k-planetransform 95
7.1 Measureequivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.2 Estimatesbyrearrangement . . . . . . . . . . . . . . . . . . . . . . . . 97
7.3 Estimatesbyinductiononk . . . . . . . . . . . . . . . . . . . . . . . . 101
7.4 EstimatesusingtheHardyspace. . . . . . . . . . . . . . . . . . . . . . 103
7.5 Thecomplexk-planetransform . . . . . . . . . . . . . . . . . . . . . . 108
8 Convolutionkernelssupportedonsubmanifolds 111
8.1 Sobolevspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2 Transportofmeasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.3 Lp improvement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Zusammenfassung 119
Index 123
iv
Introduction
Theinterpolationtheorydealswiththequestionwhatisagoodmethodtodefine
an interpolation space “between” two given Banach spaces both contained in a
largertopologicalvectorspace(e.g.two Lp spacesinsidethespaceofmeasurable
functions). Themethodshouldhavetheinterpolationproperty: givencompatible
(i.e.agreeingontheintersection)continuousoperatorsonbothspaces,onewould
likethemtoinduceacontinuousoperatorontheinterpolationspace. Thehopeis
thattheseoperatorsareeasiertoanalyzewhenconsideredontheboundaryspaces.
Theapplicationswehaveinmindareto Lp continuityofintegraloperatorsof
theform ˆ
Tf(y)= K(y,x)f(x),
M
where M is a manifold and the kernel K(y,·) is a distribution supported on a
submanifoldofstrictlypositivecodimension,e.g.onalinein(cid:82)n,n≥2.
Thefirstchaptersummarizesthestandardresultsonlinearrealinterpolation
byPeetre’sK-method(1963)andrelatedresultsforinterpolationofestimatesfor
multilinear forms. The first notable application of the abstract theory is the Lp
continuityoftheHardy-Littlewoodmaximaloperator, p>1.
Inchapter2wediscussCalderón’scomplexinterpolationmethod(1964)along
withprerequisitesfromcomplexanalysis,includingthecharacterizationofanalyt-
icityofvectorvaluedfunctionsonthecomplexplane. Wethenrelaxthehypothesis
oftheSteininterpolationtheorem(ageneralizationoftheRiesz-Thorintheorem)
as to include operators with small initial domain. This straightforward step is
essentialinwhatfollows. Weimmediatelyverifythatthisversionisapplicableto
thecomplexinterpolationspace[Lp0,Lp1]θ over(cid:82)n.
ThethirdchapterisabriefaccountofpropertiesofRiesztransforms. Theycan
bethoughtofasdifferentialoperatorsofnon-integerorder,inthesensethatthey
constituteananalyticfamilyofoperatorsandhappentobeordinarydifferential
operatorsforsomeintegerarguments.
The fourth chapter begins with the basic properties of the classical Radon
transform as an operator on the Schwartz space of test functions. We then find
therangeofexponentsinwhichtheRadontransformis Lp-continuous. Theproof
requires complex interpolation and we address the technical issues which were
leftimplicitintheoriginalliterature. WealsoconnecttheRadontransformtoa
convolutionoperatorontheHeisenberggroup.
In chapter 5 we clarify in which sense it is possible to transform a bounded
measurablesubset T of(cid:82)n intoaballbymeansofrearrangement. Thestandard
referenceforthistrickseemstobeFederer’sbook,whichonlyprovidesconvergence
tosomeballintheHausdorffdistance. Ourquantitativeargumentshowsthatthis
ballmusthavethesamemeasureas T. TheBrunn-Minkowskiinequality,themain
v
ingredientintheproofofarearrangementinequalityduetoBrascamp,Lieband
Luttinger(1974),isanimmediatecorollary.
ThenextchapterdealswiththeHardyspaceH1,whichisausefulsubstitute
for L1 ininterpolationtheory. Wearemostlyinterestedintheatomicstructureof
H1,i.e.thefactthateveryfunctioninH1 isalinearcombinationoffunctionswith
particularlyniceproperties. Weprovidethemostrefinedversionofthisdecompo-
sition. Therequiredmodificationstotheoriginalproofseemtobeknowntothe
expertsbuthavenotbeenwrittendownanywhere. Wementiontherecentresult
ofMeda,Sjögren,andVallarino(2008)clarifyinghowtheatomicdecomposition
isrelatedtothecontinuityofoperatorsonH1. TheclassicalproofthatH1 isthe
dualofVMO,thespaceoffunctionswithvanishingmeanoscillation,ispresented
in a simplified form. Our central result is the Proposition 6.37, which allows to
interpolatebetweenH1((cid:82)n)and Lp((cid:82)n)bymeansofSchwartzfunctions.
Chapter7containsapplicationsofrearrangementandinterpolationmethodsto
thek-planetransform. Wesimplifysomeargumentsandcarryoutanextensionto
thecomplexcase.
Thelastchapterdealswithcontinuityofconvolutionoperatorswithkernels
supportedonsubmanifoldsofLiegroups. Herethecentrallemmaregardstransport
ofmeasurebyasmoothmap. Werecastitinthelanguageofinterpolationtheory.
Acknowledgment
TheworkonthisthesisstartedattheScuolaNormaleSuperiorediPisawhereI
havespenttheacademicyear2009–2010thankstoanexchangeprogramofthe
UniversityofTübingen. IamgratefultoProf.FulvioRicciforthemotivationand
theguidancehehasprovidedmewithaswellasforhispatience. Thesupportof
Prof.RainerNagelwasinvaluablenotonlyinrelationtothistext.
Online version
InthepresentversionIhavecorrectedsometypographicalerrorspresentinthe
originalandcleaneduptheLATEXcode. Ihavealsotakenthelibertytoremovesome
unnecessaryfluffandaddanabstractinanefforttomakethetextmoreuseful.
vi
Chapter 1
Real interpolation
We review two equivalent real-variable methods for constructing interpolation
spaces between an appropriate couple of Banach spaces, mostly following the
expositionin[BS88].
TheMarcinkiewiczinterpolationtheoremthenallowsonetotransportestimates
onoperatorsontheendpointspacestointerpolationspaces. Itismostusefulin
conjunctionwiththeknowledgeofexplicitexpressionsforthenormsofthespaces
in question. These norms will be computed for interpolation spaces between
various Lp’s.
1.1 The K-method
Whenappliedto Lp spaces,theK-methodultimatelyboilsdowntodecomposition
ofafunctionintwopartsbyabsolutevalue. Theabstractapproachhereisdueto
Peetre[Pee63]. Itwillcomeinhandyintheproofsofthereiterationtheoremsfor
interpolation.
LetX andX beBanachspacescontainedinatopologicalvectorspace. The
0 1
K-functionalisdefinedby
K(f,t,X ,X )=inf{||f || +t||f || ,f = f +f }, for f ∈X +X .
0 1 0 X0 1 X1 0 1 0 1
Forevery0<θ <1and1≤q≤∞,the(θ,q;K,X ,X )-normonX +X isdefined
0 1 0 1
by
´
(cid:40)(cid:128) ∞(t−θK(f,t,X ,X ))qdt(cid:138)1/q, q<∞,
||f||θ,q;K,X0,X1 = sup0 t−θK(f,t,X0 ,X1 ), t q=∞.
t>0 0 1
We will call it just K-norm if the supplementary information is clear from the
context.
Theusefulnessofthisdefinitionstemsfromthefollowinginterpolationtheorem
foroperators.
Theorem1.1(Marcinkiewicz). Let T :X +X →Y +Y bealinearoperatorsuch
0 1 0 1
that
||Tf|| ≤M ||f|| ,j=0,1.
Y j X
j j
Then,forevery0<θ <1and1≤q≤∞,
||Tf||θ,q;K,Y0,Y1 ≤M01−θM1θ||Tf||θ,q;K,X0,X1.
1
2 CHAPTER1. REALINTERPOLATION
Proof. Bylinearitywehavethat
K(Tf,t,Y ,Y )≤ inf ||Tg|| +t||Th||
0 1 f=g+h Y0 Y1
≤ inf M ||g|| +M t||h|| =M K(f,tM /M ,Y ,Y ).
f=g+h 0 X0 1 X1 0 1 0 0 1
Insertingthisintothedefinitionofthe(θ,q;K,Y ,Y )-normyields
0 1
ˆ
(cid:18) ∞ dt(cid:19)1/q
||Tf||θ,q;K,Y0,Y1 ≤ (t−θM0K(f,tM1/M0,X0,X1))q t
0 ˆ
(cid:18) ∞ dt(cid:19)1/q
=M (M /M )−θ (t−θK(f,t,X ,X ))q
0 1 0 0 1 t
0
=M01−θM1θ||f||θ,q;K,X0,X1.
IfY andY areordered(say,Banachfunctionspaces),thentheassumptionsof
0 1
thetheoremmaybeweakenedastoincludesubadditiveoperators T. Thisstronger
versionwillbeusefulintheproofoftheHardy-Littlewoodmaximalinequality.
1.2 The J-method and the equivalence theorem
The J-method is modeled on dyadic decomposition by absolute value. Let X
0
and X beBanachspacescontainedinatopologicalvectorspaceanddefinethe
1
J-functionalby
J(f,t,X ,X )=max{||f|| ,t||f || }, for f ∈X ∩X .
0 1 X0 1 X1 0 1
Forevery0<θ <1and1≤q≤∞,the(θ,q;J,X ,X )-norm(orjustJ-norm)on
0 1
X +X isdefinedby
0 1
´
(cid:40)(cid:128) ∞(t−θJ(u(t),t,X ,X ))qdt(cid:138)1/q, q<∞,
||f||θ,q;J,X0,X1 =inuf sup0 t−θJ(u(t),t,X0 ,X1 ), t q=∞,
t>0 0 1
wher´etheinfimumistakenovermeasurablefunctionsu:(0,∞)→X ∩X such
that ∞u(t)dt/t= f withconvergenceinX +X . 0 1
0 0 1
Wenowshowthatthe K-andthe J-normareequivalent. Thisfactfurnishes
powerfulestimatesneededtoprovethereiterationtheorems. Theestimatesbelow
for f ∈X ∩X followimmediatelyfromthedefinitions.
0 1
K(f,t,X ,X )≤||f|| ≤J(f,s,X ,X ) forall t,s, (1.2)
0 1 X0 0 1
K(f,t,X ,X )≤t||f|| ≤t/sJ(f,s,X ,X ) forall t,s, (1.3)
0 1 X1 0 1
J(f,t,X ,X )≤J(f,s,X ,X ) for t≤s, (1.4)
0 1 0 1
J(f,t,X ,X )≤t/sJ(f,s,X ,X ) for t≥s. (1.5)
0 1 0 1
TheotheringredientsintheproofareHardy’sinequalitiesanddyadicversionsof
theK-andtheJ-norm.
1.2. THEJ-METHODANDTHEEQUIVALENCETHEOREM 3
Lemma1.6(Hardy’sinequalities). Letλ>0,1≤q<∞and f beameasurable
functionon[0,∞). Then
ˆ ˆ ˆ
(cid:130) ∞(cid:18) ∞ ds(cid:19)q dt(cid:140)1/q 1(cid:18) ∞ dt(cid:19)1/q
tλ f(s) ≤ (tλf(t))q , (1.7)
s t λ t
0 t 0
ˆ ˆ ˆ
(cid:130) ∞(cid:18) t (cid:19)q dt(cid:140)1/q 1(cid:18) ∞ dt(cid:19)1/q
t−λ f(s)ds ≤ (t1−λf(t))q . (1.8)
t λ t
0 0 0
Proof. Toshow(1.7)inthecaseq>1,write f(s)s−1=s−(λ+1)/q(cid:48)(s(λ+1)/q(cid:48)f(s)s−1).
ApplyingtheHölderinequalitytotheinnerintegralweobtain
ˆ ˆ ˆ
∞ ds (cid:18) ∞ (cid:19)1/q(cid:48)(cid:18) ∞ (cid:19)1/q
f(s) ≤ s−λ−1ds [s(λ+1)/q(cid:48)f(s)s−1]qds
s
t t ˆ t
(cid:18) ∞ (cid:19)1/q
=λ−1/q(cid:48)t−λ/q(cid:48) s(λ+1)(q−1)−qf(s)qds .
t
Theleft-handsideof(1.7)maythereforebeestimatedby
ˆ ˆ
(cid:18) ∞ ∞ dt(cid:19)1/q
...≤ tqλλ−q/q(cid:48)t−qλ/q(cid:48) s(λ+1)(q−1)−qf(s)qds
t
0 ˆ ˆ t
(cid:18) ∞ ∞ (cid:19)1/q
=λ−1/q(cid:48) tλ−1 s(λ+1)(q−1)−qf(s)qdsdt
ˆt=0ˆ s=t
(cid:18) ∞ s (cid:19)1/q
=λ−1/q(cid:48) tλ−1dts(λ+1)(q−1)−qf(s)qds
ˆ s=0 t=0
(cid:18) ∞ (cid:19)1/q
=λ−1 sλs(λ+1)(q−1)−qf(s)qds
ˆ s=0
1(cid:18) ∞ ds(cid:19)1/q
= [sλf(s)]q .
λ s=0 s
Thecaseq=1issimilarbuttheHölderinequalityisnotneeded. Theproofof(1.8)
isanalogousifwedecompose f as f(s)=s(λ−1)/q(cid:48)(s(1−λ)/q(cid:48)f(s)).
WearenowreadytoestimatetheK-normwiththeJ-norm. Letubeasabove.
Then
ˆ ˆ
∞ ∞
K(f,t,X ,X )≤ K(u(s),t,X ,X )ds/s≤ min{1,t/s}J(u(s),s,X ,X )ds/s
0 1 0 1 0 1
0 0
by(1.2)and(1.3). Inthecaseq<∞thisimpliesthat
ˆ ˆ
(cid:130) ∞(cid:18) t (cid:19)q dt(cid:140)1/q
||f||θ,q;K,X0,X1 ≤ t−θ J(u(s),s,X0,X1)ds/s t
0 0
ˆ ˆ
(cid:18) ∞(cid:18) ∞ (cid:19)q dt(cid:19)1/q
+ t−θ t/sJ(u(s),s,X ,X )ds/s
0 1 t
0 tˆ
(cid:18)1 1 (cid:19)(cid:18) ∞ dt(cid:19)1/q
≤ + (t−θJ(u(t),t,X ,X ))q
θ 1−θ 0 1 t
0
4 CHAPTER1. REALINTERPOLATION
by the Hardy inequalities (1.8) and (1.7) for the former and the latter term,
respectively. Ontheotherhand,inthecaseq=∞wehavethat
||f||θ,q;K,X0,X1 =stu>p0 t−θK(ˆf,t,X0,X1) ˆ
(cid:20) t ∞ t (cid:21)
≤supt−θ J(u(s),s,X ,X )ds/s+ J(u(s),s,X ,X )ds/s
t>0 0 0 1 ˆ t s ˆ 0 1
(cid:20) t ∞ t (cid:21)
≤supr−θJ(u(r),r,X ,X )supt−θ sθds/s+ sθds/s
r 0 1 t>0 0 t s
(cid:18)1 1 (cid:19)
≤ + supr−θJ(u(r),r,X ,X ).
θ 1−θ 0 1
r
Takingtheinfimumoveruyieldstheclaimedestimateinbothcases.
InordertoobtaintheconverseweconsiderthedyadicversionsoftheK-and
theJ-norm. Letλθ,q denotethespaceofsequences(aν)∞ν=−∞ suchthat
||(aν)ν||θ,q=(cid:40)(cid:128)su(cid:80)pν∞ν=2−−∞θν(a2ν−,θνaν)q(cid:138)1/q, qq=<∞∞, (1.9)
isfiniteanddefinethedyadicK-andJ-normsby
||f||θd,q;K,X0,X1 =||(K(f,2ν,X0,X1))ν||θ,q and
||f||θd,q;J,X0,X1 =infνf||(J(fν,2ν,X0,X1))ν||θ,q,
theinfimumthistimebeingtakenoverdecompositions f =(cid:80)ν fν with fν ∈X0∩X1
andconvergenceinX +X . Theequivalence
0 1
||f||θ,q;K,X0,X1 ≤C||f||θd,q;K,X0,X1 ≤C(cid:48)||f||θ,q;K,X0,X1
isclearfromthefactthatK(f,t)ismonotonousinitssecondargument. Further-
more,restrictingtheinfimumontheleft-handsidetopiecewiseconstantfunctions,
weobtain
||f||θ,q;J,X0,X1 ≤||f||θd,q;J,X0,X1.
InordertoestimatethedyadicJ-normandthuscompletetheproofofequivalence
of the norms we need to establish the existence of a particular decomposition
f =(cid:80)ν fν.
Lemma 1.10 (Fundamentallemmaofinterpolationtheory). Let f ∈X +X be
0 1
suchthatK(f,t)→0as t →0andK(f,t)/t →0as t →∞. Thenforeveryε>0
thereexistsadecomposition f =(cid:80)∞ν=−∞ fν withconvergencein X0+X1 suchthat
J(fν,2ν)≤3(1+ε)K(f,2ν)foreveryν ∈(cid:90).
Proof. BydefinitionoftheK-functionalthereexist fj,ν, j=0,1suchthat
||f0,ν||X0+2ν||f1,ν||X1 ≤(1+ε)K(f,2ν).
Bytheassumptions||f0,ν||X0 →0asν →−∞and||f1,ν||X1 →0asν →∞. Let
fν := f0,ν−f0,ν−1= f1,ν−1−f1,ν ∈X0∩X1.
