Wolfe’s
theorem
for
weakly
differentiable
cochains
✩Camille Petita, Kai Rajalaa,∗, Stefan Wengerb
a UniversityofJyväskylä,DepartmentofMathematicsandStatistics,P.O.Box35,
FI-40014UniversityofJyväskylä,Finland
bUniversitédeFribourg,Mathématiques,Ch.duMusée23,1700Fribourg,
Switzerland
Afundamental theoremof Wolfeisometrically identifies the spaceofflatdifferentialformsofdimensionm inRnwiththe spaceofflatm-cochains,thatis,thedualspaceofflatchains ofdimensionm inRn.Themainpurposeofthepresentpaper is to generalize Wolfe’s theorem to the setting of Sobolev differential forms and Sobolev cochains in Rn. A suitable theory of Sobolev cochains has recently been initiated by the second andthirdauthor. It isbased on the concept of upper normandupper gradientof acochain, introduced in analogy withHeinonen–Koskela’sconcept ofupper gradient ofafunction.
✩ C.P.andK.R.weresupportedbytheAcademyofFinland,project#257482.Partsofthisresearchwere carriedoutwhenK.R.wasvisitingUniversitédeFribourg.Hewouldliketothankthedepartmentforits hospitality.PartsofthisresearchwerecarriedoutwhenS.W.wasvisitingtheDepartmentofMathematics andStatisticsattheUniversityofJyväskylä.Hewouldliketothankthedepartmentforitshospitality.
* Correspondingauthor.
E-mailaddresses:[email protected](C. Petit),kai.i.rajala@jyu.fi(K. Rajala),
[email protected](S. Wenger).
Published in -RXUQDORI)XQFWLRQDO$QDO\VLV ± which should be cited to refer to this work.
1. Introduction
Inthe1940’s, Whitneyinitiatedageometricintegrationtheory,see[18],thepurpose of whichwasto integrate aquantityover “m-dimensionalsets” insuchaway thatthe integral depends on the position of the set in Rn. The quantities one integrates are
m-dimensionalflatformsand thesetsover whichoneintegratesare m-dimensionalflat chains.FlatformsareL∞-differentialformswithL∞-exteriorderivatives.Theflatnorm of suchaformis definedas the maximum ofthe L∞-norm of theform andthatof its derivative.Inorder to defineflatchains,onefirstconsiders thespacePm=Pm(Rn) of
polyhedralchainsinRn,thatisfinite formalsumsoforientedm-dimensionalpolyhedra
inRn withreal multiplicities,seeSection2forprecisedefinitions.Oneequips Pm with
theflatnorm,givenforT ∈ Pmby
|T |:= inf
M(R) + M(S) : R ∈ Pm, S∈ Pm+1, R + ∂S = T
,
where∂S istheboundaryofS andM(R) isthemassofR.ThecompletionofPmunder the flatnorm is called the space of flatm-chains inRn and denotedFm. Elements of
thedualspace ofFm arecalledflatm-cochains. In[19],seealso[18],Wolfeprovedthe
followingfundamentaltheorem:thespaceofflatm-forms,endowedwiththeflatnorm,is isometrictothespaceofflatm-cochains.Wolfe’stheoremhasrecentlybeen generalized tothesetting ofBanachspacesbySnipes in[17], whereshedefinesaflatpartial differ-ential forminaBanachspace andshows thatthespace of these forms isisometrically the dual space of the space of flat chains as defined by Adams [1]. Moreover, Wolfe’s theorem hasrecentlybeen usedbyHeinonen–Sullivan[13] and Heinonen–Keith[9], see also Heinonen–Rickman [12], to give conditions under which ametric space is locally bi-LipschitzequivalenttoRn.
ThemainpurposeofthepresentpaperistogeneralizetheclassicalWolfe’stheoremto thesettingof Sobolevdifferentialformsand SobolevcochainsinRn.A suitabletheory,
based on upper gradients, of Sobolev cochains incomplete metric measure spaces has recently been initiated by the second and the third author in [16]. Before stating our main results webriefly recall the relevantdefinitionsfrom [16], restrictingourselves to thesettingofRn.WerefertoSection2.3forprecisedefinitions.Asubadditivem-cochain
onPm isafunctionX :Pm→ R which satisfiesX(0)= 0 andwhichis subadditivein
thesensethat
X(T ) ≤ X(T + R) + X(R),
for all T,R ∈ Pm. If furthermore X(T + R) = X(T )+ X(R) whenever each term
is finite then X is called a (additive) cochain. In [16] a notion of upper gradient of a subadditive cochain is defined in analogy with Heinonen–Koskela’s notion of upper gradientofafunction[10,11].ABorelfunctiong :Rn → [0,∞] is calleduppergradient
ofX if
X(T ) ≤ˆ
Rn
g dS
for allT ∈ Pmand S∈ Pm+1 satisfying∂S = T ,whereT denotes themassmeasure
ofT ,see Section2.2.Similarly,aBorelfunctionh:Rn→ [0,∞] isanuppernorm ofX
if
X(T ) ≤ˆ
Rn
h dT
for all T ∈ Pm. Given an additive cochain X with upper norm in Lq(Rn) and upper
gradientinLp(Rn),wedefineitsSobolevnormby
Xq,p= max
infhq, infgp
,
where the infima are taken with respect to upper norms h and upper gradients
g of X, respectively. Note that this norm is different from but equivalent to the norm introduced in [16]. The Sobolev space Wq,p(Pm) of additive cochains is the
set of equivalence classes of additive cochains with upper norm in Lq(Rn) and
up-per gradient in Lp(Rn), under the equivalence relation defined by X
1 ∼ X2 if X1− X2q,p= 0.
The main resultof thepresent paper is the following generalization of the classical Wolfe’s theorem.
Theorem 1.1. Let 1 ≤ m ≤ n and 1 < q,p< ∞. If p > n− m or q ≤ n−ppn then the spaceWdq,p(Rn,∧m) ofSobolevdifferentialformsisisometricallyisomorphictothespace
Wq,p(Pm).
Recall thatthespace Wdq,p(Rn,∧m) ofSobolevdifferential m-formsconsists ofthose
Lq-integrable differential m-forms ω whose distributional exterior derivatives dω are
Lp-integrable.It isendowedwiththenorm
ωq,p= max ˆ ω(x)q dx 1 q ,ˆ dω(x)pdx 1 p ,
whereω(x) denotesthecomassnormofω(x).SeeSection2fortheprecisedefinitions. It is well-known that the Sobolev space W1,p of functions is (for all p ≥ 1)
isomor-phicto theNewtonian Sobolev spaceN1,p definedusing uppergradients,cf. [11].This remains true in the setting of Theorem 1.1 when m = 0, cf. [16, Proposition 3.11].
Theorem 1.1 gives apartial answer to the question whether thesame result holds for
m≥ 1.
TheclassicalWolfe’s theoremisaneasyconsequenceofTheorem 1.1.
Corollary 1.2. For1≤ m≤ n the spaceWd∞,∞(Rn,∧m) is isometrically isomorphic to
thedual spaceofthespaceFmof flat chains inRn.
Here,Wd∞,∞(Rn,∧m) isendowed with the norm ω
∞,∞ which is defined similarly
toωq,p butusingtheessentialsupremumofthepointwisecomassnorms.
Asmentionedabove,Heinonen–Sullivan[13]andHeinonen–Keith[9]appliedWolfe’s theorem inorder to give conditions underwhich a metric space is locally bi-Lipschitz equivalenttoRn.Findingsimilarconditionsforquasiconformalequivalenceisan
interest-ingopenproblem.Toattackthisproblem,itisdesirabletofindgeneralizationsofWolfe’s theoremforSobolevforms.Theorem 1.1waspartiallymotivatedbythisapplication.
WedonotknowwhetherTheorem 1.1holdsforallvaluesofq andp.However,wehave an unconditional result forcochains on P0
m, the space of polyhedral m-chainswithout
boundary. In order to state our result, define the norm of a cochain X on P0
m with
uppergradientinLp(Rn) byX
p= infgp,wheretheinfimumistakenwith respect
to upper gradients g of X. Let Wp(Pm0) be the set of equivalence classes of additive
cochains onP0
m with upper gradientinLp(Rn),under theequivalence relationdefined
by X1 ∼ X2 if X1 − X2p = 0. Denote furthermore by Wpd(Rn,∧m) the quotient
spaceofthespaceofm-formsonRn withcoefficientsinL1
loc(Rn) andcoefficientsofthe
distributional exteriorderivative inLp(Rn) by the subspace of those elements ω with
dω = 0.For[ω]∈ Wpd(Rn,∧m) define[ω]
p :=dωp, where·p denotes theLp-norm
ofthepointwisecomassnorm.ThisdefinesanormonWpd(Rn,∧m) whichisboundedby thequotientnorm.Ourresultcan nowbestatedas follows.
Theorem1.3.For1≤ m≤ n− 1 and1< p<∞ thespaceWpd(Rn,∧m) isisometrically
isomorphic toWp(Pm0).
WebrieflyoutlinetheproofsofTheorems 1.1 and1.3.Asmoothcompactlysupported differentialformnaturallyinducesaSobolevcochainbyintegration.Usingthis observa-tionandapproximationof Sobolevforms bysmoothforms,weshowthatthereexists a linear,norm-preservingmapmappingthespaceofSobolevformstothespaceofSobolev cochainswithcorrespondingexponents.Ontheotherhand,weconstructaSobolevform fromaSobolev cochain as follows: we restrict thecochain to the m-planes inducedby coordinate vectors, and then use Lebesgue differentiation to construct the coefficients ofthe resultingform.Themap definedthis wayis alsolinearand norm-preserving.To proveTheorem 1.1,weshowthatthetwomapsareactuallyinversestoeachother.The main problem inshowing this is thatit isdifficult to see whythe restrictionof a non-zero Sobolev cochain to the coordinate m-planes should be non-zero. In other words, whyshould anon-zero Sobolevcochain induce anon-zero Sobolevform? To overcome thisproblem,we“smoothen” thecochains byapplyingaverages.Given acochainX on Pm andr > 0 weset
Xr(T ) := B(0,r)
X(ϕx#T )dx
for every T ∈ Pm, where ϕx : Rn → Rn is thetranslation map ϕx(y) = x+ y. Here,
ffl
E denotes the integral averageLn(E)−1
´
E and Ln denotes Lebesgue measure.The
integrand is measurable and locally integrableunder ourassumptions, see Lemma 2.6. Using the Federer–Fleming deformation theorem, we show that the cochains Xr are
determined by their action inthe coordinate m-planes. Theorem 1.1 follows if we can showthatthecochainX canbeapproximatedbythecochainsXr.Thisisgivenbythe
following continuityresult,whichis ofindependentinterest.
Theorem 1.4.Let0≤ m≤ n and1< p,q≤ ∞. If X ∈ Wq,p(Pm) then
Xr(T )− X(T ) → 0 as r → 0
for every T ∈ Pm\ Λ for some family Λ ⊂ Pm of zero ν-modulus, where ν = q if
p> n− m and
ν = minq, pn/(n− p) otherwise.
Proving Theorem 1.1 for all exponents p and q would require a stronger form of
Theorem 1.4.Themodulusappearinginthestatementmeasures thesize ofexceptional sets,seeSection2.4forthedefinition.AsimilarstatementholdsforcochainsonP0
m,see
Theorem 3.1.This,togetherwiththeargumentsaboveandconsiderationsinvolvingthe so-calledcoboundaryofacochain,areusedto proveTheorem 1.3.
Wefinally mentionthatadifferentvariantofWolfe’stheoremforSobolevforms was givenin[6].Inthatpaper,theauthorsprovideaone-to-onecorrespondencebetweenthe forms inWdq,p(Rn,∧m) andlinearfunctionalsonP
mwhich,togetherwiththeirexterior
derivatives,satisfycertainboundednessconditionswithrespecttoaso-calledq-massand
p-mass.WebelievethatthenotionofSobolevcochainusedinthepresentpaperismore naturalthantheonedefinedin[6].
Our paperis structuredas follows. InSection2,we recallthe definitionsof Sobolev forms, polyhedral chains, and Sobolev cochains. In Section 3, which is the most sub-stantial partof the paper, we prove the main continuity result, Theorem 1.4, and an analogous version for cochains on P0
m, see Theorem 3.1. In Section 4, we construct a
linearmap fromthespaceof Sobolevforms tothe spaceofSobolev cochainsandshow thatthismappreservesnorms.InSection5,weconstructacontinuouslinearmapfrom thespaceofSobolevcochainsto thespaceofSobolevforms.Finally,Section6contains theproofsofTheorems 1.1 and1.3.
2. Preliminaries
Inthissectionwecollectthedefinitionsofthebasicobjectsofthepresentpaper.
2.1. Sobolevdifferential formsin Rn
We recall the definition of Wdq,p(Rn,∧m), the space of (weak) Sobolev differential
m-formsonRn. Wereferto[14] fordetails.Let1≤ m≤ n andset
Λ(m, n) :=α :{1, . . . , m} → {1, . . . , n} strictly increasing.
Letω beanm-formonRn, giveninEuclidean coordinatesby
ω =
α∈Λ(m,n)
ω(·, α) dxα,
withlocallyintegrablecoefficientsω(·,α).An(m+ 1)-formdω onRn,giveninEuclidean coordinatesby
dω =
β∈Λ(m+1,n)
dω(·, β) dxβ
and withlocally integrablecoefficientsdω(·,β),is saidto be thedistributionalexterior derivativeofω,if ˆ Rn dω∧ ν = (−1)m+1 ˆ Rn ω∧ dν
for every C∞-smooth compactly supported(n− m− 1)-form ν.Given 1≤ p,q ≤ ∞,
the space Wdq,p(Rn,∧m) consists of (equivalence classesof) m-forms ω on Rn with
co-efficients ω(·,α) inLq(Rn) and suchthat ω has adistributionalexterior derivativedω
withcoefficientsdω(·,β) inLp(Rn).
WeendowWdq,p(Rn,∧m) withthefollowingnorm,whichisdifferentbutequivalentto
thenormconsideredin[14]and[16].Forthis,denoteby|·| thenormonthespace∧mRnof
m-vectorsassociatedwiththeinnerproductforwhich{eα(1)∧ · · · ∧eα(m): α∈ Λ(m,n)}
is an orthonormal basis. Here and throughout the text, ej denotes the j-th standard
unit vector inRn. Anm-vector ξ ∈ ∧
mRn is called simple if it can be writtenin the
formξ = ξ1∧ · · · ∧ ξm for vectorsξi ∈ Rn,i = 1,. . . ,m. Thecomass of anm-covector
ν ∈ ∧mRn isdefinedby
ν = sup ν, ξ : ξ ∈ ∧mRn simple, |ξ| ≤ 1
,
where ·,· denotesthe naturalpairingof m-covectorsandm-vectors. Givenan m-form ω onRn,withcoefficientsinLq(Rn),weset
ωq:=
ˆ ω(x)q
dx
1
q
ifq <∞,whereω(x) denotesthecomassofω(x).Wedefineωqanalogouslyincase
q =∞.TheSobolevnorm ofanelementω∈ Wdq,p(Rn,∧m) isthendefinedby
ωq,p= max
ωq,dωp
.
Finally,wedenotebyWd,loc1,p (Rn,∧m) the spaceofm-formsonRn withcoefficientsin
L1loc(Rn) andcoefficients ofthe distributionalexteriorderivativeinLp(Rn). Thespace Wdp(Rn,∧m) isthequotientofWd,loc1,p (Rn,∧m) bythesubspaceofthoseelementsω with dω = 0. Thenormofanelement[ω]∈ Wdp(Rn,∧m) isdefinedby
[ω]
p:=dωp.
This clearly defines a norm on Wpd(Rn,∧m). Note that if m = 0 and f is a Sobolev
functionthen[f]p=|∇f|p.
2.2. Polyhedralchains in Rn
We recall thebasicdefinitions relatedto polyhedralchains. Werefer to [18] and[8]
for furtherdetails.Let 0≤ m≤ n.Formally, apolyhedral m-chainT inRn isaformal
finite sum T = N i=1 aiTi, (2.1)
where ai ∈ R and Ti is an oriented m-dimensional polyhedron in Rn in case m ≥ 1
and Ti is a point in Rn in case m = 0. More precisely, consider the additive groupof
formal finite sums (withcoefficients inR)of compact, convex, oriented m-dimensional
polyhedra (respectively, points if m = 0). Then, quotient by the equivalence relation identifying −T withT ,˜ where T is˜ T with the oppositeorientation, and identifying T
withT1+ T2ifT isformedbygluingT1andT2alongafacewiththecorrectorientation.
Thequotientgroupisthesetofpolyhedralm-chains inRn,whichwedenotebyP
m(Rn)
or Pm forshort.
It is easily seenthat forevery T ∈ Pm there exists arepresentation (2.1)such that
the Ti havenon-overlappinginteriors. Weassociateto apolyhedral m-chainT ∈ Pm a
finite measure,denotedbyT anddefinedby
T :=
N
i=1
|ai|HmTi,
where aiTi isanon-overlappingrepresentationofT .Here,Hmdenotesthe
m-dimen-sionalHaudorffmeasure.ThenumberT (Rn) is called themass ofT and denotedby
M(T ).Itisworthmentioningthatapolyhedralm-chainT givesrisetoanm-dimensional
normalcurrentinRnbyintegratingsmoothcompactlysupportedm-formsoverT ,see[4],
andthusalsoto anm-dimensionalmetriccurrentinthesenseof[3].
Theboundary ∂T ofT ∈ Pm,1≤ m≤ n,isapolyhedral(m− 1)-chaindefinedinthe
usual way, namely theboundary of apolygonis the sumof its faces withthe induced orientations. IfT = aiTi is apolyhedral 0-chainthen wewrite ∂T = 0 ifand onlyif
ai= 0.Notethatwehave∂∂T = 0 foreveryT ∈ Pmwithm≥ 2.DenotebyPm0 the
setofpolyhedralm-chainsT ∈ Pmwith∂T = 0,andPm+ thesetofpolyhedralm-chains
T such thatforthenon-overlapping representationT = aiTi mentionedabove,each
polyhedron Ti is parallel to one of the m-dimensional coordinate planes. Note thatif
T ∈ Pm+ then,ingeneral,∂T neednotbe inPm−1+ .
If T = Ni=1aiTi is apolyhedral m-chain and ϕ is an affinemap, then ϕ#T is the
polyhedralm-chaindefinedby
ϕ#T =
N
i=1
aiϕ(Ti)
and is called the push-forward of T bythe map ϕ. If x∈ Rn, we use the notation ϕ x
forthetranslationmapy→ x+ y.Thepush-forwardϕx#T isthussimplythetranslate
of T .
2.3. Sobolevcochains inRn
Werecall thebasic definitionsfromthetheory of weaklydifferentiable cochains ini-tiated bythesecond and thirdauthor in[16]. Whereas thedefinitionsin[16]are given forarbitrary complete metricspacesX and metricnormalor integralcurrents inX in
thesense ofAmbrosio–Kirchheim [3]wewill restrictourselvestothe settingofRn and
thespacesPmand Pm0 ofpolyhedralchainsinRn inthis paper.Wewill thereforeonly
givetherelevantdefinitionsinthissetting.
Definition 2.1. Let0≤ m≤ n. Afunction X :Pm→ R is called asubadditive cochain
on Pm ifX(0)= 0 and
X(T ) ≤ X(T + R) + X(R) forallT,R∈ Pm.Iffurthermore
X(T + R) = X(T ) + X(R)
whenevereachtermis finite,then X iscalledan additivecochain,or simplyacochain, onPm.
CochainsonPm0 aredefinedbysimplyreplacingPmbyPm0 everywhereinthedefinition
above.
A largeclassof additivecochainscomesfromdifferentialforms.
Example 2.2. Given a(smooth) differential m-form ω on Rn, we can define acochain
Xω:P
m→ R bysettingXω(T )=
´
Tω foreveryT ∈ Pm.
The following notions of upper norm and upper gradient of a subadditive cochain definedin[16]areinanalogy withthedefinitionofuppergradientof afunction. Definition 2.3.LetX beasubadditivecochain onPm.
(i) ABorelfunctionh:Rn→ [0,∞] iscalledupper normof X if
X(T ) ≤
ˆ
Rn
h dT (2.2)
foreveryT ∈ Pm.
(ii) ABorelfunctiong :Rn→ [0,∞] iscalledupper gradientof X if
X(∂S) ≤
ˆ
Rn
g dS (2.3)
forallS∈ Pm+1.
The uppernorm andupper gradientofcochains onP0
mare definedanalogously. We
note here thatthroughoutthis paper, when dealing with cochainson P0
m we will only
use uppergradients. In[16], theauthors provedthatupper gradientsof 0-cochainsare exactlyuppergradientsoffunctions.
For 1 ≤ p,q ≤ ∞ denote by Wq,p(Pm) the set of subadditive m-cochains which
have anupper norm in Lq(Rn) and anupper gradient inLp(Rn). In[16] thenotation
Wq,p(Pm,Pm+1) wasused. Wedefine the Sobolev norm of asubadditive cochain X ∈
Wq,p(Pm) by Xq,p= max infhq, infgp ,
where theinfimaaretakenwith respectto uppernormsh anduppergradientsg ofX,
respectively.Thisnorm isdifferentfrombutequivalenttothenormintroduced in[16]. Given two (additive) cochains X1,X2 : Pm → R the cochain X1+ X2 is defined
by (X1+ X2)(T ) = X1(T )+ X2(T ) if |X1(T )|+|X2(T )| < ∞, and (X1+ X2)(T ) = ∞ otherwise. The space Wq,p(Pm) is then defined as the set of equivalence classes of
additive cochains in Wq,p(Pm) under the equivalence relation defined by X1 ∼ X2 if X1− X2q,p= 0.In[16]thenotationWq,p(Pm,Pm+1) wasusedinstead.Itisclearthat
Wq,p(Pm) isavectorspace.Notethattheclassicalspaceofflatcochainsisisometrically
isomorphictoW∞,∞(Pm),see Lemma 6.2.
For1≤ p≤ ∞ denotebyWp(P0
m) thesetofsubadditivecochainsonPm0 whichhave
anuppergradientinLp(Rn).TheSobolevnorm ofanelement X ∈ W
p(Pm0) isdefined
by
Xp= infgp,
wheretheinfimumistakenwithrespect touppergradientsg ofX.ThespaceWp(Pm0)
isdefinedtobetheset ofequivalenceclassesofadditivecochainsinWp(Pm0) underthe
equivalencerelationdefinedbyX1∼ X2 ifX1− X2p = 0.
Thecoboundary dX ofasubadditivem-cochainX isthesubadditive(m+ 1)-cochain definedbydX(S)= X(∂S) forallS∈ Pm+1.ItfollowsfromthedefinitionthataBorel
functionis anuppergradient ofX ifand onlyifit isanupper normof dX.Notethat forall1≤ q,p≤ ∞ andall1≤ s≤ ∞ thecoboundaryoperatoryields alinearmap
d : Wq,p(Pm)→ Wp,s(Pm+1).
2.4. Modulus andcapacityforpolyhedral chains
LetnowΛ⊂ Pm beafamilyofpolyhedral m-chainsand 1≤ p<∞. Thep-modulus
Mp(Λ) isdefined as inf
´
RnfpdLn, where the infimum is taken over all non-negative Borelfunctions f such that´Rnf dT ≥ 1 foreveryT ∈ Λ. Thetheory of p-modulus ofgeneralmeasures wasinitiatedbyFuglede[5]. InFuglede’sdefinition,thep-modulus
is defined fora familyof measures inametric measure space. The abovedefinition of
p-modulusisexactlytheoneofFugledeforthefamilyofmeasures {T : T ∈ Λ}. Note that thep-modulus is anouter measure on the set of polyhedral m-chains Pm. Given
Λ⊂ P0
m,thep-capacity capp(Λ) isdefinedby
capp(Λ) := Mp(Γ ),
where Γ = {S ∈ Pm+1 : ∂S = T for some T ∈ Λ}. We define the (q,p)-capacity of a
family Λ⊂ Pmby capq,p(Λ) := inf ˆ f1qdLn+ ˆ f2pdLn ,
where the infimum is taken over all non-negative Borel functions f1 ∈ Lq(Rn) and f2 ∈ Lp(Rn) satisfying´ f1dR+´ f2dS≥ 1 for everydecomposition R + ∂S∈ Λ
withR∈ PmandS ∈ Pm+1.Noticethatcapq,p(Λ)= 0 ifandonlyifthereexistΛ1⊂ Pm
and Λ2 ⊂ Pm0 with Mq(Λ1)= capp(Λ2) = 0 and such thatif T = R + ∂S ∈ Λ, then R ∈ Λ1 or ∂S ∈ Λ2. This notion of (q,p)-capacity is adapted to the set of cochains
Wq,p(Pm) inthesense thatXq,p= 0 if and onlyif X(T )= 0 foreveryT ∈ Pm\ Λ,
where capq,p(Λ)= 0.
Withthenotionofmodulusavailable,onedefinesweak versionsofuppernormsand upper gradientsasfollows. Given acochainX onPmaBorelfunctionh:Rn → [0,∞]
is said to be aq-weak upper norm of X if there exists Λ ⊂ Pm with Mq(Λ) = 0 such
that(2.2)holdsforeveryT ∈ Pm\ Λ.Similarly,aBorelfunctiong :Rn→ [0,∞] issaid
to beap-weakupper gradientof X ifthereexistsΓ ⊂ Pm+1withMp(Γ )= 0 suchthat
(2.3)holdsforeveryS∈ Pm+1\ Γ .
We will make frequent use of Fuglede’s lemma [5] which, in our setting, reads as follows.
Lemma2.4 (Fuglede’slemma).Let1≤ p<∞,andletf beaBorelfunction.Moreover, let (fj) be asequenceof Borel functionsconverging to f inLp(Rn). Then thereexist a
subsequence(fjk) andΛ⊂ Pm withMp(Λ)= 0 such that
ˆ
Rn
|fjk− f| dT → 0
forevery T ∈ Pm\ Λ.
As aconsequence,oneobtainsthatif1< q,p<∞ thenLq(Rn)-boundedsequences
of upper norms converge,up to a subsequence, to q-weak upper norms, and similarly,
Lp(Rn)-boundedsequencesofuppergradientsconverge,upto asubsequence,top-weak
upper gradients(see [16]fordetails).Inparticular,theinfimuminthedefinitionof the Sobolev norm of acochain is attained by some q-weak upper norm and some p-weak
uppergradient.
We end this section with the following useful observation proved in [16, Proposi-tion 4.17].
Lemma 2.5. Let0≤ m≤ n and T ∈ Pm with T = 0. LetB ⊂ Rn be a Borelset with
Ln(B)> 0. ThenthesetΛ:={ϕ
x#T : x∈ B} hasMq(Λ)> 0 forevery q≥ 1.
2.5. Averagesof cochains
Let 0 ≤ m ≤ n and 1 ≤ p,q ≤ ∞. Given an additive cochain X ∈ Wq,p(Pm) and
r > 0,define anadditivecochainXr:Pm→ R by
Xr(T ) := B(0,r)
X(ϕx#T )dx
for every T ∈ Pm, where ϕx : Rn → Rn is thetranslation map ϕx(y) = x+ y. Here,
ffl
E denotes the integral average Ln(E)−1
´
E and B(0,r) is the open ball of radius r
centeredat 0. Foradditive m-cochains onintegral or normalcurrents with m≤ n− 1
themeasurabilityandthe localintegrabilityof thefunctionx→ X(ϕx#T ) wasproved
in[16, Lemma4.14(i)].Wehavethefollowinganalogforcochainsonpolyhedralchains. Lemma 2.6. Let 0 ≤ m≤ n and 1≤ p,q ≤ ∞ and let X ∈ Wq,p(Pm) be an additive
cochain.Thenforevery T ∈ Pm thefunctionu:Rn→ R given by u(x):= X(ϕx#T ) is
Lebesguemeasurable andlocallyintegrable.
ThesameresultholdswhenX isanadditivecochaininWp(P0
m) andT ∈ Pm0 andwe
can thus define Xr inthis case as well.It isnot difficultto show thatXr ∈ Wq,p(Pm)
foreveryr > 0 andfurthermore Xr∈ W∞,∞(Pm).
Proof. If m = n then it is straight-forward to check that u is continuous. We may therefore assumethatm ≤ n− 1.In this case,the proof of[16, Lemma 4.14(i)]shows thatthereexistsanon-negativeBorelmeasurableandlocallyintegrablefunctionν which
isanuppergradientofu withrespecttopolygonal curves,thatis,suchthat
u(b)− u(a) ≤ˆ1
0
ν◦ γ(t)˙γ(t)dt
foralla,b∈ Rn andeverypolygonalcurveγ connectinga andb.Here,itisunderstood
thattheright-handsidemustequal∞ incaseu(a)=∞ oru(b)=∞.(Weremarkhere that the proof of [16, Lemma 4.14(i)] is stated only for p,q < ∞. However, the same arguments applyinthe casethatq =∞ or p=∞.) Itnow follows fromawell-known argument(seee.g. p. 28in[7])that
u(b)− u(a) ≤ C|b − a|Mν(b) + Mν(a)
for all a,b ∈ Rn, where M ν is the maximal function of ν, and where C is aconstant
onlydependingonn.Thisimpliesthatu isLipschitzcontinuouson{Mν ≤ k} forevery
k ∈ N and, since {Mν ≥ k} is negligible, it follows that u is Lebesgue measurable. Finally,theuppernorminequality,Fubini’stheoremand(in casethatq <∞)Hölder’s inequalityyieldthatu islocallyintegrable. 2
In the proof of Theorem 3.1 we will need the following two crucial facts from [16]
about the averages Xr(T ). The statements given in [16] are slightly stronger and are
provedinthegenerality of normalandintegralcurrents inLie groupsequipped with a left-invariantFinslermetric.InthesettingofRntheresultscanbestatedinasomewhat
simplerform.Wethusprovidethemherefortheconvenienceofthereader.Thefollowing isarestatementof[16,Proposition4.15(ii)].
Proposition2.7.Let0≤ m≤ n− 1 and1≤ p<∞.LetfurthermoreX ∈ Wp(Pm0) bean
additive cochainwith uppergradientg inLp(Rn). Thenthereexists C = C(n,m,p)> 0
suchthat forevery T ∈ P0
m,everyS ∈ Pm+1 with ∂S = T ,and everyr > 0, wehave
Xr(T ) ≤ Cr−n/pM(S)g
p.
Inordertostatethesecondpropositionwedefinethemaximalgrowthofapolyhedral chainT ∈ Pmby
Θm(T ) := supT (B(x, r))rm ,
where thesupremumis takenoverallx∈ Rn and allr > 0.NotethatΘ
m(T )<∞ for
everyT ∈ Pm.Wenowgivearestatementof[16,Proposition4.16].
Proposition 2.8.Let0≤ m≤ n− 1 andletn− m+ 1< q <∞ andn− m< p<∞.
(i) Let X ∈ Wp(Pm0) be an additive cochain with upper gradient g in Lp(Rn). Then
thereexists a constant D = D(p,m,n)> 0 suchthat forevery T ∈ P0
m and every r > 0 wehave Xr(T )− X(T ) ≤ DΘ1/p m (T )r1+ m−n p M(T )p−1p gp.
(ii) Let X ∈ Wq,p(Pm) be an additivecochain with uppernormh in Lq(Rn) andupper
gradientg in Lp(Rn). Then there exists E = E(q,p,m,n)> 0 such that for every
T ∈ Pm andevery r > 0 wehave
Xr(T )− X(T ) ≤ EΘ1/p
m (T )r
1+m−np M(T )p−1p g
p
+ Θm−11/q (∂T )r1+m−1−nq M(∂T )q−1q hq.
The proofof this propositionisexactly asthe proofof [16, Proposition4.16]except that the reference to [16, Lemma 4.14(i)] therein shouldbe replaced by areference to
Lemma 2.6above. 3. Continuityofaverages
The aim of this section is to prove Theorem 3.1 below, which provides the main continuity result needed in the proof of our generalizations of Wolfe’s theorem. The second partofTheorem 3.1 wasstatedasTheorem 1.4intheintroduction.
Theorem3.1. Let0≤ m≤ n and1< p,q≤ ∞. Thenwehave:
(i) ifX ∈ Wp(Pm0) isan additivecochain then
Xr(T )− X(T ) → 0 as r → 0
foreveryT ∈ P0
m\ Λ1,where Λ1⊂ Pm0 hasp-capacity 0.Ifp> n− m thenwemay
take Λ1=∅;
(ii) if X ∈ Wq,p(Pm) isanadditive cochainthen
Xr(T )− X(T ) → 0 as r → 0
forevery T ∈ Pm\ Λ2 forsome family Λ2⊂ Pm ofzero ν-modulus, whereν = q if
p> n− m and
ν = minq, pn/(n− p)
otherwise.If q > n− m+ 1 andp> n− m thenwemay take Λ2=∅.
TheproofofthistheoremwillbegiveninSection3.2.Wefirstestablishsome prelim-inaryresultswhichwill beusedinitsproof.
3.1. Auxiliaryresults
Wewillneedthefollowingresultsintheproof ofTheorem 3.1.
Lemma 3.2. Let0≤ m≤ n− 1 and letT ∈ Pm.Let furthermore u:Rn → [0,∞] be a
Borelfunction. Forx∈ Rn define Sx:= ψx#
[0, 1]× T∈ Pm+1,
where
ψx: [0, 1]× Rn→ Rn, ψx(t, z) = z + tx.
Thenforevery r > 0 wehave
B(0,r) ˆ Rn u(y) dSx(y)dx ≤ (n − 1)−1Ln B(0, 1)−1 ˆ Rn Ir(u)(y) dT (y),
whereIr isthetruncated Riesz potential
Ir(u)(y) := ˆ B(y,r) u(x) |x − y|n−1dx.
http://doc.rero.ch
Proof. By[4,4.1.9],wehave
Sx ≤ rψx#
L1× T
foreveryx∈ B(0,r).UsingpolarcoordinatesandFubini’stheoremwe calculatethat ˆ B(0,r) ˆ Rn u(y) dSx(y)dx ≤ r ˆ B(0,r) 1 ˆ 0 ˆ Rn u(z + tx) dT (z)dtdx = r r ˆ 0 τn−1 ˆ Sn−1 1 ˆ 0 ˆ Rn u(z + tτ θ) dT (z)dtdθdτ ≤ r r ˆ 0 τn−2 ˆ Sn−1 r ˆ 0 ˆ Rn u(z + tθ) dT (z)dtdθdτ = r n n− 1 ˆ B(0,r) ˆ Rn u(z + x) |x|n−1 dT (z)dx = r n n− 1 ˆ Rn Ir(u)(y) dT (y),
whichfinishestheproof. 2
Thefollowingnotationwill beusefulinthesequel.ForT ∈ Pmand0< r < R set
N (T, R, r) = N (T, R)\ N(T, r),
where N (T,s) denotestheopens-neighborhoodof thesupportofT .
Proposition 3.3. Let0≤ m≤ n− 1 andT ∈ P0
m.Thenthere existr0,A> 0 and t≥ 1 (dependingonT )withthefollowingproperty.ForeveryS∈ Pm+1with∂S = T wehave
SB(y, r)\ B(y, r/2)∩ NT, r, t−1r≥ ArT B(y, 2r) forevery 0< r < r0 andeveryy∈ spt(T ).
Wepostpone theproof ofthis propositionuntil Section3.3 sinceitis quitedifferent inspiritfromtherestoftheproofsinthissection.From Proposition 3.3wededucethe following fact.
Proposition 3.4. Let 0≤ m≤ n− 1 and T ∈ Pm0.Then there exist C,r0 > 0 with the followingproperty. ForeveryS ∈ Pm+1 with ∂S = T wehave
ˆ Rn Ir(h)(y) dT (y) ≤ C ˆ N (T,r) M h(y) dS(y)
for every 0 < r < r0 and every Borel function h : Rn → [0,∞]. Here M h is the Hardy–Littlewood maximalfunctionof h.
Proof. LetA,t andr0beasinProposition 3.3forT ,andfix0< r < r0.Forj = 0,1,. . .
define Aj(y) = B y, 2−jr\ By, 2−(j+1)r and Nj(T ) = N T, 2−jr, t−12−jr. Set Hj(y) := ˆ Aj(y) h(x) |x − y|n−1dx,
andnote thatIr(h)(y)=
jHj(y).Ify∈ spt(T ) thenwehave
Hj(y)≤ D2−jrM h(w) for every w∈ Aj(y)∩ Nj(T ) =: Qj(y),
whereD isaconstantdependingonlyonn.Consequently,
Hj(y)≤ D2−jr SQj(y) −1ˆ Rn M h(w)χQj(y)(w) dS(w).
IntegrationandFubini’stheoremthenyield ˆ Rn Hj(y) dT (y) ≤ D ˆ Rn M h(w)2−jrχNj(T )(w) ˆ Rn χB(w,2−jr)(y) S(Qj(y)) dT (y)dS(w).
WeapplyProposition 3.3tobound theright-hand-sidefromaboveby
A−1D
ˆ
Rn
M h(w)χNj(T )(w) dS(w).
Summingoverj yields
ˆ
Rn
Ir(h)(y) dT (y) ≤ C
ˆ
Rn
M h(y)χN (T,r)(y) dS(y),
withaconstantC dependingonT butnotonr. 2
3.2. Proofofthemain continuityresult
Wenowturn totheproofofTheorem 3.1.
Proof of Theorem 3.1. Wefirst prove statement (i). For this, let X ∈ Wp(Pm0) be an
additive cochain with p-integrable upper gradient g.We may assume thatm ≤ n− 1
because P0
n ={0} andthusX ≡ 0 whenm= n.Set
Λ1:=
T ∈ Pm0 : ˆ
Rn
M g dS = ∞ for all S ∈ Pm+1 such that ∂S = T
.
If p= ∞ then it follows that Λ1 = ∅. If p < ∞ then M g ∈ Lp(Rn) by the maximal
functiontheoremandhencecapp(Λ1)= 0.LetT ∈ Pm0 \Λ1.ThenthereexistsS∈ Pm+1
with ∂S = T andsuchthat
ˆ
Rn
M g dS < ∞. (3.1)
For x ∈ Rn set V
x := ψx#([0,1]× T ), where ψx is as in Lemma 3.2, and note that
ϕx#T − T = −∂Vx.BytheuppergradientinequalityandLemma 3.2wehave
B(0,r)
X(∂Vx)dx≤ D
ˆ
Rn
Ir(g)(y) dT (y)
foreveryr > 0,where D isaconstantonlydependingonn. Thistogetherwith Propo-sition 3.4yields B(0,r) X(∂Vx)dx≤ C ˆ N (T,r) M g(y) dS(y)
for every 0< r < r0, where C isa constantdepending onT and n but noton r, and
where r0> 0 isas inProposition 3.4. Togetherwith(3.1)thisyields |X(∂Vx)|<∞ for
almost everyx∈ B(0,r). Note alsothat|X(ϕx#T )|<∞ for almost everyx∈ B(0,r)
bytheuppergradientinequalityandLemma 2.5.ThesubadditivitypropertyofX thus
impliesthat|X(T )|<∞ andhence
Xr(T )− X(T ) ≤
B(0,r)
X(∂Vx)dx
bytheadditivitypropertyofX.Consequently, Xr(T )− X(T ) ≤ C ˆ
N (T,r)
M g(y) dS(y).
Given(3.1)wehave ˆ
Rn
M gχN (T,r)dS → 0 as r → 0
byabsolutecontinuityofintegrals,andthusweobtain|Xr(T )− X(T )|→ 0 asr→ 0.
Inorderto completetheproof ofstatement(i)itremainsto showthatwemay take
Λ1 =∅ whenn− m< p<∞.Inthis caseitfollowsdirectlyfromProposition 2.8that
foreveryT ∈ P0
mwe have|Xr(T )− X(T )|→ 0 asr→ 0.(Alternatively,onecan show
thatthefamilyΛ1 definedaboveisinfactempty.Indeed, foreveryT ∈ Pm0 andevery
n− m< p<∞ wehavecapp({T })> 0 bytheproofof[16,Proposition 4.17]andhence
Λ1 = ∅. We remark that [16, Proposition 4.17] is stated in the setting of integral or
normalcurrentsbutthesameargumentsapplyinthesettingofpolyhedralchains.)This completestheproofofstatement(i).
Weturntostatement(ii).LetX ∈ Wq,p(Pm) beanadditivecochainwithq-integrable
upper norm h and p-integrable upper gradient g. If m = n then the function x → X(ϕx#T ) iscontinuousandthusthestatementholdstrivially.Wemaythereforeassume
fromnowonthatm≤ n− 1.Given T ∈ Pm andx∈ Rn wecanwrite
ϕx#T− T = ψx#
[0, 1]× ∂T− ∂ψx#
[0, 1]× T=: Ux− ∂Vx,
where ψx is as inLemma 3.2. Note thatifm = 0 then Ux = 0 by definition. We now
distinguish twocases. Firstassumethat1< p≤ n− m or p=∞. SetΛ2:= Λ2∪ Λ2,
where Λ2:= T ∈ Pm: ˆ Rn M h dT = ∞ and Λ2 := T ∈ Pm: ˆ Rn I1(g) dT = ∞ .
Here,I1(g) denotesthetruncatedRieszpotentialwithr = 1 definedinLemma 3.2.Note
that ifp = ∞ then Λ2 =∅, and if q = ∞ then Λ2 = ∅. In particular,it follows that Λ2 =∅ in thecasethatq = p=∞. Ifq <∞ thenMq(Λ2)= 0 sinceM h∈ Lq(Rn) by
themaximal functiontheorem.Also, if1< p≤ n− m thenMpn/(n−p)(Λ2)= 0 sinceI
maps Lp(Rn) toLnp/(n−p)(Rn),cf. [7,page 20].Since Mp0(Λ)= 0 implies Mq0(Λ)= 0
forq0< p0,wehaveMν(Λ2)= 0.Now,letT ∈ Pm\ Λ2.Clearly,
ˆ Rn M h(y) + I1(g)(y) dT (y) < ∞. (3.2)
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From theuppergradientinequalityandLemma 3.2weinfer B(0,r) X(∂Vx)dx≤ D ˆ Rn Ir(g)(y) dT (y) (3.3)
foreveryr > 0,whereD isaconstantonlydependingonn.Analogously,fromtheupper norm inequalityandLemma 3.2weobtain
B(0,r)
X(Ux)dx≤ D
ˆ
Rn
Ir(h)(y) d∂T (y)
foreveryr > 0.Proposition 3.4thusimplies
B(0,r)
X(Ux)dx≤ C
ˆ
Rn
M h(y)χN (∂T,r)(y) dT (y) (3.4)
for every 0< r < r0, where C isa constantdepending on∂T and n, butnoton r. It
thus followstogether with(3.2)that|X(Ux)|<∞ and |X(∂Vx)|<∞ for almost every
x∈ B(0,r).Moreover,|X(ϕx#T )|<∞ foralmost everyx∈ B(0,r) bytheuppernorm
inequality and Lemma 2.5. This together with the subadditivity property of X shows
that|X(T )|<∞.Now, theadditivityproperty ofX yields
Xr(T )− X(T ) ≤
B(0,r)
X(Ux)dx +
B(0,r)
X(∂Vx)dx
and therefore,inviewof(3.3)and(3.4), Xr(T )− X(T ) ≤ C
ˆ
Rn
M h(y)χN (∂T,r)(y) + Ir(g)(y)
dT (y) (3.5)
forevery0< r < r0,whereC isaconstantdependingon∂T andn,butnotonr.Given (3.2),itfollowsfromabsolutecontinuityoftheintegralandmonotoneconvergencethat theright sidein(3.5)converges to0 asr→ 0 and thus|Xr(T )− X(T )|→ 0 asr→ 0.
This provesstatement(ii)when1< p≤ n− m orp=∞. Nowassumethatn− m< p<∞ andset
Λ2:= T ∈ Pm: ˆ Rn M h dT = ∞ .
Observethatifq =∞ thenΛ2=∅.LetT ∈ Pm\ Λ2.Firstly,asabove,theuppernorm
inequalitytogetherwithLemma 3.2andProposition 3.4yields
B(0,r)
X(Ux)dx≤ C
ˆ
Rn
M h(y)χN (∂T,r)(y) dT (y) (3.6)
for every0< r < r0,where C is aconstantdepending on∂T and n, butnot onr. In
particular,|X(Ux)|<∞ for almost everyx∈ B(0,r). Moreover,as above,theintegral ontherighthandsideof(3.6)convergesto0 asr→ 0 andhence
B(0,r)
X(Ux)dx→ 0
as r→ 0. Secondly,Propositions 2.7 and2.8 yield,with s= rαfor0< α < p
n,thatfor
suitableconstantsC,D,andC dependingonlyonp,n,m wehave X(∂Vx) ≤ X(∂Vx)− Xs(∂Vx)+Xs(∂Vx) ≤ CΘ1p m(∂Vx)s1+ m−n p M(∂Vx)p−1p gp+ Ds−npM(Vx)gp ≤ CΘ m(T ) + Θm−1(∂T ) 1 prα(1+m−np )M(T ) + rM(∂T )p−1p g p + Dr1−αnpM(T )gp
for every x ∈ B(0,r), where we have used M(Vx) ≤ |x|M(T ) as well as M(∂Vx) ≤
2M(T )+|x|M(∂T ) andΘm(∂Vx)≤ 2Θm(T )+2m−1Θm−1(∂T ).Inparticular,|X(∂Vx)|<
∞ foreveryx∈ B(0,r) and,moreover,
B(0,r)
X(∂Vx)dx→ 0
as r → 0. Thus, we see the sameway as abovethat|X(T )| < ∞ andhence, with the additivitypropertyofX,that
Xr(T )− X(T ) ≤
B(0,r)
X(Ux)dx +
B(0,r)
X(∂Vx)dx→ 0
asr→ 0.Thisshowsthat|Xr(T )− X(T )|→ 0 asr→ 0.
ItremainstoshowthatwemaytakeΛ2=∅ inthecasethatn− m+ 1< q <∞ and n− m< p<∞. Inthis case, Proposition 2.8infactyields thatfor everyT ∈ Pm we
have|Xr(T )− X(T )|→ 0 asr→ 0.Thisconcludestheproofofstatement(ii)andthus ofthetheorem. 2
3.3. Proof ofProposition 3.3
InordertoproveProposition 3.3wewillneedthefollowingresult.
Lemma 3.5. Let X ⊂ Rn be a finite simplicial complex. Then X is a local Lipschitz neighborhoodretract.
Proof. Let X ⊂ Rn beafinite simplicialcomplex.By[2,Theorem 1.2],it isenoughto show thatthere exist C > 0 andε> 0 suchthateveryLipschitzmap f : Sr→ X with
image inanε-balladmitsaLipschitzextensionf : Br+1→ X whoseLipschitzconstant
isbounded byC timestheLipschitzconstantoff .
We startwith the following auxiliaryconstruction. LetΣ ⊂ Rn be a k-simplexand
F ⊂ Σ an -face,possiblyF = Σ. Wecan writeΣ = [v0,. . . ,vk] and F = [v0,. . . ,v ].
Set ΣF := k i=0 tivi∈ Σ : i=0 ti≥ 2−1 and define ψΣF : [0, 1]× ΣF → ΣF by ψΣ F(s, x) = sx + (1− s) i=0 ti −1 i=0 tivi,
where x= ki=0tivi.Itisclearthatthefollowing propertieshold:
(i) ψΣ
F(1,x)= x andψΣF(0,x)∈ F forallx∈ ΣF;
(ii) ψΣ
F(s,x)= x forallx∈ F ands∈ [0,1];
(iii) ψΣ
F isLipschitzwithconstantdependingonΣ;
(iv) IfG⊂ Σ isafacewith F∩ G= ∅ thenΣF∩ G= GG∩F and
ψΣF(s, x) = ψGF ∩G(s, x) forallx∈ GG∩F andalls∈ [0,1].
Next,fixasimplexΣ0 inX anddefine asubsetA⊂ X by A :={ΣΣ∩Σ0 : Σ simplex in X}.
There existsε0> 0 dependingonlyonX such that N (Σ0, ε0)∩ X ⊂ A,
where N (Σ0,ε0) denotestheε0-neighborhoodofΣ0inRn.Defineamap ϕ : [0, 1]× A → A
byϕ(s,x)= ψΣΣ∩Σ0(s,x) forx∈ ΣΣ∩Σ0.Byproperty(iv)above,thismapiswell-defined,
thatis,independentofthechoiceofΣ.Fromproperties(i)–(iii)itfollowsthatϕ(1,x)= x forallx∈ A,ϕ(s,x)= x forallx∈ Σ0andalls∈ [0,1],andϕ(0,x)∈ Σ0forallx∈ A.
Moreover,ϕ is“piecewiseLipschitz”.
Finally,letf : Sr→ X beaLipschitzmap withimageinanε-ballcenteredat some
x0 ∈ X, where 0< ε < ε0 is so small thatevery ε-ball inX is quasi-convex. We will
show that f admits a Lipschitz extension f : Br+1 → X whose Lipschitz constant is
bounded byC timestheLipschitzconstantof f ,where C onlydependsonX.Ifr = 0
then this follows immediately from thequasi-convexity of ε-balls. Lettherefore r ≥ 1.
LetΣ0⊂ X beasimplexsuchthatx0∈ Σ0.WiththedefinitionofA above,weclearly
have f (Sr) ⊂ A. Using ϕ we easily construct a Lipschitz extension f : Br+1 → X of
f withLipschitz constantLip(f )≤ C Lip(f), where C only dependson X.Indeed,let
: [0,1]× Σ0→ Σ0 beaLipschitzmapwhichcontractsΣ0to apoint.Define f (sz) :=
ϕ(2s− 1, f(z)) s∈ [1/2, 1] (2s, ϕ(0, f (z))) s∈ [0, 1/2)
wheneverz∈ Sr ands∈ [0,1].Thenf extendsf andis Lipschitzwithaconstantonly
depending on the“piecewise” Lipschitz constant of ϕ andthe Lipschitz constantof .
This proves theclaimand thusshows, by[2, Theorem 1.2],thatX is alocal Lipschitz neighborhoodretract. 2
Wearefinallyready toproveProposition 3.3.
ProofofProposition 3.3. SinceT is apolyhedralcycle itfollowsfromLemma 3.5that spt T is alocal Lipschitz neighborhood retract. There thus exist r1 > 0, λ ≥ 1 and a λ-Lipschitz retraction : N (T,r1) → spt T . Let y ∈ spt T and let u be the distance
functionto the point y.By[4, 4.2.1and 4.3.2], almost everys∈ (0,r1/λ) issuchthat
theslice
S, u, s = ∂S{u ≤ s}− (∂S){u ≤ s}
isanormalm-currentsupportedin{x: u(x)= s}.Clearly,B(y,s)⊂ N(T,r1) andhence S,u,s issupportedinN (T,r1).Weclaimthat
# S, u, s = −T B(y, s). (3.7)
Inordertosee this,setV := # S,u,s + TB(y,s) andnote firstthat ∂V = #
∂ S, u, s + ∂TB(y, s)=−# T, u, s + ∂
TB(y, s)= 0,
henceV is acycle. Note thatV is supportedin spt T ∩ B(y,s/λ). Let W be an(m+ 1)-dimensionalnormalcurrentwith∂W = V .AfterpossiblyprojectingW ontoB(y,s/λ)
wemayassumethatW issupportedintheballB(y,s/λ) andthusinN (T,r1).Itfollows
that #W satisfies ∂#W = #V = V . Since #W is supported inan m-dimensional
simplicial complex it follows that #W = 0 and hence thatV = ∂(#W ) = 0. This
proves (3.7).
Nowset t:= 400λ and r0:= r1/λ.If0< r < r0 ands∈ (r/2,r) thenitfollowsthat
#
S, u, s NT, t−1rB(y, 99s/100)= 0 and thus,with(3.7),that
T B(y, 99s/100)=#
S, u, s NT, t−1rcB(y, 99s/100) ≤ λm S,u,s NT,t−1rc
.
Now, integrationandtheslicinginequality(see[4,4.2.1]or [15,Theorem 6.2])yield
r 2T B(y, 99r/200)≤ λm r ˆ r/2 S,u,s NT,t−1rc ds ≤ λmSB(y, r)\B(y, r/2)∩ NT, t−1rc .
Since T ispolyhedral therefurthermoreexists D≥ 1 suchthat
T B(y, 2r)≤ DT B(y, 99r/200)
foreveryy∈ spt T andeveryr > 0.Wethusconcludethat
ArT B(y, 2r)≤ SB(y, r)\B(y, r/2)∩ NT, t−1rc
foreveryy∈ spt T andall0< r < r0,whereA:= (2Dλm)−1.Since
B(y, r)\B(y, r/2)∩ NT, t−1rc=B(y, r)\B(y, r/2)∩ NT, r, t−1r
this concludestheproof. 2
4. FromSobolevdifferentialformsto Sobolevcochains Theaimofthis sectionistoconstructalinearmap
Ψm: Wdq,p
Rn,∧m→ W
q,p(Pm)
forall1≤ m≤ n whenever1≤ q,p<∞ orq = p=∞,andtoshowthatΨmisisometric,
i.e. itpreservesnorms.This constructionalreadyappearedin[16]. Theauthorsdidnot show, however, thatthemap isisometric. In fact,as mentionedearlier, adifferent but equivalent normonWdq,p(Rn,∧m) wasusedin[16].
4.1. Construction ofΨm whenp,q <∞
Let1≤ p,q <∞ and ω ∈ Wdq,p(Rn,∧m). There is asequence ofsmooth compactly supported m-forms ωj converging to ω in Wdq,p(Rn,∧m). We may assume, of course,
that the coefficients ω(·,α) of ω and dω(·,β) of dω are Borel functions. Therefore, by Fuglede’s lemma (Lemma 2.4),there isa subsequenceof (ωj),also denoted (ωj), such
thatforeveryα∈ Λ(m,n) andeveryβ∈ Λ(m+ 1,n),
ˆ
Rn
ωj(·, α) − ω(·, α)dT → 0 (4.1) foreveryT ∈ Pm\ Λ,where Mq(Λ)= 0,and
ˆ
Rn
dωj(·, β) − dω(·, β)dS → 0 (4.2) foreveryS∈ Pm+1\ Γ , whereMp(Γ )= 0.
ForT ∈ Pm andj∈ N,define
Xωj(T ) := ˆ
T
ωj.
StokestheoremimpliesthatXωj(∂S)= Xdωj(S) andhencethatXdωj = dXωj,where
dXdωj isthecoboundaryofXωj.ItisimmediatethatXωj isanadditivecochain.Define
Xω : P
m → R by Xω(T ):= limj→∞Xωj(T ) when the limitexists, and ∞ otherwise.
Note that, by Fuglede’s lemma, the limit exists for Mq-almost every T ∈ Pm and,
moreover,adifferent choiceof(ωj) satisfying(4.1)yields acochainwhichisMq-almost
everywhereequaltoXω.ItisclearthatXω isadditiveandanelement ofW
q,p(Pm).In
fact,wehavethefollowing proposition.
Proposition4.1. Let1≤ m≤ n and1≤ p,q <∞.Thenthemap Ψm: Wdq,p
Rn,∧m→ W
q,p(Pm)
givenby Ψm(ω)= Xω is linearandisometric.Moreover, if m< n, wehave
Ψm+1◦ d = d ◦ Ψm, (4.3)
that is, forevery ω∈ Wdq,p(Rn,∧m) and every s≥ 1 wehave Xdω = dXω as elements of Wp,s(Pm+1).
Proof. Let ω ∈ Wdq,p(Rn,∧m). Observe thatthe functions ω(x) and dω(x) are in
Lq(Rn) andLp(Rn),respectively,where · denotes comass. Itis notdifficult to check
thatω(x) is aq-weakupper normof Xω and dω(x) isap-weakupper gradientof
Xω.Indeed,let(ω
j) beasequenceofsmoothcompactlysupportedm-formsconverging
toω inWdq,p(Rn,∧m) andsatisfying(4.1)and(4.2).LetT ∈ P
m\Λ,whereΛ isasabove.
WriteT = aiTi asinSection2.2andletτi bethesimpleunitm-vectororientingTi.
It followsthat Xωj(T ) ≤ i |ai| ˆ Ti ωj, τi dHm ≤ i |ai| ˆ Ti ω,τi dHm+ i |ai| ˆ Ti ωj− ω, τi dHm.
By (4.1), each term in the second sum in the last line converges to 0 as j → ∞ and
henceweobtainthat
Xω(T ) ≤ i |ai| ˆ Ti ω,τi dHm≤ ˆ Rn ω(x)dT (x).
This shows that ω(x) is a q-weak upper norm of Xω. One shows analogously that
dω(x) isap-weakuppergradientofXω.Consequently,wehavethat
Xω
q,p≤ ωq,p. (4.4)
Weclaimthatequalityholdsin(4.4).Forthisleth∈ Lq(Rn) beanuppernormofXω. Fixτ = ξ1∧· · ·∧ξm,whereξ1,. . . ,ξm∈ Rnarepairwiseorthonormal,anddefineforeach
x∈ Rn amapϕ
τ,x :Rm→ Rn by ϕτ,x(z):= x+ mi=1ziξi forz = (z1,. . . ,zm)∈ Rm.
Wewill showthatforalmost everyx∈ Rn andeverypolyhedronΔ,wehave
Xωϕ τ,x#JχΔK = ˆ Δ ω◦ ϕτ,x(z), τ dz, (4.5)
whereJχΔK denotesthepolyhedralm-chaininRminducedbythesimplefunctionχΔ.By Lemma 2.5,Xωj(ϕ
τ,x#JχΔK) convergesasj→ ∞ foralmosteveryx∈ Rn.LetVτ⊂ Rn
bethespanofthevectorsξ1,. . . ,ξmandletVτ⊥⊂ Rndenotetheorthogonalcomplement.
Sinceωj(·,α) convergesinLq(Rn) toω(·,α) foreveryα itfollowsthat ωj,τ converges
inLq(Rn) to ω,τ .Fromthisweobtainthatthereexistsasubsequence(ω
jk) suchthat
ωjk◦ϕτ,y,τ convergesinLq(Rm) to ω ◦ϕτ,y,τ foralmosteveryy∈ Vτ⊥.Inparticular,
forallsuchy andeveryz0∈ Rmwehave
Xωjkϕτ,y #Jχz 0+ΔK = ˆ z0+Δ ωjk◦ ϕτ,y(z), τ dz−→ ˆ z0+Δ ω◦ ϕτ,y(z), τ dz