Differential
of
metric
valued
Sobolev
maps
✩Nicola Giglia,∗, Enrico Pasqualettob,Elefterios Soultanisc
aSISSA,ViaBonomea265,34136Trieste,Italy
bUniversityofJyväskylä,POBox35,FI-40014Jyväskylä,Finland
cUniversityofFribourg,Av.del’Europe20,CH-1700Fribourg,Switzerland
Keywords:
Functionspaces Metricmeasurespaces Sobolevspaces
WeintroduceanotionofdifferentialofaSobolevmapbetween metric spaces. Thedifferential isgiven intheframework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim’s metric differential when the sourceisaEuclideanspace,andwiththeabstractdifferential provided by the first author when the target is R. We also show compatibility with the concept of co-local weak differential introducedbyConventandVanSchaftingen.
Contents
1. Introductionandmainresults . . . 2
2. Preliminaries . . . 4
3. Differentialofmetric-valuedSobolevmaps . . . 7
4. Consistencywithpreviouslyknownnotions . . . 10
4.1. ThecaseY=R . . . 10
4.2. Thecaseu ofboundeddeformation . . . 12
4.3. ThecaseX=Rd andu Lipschitz . . . 15
4.4. Thecaseofsmoothspacesandco-localweakdifferential . . . 16
✩ ThankstoMIURSIR-grant‘NonsmoothDifferentialGeometry’(RBSI147UG4).
* Correspondingauthor.
E-mailaddresses:[email protected](N. Gigli),enpasqua@jyu.fi(E. Pasqualetto),
[email protected](E. Soultanis).
http://doc.rero.ch
Published in "Journal of Functional Analysis 278(6): 108403, 2020"
which should be cited to refer to this work.
5. DifferentialoflocallySobolevmapsbetweenmetricspaces . . . 19
5.1. Inverselimitsofmodules . . . 19
5.2. LocallySobolevmapsandtheirdifferential . . . 21
References . . . 23
1. Introductionandmainresults
Background and setting The concept of real valued Sobolev functions defined on a metric measure space (X,dX,mX) is by now well understood. Given an exponent p∈
[1,∞) the space of functionsf : X → R having ‘distributionaldifferential inLp(X) in
a suitable sense’ is denoted by Sp(X). To each f ∈ Sp(X) one associates the function
|Df|∈ Lp(X),calledminimalweakuppergradient,whichinthesmoothsettingcoincides
with themodulusofthedistributionaldifferential(see [6] and [18],[4]).
Inspired by the work of Weaver [20], in [9] the first author built the theory of
Lp-normed modules and gave a notion of differential df for maps f ∈ Sp(X) in that
framework:bydefinition,df isanelementofthesocalledcotangent Lp-normedmodule
Lp(T∗X) andhastheproperty thatitspointwisenormcoincidesm
X-a.e.with|Df|. We
remark thatthelinearstructure ofthespace Sp(X),aconsequenceof thefactthatthe target spaceR isavectorspace,plays akeyroleintheconstruction.
Wenowturn tothecaseofmetric-valuedSobolevmaps. Let(X,dX,mX) beametric
measure spaceasbeforeandlet(Y,dY) beametricspacewhichshallbeassumedtobe
completeandseparable.Weshallalsofixp= 2 forsimplicity.Therearevariouspossible definitionsoftheconceptofSobolevmapsfromX toY;hereweshallworkwiththeone based on post-composition (see [13] for historical remarks): we say thatf ∈ S2(X;Y)
providedthereisG∈ L2(X) suchthatforanyϕ: Y→ R Lipschitzwehaveϕ◦f ∈ S2(X)
with
|D(ϕ ◦ f)| ≤ Lip(ϕ) G mX− a.e.
The least such G isthen denoted |Df| andcalled theminimal weak upper gradient of
themap f.NoticethatsinceY hasnolinearstructure,theset S2(X;Y) isnotavector
space ingeneral.
Thequestionwe addressinthispaper isthefollowing:inanalogy withthefactthat
‘behind’ the minimal weak upper gradient |Df| of areal-valued Sobolevmap there is
an abstract differential df,does there exist a notion of differential fora metric-valued Sobolevmap?
Before turningto the(positive) answerto thisquestion,letus motivateour interest inthe problem,whichgoes beyondthe meredesireof generalization.In thecelebrated
paper [7], Eells and Sampson proved Lipschitz regularity for harmonic maps between
RiemannianmanifoldswhenthetargetN hasnon-positivecurvatureandissimply
con-nected,andtheLipschitzestimateisgivenintermsofalowerRiccicurvatureboundand
anupper dimensionboundonthesourcemanifoldM. Akeypoint intheirproof isthe
establishmentof the now-called Bochner-Eells-Sampsonformula for maps f : M → N whichweshallwriteas
Δ|df|
2
2 ≥ ∇f(Δf) + K|df|
2, (1.1)
where |df| is the Hilbert-Schmidt norm of the differential of f and K ∈ R is alower
boundfortheRiccicurvatureofM (letusremainvagueaboutthemeaningof∇f(Δf)).
Adirectconsequenceof(1.1) isthatiff isharmonic,then
Δ|df|
2
2 ≥ K|df|
2. (1.2)
This bound and Moser’s iteration technique are sufficient to show that |df| is locally
bounded fromaboveinthe domain ofdefinition off, thus showingthe local Lipschitz
regularity of f (the upper dimension bound for M enters into play in the constants
appearinginMoser’sargument).
SincetheLipschitzregularityofharmonicfunctionsdoesnotdepend onthe
smooth-nessofM andN but onlyinthe statedcurvaturebounds, itisnaturalto askwhether thesameresultsholdassumingonlytheappropriatecurvatureboundsonthesourceand
target space, without any reference to smoothness. Efforts in this direction have been
madeby Gromov-Schoenin[12], by Korevaar-Shoenin[15] andbyZhang-Zhu in[21]. Themostgeneralresultisin[21], where theauthorsconsider thecaseofsource spaces
whicharefinite-dimensionalAlexandrovspaceswith(sectional)curvatureboundedfrom
belowandtargetswhichare CAT(0) spaces.Still,givenEells-Sampson’sresultthe
nat-uralsyntheticsettingappearstobethatofmapsfromaRCD(K,N) spaceto aCAT(0)
space;asoftoday,thisappearstobeoutofreach.Letusremarkthatinnoneofthese3 papershasinequality(1.1) been writtendownexplicitly; in[15] and[21] “only”aform of(1.2) forharmonicmapshasbeen established(in[12] theargumentwasdifferentand
basedonAlmgren’s frequencyfunction).
Thepresentmanuscriptaimsatbeingafirststepinthedirectionofobtaining(1.1) for mapsfromRCD spacestoCAT(0) ones(seealso[11]):ifsuccessful,thisresearchproject easilyimplies thedesiredLipschitz regularityfor harmonicmapsand at thesametime
improves the understandingof the subject even inpreviously studiednon-smooth
set-tings.Theoverallprogramisdefinitelyambitious,butwebelievethatevenintermediate stepslikethecurrentmanuscripthaveanintrinsicinterest:seeinparticularthe‘review’ ofKirchheim’snotionofmetricdifferential inSection4.3.
Theveryfirststeptotackleinordertowritedown(1.1) isto understandwhat“df” is. As stated, this is our goal in this manuscript. Let us informally describe the key conceptinthis work(theprecisedefinitionswillbe giveninSections 2and 3).
Differential of Sobolev maps Given a Sobolev map u ∈ S2(X;Y) between a metric
measurespace(X,dX,m) andacompleteseparablemetricspace(Y,dY),weconsiderthe
metricmeasurespace(Y,dY,μ),whereμ:=u∗(|Du|2m).Thenwedefinethedifferential
du ofu asanoperator
du : L0(T X) → (u∗L0μ(T∗Y))∗ satisfying
u∗df, du(V ) = V (d(f ◦ u)) m-a.e. (1.3)
foreveryf ∈ S2(Y,d
Y,μ) and V ∈ L0(T X) (Definition3.4).
Theparticular choiceofmeasure μ isimportant:it ensuresthatfor f ∈ S2(Y,dY,μ)
thepullbackfunctionu∗f := f ◦ u belongsto S2(X,d
X,m) with
|D(f ◦ u)| ≤ |Df| ◦ u |Du|, (1.4) see Proposition3.3fortheprecise formulation.Once thisisestablished, thedifferential of u canbe definedby taking theappropriate adjointof themap df → d(f ◦ u), as in (1.3). Let us emphasise thatonthe right handside of the crucial bound (1.4) thereis theproduct oftwo‘weak’objects:thismakestheinequalitynon-trivial.
Once the definition is given we verify that it is compatible, and thus generalizes,
previouslyexistingnotionsofdifferentialsinthenon-smoothsetting. Allourdiscussion
is made for the Sobolev exponent p = 2, but obvious modifications generalise all the
resultsto thecasep∈ (1,∞). 2. Preliminaries
To keep the presentation short we assume the reader is familiar with the concept
of Sobolev functions on a metric measure space ([6], [18], [4], [3]) and with that of
L0-normed modulesanddifferentialsofrealvaluedSobolevmaps([9],[8]).
Hereweonlyrecallthoseconceptsweshallusemostfrequently.Letusfixacomplete, separablemetricspace(X,dX) andanon-negativeandnon-zeroRadonmeasurem giving
finite mass to boundedsets. We shalldenote by Lip(f) the (global)Lipschitz constant of afunction,byLIP(X),LIPbs(X),LIPbd(X) thespace ofLipschitzfunctions,Lipschitz
functions with bounded support, and functions which are Lipschitz on bounded sets,
respectively.Wealso denotebylipa(f): X→ [0,∞] theasymptotic Lipschitzconstant,
definedby
lipa(f)(x) := lim
y,z→x
|f(y) − f(z)|
dX(y, z)
ifx is not isolated, 0 otherwise.
Then wedefine:
Definition 2.1 (The Sobolev class S2(X)).We say that f ∈ S2(X) provided there is a
functionG∈ L2(m) andasequence(f
n)⊂ LIPbd(X) convergingtof inL0(m) suchthat
(lipa(fn)) weaklyconvergesto G inL2(m).
Withrespecttotheapproachin[4], [3] herethedifferenceis inthetopologyusedin therelaxationprocedure.Thefactthatourapproachisequivalent totheonein[4],[3] follows fromthe L0-stability ofweak uppergradients grantedbytheapproach viatest
plansinconjunctionwithacut-offargument.
For f ∈ S2(X) we recall that there is a minimal, in the m-a.e. sense, non-negative
functionG∈ L2(m) forwhichthe situationinDefinition2.1occurs.Such G isdenoted
|Df| andcalled minimalweakuppergradient.It istheneasytocheck that:
∀f ∈ S2(X) there is (f
n)⊂ LIPbd(X)m − a.e. converging to f such that
lipa(fn)→ |Df| in L2(m). (2.1)
Fromtheminimalweak uppergradientsonecan ‘extract’anotionofdifferential:
Theorem2.2(Cotangentmoduleanddifferential). Withtheabove notationand
assump-tions, there isaunique (up tounique isomorphism)couple (L0(T∗X),d) with L0(T∗X)
beinga L0(m) normed module, d: S2(X)→ L0(T∗X) linear andsuchthat: |df|=|Df|
m-a.e.forevery f ∈ S2(X) and{df : f ∈ S2(X)} generatesL0(T∗X).
When we want to emphasise the role of the chosen measure, we shall write
(L0m(T∗X),dm) inplaceof(L0(T∗X),d).Amongthevariouspropertiesofthedifferential, weshallfrequentlyuseitslocality:
df = dg m − a.e. on {f = g}, ∀f, g ∈ S2(X). Letusnowrecallfewfactsaboutpullback ofmodules:
Theorem2.3(Pullback).Let(X,dX,mX),(Y,dY,mY) bemetricmeasurespacesasabove,
u : X → Y such that u∗mX mY and M an L0(mY)-normed module. Then
there is a unique (up to unique isomorphism) couple (u∗M,[u∗]) such that u∗M is a L0(mX)-normed module and [u∗] : M → u∗M is linear, continuous and such that
|[u∗v]|=|v|◦ u m
X-a.e. foreveryv ∈M and {[u∗v]:v ∈M} generates u∗M.
Themoduleu∗M iscalledthepullbackmoduleand[u∗] thepullbackmap.Itcanbe
directlychecked bytheuniquenesspartofTheorem 2.3that
if u∗mX mY then u∗L0(mY)∼ L0(mX) via the map [u∗f] → f ◦ u. (2.2)
Thepullbackhasthefollowing universalproperty,whichweshallfrequentlyuse:
Proposition 2.4 (Universal property of the pullback).With the same notation and assumptions as in Theorem 2.3 above, let V ⊂ M a generating subspace, N a L0(m
X)-normed module and T : V → N a linear map such that |T (v)| ≤ f|v|◦ u
mX-a.e. ∀v ∈ V for some f ∈ L0(mX). Then there exists a unique L0(mX)-linear and
continuousmapT : u˜ ∗M →N suchthat T ([u˜ ∗v])=T (v) foreveryv ∈ V andthismap satisfies
| ˜T (w)| ≤ f|w| mX− a.e. ∀w ∈ u∗M. (2.3)
Inparticular,ifT :M1→M2isaL0(mY)-linearandcontinuousmapsatisfying|T (v)|≤
g|v| mY-a.e. ∀v ∈M1,forsome g ∈ L0(mY), applying theabove tothemapM1 v →
[u∗T (v)] ∈ u∗M2 we deduce that there exists a unique L0(mX)-linear and continuous
mapu∗T : u∗M1→ u∗M2 makingthediagram
M1 M2 u∗M 1 u∗M2 T [u∗] [u∗] u∗T
commute andsuchmapsatisfies
|u∗T (w)| ≤ g ◦ u|w| m
X− a.e. ∀w ∈ u∗M1. (2.4)
Thesepropertiesofpullbackshavebeenstudiedin[9],[8] formapssatisfyingu∗mX≤
CmY, butifoneisonly interestedinL0-modulesthetheorems aboveareeasilyseento
hold withsmallmodifications.
Finally, letus presentasimpleconstructionthatweshallfrequentlyuse. LetE ⊂ X beBorel,putν := m|
E andletM beaL
0(ν)-normedmodule.Tosuchamodulewecan
canonically associate aL0(m)-normed module, called extension of M and denoted by
Ext(M),inthefollowingway.Firstofallwenoticethatwehaveanatural projection/re-striction operatorproj :L0(m)→ L0(ν) givenbypassagetothequotientuptoequality
ν-a.e.andanatural‘extension’operatorext :L0(ν)→ L0(m) which sendsf ∈ L0(ν) to
thefunctionequaltof m-a.e.onE andto0 onX\ E.ThenforagenericL0(ν)-normed
moduleM weputExt(M):=M asaset,multiplicationof v ∈ Ext(M) byf ∈ L0(m) is definedas proj(f)v ∈M = Ext(M) andthepointwise normasext(|v|)∈ L0(m).We
shall denotebyext :M → Ext(M) theidentitymap andnoticethatinarathertrivial
way wehave
Ext(M∗)∼ Ext(M)∗ via the coupling ext(L)ext(v):= ext(L(v)). (2.5) Inwhatfollowsweshallalwaysimplicitlymakethis identification.
3. Differentialofmetric-valuedSobolevmaps
Throughoutthismanuscript(X,dX,m) willalwaysdenoteacompleteseparablemetric
space endowed with a non-negative and non-zero Radon measure which is finite on
boundedsets; (Y,dY) denotesacompleteseparablemetricspace.
Definition3.1(MetricvaluedSobolevmap).ThesetS2(X,Y) isthecollectionofallBorel mapsu: X→ Y for whichthere isG∈ L2(X,m),G≥ 0 suchthatforany f ∈ LIP(Y)
itholdsf ◦ u∈ S2(X) and
|d(f ◦ u)| ≤ Lip(f)G m − a.e. (3.1) Theleast,inthem-a.e.sense,functionG forwhichtheaboveholdswillbedenoted|Du|. Noticethatforu∈ S2(X,Y) theclassof G∈ L2(X) for which(3.1) holdsis aclosed lattice,henceam-a.e.minimaloneexistsandthedefinitionof|Du| iswellposed.
OurstudyoffunctionsinS2(X,Y) begins withthefollowingbasiclemma:
Lemma3.2. Letu∈ S2(X,Y) andf ∈ LIP(Y).Thenf ◦ u∈ S2(X) with
|d(f ◦ u)| ≤ lipa(f) ◦ u |Du| m − a.e. (3.2)
Proof. Let (yn) ⊂ Y be countableand dense and for r ∈ Q, r > 0, letfr,n ∈ LIP(Y)
beaMcShaneextensionof f|
Br(yn), i.e.aLipschitzmap definedonthe wholeY which
coincides with f on Br(yn) and such that Lip(fr,n) = Lip(f|B
r(yn)). Then from (3.1)
andthelocalityofthedifferentialweseethat
|d(f ◦ u)| ≤ Lip(f|Br(yn))|Du| m − a.e. on u
−1(B
r(yn)).
Since for every y ∈ Y we have lipa(f)(y) = inf Lip(f|
Br(yn)), where the inf is taken
amongalln,r suchthaty ∈ Br(yn),theconclusionfollows. 2
Letusfixu∈ S2(X,Y) andequip thetarget spaceY withthefinite Radonmeasure
μ := u∗(|Du|2m).
Noticethatforf ∈ L0(Y,μ) thefunctionf ◦u isnotwell-defineduptoequalitym-a.e.in
thesensethatiff = ˜f μ-a.e.,thennotnecessarilyf ◦ u= ˜f ◦ u m-a.e. Still,wecertainly havef ◦u= ˜f ◦u m-a.e.on{|Du|> 0} andforthisreasonwehavef ◦u|Du|= ˜f ◦u|Du| m-a.e.,i.e.themap f → f ◦ u|Du| iswelldefinedfromL0(Y,μ) toL0(X,m).Thenthe
identity f ◦ u|Du|2dm=|f|2dμ showsthat
L2(Y, μ) f → f ◦ u |Du| ∈ L2(X, m) is linear and continuous. (3.3)
Wenow turntoourkeybasicresultaboutpullbackof Sobolevfunctions: Proposition 3.3. Letu∈ S2(X,Y),put μ:=u
∗(|Du|2m) andlet f ∈ S2(Y,dY,μ). Then
there isg ∈ S2(X) suchthat g = f ◦ u m-a.e. on{|Du|> 0} and
|dg| ≤ |dμf| ◦ u|Du| m − a.e. (3.4)
Moreprecisely,there isg ∈ S2(X) andasequence(f
n)⊂ LIPbd(Y) suchthat
fn → f μ − a.e. lipa(fn) → |dμf| inL2(μ),
fn◦ u → g m − a.e. lipa(fn)◦ u|Du| → |dμf| ◦ u|Du| in L2(m).
(3.5)
Proof. Up to a truncation and diagonalization argument we can assume that f ∈
L∞(Y,μ). Then let (f
n) ⊂ LIPbd(Y) be as in (2.1) for f and observe that since f is
bounded,bytruncationwecanassumethefn’stobeuniformlybounded.Thusthefirst
twoconvergencesin(3.5) holdand,taking(3.3) intoaccountweseethatalsothelastin (3.5) holds.Now observe thatifwecan provethat(fn◦ u) hasalimitm-a.e.,call itg,
then (3.4) wouldfollowfromLemma3.2above,(3.3) andtheclosureofthedifferential. LetB ⊂ X beboundedandBorel.Thefunctionsfn◦ u areequiboundedandm(B)<
∞, hence(fn◦ u) is bounded in L2(B,m|B). Thus by passingto an appropriate - not
relabeled - sequence of convex combinations (which do not affect the already proven
convergences in(3.5))weobtainthat(fn◦ u) hasastronglimitinL2(B,m|B). Thusa
subsequenceconvergesm-a.e.onB andbyconsideringasequence(Bk) ofboundedsets
suchthatX=∪kBk,byadiagonalizationargumentweconcludetheproof. 2
Let us notice thatsince μ is afinite measure on Y we have LIP(Y)⊂ S2(Y,
dY,μ).
Also,
forf ∈ LIP(Y) and g ∈ S2(X) as in Proposition3.3we have d(f ◦ u) = dg. (3.6) Indeed,thelocalityofthedifferentialgivesd(f ◦ u)= dg on{|Du|> 0} andthebounds (3.2) and (3.4) give|d(f ◦ u)|=|dg|= 0m-a.e.on{|Du|= 0}.
Observe that for ν := m|
{|Du|>0} we have u∗ν μ, thus u∗L0μ(T∗Y) is a well
de-fined L0(ν)-normed module.Recalling the‘extension’ functorintroduced at theend of
Section2,ourdefinitionofdu is:
Definition 3.4.Thedifferential du ofu∈ S2(X,Y) istheoperator du : L0(T X) → Ext(u∗L0μ(T∗Y))∗ given as follows. For v ∈ L0(T X), the object du(v) ∈ Ext(u∗L0
μ(T∗Y))∗
is charac-terized by the property: for every f ∈ S2(Y,d
Y,μ) and every g ∈ S2(X,dX,m) as in
Proposition3.3wehave
ext[u∗dμf]
du(v)= dg(v) m − a.e. (3.7) Wenowverifythatthisisagooddefinitionandcheck theverybasicproperties: Proposition3.5(Wellposedness ofthedefinition).The differentialdu(v) ofu in Defini-tion3.4iswell-definedandthemapdu: L0(T X)→ Ext(u∗L0μ(T∗Y))∗isL0(m)-linear
andcontinuous.Moreover, itholdsthat
|du| = |Du| m − a.e. (3.8) Proof. Let f ∈ S2(Y,d
Y,μ) and observe that if g,g ∈ S2(X,dX,m) both satisfy the
propertieslisted inProposition 3.3 then the locality of the differential and the bound (3.4) show that dg = dg. Hence the right handside of (3.7) depends only on f,u,v. Thennoticethatagainthebound(3.4) gives
ext[u∗dμf]
du(v)(3.7)= |dg(v)| ≤ |dg| |v|(3.4)≤ |dμf|◦u|Du| |v| =ext
[u∗dμf]|Du||v|
andthusthearbitrarinessoff ∈ S2(Y,d
Y,μ),Proposition2.4andproperty(2.5) ensure
thatdu(v) is a well defined element of Ext(u∗L0
μ(T∗Y)) ∗ ∼ Ext(u∗L0 μ(T∗Y))∗ , as desired,with |du(v)| ≤ |Du| |v|. (3.9) Thefactthatdu(v) isL0(m)-linearinv istrivialandthebound(3.9) givesboth
continu-ityandtheinequality≤ in(3.8).Togettheotherinequalityletf : Y → R be1-Lipschitz and notice thatsinceμ(Y)< ∞ we alsohave f ∈ S2(Y,d
Y,μ). Since u∈ S2(X,Y) we
havef ◦ u∈ S2(X) andcanfindv ∈ L0(T X) suchthat
|v| = 1 and d(f ◦ u)(v) = |d(f ◦ u)| m-a.e. (3.10) (theexistenceofsuchv followsbyBanach-Alaoglu’stheorem,see[9,Corollary1.2.16]). Moreover,letg ∈ S2(X) beasinProposition3.3andnoticethat
|d(f ◦ u)|(3.10)= |d(f ◦ u)(v)|(3.6)= |dg(v)|
(3.7)
= ext[u∗dμf]
du(v) ≤ ext[u∗dμf]|du||v|
(3.10)
= |dμf| ◦ u |du| ≤ |du|,
havingusedthefactthatf is1-Lipschitzinthelast step.Bythearbitrarinessoff and theverydefinitionof|Du| giveninDefinition3.1,thisestablishes ≥ in(3.8). 2
4. Consistencywithpreviouslyknownnotions
4.1. Thecase Y=R
InthissectionweassumeY=R andprovethatonceafewnaturalidentificationsare taken into account,thenewly defineddifferentialdu: L0(T X)→ Extu∗L0
μ(T∗R)
∗
is
‘thesame’astheonedefinedbyTheorem2.2,whichforthemomentweshalldenoteby
du∈ L0(T∗X).
To startwith, letus observethatdirectlyfromthedefinitionsandthechainrule d(f ◦ u) = f◦ u du m − a.e. ∀u ∈ S2(X), f ∈ C1∩ LIP(R) (4.1) (see [8, Corollary 2.2.8]), we have that the class S2(X,Y) coincides with S2(X) when
Y=R andthatthetwonotionsofminimalweak uppergradientscoincide.
For later use it will be convenient to consider the case where the target space is
a generic Riemannian manifold rather than R. Thus letN be acomplete Riemannian
manifold, dN the distance induced by the metric tensor and fix u ∈ S2(X,N). Also,
let μ beanon-negative Radonmeasure onN. Weshall denote byL0(N,T∗N;μ) (resp.
L0(N,T N;μ)) the L0(μ)-normed module of sections of the cotangent (resp. tangent)
bundleidentifieduptoequalityμ-a.e.(thenotationisunusual,buthopefullyhelps com-paringthese‘concrete’notionswiththemoreabstractoneswearediscussinghere).We shall insteaddenote byL0
μ(T∗N) (resp. L0μ(T N))thecotangent(resp. tangent) module
associated tothespace(N,dN,μ).
The nextresult hasbeen provedin [10] forthecase N=Rd and generalisedin[17]
to themanifoldcase:
Theorem 4.1. Let N be a complete Riemannian manifold and μ be a non-negative
Radon measure on it. Then there is a unique L0(μ)-linear and continuous map P :
L0(N,T∗N;μ)→ L0
μ(T∗N) suchthat
P (Df) = dμf ∀f ∈ C1(N), (4.2)
whereDf : N→ T∗N isthedifferentialoff.Itsadjointmapι:L0
μ(T N)→ L0(N,T N;μ)
is anisometry.In particular,L0
μ(T N) isseparable.
Let now u∈ S2(X,N), putμ:=u
∗(|du|2m) andconsider the L0(m)-normedmodule
Ext(u∗L0
μ(T∗N)).TheseparabilityofL0μ(T N) providedbyTheorem4.1above,the
char-acterisation of the dual of thepullback obtained in[8, Theorem 1.6.7] and(2.5) grant that
Ext(u∗L0μ(T N)) ∼ Ext(u∗L0μ(T∗N))∗ via the coupling
ext([u∗L])ext([u∗v]):= ext(L(v) ◦ u). (4.3)
Hence in the present situation we shall think of du as a map from L0(T X) to
Ext(u∗L0
μ(T N)).
Now put ν := χ{|Du|>0}m as before and consider the L0(ν)-linear and continuous operators
u∗P : u∗L0(N, T∗N;μ) −→ u∗L0
μ(T∗N),
u∗ι : u∗L0
μ(T N) −→ u∗L0(N, T N; μ)
definedviatheuniversalproperty ofthepullbackmodulegiveninProposition2.4.It is thenclearthatu∗ι istheadjointofu∗P ,thusfrom(4.2) weseethat
[u∗Df]u∗ι(V )= [u∗dμf](V ) ν − a.e.
for everyV ∈ u∗L0μ(T N), f ∈ C1(N). (4.4)
Finally, noticing that ext : u∗L0
μ(T N) → Ext(u∗L0μ(T N)) is invertible, we define I :
Ext(u∗L0
μ(T N))→ Ext(u∗L0(N,T N;μ)) as
I := ext ◦ u∗ι ◦ ext−1. (4.5)
LetusnowconsiderthecaseN=R.InthiscasethecanonicalisomorphismTxR∼ R
valid forany x∈ R givesL0(R,T R;μ) ∼ L0(μ). With this identification and recalling
(2.2) andtheverydefinitionoftheextensionfunctorweseethatthemapI takesvalues inL0(m).Thenwehave:
Theorem4.2. Withtheabove notation andassumptionswehave |du|=|du| m-a.e. and
I(du(v)) = du(v) m − a.e. ∀v ∈ L0(T X). (4.6)
Proof. Theidentity |du|=|du| followsfrom(3.8) andthe alreadynoticedfactthatfor
u∈ S2(X)= S2(X,R) thetwo notionsofminimal weak uppergradientsunderlyingthe
twospacescoincide.
Weturnto(4.6).Forf ∈ C1
c(R) letusdenotebyDf : R→ R∗itsdifferentialandby
f:R→ R itsderivative.Clearly,uptoidentifyingR andR∗viatheRieszisomorphism
thesetwo objectscoincideandthuschecking firstthecaseh= [u∗g] weeasily getthat
f◦ u h = ext[u∗Df](h) m − a.e.
∀h ∈ Extu∗L0(μ)(2.2)∼ ExtL0(ν)⊂ L0(m). (4.7)
Thenforg asinProposition3.3we have
f◦ u I(du(v))(4.7)
= ext[u∗Df] I(du(v))(4.5),(2.5)= ext[u∗Df]u∗ιext−1(du(v))
(4.4)
= ext[u∗dμf]
ext−1(du(v))(2.5)= ext([u∗dμf])(du(v))
(3.7)
= dg(v)
(3.6)
= d(f ◦ u)(v)(4.1)= f◦ u du(v). Since the space {f◦ u : f ∈ C1
c(R)} generates L0(m), this is sufficient to establish
(4.6). 2
4.2. Thecase u ofboundeddeformation
Inthissectionweshallassumethatalso(Y,dY) carriesanon-negativeRadonmeasure
mY which gives finite mass to bounded sets and study the differential of a map u ∈
S2(X,Y) which is also of bounded deformation. Recall that the latter means that u is
Lipschitz andforsomeC > 0 itholdsu∗mX≤ CmY, wherewe denotemX:=m forthe
sakeofclarity.Forsuchu itiseasyto provethat
f ∈ S2(Y) ⇒ f ◦ u ∈ S2(X) with |d(f ◦ u)| ≤ Lip(u)|df| ◦ u m
X− a.e.
Then a notion of differential ˆdu : L2(T X) → u∗L2
mY(T∗Y)
∗
can be defined by the
formula
[u∗dmYf](ˆdu(v)) := d(f ◦ u)(v) mX− a.e. ∀f ∈ S2(Y, dY, mY), v ∈ L2(T X), (4.8)
see [8,Proposition2.4.6]. Inthis sectionwe studythe relationbetweendˆu anddu. We start noticing thatthe definition of |Du| trivially gives |Du| ≤ Lip(u) mX-a.e., so we
have
μ = u∗(|Du|2mX)≤ Lip2(u)u∗mX≤ CLip2(u)mY. (4.9)
Also,letusprovethefollowinggeneralstatement:
Lemma4.3. Letμ1,μ2betwonon-negativeandnon-zeroRadonmeasuresonthecomplete
space (Y,dY) with μ1 ≤ μ2. Then S2(Y,dY,μ2) ⊂ S2(Y,dY,μ1) and there is aunique
L0(μ
2)-linear andcontinuousmapP : L0μ2(T∗Y)→ Ext(L 0
μ1(T∗Y)) suchthat
P (dμ2f) = ext(dμ1f) ∀f ∈ S2(Y, dY, μ2),
and itsatisfies|P (ω)|≤ |ω| μ2-a.e. forevery ω ∈ L0μ2(T∗Y),where here the‘extension’
operator actsfromL0(μ
1)- toL0(μ2)- normedmodules.
Proof. The assumption μ1 ≤ μ2 ensures that the topologies of L2(μ2),L0(μ2)
are stronger than those of L2(μ
1),L0(μ1) respectively. Thus both the inclusion
S2(Y,d
Y,μ2) ⊂ S2(Y,dY,μ1) and the bound ext(|dμ1f|) ≤ |dμ2f| μ2-a.e. for every
f ∈ S2(Y,d
Y,μ2) follow from Definition 2.1. To conclude apply, e.g., Proposition 2.4
withu:= Identity andT (dμ2f):= ext(dμ1f)∈ Ext(L0μ1(T∗Y)). 2
Applyingthislemmato thecaseunderconsiderationweget:
Proposition 4.4. Assume that u : X → Y is of bounded compression. Then with the
above notation there is a unique L0(m
Y)-linear and continuous map π : L0mY(T∗Y) →
Ext(L0μ(T∗Y)) suchthatπ(dmYf)= ext(dμf) foreveryf ∈ S2(Y,dY,mY) (theextension
operatorbeingintendedfromL0(μ)- toL0(m
Y)- normedmodules)anditsatisfies|π(ω)|≤
|ω| mY-a.e.forevery ω ∈ L0mY(T∗Y).
Moreover,forany f ∈ S2(Y,d
Y,mY) andg ∈ S2(X) asin Proposition 3.3wehave
dg = d(f ◦ u). (4.10)
Proof. ThefirstpartofthestatementfollowsfromLemma4.3and(4.9).Toprove(4.10) notice thatthanksto thelocality of thedifferential weknow that(4.10) holdsmX-a.e.
on{|Du|> 0},while(3.4) showsthatdg = 0 mX-a.e. on{|Du|= 0},hencetoconclude
itissufficienttoprovethat|d(f ◦ u)|= 0mX-a.e.on{|Du|= 0}.Toseethis,let(fn)⊂
LIPbd(Y) be such that (fn),(lipa(fn)) converge to f,|dmYf| mY-a.e. and in L2(mY)
respectively. Then the assumption u∗mX ≤ CmY grants that (fn ◦ u),
lipa(fn)◦ u
converge to f ◦ u,|dmYf|◦ u mX-a.e. and inL2(mX) respectively.Hence passingto the
limitin(3.2) weconcludethat|d(f ◦ u)|= 0 mX-a.e.on{|Du|= 0},asdesired. 2
Itisreadilyverifiedthatthemapsending[u∗ext(ω)] toext([u∗ω]) isanisomorphism fromu∗Ext(L0
μ(T∗Y)) to Ext(u∗L0μ(T∗Y)), hence from Proposition 4.4 above and the
universalpropertyofthepullbackstatedinProposition2.4weseethatthereisaunique
L0(m
X)-linearandcontinuous mapu∗π : u∗L0mY(T∗Y)→ Ext(u∗L0μ(T∗Y)) suchthat
u∗π([u∗d
mYf]) = ext[u∗dμf] ∀f ∈ S2(Y, dY, mY) (4.11)
andsuchmapsatisfies
|u∗π(ω)| ≤ |ω| m
X− a.e. ∀ω ∈ u∗L0mY(T∗Y). (4.12)
Then denoting by (u∗π)∗ : Ext(u∗L0
μ(T∗Y)) ∗ → u∗L0 mY(T∗Y) ∗ the adjoint of u∗π wehave:
Theorem4.5. Withtheabove notation andassumptionswehave
ˆ
du(v) = (u∗π)∗du(v) ∀v ∈ L0(T X) (4.13)
and
ˆdu(v) ≤ du(v) mX-a.e. on X ∀v ∈ L0(T X). (4.14)
Proof. Let f ∈ S2(Y,d
Y,mY) andnoticethat
[u∗dmYf](ˆdu(v))(4.8)= d(f ◦ u)(v)(4.10)= dg(v)(3.7)= ext[u∗dμf](du(v))
(4.11)
= (u∗π)([u∗dmYf])(du(v)). Since elements of the form [u∗dmYf] generate u∗L0
mY(T∗Y), this is sufficient to prove
(4.13). Now observe that by duality (4.12) yields |(u∗π)∗(V )| ≤ |V | mX-a.e. for every
V ∈Ext(u∗L2μ(T∗Y))∗,hence(4.14) followsfrom(4.13). 2
Equalityin(4.14) canbe obtainedunderappropriateassumptionsoneitherX orY: Proposition4.6. SupposethateitherW1,2(X,d
X,mX) orW1,2(Y,dY,μ) isreflexive.Then
ˆdu(v)=du(v) holds mX-a.e. for every v ∈ L0(T X).
Proof. W1,2(X,d
X,m) is reflexive. Byinequality(4.14) andadensityargument to
con-clude it is sufficient to show that for any f ∈ L∞ ∩ S2(Y,d
Y,μ), g ∈ S2(X) as in
Proposition3.3and v ∈ L∞(T X) with boundedsupportitholds
dg(v) ≤ |dμf| ◦ u|ˆdu(v)| m − a.e. (4.15)
Letusobservethat(4.14) andtheverydefinitionof|du| give|ˆdu(v)|≤ |du(v)|≤ |du||v| m-a.e.,hencethem-a.e.valueofG◦ u|ˆdu(v)| isindependentontheμ-a.e.representative of G, andthe right handsideof (4.15) is welldefined m-a.e.(and equalto 0m-a.e. on
{|du|= 0}).Thetrivialbound |G|2◦ u|ˆdu(v)|2dm ≤ |G|2◦ u|du|2|v|2dm ≤ |v|2 ∞ |G|2du ∗(|du|2m) shows that
L2(Y, μ) G → G ◦ u|ˆdu(v)| ∈ L2(X, m) is linear and continuous. (4.16)
Now fix f,v asin(4.15),letη ∈ LIP(X) be identically1onthe supportofv and with boundedsupport,(fn)⊂ LIPbd(Y) beasin(2.1) forthespace(Y,dY,μ) andnoticethat
sinceweassumedf tobebounded,uptoatruncationargumentwecanassumethefn’s
to be equibounded.Thus thefunctions fn◦ u are equibounded as welland taking into
account theLeibnizrulewe seethatηfn◦ u∈ W1,2(X,dX,m) withequibounded norm.
Since weassumedsuchspaceto bereflexive,uptopasstoanon-relabeledsubsequence
wecanassumethat(ηfn◦ u)n hasaW1,2-weaklimitanditisthenclearthatsuchlimit
is ηg. Thus wehave that(d(ηfn◦ u))n converges to d(ηg) weaklyinL2(T∗X) and, by
thechoicesofv,η,thisimpliesthat(d(fn◦ u)(v))n weaklyconvergestodg(v) inL2(X).
Now noticethat
d(fn◦ u)(v) = [u∗dmYfn](ˆdu(v)) ≤ |dmYfn| ◦ u|ˆdu(v)| ≤ lipa(fn)◦ u|ˆdu(v)|.
This, (4.16) and the choiceof (fn) give thatthe rightmost side of the estimate above
convergestotheright handsideof(4.15) inL2(X).Thisconcludestheargument.
W1,2(Y,
dY,μ) isreflexive. Accordingto[1,Proposition7.6] anditsproof,inthiscase
forany f ∈ W1,2(Y,dY,μ) we can find(fn)⊂ LIPbs(Y)⊂ W1,2(Y,dY,mY) converging
tof inW1,2(Y,d
Y,μ) andsuchthatlipa(fn)→ |dμf| inL2(μ).Thedefinitionsofdu,ˆdu
give
ext[u∗dμfn](du(v))
(3.6)
= d(fn◦ u)(v)
= [u∗dmYfn](ˆdu(v)) ≤ |ˆdu(v)| |dmYfn| ◦ u ≤ |ˆdu(v)|lipa(fn)◦ u
andsincetheconstruction alsoensuresthat[u∗dμfn]→ [u∗dμf] asn→ ∞,bypassing
tothelimitwegetthat
ext([u∗dμf])(du(v)) ≤ |ˆdu(v)| |dμf| ◦ u = |ˆdu(v)| |ext([u∗dμf])|, m − a.e.
Bythearbitrarinessoff ∈ W1,2(Y,dY,μ),thisissufficientto concludetheproof. 2
4.3. The caseX=Rd andu Lipschitz
InthissectionweassumethatoursourcespaceX is(Rd,d
Eucl,Ld) andthatthemap
u∈ S2(Rd,Y) is also Lipschitz. In this case Kirchheim proved in [14] that for Ld-a.e.
x∈ Rd there isaseminorm md(u,x) onRd suchthat:
forLd-a.e.x we have md(u, x)(v) = lim
t↓0
dY
u(x + tv), u(x)
t for everyv ∈ Rd,
where it is part of the claim the fact that the limit in the right hand side exists for
Ld-a.e. x.
Wenow showthatsuchconceptisfullycompatiblewiththenotionofdifferentialwe
introduced:
Theorem4.7. Letu:Rd→ Y beaLipschitzmapthatisalsoinS2(Rd,Y) andv ∈ Rd∼
T Rd.Denote byv ∈ L¯ 0(T Rd) the vectorfieldconstantlyequaltov. Then
du(¯v)(x) = md(u, x)(v) forLd-a.e.x ∈ Rd. (4.17)
Proof. ≥ Let (yn)n be countable and dense in u(Rd) ⊂ Y and, for any n ∈ N, put
fn(·):= dY(·,yn).From the compatibilityof theabstract differential with theclassical
distributional notion in the case X = Rd (see [8, Remark 2.2.4]) and Rademacher’s
theoremwesee that
d(fn◦ u)(¯v) = lim
h→0
fn◦ u(· + hv) − fn◦ u(·)
h Ld− a.e. (4.18)
Forx∈ Rdletγx: [0,1]→ Y betheLipschitzcurvedefinedbyγx
t :=u(x+tv) andput
gx
n,t:=fn◦ γtx. By[2, Theorem 1.1.2] anditsproof we knowthatfor themetricspeed
| ˙γx
t| it holds | ˙γtx| = supn∂tgxn,t for every x∈ Rd and a.e. t,so thattaking (4.18) into
account weobtain md(u, x + tv)(v) = | ˙γxt| = sup n ∂tg x n,t= sup n
d(fn◦ u)(¯v)(x + tv) Ld− a.e. x, a.e. t.
HenceFubini’stheoremyields
md(u, ·)(v) = sup n d(fn◦ u)(¯v) (3.6) = sup n
ext([u∗dμfn])(du(¯v)) ≤ |du(¯v)| Ld− a.e.,
having usedthetrivialbound|dμfn|≤ 1 μ-a.e.inthelast step.
≤ Let f ∈ S2(Y,d
Y,μ) be arbitrary and g ∈ S2(X) as inProposition 3.3. We will
show that
dg(v) ≤ |dμf| ◦ u md(u, ·)(v) Ld− a.e., (4.19)
which is sufficientto conclude.The bound≥ in (4.17) thatwe already provedand the
same argumentsused instudying(4.15) show thattheright handsideof (4.19) is well definedLd-a.e. andthat
L2(Y, μ) G → G ◦ u md(u, ·)(v) ∈ L2(Rd) is linear and continuous. (4.20)
Now let(fn)⊂ LIPbd(Y) be asinProposition 3.3and notice thatforeveryn∈ N the
identity (4.18) yields, forLd-a.e.x:
|d(fn◦ u)(¯v)|(x) ≤ lipa(fn)(u(x)) lim h→0 dY u(x + hv), u(x) |h| =lipa(fn)◦ u (x) md(u, x)(v).
By(4.20) andthechoiceof (fn) we see thattherightmost sideof theaboveconverges
to therighthandsideof(4.19) inL2(Rd) andagainfollowingtheargumentsinthefirst
partoftheproofofProposition4.6(applicable,asW1,2(Rd) iscertainlyreflexive)wesee
thatd(fn◦ u)(¯v)
n convergestodg(v) weaklyinL
2(Rd).Hence(4.19) isobtained. 2
4.4. Thecase ofsmoothspaces andco-localweakdifferential
Inthissectionweassumethatthesourceandtargetspacesarecompleteandsmooth
Riemannian manifoldsM,N respectivelyandprovethe compatibilityof ourconceptof
differential with the oneintroduced in[19]. We stressthat alsothe approachin [19] is
basedonpost-composition;assuch,provingcompatibilityofthetwonotionsamountsto alargeextenttotranslateonevocabularyintotheother,themoretechnicalpartabout therelationbetween‘abstract’and‘concrete’(co)vectorfieldsbeingalreadycoveredby Theorem4.1.
Let us denote by πM : T M → M the canonical projection. Recall that a bundle
morphismU : T M→ T N thatcoversu: M→ N isamapmakingthediagram
T M T N
M N
U
πM πN
u
commuteand linearonfibres. Letus recallthe notionofco-localweak differential and Sobolevspaceintroducedin[19]:
Definition4.8. Amap u: M→ N issaidtobe co-locallyweaklydifferentiableprovided
for any f ∈ C1
c(N,R) we have that f ◦ u ∈ Wloc1,1(M). In this case, the co-local weak
differentialDu isthebundlemorphism thatcoversu characterisedbytheidentity
D(f ◦ u)(x) = Dfu(x)◦ Du(x) VolM− a.e. x ∈ M ∀f ∈ Cc1(N). (4.21)
Finally,W˙ 1,2(M,N) isthe collectionof allmaps u: M→ N co-locallyweakly
differen-tiableforwhich|Du|g∗M⊗gN ∈ L2(M,VolM).
Wereferto[19,Proposition1.5] fortheproofthatanyco-locallyweaklydifferentiable maphasa,uniquelydefineduptoVolM-a.e. equality,co-localweak differential.
Itiseasytosee that
S2(M, N) = ˙W1,2(M, N) and |Du| ≤ |Du|gM∗⊗gN ≤ k|Du|
VolM− a.e. ∀u ∈ S2(M, N), (4.22)
wherek := min{dim M,dim N}.Indeed,ifu∈ S2(M,N),thenbydefinitionwehavethat
f ◦ u∈ W1,2(M) ⊂ W1,1
loc(M) foreveryf ∈ Cc1(N). Thenfor f = (f1,. . . ,fk): N→ Rk
wehave |D(f ◦ u)|gM∗⊗gRk ≤ k j=1 |D(fj◦ u)|gM∗⊗gR= k j=1 |D(fj◦ u)| ≤ k j=1
Lip(fj)|Du| ≤ kLip(f)|Du|,
having used the compatibility between the ‘metric’ and ‘classical’ Sobolev spaces in
theequality. Thus by[19,Proposition 2.2] we concludethatu∈ ˙W1,2(M,N) and that
|Du|gM∗⊗gN ≤ k|Du| holdsalmost everywhere.
Conversely,letu∈ ˙W1,2(M,N) andforanyf ∈ C1
c(N) letf := (f,˜ 0,. . . ,0): N→ Rk.
Then VolM-a.e.we have
|D(f ◦u)| = |D(f ◦u)|gM∗⊗gR=|D( ˜f ◦u)|g∗M⊗gRk ≤ Lip( ˜f)|Du|g∗M⊗gN = Lip(f)|Du|gM∗⊗gN.
Nowanargumentbasedontheapproximationlemma[19,Lemma2.3] andontheclosure
of (classical) weak differentials shows that the above holds for any f ∈ LIP(N). By
definition, this proves that u∈ S2(M,N) and thatthe bound |Du|≤ |Du|
g∗M⊗gN holds
VolM-a.e.,concludingtheproof of(4.22).
We turn to the statement and proof of the compatibility of the two notions of
differential. Weshallmake useofthe terminologyandresultsalreadydiscussed in Sec-tion 4.1. In particular,recalling (4.3) we shall think of du(v) as a map from L0(T X)
to Ext(u∗L0
μ(T N)).Also,weshallusethemapsP,ι introducedinTheorem4.1and the
map I : Ext(u∗L0μ(T N))→ Ext(u∗L0(N,T N;μ)) definedin(4.5).
Then wehave:
Theorem 4.9.LetM and N becomplete Riemannianmanifolds, andletu∈ S2(M,N)=
˙
W1,2(M,N).
Then foreveryv ∈ L0(T M) wehave
I(du(v)) = Du(ι(v)) VolM− a.e.
Proof. Using charts it is easy to see that the L0(μ)-normed module L0(N,T N;μ) is
generated by {Df : f ∈ Cc1(N)}. It follows that Ext(u∗L0(N,T N;μ)) is generated by
{ext([u∗Df]) : f ∈ C1
c(N)} and thus the conclusion follows if we show that for any
f ∈ C1
c(N) itholds
ext([u∗Df])I(du(v))= ext([u∗Dμf])
Du(ι(v)) VolM− a.e. (4.23)
To seethisstartnoticingthat
ext([u∗Df])I(du(v))(4.5),(2.5)= ext[u∗Df](u∗ι(ext−1(du(v)))
(4.4),(2.5)
= ext([u∗dμf])(du(v)) VolM− a.e.
Nowobserve thatthefunctionf ◦ u belongstoS2(X) andthus,bythelocalityproperty of minimal weak upper gradientsand (3.2), satisfies theconclusion of Proposition3.3. Hencetakinginto accounttheverydefinitionofdu(v) weobtain
ext([u∗Df])I(du(v))= d(f ◦ u)(v) VolM− a.e. (4.24)
Now put, as usual, ν := χ{|Du|>0}VolM and notice that a direct verification of the
property stated in Theorem 2.3 shows thatthe pullback module u∗L0(N,T∗N;μ) can
beidentified withthe spaceof(equivalenceclassesupto ν-a.e.equalityof) Borelmaps
ω : M → T∗N such thatω(x) ∈ T∗
u(x)N for ν-a.e. x, the correspondingpullback map
beingtherightcompositionwithu.Withthisinmindandrecallingthedefinitionofthe extensionfunctorweget
ext([u∗Df])(x) =
Dfu(x) for VolM− a.e. x ∈ {|Du| > 0},
0 for VolM− a.e. x ∈ {|Du| = 0},
andthus ext([u∗Df])Du(ι(v))(x) = Dfu(x) Dux(ι(v)(x))
for VolM− a.e. x ∈ {|Du| > 0},
0 for VolM− a.e. x ∈ {|Du| = 0}.
Noticingthat(4.22) yieldsthat{|Du|= 0}={|Du|= 0},theabovegives ext([u∗Df])Du(ι(v))(x) = Dfu(x)
Dux(ι(v)(x))
for VolM− a.e. x ∈ M.
Hencefromthedefinition(4.21) weconcludethat
ext([u∗Df])Du(ι(v))=D(f ◦ u)(ι(v)) VolM− a.e. (4.25)
Theclaim(4.23) is thenaconsequenceof (4.24),(4.25),(4.2) andthefactthatι is the
adjointofP . 2
5. Differentialoflocally Sobolevmapsbetweenmetricspaces
5.1. Inverse limits ofmodules
Here we briefly discussproperties ofinverse limitsinthe category of L0(m)-normed
modules,where morphisms are L0(m)-linear contractions, i.e.maps T : M → N such
that|T (v)|≤ |v| m-a.e. Westartwith:
Proposition 5.1.Let ({Mi}i∈I,{Pji}i≤j∈I) be aninverse system of L0(m)-normed
mod-ules. Then there exists the inverse limit (M,{Pi}
i∈I). Moreover, for every family
I i → vi∈M
i suchthat
Pi
j(vj) =vi and ess sup
i∈I |v
i| ∈ L0(m) (5.1)
there is a unique v ∈ M such that vi = Pi(v) for every i ∈ I and it satisfies |v| =
ess supi|vi|.
Proof. The system ({Mi}i∈I,{Pji}i≤j∈I) is also an inverse system in the category of
algebraic modulesover the ring L0(m) in the sense of [16, Chapter III.§10]. Hence
ac-cordingto[16,ChapterIII,Theorem10.2] anditsproofthereexiststhealgebraicinverse limit (MAlg,PAlgi ) and for everyfamily i → vi ∈ Mi there is aunique v ∈ MAlg such
thatPi
Alg(v)=vi foreveryi∈ I.Nowdefine|v| for anyv ∈MAlg as
|v| := ess sup
i∈I |P
i
Alg(v)| (5.2)
so that|v|: X→ [0,+∞] isthe equivalenceclass ofaBorelmap upto m-a.e.equality,
and put M := v ∈MAlg : |v| ∈ L0(m) = v ∈MAlg : |v| < +∞ m−a.e. , Pi:=Pi Alg|M.
We claimthat(M,Pi) isthedesired inverselimit.Startbynoticingthat(5.2) ensures
that|Pi(v)|≤ |v| m-a.e.,i.e.thePi’sarecontractions,asrequired.Letusnowcheckthat
M isaL0(m)-normedmodule:theonlynon-trivialthingtoverifyisthatitiscomplete,
i.e.thatif(vn) isCauchyinM,thenithasalimit.SincethePi’sarecontractions,wesee
thatn → Pi(v
n) isCauchyinMiandthushasalimitvi foreveryi∈ I.Passingto the
limitintheidentityPi(v
n)=Pji(Pj(vn)) validforeveryi≤ j andusing thecontinuity
of Pi
j wededucethatvi =Pji(vj),i.e.thereisv = (vi)i∈I ∈MAlg.Since (vn) isCauchy
and, trivially, the pointwise norm in M satisfies the triangle inequality, we see that
(|vn|) hasalimitf inL0(m).Thenfromthebound|vi|= limn|Pi(vn)|≤ limn|vn|=:f
valid foreveryi∈ I wededuce |v|≤ f andthusv ∈M.Similarly,from|vi− Pi(v
n)|=
limm|Pi(vm)−Pi(vn)|≤ limm|vm−vn| wededuce|v −vn|≤ limm|vm−vn| andpassing
to the L0(m)-limitin n andusing that(v
n) is M-Cauchywe conclude thatvn → v in
M,thusprovingcompleteness.
Now thefactthatforvi’sas in(5.1) thereisauniquev ∈M projectingon themis
atrivialconsequenceoftheconstructionandfromthisfacttheuniversalityproperty of (M,Pi) follows. 2
It isnow easyto check thatthere exists the inverselimit of acompatible familyof maps:
Proposition5.2. Let({Mi}
i∈I,{Pji}i≤j∈I) and({N i}i∈I,{Qij}i≤j∈I) betwoinverse
sys-temsofL0(m)-normedmodulesand(M,Pi),(N,Qi) theirinverselimits.Also,forevery
i∈ I letTi:Mi→N i beL0(m)-linearandcontinuousand suchthat
Ti◦ Pi
j =Qij◦ Tj ∀i ≤ j ∈ I (5.3)
and sothatforsome ∈ L0(m) wehave
|Ti(vi)| ≤ |vi| m − a.e. ∀i ∈ I, vi∈Mi. (5.4)
Then there exists a unique L0(m)-linear and continuous map T : M → N such that
Qi◦ T = Ti◦ Pi foreveryi∈ I anditsatisfies|T (v)|≤ |v| m-a.e. forevery v ∈M.
Proof. Letv ∈M,putwi:=Ti(Pi(v))∈N i andnoticethat(5.3) yieldsQi
j(wj)=wi
and(5.4) that|wi|≤ |v| m-a.e.foreveryi≤ j ∈ I.ThusProposition5.1aboveensures
thatthereisauniqueT (v)∈N suchthatQi(T (v))=wiforeveryi∈ I anditsatisfies
|T (v)|≤ |v| m-a.e. Sincetheassignmentv → T (v) istriviallyL0(m)-linear,theproofis
completed. 2
5.2. LocallySobolevmapsandtheir differential
Inthissectionwecomebacktothecaseofgeneral(X,dX,m),(Y,dY) asinSection3
and studythe caseof u∈ S2
loc(X,Y), this being thecollectionof functionsu suchthat
every x∈ X has aneighbourhood Ux such thatu coincides with some ux ∈ S2(X,Y)
m-a.e. in Ux. Then for u ∈ Sloc2 (X,Y) the locality of the differential ensures that the
formula
|Du| := |Dux| m − a.e. on Ux ∀x ∈ X
givesawell-defined function|Du|∈ L2
loc(X). Here L2loc(X) denotesthe space oflocally
square-integrablefunctionsonX.
Forthiskindofu themeasureu∗(|Du|2m) isingeneralnotσ-finiteanylonger.Hence,
to define the differential du we need to suitably adaptthe definition previously given. Thisisthescopeof thecurrent section.
Fixu∈ S2
loc(X,Y). ByF(u) we denote the collectionof open sets Ω⊂ X suchthat
Ω|Du|
2dm < ∞. Since u ∈ S2
loc(X,Y) we see that F(u) is a cover of X. We shall
now build twoinverse limitsof L0(m)-normed modulesindexed over F(u),directedby
inclusion.Forthefirstdefine,forΩ∈ F(u),themeasure μΩonY as
μΩ:=u∗(|Du|2m|Ω).
ThusμΩ isRadonand wecan consider thecotangentmoduleLμ0Ω(T∗Y) of(Y,dY,μΩ)
and its pullback u∗L0
μΩ(T∗Y) which is a L0(m|Ω∩{|Du|>0})-normed module. Then put
u∗L0
Ω(T∗Y) := Ext(u∗L0μΩ(T∗Y)), which is L
0(m)-normed.Observe thatfor Ω ⊂ Ω ∈
F(u) we have μΩ ≤ μΩ and thus Lemma 4.3 provides a canonical ‘projection’ map
PΩ
Ω :L0μΩ(T∗Y)→ Ext(L 0
μΩ(T∗Y)).Thenwecanconsiderthe(extended)pullbackmap
u∗PΩ
Ω : u∗L0Ω(T∗Y)→ u∗L0Ω(T∗Y) and notice thatsince PΩΩ21 ◦ PΩΩ32 =PΩΩ31 for every
Ω3⊂ Ω2⊂ Ω1∈ F(u),thefunctorialityofthepullbackgrantsthat(u∗L0Ω(T∗Y),u∗PΩ
Ω )
isaninversesystemofL0(m)-normedmodules.Wethencall(u∗L0u(T∗Y),PΩ) itsinverse limit(recallProposition5.1).
