Wolfe's theorem for weakly differentiable cochains

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by  Camille Petit

WOLFE’S THEOREM FOR WEAKLY DIFFERENTIABLE COCHAINS
CAMILLEPETIT,KAIRAJALA,ANDSTEFANWENGER
Abstract. AfundamentaltheoremofWolfeisometricallyidentifiesthespaceof
flatdifferential formsof dimension m inRn withthespaceof flatm-cochains,
that is, thedual space of flatchains of dimension m inRn. Themainpurpose
4
ofthepresent paper istogeneralizeWolfe’stheoremtothesettingofSobolev
1
differential forms and Sobolev cochains in Rn. A suitable theory of Sobolev
0
cochainshasrecentlybeeninitiatedbythesecondandthirdauthor. Itisbased
2
on the concept of upper norm and upper gradient of a cochain, introduced in
n
analogywithHeinonen-Koskela’sconceptofuppergradientofafunction.
a
J

0
3
1. Introduction

]
P Inthe1940’s,Whitneyinitiatedageometricintegrationtheory,see[Whi57],the
A
purpose of which was to integrate a quantity over “m-dimensional sets” in such a
. way that the integral depends on the position of the set in Rn. The quantities one
h
t integrates are m-dimensional flat forms and the sets over which one integrates are
a
m m-dimensional flat chains. Flat forms are L∞-differential forms with L∞-exterior
derivatives. The flat norm of such a form is defined as the maximum of the L∞-
[
norm of the form and that of its derivative. In order to define flat chains, one

1 first considers the space P = P (Rn) of polyhedral chains in Rn, that is finite
v m m
6 formal sums of oriented m-dimensional polyhedra in Rn with real multiplicities,
5 see Section 2 for precise definitions. One equips P with the flat norm, given for
m
9
T ∈ P by
7 m
1. |T|♭ := inf{M(R)+M(S) :R ∈ Pm,S ∈ Pm+1,R+∂S = T},
0
where ∂S is the boundary of S and M(R) is the mass of R. The completion of
4
1 Pm under the flatnorm iscalled thespace of flatm-chains inRn and denoted Fm.
: Elements ofthe dual space of F arecalled flatm-cochains. In[Wol48], seealso
v m
i [Whi57], Wolfe proved the following fundamental theorem: the space of flat m-
X
forms, endowed with the flat norm, is isometric to the space of flat m-cochains.
r
a Wolfe’s theorem has recently been generalized to the setting of Banach spaces by
Snipes in [Sni13], where she defines a flat partial differential form in a Banach
spaceandshowsthatthespaceoftheseformsisisometricallythedualspaceofthe
spaceofflatchainsasdefinedbyAdams[Ada08]. Moreover,Wolfe’stheoremhas
recently been used by Heinonen-Sullivan [HS02] and Heinonen-Keith [HKe11],
seealsoHeinonen-Rickman[HR02],togiveconditionsunderwhichametricspace
islocallybi-Lipschitz equivalenttoRn.
2010MathematicsSubjectClassification. 49Q15,46E35,53C65,49J52.
C.P.andK.R.weresupportedbytheAcademyofFinland,project#257482. Partsofthisresearch
werecarriedoutwhenK.R.wasvisitingUniversite´ deFribourg. Hewouldliketothankthedepart-
mentforitshospitality.PartsofthisresearchwerecarriedoutwhenS.W.wasvisitingtheDepartment
ofMathematicsandStatisticsattheUniversityofJyva¨skyla¨. Hewouldliketothankthedepartment
foritshospitality.
1
2 CAMILLEPETIT,KAIRAJALA,ANDSTEFANWENGER
ThemainpurposeofthepresentpaperistogeneralizetheclassicalWolfe’sthe-
orem to the setting of Sobolev differential forms and Sobolev cochains in Rn. A
suitable theory, based on upper gradients, of Sobolev cochains in complete met-
ric measure spaces has recently been initiated by the second and third authors in
[RW13]. Before stating our main results we briefly recall the relevant definitions
from [RW13], restricting ourselves to the setting of Rn. We refer to Section 2.3
for precise definitions. Asubadditive m-cochain onP isafunction X : P → R
m m
whichsatisfies X(0) = 0andwhichissubadditive inthesensethat
|X(T)| ≤ |X(T +R)|+|X(R)|,
for all T,R ∈ P . If furthermore X(T +R) = X(T)+X(R)whenever each term is
m
finite then X is called a (additive) cochain. In [RW13] a notion of upper gradient
of a subadditive cochain is defined in analogy with Heinonen-Koskela’s notion of
upper gradient of a function [HKST]. A Borel function g : Rn → [0,∞] is called
uppergradientof X if
|X(T)| ≤ gdkSk
ZRn
for all T ∈ Pm and S ∈ Pm+1 satisfying ∂S = T, where kTk denotes the mass
measure of T, see Section 2.2. Similarly, a Borel function h : Rn → [0,∞] is an
uppernormof X if
|X(T)| ≤ hdkTk
ZRn
forallT ∈ P . Givenanadditive cochain X withupper normin Lq(Rn)andupper
m
gradient inLp(Rn),wedefineitsSobolevnormby
kXk = max infkhk ,infkgk ,
q,p q p
n o
wheretheinfimaaretakenwithrespect toupper normshandupper gradients gof
X, respectively. Note that this norm is different from but equivalent to the norm
introduced in [RW13]. The Sobolev space W (P ) of additive cochains is the
q,p m
set of equivalence classes of additive cochains with upper norm in Lq(Rn) and
upper gradient in Lp(Rn), under the equivalence relation defined by X ∼ X if
1 2
kX −X k = 0.
1 2 q,p
Themainresult ofthepresent paper isthe following generalization oftheclas-
sicalWolfe’stheorem.
Theorem 1.1. Let 1 ≤ m ≤ n and 1 < q,p < ∞. If p > n−m or q ≤ pn then
n−p
the space Wq,p(Rn, m) of Sobolev differential forms is isometrically isomorphic
d
tothespaceW (P ).
q,p Vm
Recall that the space Wq,p(Rn, m) of Sobolev differential m-forms consists of
d
thoseLq-integrabledifferentialm-formsωwhosedistributionalexteriorderivatives
V
dωare Lp-integrable. Itisendowedwiththenorm
1 1
q p
kωk = max kω(x)kqdx , kdω(x)kpdx ,
q,p  Z ! Z ! 
 
where kω(x)k denotes the comass norm of ω(x). See Section 2for the precise
definitions. Itiswell-knownthattheSobolevspaceW1,poffunctionsis(forall p ≥
1)isomorphictotheNewtonianSobolevspaceN1,p definedusinguppergradients,
cf. [HKST]. This remains true in the setting of Theorem 1.1 when m = 0, cf.
WOLFE’STHEOREMFORWEAKLYDIFFERENTIABLECOCHAINS 3
[RW13, Proposition 3.11]. Theorem 1.1 gives a partial answer to the question
whetherthesameresultholdsform ≥ 1.
TheclassicalWolfe’stheoremisaneasyconsequence ofTheorem1.1.
Corollary 1.2. For 1 ≤ m ≤ n the space W∞,∞(Rn, m) is isometrically isomor-
d
phictothedualspaceofthespaceF offlatchainsinRn.
m V
Here, W∞,∞(Rn, m) is endowed with the norm kωk which is defined simi-
d ∞,∞
larlytokωk butusingtheessential supremumofthepointwise comassnorms.
q,p V
Asmentioned above, Heinonen-Sullivan [HS02] and Heinonen-Keith [HKe11]
applied Wolfe’s theorem in order to give conditions under which a metric space
is locally bi-Lipschitz equivalent to Rn. Finding similar conditions for quasicon-
formal equivalence is an interesting open problem. To attack this problem, it is
desirable to find generalizations of Wolfe’s theorem for Sobolev forms. Theorem
1.1waspartially motivatedbythisapplication.
WedonotknowwhetherTheorem1.1holdsforallvaluesofqand p. However,
we have an unconditional result for cochains on P0, the space of polyhedral m-
m
chains without boundary. Inordertostate ourresult, definethenorm ofacochain
X on P0 with upper gradient in Lp(Rn) by kXk = infkgk , where the infimum
m p p
is taken with respect to upper gradients g of X. Let W (P0) be the set of equiva-
p m
lence classes ofadditive cochains on P0 withupper gradient in Lp(Rn), under the
m
equivalence relation defined by X ∼ X if kX − X k = 0. Denote furthermore
1 2 1 2 p
byWp(Rn, m)thequotient spaceofthespaceofm-formsonRn withcoefficients
d
in L1 (Rn)Vand coefficients of the distributional exterior derivative in Lp(Rn) by
loc
the subspace of those elements ω with dω = 0. For [ω] ∈ Wp(Rn, m) define
d
k[ω]k := kdωk ,wherek·k denotes the Lp-norm ofthepointwise comassnorm.
p p p V
This defines a norm on Wp(Rn, m) which is bounded by the quotient norm. Our
d
resultcannowbestatedasfollows.
V
Theorem 1.3. For 1 ≤ m ≤ n − 1 and 1 < p < ∞ the space Wp(Rn, m) is
d
isometrically isomorphic toWp(P0m). V
WebrieflyoutlinetheproofsofTheorems1.1and1.3. Asmoothcompactlysup-
porteddifferential formnaturally induces aSobolevcochain byintegration. Using
this observation and approximation of Sobolev forms by smooth forms, we show
thatthereexistsalinear,norm-preservingmapmappingthespaceofSobolevforms
tothespaceofSobolevcochainswithcorrespondingexponents. Ontheotherhand,
we construct a Sobolev form from a Sobolev cochain as follows: we restrict the
cochaintothem-planesinducedbycoordinatevectors,andthenuseLebesguedif-
ferentiation to construct the coefficients of the resulting form. The map defined
this way is also linear and norm-preserving. Toprove Theorem 1.1, we show that
the two maps are actually inverses to each other. The main problem in showing
this is that it is difficult to see why the restriction of a non-zero Sobolev cochain
tothecoordinate m-planes shouldbenon-zero. Inotherwords,whyshouldanon-
zeroSobolevcochaininduceanon-zeroSobolevform? Toovercomethisproblem,
we“smoothen” thecochains by applying averages. Given acochain X onP and
m
r > 0weset
X (T) := X(ϕ T)dx
r ? x#
B(0,r)
4 CAMILLEPETIT,KAIRAJALA,ANDSTEFANWENGER
for every T ∈ P , where ϕ : Rn → Rn is the translation map ϕ (y) = x + y.
m x x
Here, denotestheintegralaverageLn(E)−1 andLn denotes Lebesguemea-
E E
sure. >The integrand is measurable and locallyRintegrable under our assumptions,
see Lemma 2.6. Using the Federer-Fleming deformation theorem, we show that
the cochains X are determined by their action in the coordinate m-planes. The-
r
orem 1.1 follows if we can show that the cochain X can be approximated by the
cochains X . Thisisgivenbythefollowing continuity result, whichisofindepen-
r
dentinterest.
Theorem1.4. Let0 ≤ m ≤ nand1 < p,q ≤ ∞. If X ∈ W (P )then
q,p m
|X (T)−X(T)| → 0asr → 0
r
for every T ∈ P \Λ for some family Λ ⊂ P of zero ν-modulus, where ν = q if
m m
p> n−mand
ν = min{q,pn/(n− p)}
otherwise.
Proving Theorem 1.1 for all exponents p and q would require a stronger form
of Theorem 1.4. The modulus appearing in the statement measures the size of
exceptional sets, see Section 2.4 for the definition. A similar statement holds for
cochains on P0, see Theorem 3.1. This, together with the arguments above and
m
considerations involving the so-called coboundary of acochain, are used toprove
Theorem1.3.
WefinallymentionthatadifferentvariantofWolfe’stheoremforSobolevforms
wasgiven in [GKS83]. In that paper, the authors provide aone-to-one correspon-
dence betweentheformsinWq,p(Rn, m)andlinear functionals onP which, to-
d m
gether with their exterior derivatives,Vsatisfy certain boundedness conditions with
respect to a so-called q-mass and p-mass. We believe that the notion of Sobolev
cochainusedinthepresentpaperismorenaturalthantheonedefinedin[GKS83].
Our paper is structured as follows. In Section 2, we recall the definitions of
Sobolev forms, polyhedral chains, and Sobolev cochains. In Section 3, which is
the most substantial part of the paper, we prove the main continuity result, The-
orem 1.4, and an analogous version for cochains on P0, see Theorem 3.1. In
m
Section4,weconstruct alinear mapfromthespace ofSobolevformstothespace
of Sobolev cochains and show that this map preserves norms. In Section 5, we
constructacontinuous linearmapfromthespaceofSobolevcochainstothespace
ofSobolevforms. Finally,Section6containstheproofsofTheorems1.1and1.3.
2. Preliminaries
Inthissectionwecollectthedefinitionsofthebasicobjectsofthepresentpaper.
2.1. Sobolev differential forms in Rn. Werecall the definition of Wq,p(Rn, m),
d
the space of (weak) Sobolev differential m-forms on Rn. We refer to [IL93V] for
details. Let1 ≤ m ≤ nandset
Λ(m,n):= {α :{1,...,m} → {1,...,n}strictlyincreasing}.
Letωbeanm-formonRn,giveninEuclidean coordinates by
ω = ω(·,α)dxα,
α∈XΛ(m,n)
WOLFE’STHEOREMFORWEAKLYDIFFERENTIABLECOCHAINS 5
with locally integrable coefficients ω(·,α). An (m+1)-form dω on Rn, given in
Euclideancoordinates by
dω = dω(·,β)dxβ
β∈ΛX(m+1,n)
and with locally integrable coefficients dω(·,β), is said to be the distributional
exteriorderivativeofω,if
dω∧ν = (−1)m+1 ω∧dν
ZRn ZRn
foreveryC∞-smoothcompactlysupported(n−m−1)-formν. Given1≤ p,q≤ ∞,
thespaceWq,p(Rn, m)consists of(equivalence classesof)m-formsωonRn with
d
coefficientsω(·,α)inLq(Rn)andsuchthatωhasadistributionalexteriorderivative
V
dωwithcoefficients dω(·,β)inLp(Rn).
WeendowWq,p(Rn, m)withthefollowingnorm,whichisdifferentbutequiv-
d
alent to the norm considered in [IL93] and [RW13]. For this, denote by | · | the
V
normonthespace Rn ofm-vectorsassociatedwiththeinnerproductforwhich
m
{e ∧···∧e : α ∈ Λ(m,n)} is an orthonormal basis. Here and throughout
α(1) α(m)V
the text, e denotes the j-th standard unit vector in Rn. Anm-vector ξ ∈ Rn is
j m
called simple ifitcanbewritten intheform ξ = ξ ∧···∧ξ forvectors ξ ∈ Rn,
1 m Vi
i= 1,...,m. Thecomassofanm-covectorν ∈ mRn isdefinedby
kνk = sup{hν,ξi :ξ ∈ RnVsimple,|ξ|≤ 1},
m
^
where h·,·i denotes the natural pairing of m-covectors and m-vectors. Given an
m-formωonRn,withcoefficients in Lq(Rn),weset
1
q
kωk := kω(x)kqdx
q
Z !
if q < ∞, where kω(x)k denotes the comass of ω(x). We define kωk analogously
q
incaseq = ∞. TheSobolevnormofanelementω ∈ Wq,p(Rn, m)isthendefined
d
by
V
kωk = max kωk ,kdωk .
q,p q p
n o
Finally, we denote by W1,p (Rn, m) the space of m-forms on Rn with coeffi-
d,loc
cientsinL1 (Rn)andcoefficientsofVthedistributionalexteriorderivativeinLp(Rn).
loc
The space Wp(Rn, m) is the quotient of W1,p (Rn, m) by the subspace of those
d d,loc
elementsωwithdωV= 0. Thenormofanelement[ωV] ∈ Wp(Rn, m)isdefinedby
d
k[ω]k := kdωk . V
p p
ThisclearlydefinesanormonWp(Rn, m). Notethatifm = 0and f isaSobolev
d
function thenk[f]k = k|∇f|k .
p p V
2.2. PolyhedralchainsinRn. Werecallthebasicdefinitionsrelatedtopolyhedral
chains. We refer to [Whi57] and [Hei05] for further details. Let 0 ≤ m ≤ n.
Formally,apolyhedral m-chainT inRn isaformalfinitesum
N
(2.1) T = aT ,
i i
Xi=1
6 CAMILLEPETIT,KAIRAJALA,ANDSTEFANWENGER
wherea ∈ RandT isanoriented m-dimensional polyhedron inRn incasem ≥ 1
i i
and T is a point in Rn in case m = 0. More precisely, consider the additive
i
group of formal finite sums (with coefficients in R) of compact, convex, oriented
m-dimensional polyhedra (respectively, points if m = 0). Then, quotient by the
equivalence relation identifying −T withT˜,where T˜ isT withtheopposite orien-
tation, and identifying T with T + T if T is formed by gluing T and T along
1 2 1 2
a face with the correct orientation. The quotient group is the set of polyhedral
m-chainsinRn,whichwedenotebyP (Rn)orP forshort.
m m
It is easily seen that for every T ∈ P there exists a representation (2.1) such
m
that the T have non-overlapping interiors. We associate to a polyhedral m-chain
i
T ∈ P afinitemeasure, denotedbykTkanddefinedby
m
N
kTk := |a|Hm T ,
i i
Xi=1
where aT is a non-overlapping representation of T. Here, Hm denotes the m-
i i
dimensPional Haudorff measure. The number kTk(Rn) is called the mass of T and
denoted byM(T). Itisworthmentioning thatapolyhedral m-chainT givesriseto
anm-dimensionalnormalcurrentinRnbyintegratingsmoothcompactlysupported
m-forms over T, see [Fed69], and thus also to anm-dimensional metric current in
thesenseof[AK00].
Theboundary ∂T ofT ∈ P ,1 ≤ m ≤ n, isapolyhedral (m−1)-chain defined
m
intheusualway,namelytheboundaryofapolygonisthesumofitsfaceswiththe
induced orientations. If T = aT is a polyhedral 0-chain then we write ∂T = 0
i i
if and only if a = 0. Note that we have ∂∂T = 0 for every T ∈ P with
i P m
m ≥ 2. DenotebyP0 thesetofpolyhedral m-chainsT ∈ P with∂T = 0,andP+
P m m m
the set of polyhedral m-chains T such that for the non-overlapping representation
T = aT mentioned above, each polyhedron T is parallel to one of the m-
i i i
dimenPsional coordinate planes. Note that if T ∈ P+m then, in general, ∂T need not
beinP+ .
m−1
IfT = N aT isapolyhedral m-chainandϕisanaffinemap,thenϕ T isthe
i=1 i i #
polyhedral m-chaindefinedby
P
N
ϕ T = aϕ(T )
# i i
Xi=1
and is called the push-forward of T by the map ϕ. If x ∈ Rn, we use the notation
ϕ for the translation map y 7→ x+y. The push-forward ϕ T is thus simply the
x x#
translate ofT.
2.3. Sobolev cochains in Rn. We recall the basic definitions from the theory of
weaklydifferentiable cochains initiatedbythesecondandthirdauthorin[RW13].
Whereas the definitions in [RW13] are given for arbitrary complete metric spaces
X andmetricnormalorintegralcurrents inX inthesenseofAmbrosio-Kirchheim
[AK00] we will restrict ourselves to the setting of Rn and the spaces P and P0
m m
of polyhedral chains in Rn in this paper. We will therefore only give the relevant
definitions inthissetting.
Definition 2.1. Let 0 ≤ m ≤ n. A function X : P → R is called a subadditive
m
cochainonP ifX(0) = 0and
m
|X(T)| ≤ |X(T +R)|+|X(R)|
WOLFE’STHEOREMFORWEAKLYDIFFERENTIABLECOCHAINS 7
forallT,R ∈ P . Iffurthermore
m
X(T +R)= X(T)+X(R)
whenever each term is finite, then X is called an additive cochain, or simply a
cochain, onP .
m
Cochains on P0 are defined by simply replacing P by P0 everywhere in the
m m m
definitionabove.
Alargeclassofadditivecochainscomesfromdifferential forms.
Example 2.2. Given a (smooth) differential m-form ω on Rn, we can define a
cochain Xω : P →Rbysetting Xω(T) = ωforeveryT ∈ P .
m T m
R
Thefollowingnotionsofuppernormanduppergradientofasubadditivecochain
defined in [RW13] are in analogy with the definition of upper gradient of a func-
tion.
Definition2.3. LetX beasubadditive cochainonP .
m
(i) ABorelfunctionh :Rn → [0,∞]iscalleduppernormof X if
(2.2) |X(T)| ≤ h dkTk
ZRn
foreveryT ∈ P .
m
(ii) ABorelfunctiong :Rn → [0,∞]iscalleduppergradientof X if
(2.3) |X(∂S)|≤ g dkSk
ZRn
forallS ∈ Pm+1.
Theuppernormanduppergradient ofcochains onP0 aredefinedanalogously.
m
We note here that throughout this paper, when dealing with cochains on P0 we
m
willonly use upper gradients. In [RW13], theauthors proved that upper gradients
of0-cochains areexactlyuppergradients offunctions.
For1 ≤ p,q ≤ ∞denotebyW (P )thesetofsubadditive m-cochains which
q,p m
have an upper norm in Lq(Rn) and an upper gradient in Lp(Rn). In [RW13] the
notation Wq,p(Pm,Pm+1)wasused. Wedefine the Sobolev norm ofasubadditive
cochain X ∈ W (P )by
q,p m
kXk = max infkhk ,infkgk ,
q,p q p
n o
wheretheinfimaaretakenwithrespect toupper normshandupper gradients gof
X,respectively. Thisnorm isdifferent from but equivalent tothe norm introduced
in[RW13].
Giventwo(additive) cochains X ,X : P → Rthe cochain X +X isdefined
1 2 m 1 2
by(X +X )(T) = X (T)+X (T)if|X (T)|+|X (T)| < ∞,and(X +X )(T) = ∞
1 2 1 2 1 2 1 2
otherwise. Thespace W (P )isthen definedasthesetofequivalence classes of
q,p m
additive cochains inW (P )under the equivalence relation defined by X ∼ X
q,p m 1 2
ifkX1−X2kq,p = 0. In[RW13]thenotationWq,p(Pm,Pm+1)wasusedinstead. Itis
clearthatW (P )isavectorspace. Notethattheclassical space offlatcochains
q,p m
isisometrically isomorphic toW (P ),seeLemma6.2.
∞,∞ m
For1 ≤ p≤ ∞denotebyW (P0)thesetofsubadditivecochainsonP0 which
p m m
have an upper gradient in Lp(Rn). TheSobolev norm ofan element X ∈ W (P0)
p m
isdefinedby
kXk = infkgk ,
p p
8 CAMILLEPETIT,KAIRAJALA,ANDSTEFANWENGER
where the infimum is taken with respect to upper gradients g of X. The space
W (P0) is defined to be the set of equivalence classes of additive cochains in
p m
W (P0)undertheequivalence relation definedby X ∼ X ifkX −X k = 0.
p m 1 2 1 2 p
The coboundary dX of a subadditive m-cochain X is the subadditive (m + 1)-
cochaindefinedbydX(S) = X(∂S)forallS ∈ Pm+1. Itfollowsfromthedefinition
thataBorelfunction isanuppergradient of X ifandonlyifitisanupper normof
dX. Note that for all 1 ≤ q,p ≤ ∞ and all 1 ≤ s ≤ ∞ the coboundary operator
yieldsalinearmap
d :Wq,p(Pm)→ Wp,s(Pm+1).
2.4. Modulusandcapacityforpolyhedralchains. LetnowΛ ⊂ P beafamily
m
of polyhedral m-chains and 1 ≤ p < ∞. The p-modulus M (Λ) is defined as
p
inf fpdLn, where the infimum is taken over all non-negative Borel functions
Rn
f suRch that f dkTk ≥ 1 for every T ∈ Λ. The theory of p-modulus of general
Rn
measureswaRsinitiatedbyFuglede[Fug57]. InFuglede’sdefinition,the p-modulus
isdefinedforafamilyofmeasuresinametricmeasurespace. Theabovedefinition
of p-modulus is exactly the one of Fuglede for the family of measures {kTk : T ∈
Λ}. Notethatthe p-modulus isanoutermeasureonthesetofpolyhedral m-chains
P . GivenΛ ⊂ P0,the p-capacity cap (Λ)isdefinedby
m m p
cap (Λ) := M (Γ),
p p
whereΓ = {S ∈ Pm+1 : ∂S = T forsomeT ∈ Λ}. Wedefinethe(q,p)-capacity of
afamilyΛ ⊂ P by
m
cap (Λ):= inf fqdLn+ fpdLn ,
q,p (Z 1 Z 2 )
where the infimum is taken over all non-negative Borel functions f ∈ Lq(Rn)
1
and f ∈ Lp(Rn) satisfying f dkRk + f dkSk ≥ 1 for every decomposition
2 1 2
R+∂S ∈ Λ with R ∈ Pm andRS ∈ Pm+1.RNotice that capq,p(Λ) = 0 if and only if
there exist Λ ⊂ P and Λ ⊂ P0 with M (Λ ) = cap (Λ ) = 0 and such that if
1 m 2 m q 1 p 2
T = R+∂S ∈ Λ,thenR ∈ Λ or∂S ∈ Λ . Thisnotionof(q,p)-capacityisadapted
1 2
tothesetofcochainsW (P )inthesensethatkXk = 0ifandonlyifX(T) = 0
q,p m q,p
foreveryT ∈ P \Λ,wherecap (Λ) = 0.
m q,p
Withthenotionofmodulusavailable,onedefinesweakversionsofuppernorms
and upper gradients as follows. Given a cochain X on P a Borel function h :
m
Rn → [0,∞] is said to be a q-weak upper norm of X if there exists Λ ⊂ P with
m
M (Λ) = 0suchthat(2.2)holds foreveryT ∈ P \Λ. Similarly, aBorelfunction
q m
g :Rn → [0,∞]issaidtobea p-weakuppergradientofXifthereexistsΓ ⊂ Pm+1
with Mp(Γ)= 0suchthat(2.3)holdsforeveryS ∈ Pm+1 \Γ.
We will make frequent use of Fuglede’s lemma [Fug57] which, in our setting,
readsasfollows.
Lemma 2.4 (Fuglede’s lemma). Let 1 ≤ p < ∞, and let f be a Borel function.
Moreover, let (f ) be a sequence of Borel functions converging to f in Lp(Rn).
j
Thenthereexistasubsequence (f )andΛ⊂ P with M (Λ)= 0suchthat
jk m p
|f − f|dkTk → 0
ZRn jk
foreveryT ∈ P \Λ.
m
WOLFE’STHEOREMFORWEAKLYDIFFERENTIABLECOCHAINS 9
As a consequence, one obtains that if 1 < q,p < ∞ then Lq(Rn)-bounded se-
quences of upper norms converge, up to a subsequence, to q-weak upper norms,
andsimilarly, Lp(Rn)-boundedsequencesofuppergradientsconverge,uptoasub-
sequence, to p-weak upper gradients (see [RW13] for details). In particular, the
infimum in the definition of the Sobolev norm of a cochain is attained by some
q-weakuppernormandsome p-weakuppergradient.
We end this section with the following useful observation proved in [RW13,
Proposition 4.17].
Lemma 2.5. Let 0 ≤ m ≤ n and T ∈ P with T , 0. Let B ⊂ Rn be a Borel set
m
withLn(B)> 0. ThenthesetΛ := {ϕ T : x∈ B}has M (Λ) > 0foreveryq ≥ 1.
x# q
2.5. Averages of cochains. Let 0 ≤ m ≤ n and 1 ≤ p,q ≤ ∞. Given an additive
cochain X ∈ W (P )andr > 0,defineanadditivecochain X :P → Rby
q,p m r m
X (T) := X(ϕ T)dx
r ? x#
B(0,r)
for every T ∈ P , where ϕ : Rn → Rn is the translation map ϕ (y) = x + y.
m x x
Here, denotes the integral average Ln(E)−1 and B(0,r) is the open ball of
E E
radius>rcentered at0. Foradditivem-cochains oRnintegralornormalcurrents with
m ≤ n−1themeasurabilityandthelocalintegrabilityofthefunction x7→ X(ϕ T)
x#
wasprovedin[RW13,Lemma4.14(i)]. Wehavethefollowinganalogforcochains
onpolyhedral chains.
Lemma2.6. Let0≤ m ≤ nand1≤ p,q ≤ ∞andletX ∈ W (P )beanadditive
q,p m
cochain. ThenforeveryT ∈ P thefunctionu :Rn → Rgivenbyu(x) := X(ϕ T)
m x#
isLebesguemeasurable andlocallyintegrable.
The same result holds when X is an additive cochain in W (P0) and T ∈ P0
p m m
and we can thus define X in this case as well. It is not difficult to show that
r
X ∈W (P )foreveryr > 0andfurthermore X ∈ W (P ).
r q,p m r ∞,∞ m
Proof. If m = n then it is straight-forward to check that u is continuous. We
may therefore assume that m ≤ n−1. In this case, the proof of [RW13, Lemma
4.14(i)] shows that there exists a non-negative Borel measurable and locally inte-
grablefunctionν¯ whichisanuppergradientofuwithrespecttopolygonal curves,
thatis,suchthat
1
|u(b)−u(a)| ≤ ν¯ ◦γ(t)|γ˙(t)|dt
Z
0
for all a,b ∈ Rn and every polygonal curve γ connecting a and b. Here, it is
understood that the right-hand side must equal ∞ in case u(a) = ∞ or u(b) = ∞.
(Weremarkherethattheproofof[RW13,Lemma4.14(i)]isstatedonlyfor p,q <
∞. However, the same arguments apply inthe case that q = ∞or p = ∞.) Itnow
followsfromawell-knownargument (seee.g.page28in[Hei01])that
|u(b)−u(a)| ≤C|b−a|(Mν¯(b)+ Mν¯(a))
foralla,b ∈Rn,where Mν¯ isthemaximalfunctionofν¯,andwhereC isaconstant
only depending on n. This implies that u is Lipschitz continuous on {Mν¯ ≤ k}
for every k ∈ N and, since ∩{Mν¯ ≥ k} is negligible, it follows that u is Lebesgue
measurable. Finally, theuppernorm inequality, Fubini’s theorem and(incasethat
q < ∞)Ho¨lder’sinequality yieldthatuislocallyintegrable. (cid:3)
10 CAMILLEPETIT,KAIRAJALA,ANDSTEFANWENGER
In the proof of Theorem 3.1 we will need the following two crucial facts from
[RW13] about the averages X (T). The statements given in [RW13] are slightly
r
stronger and are proved in the generality of normal and integral currents in Lie
groupsequippedwithaleft-invariant Finslermetric. InthesettingofRntheresults
can be stated in a somewhat simpler form. We thus provide them here for the
convenience of the reader. The following is a restatement of [RW13, Proposition
4.15(ii)].
Proposition2.7. Let0 ≤ m ≤ n−1and1≤ p < ∞. Letfurthermore X ∈ W (P0)
p m
be an additive cochain with upper gradient g in Lp(Rn). Then there exists C =
C(n,m,p)> 0suchthatforeveryT ∈ P0m,everyS ∈ Pm+1 with∂S = T,andevery
r > 0,wehave
|X (T)| ≤Cr−n/pM(S)kgk .
r p
Inordertostatethesecondpropositionwedefinethemaximalgrowthofapoly-
hedralchainT ∈ P by
m
kTk(B(x,r))
Θ (T) := sup ,
m rm
where thesupremum istaken over all x ∈ Rn and allr > 0. Note that Θ (T) < ∞
m
foreveryT ∈ P . Wenowgivearestatement of[RW13,Proposition 4.16].
m
Proposition2.8. Let0 ≤ m ≤ n−1andletn−m+1 < q < ∞andn−m < p< ∞.
(i) LetX ∈ W (P0)beanadditivecochainwithuppergradientginLp(Rn).
p m
Then there exists a constant D = D(p,m,n) > 0 such that for every T ∈
P0 andeveryr > 0wehave
m
|Xr(T)−X(T)| ≤ DΘ1m/p(T)r1+m−pnM(T)p−p1kgkp.
(ii) Let X ∈ W (P ) be an additive cochain with upper norm h in Lq(Rn)
q,p m
and upper gradient g in Lp(Rn). Then there exists E = E(q,p,m,n) > 0
suchthatforeveryT ∈ P andeveryr > 0wehave
m
|Xr(T)−X(T)| ≤ E Θ1m/p(T)r1+m−pnM(T)p−p1kgkp
(cid:20)
+ Θ1m/−q1(∂T)r1+m−q1−nM(∂T)q−q1khkq .
(cid:21)
Theproofofthispropositionisexactlyastheproofof[RW13,Proposition4.16]
exceptthatthereference to[RW13,Lemma4.14(i)]therein should bereplaced by
areference toLemma2.6above.
3. Continuityofaverages
TheaimofthissectionistoproveTheorem3.1below,whichprovidesthemain
continuity result needed in the proof of our generalizations of Wolfe’s theorem.
ThesecondpartofTheorem3.1wasstatedasTheorem1.4intheintroduction.
Theorem3.1. Let0 ≤ m ≤ nand1 < p,q ≤ ∞. Thenwehave:
(i) ifX ∈ W (P0)isanadditivecochainthen
p m
|X (T)−X(T)| → 0asr → 0
r
for every T ∈ P0 \Λ , where Λ ⊂ P0 has p-capacity 0. If p > n−m
m 1 1 m
thenwemaytakeΛ = ∅.
1