On a Capacity for Modular Spaces

0.18 MB·English

by  Markus Biegert

ON A CAPACITY FOR MODULAR SPACES
MARKUSBIEGERT
9
0
ABSTRACT. Thepurposeofthisarticleistodefineacapacityoncertaintopologicalmea-
0
surespaces X withrespect tocertain function spacesV consisting ofmeasurable func-
2
tions. Inthis general theory wewill not fixthe spaceV butwe emphasize thatV can
n be the classical Sobolev space W1,p(W ), the classical Orlicz-Sobolev space W1,F (W ),
a theHajłasz-Sobolev spaceM1,p(W ),theMusielak-Orlicz-Sobolev space(orgeneralized
J Orlicz-Sobolev space) and many other spaces. Ofparticular interest is the spaceV :=
8 W˜1,p(W )givenastheclosureofW1,p(W )∩Cc(W )inW1,p(W ).Inthiscaseeveryfunction
u∈V(aprioridefinedonlyonW )hasatraceontheboundary¶ W whichisuniqueuptoa

] Capp,W -polarset.
A
F
.
h
t 1. INTRODUCTION
a
m ThenotionofcapacityisfundamentaltotheanalysisofpointwisebehaviorofSobolev
typefunctions. Dependingonthestartingpointofthestudy,the capacityofa setcanbe
[
definedin manyappropriateways. The Choquettheory[6] givesa standardapproachto

1
capacities. Capacity is a necessary tool in classical and nonlinear potential theory. One
v
purposeofthisarticleistointroduceanextensionoftheclassicalp-capacitywhichwecall
6
3 therelativep-capacity.Forexample,givenanopensetW ⊂ Ntheclassicalp-capacityand
0 the relative p-capacitycan be used to decidewhethera givRenfunctionu lies inW1,p(W )
0
1 or not. The notion of relative 2-capacity was first introduced by Wolfgang Arendt and
.
1 MahmadiWarmain[2]tostudytheLaplacianwithgeneralRobinboundaryconditionson
0 arbitrarydomainsin N. Fortheinvestigationofthe p-LaplacianwithgeneralizedRobin
9 R
boundaryconditionsonbaddomains,suchasthesnowflake(thedomainboundedbythe
0
vonKochcurve),therelative p-capacityplaysanimportantrole.
:
v For results on the classical p-capacity and other capacities we refer the reader to the
i
X following books and the references therein: David R. Adams and Lars I. Hedberg [1],
Nicolas Bouleau and Francis Hirsch [4], Gustave Choquet [6], Lawrence C. Evans and
r
a Ronald F. Gariepy [7], Juha Heinonen and Tero Kilpela¨inen and Olli Martio [11], Jan
Maly´ andWilliamP.Ziemer[14]andVladimirG.Maz’ya[15]. ForcapacitiesonOrlicz-
SobolevspacesandtheirfinebehaviorwerefertoarecentarticleofJ.Maly´,D.Swanson
andW.P.Ziemer[13]andthereferencestherein.
2. PRELIMINARIES
2.1. Setting.
Definition2.1 (TypeL ). LetV be a real vectorspace. We will calla mappingr :V →
[0,¥ ]aLuxemburgfunctionalonV andV =(V,r )aLuxemburgspaceif
(M3) r (u)=0ifandonlyifu=0;
(M2) r (−u)=r (u)forallu∈V;
2000MathematicsSubjectClassification. 31B15.
Keywordsandphrases. RelativeCapacity,TracesofSobolevtypefunctions.
1
2 MARKUSBIEGERT
(M3) r (a u+b v)≤ar (u)+br (v)whenevera +b =1,a ,b ≥0andu,v∈V;
(M4) liml →0+r (l u)=0forallu∈V;
ALuxemburgspace(V,r )isendowedwiththeLuxemburgnormk·k givenby
V
kuk :=inf{a >0:r (u/a )≤1}.
V
See[16,Theorem1.5].Ifinaddition
(L1) r :(V,k·k )→ :=[0,¥ )iscontinuous;
V +
(L2) limkukV→¥ r (u)R=¥ ;
(L3) limr (u)→0kukV =0;
thenwesaythat(V,r )isoftypeL . ALuxemburgspace(V,r )iscalledreflexive/complete
ifV withrespecttotheLuxemburgnormisreflexive/complete.
Remark2.2. If(V,r )isaLuxemburgspace,thenr (u)≤kuk forallu∈V withkuk <1
V V
[16,Theorem1.5(III)].Ifinadditionr iscontinuous[seeproperty(L1)],thenr (u/kuk )=
V
1forallu∈V\{0}. Moreover,r isweaklylower-semicontinuous[5,CorollaryIII.8],in
particular,
r (u)≤liminfr (u ).
n
un⇀u
If(V,k·k )isarealnormedvectorspace,thenr (u):=kuk isaLuxemburgfunctionalon
1 1
V,(V,r )isaLuxemburgspaceoftypeL andtheLuxemburgnormk·k isequaltok·k .
V 1
SocompletenessandreflexivityofaLuxemburgspaceoftypeL isnotautomatic.
Definition2.3(TypeQ ). Byatopologicalmeasurespace(abbreviatedbytms)wemeanthe
quadruple(X,t ,S ,m )where(X,t )isatopologicalspaceand(X,S ,m )ameasurespace.
WewillsaythatatopologicalmeasurespaceX isoftypeQ if
(M1) (X,t )isaHausdorfftopologicalspace;
(M2) (X,S ,m )isacompletemeasurespace;
(M3) IfU ⊂X isopenandS -measurablewithm (U)=0,thenU =0/;
(M4) EveryopensetU ⊂X isthecountableunionofcompactsets.
Remark2.4. IfX=(X,t ,S ,m )isoftypeQ , f,g∈C(X)aremeasurableand f =gm -a.e.
onX,then f =geverywhereonX.
Definition2.5(DominationoftypeQ ). For j=1,2letX =(X ,t ,S ,m )betmsoftype
j j j j j
Q . WewillsaythatX dominatesX ,abbreviatedbyX (cid:22)X ,if
2 1 1 2
(D1) X ∈S andS =S ∩X ={M∩X :M∈S };
1 2 1 2 1 1 2
(D2) t =t ∩X ={O ∩X :O ∈t };
1 2 1 2 1 2 2
(D3) m (A)≤m (A)forallA∈S .
1 2 1
Example 2.6. Let W ⊂W be two non-emptyopen sets in N, l be the N-dimensional
1 2
LebesguemeasureandS bethes -algebraofallLebesgue-mReasurablesetsin N.
R
(1) For j=1,2weletX :=W ,t :=t ∩X ,S :=S ∩X anddefinethemeasure
j j j N j j j
m on(X ,S )bym (B):=l (B∩X )R. Then(X ,t ,S ,m )isatmsoftypeQ and
j j j j j j j j j
X (cid:22)X .
1 2
(2) For j=1,2weletX :=W ,t :=t ∩X ,S :=S ∩X anddefinethemeasure
j j j N j j j
m on(X ,S )bym (B):=l (B∩X )R. Then(X ,t ,S ,m )isatmsoftypeQ and
j j j j j j j j j
X (cid:22)X .
1 2
Definition2.7(Class¡ ). LetX =(X,t ,S ,m )beatmsoftypeQ anddenotebyL0(X)the
vectorspaceofallreal-valued(equivalenceclassesofm -a.e.equal)measurablefunctions
onX.AsubspaceV ⊂L0(X)equippedwithaLuxemburgfunctionalr belongstotheclass
¡ =¡ (X,t ,S ,m ),brieflyV ∈¡ ,ifitsatisfiesthefollowingproperties.
ONACAPACITYFORMODULARSPACES 3
(V1) (V,r )isareflexiveandcompleteLuxemburgspaceoftypeL ;
(V2) ThespaceV∩C (X)isdenseinV;
c
(V3) Ifu ,u∈V andu →uthenasubsequenceof(u ) convergesm -a.e.tou;
n n n n
(V4) V isavectorlatticewithrespecttothem -a.e.pointwiseordering;
(V5) Thefunctionu∧cbelongstoV foreveryu∈V andc∈ andr (u∧c)≤r (u);
+
(V6) Foreveryc∈ themappingV →V,u7→u∧ciscontiRnuous.
+
R
Remark2.8. Fromproperty(V5)wegetthatr (u+)≤r (u)forallu∈V. Thisimpliesthat
ku+k ≤kuk . Similarly,wegetthatr (u−)=r ((−u)+)≤r (−u)=r (u)andku−k ≤
V V V
kuk . Therefore
V
k|u|k =ku++u−k ≤ku+k +ku−k ≤kuk +kuk =2kuk .
V V V V V V V
Definition2.9(Dominationofclass¡ ). For j=1,2letX beatmsoftypeQ and(V ,r )
j j j
of class ¡ (X ). Then we say thatV dominatesV , abbreviated byV (cid:22)V , if X (cid:22)X ,
j 2 1 1 2 1 2
u | ∈V forallu ∈V andthereisaconstantc>0suchthatr (u | )≤cr (u )for
2 X1 1 2 2 1 2 X1 2 2
allu ∈V .
2 2
Example2.10(Sobolevspaces). Forp∈(1,¥ )andW ⊂ N openweletW1,p(W )⊂Lp(W )
bethefirstorderSobolevspaceconsistingofallfunctionRsu∈Lp(W )whosedistributional
derivativesoforderonebelongtoLp(W ). Equippedwiththenormk·kW1,p(W )givenby
kukp :=kukp +k|(cid:209) u|kp
W1,p(W ) Lp(W ) Lp(W )
the space W1,p(W ) is a reflexive Banach space. Let V :=W˜1,p(W ) be the closure of
W1,p(W )∩Cc(W ) in W1,p(W ) and r 1,p,W (u):=kukW1,p(W ). Then (V,r 1,p,W ) is of class
¡ (W )wherethetmsX :=W isasinExample2.6(2). Moreover,ifW ⊂W arenon-empty
1 2
opensetsin N wegetthatW˜1,p(W )(cid:22)W˜1,p(W ).
1 2
R
Definition2.11(N -function). AmappingF : → iscalledanN -functionif
+
R R
(N1) F isevenandconvex;
(N2) F (x)=0ifandonlyifx=0;
(N3) lim x−1F (x)=0and limx−1F (x)=¥ .
x→0+ x→¥
LetY : → begivenbyY (y):=sup{x|y|−F (x):x≥0}. ThenY isanN -function,
+
calledthRecomRplementaryN -functiontoF .
Definition2.12(The△ -and▽ -condition). AnN -functionF issaidtoobeytheglobal
2 2
△ -condition if there exists a constantC>2 such that F (2x)≤C·F (x) for all x∈ ,
2
abbreviatedbyF ∈△ . WesaythatF obeystheglobal▽ -conditionifthecomplementaRry
2 2
N -functionY obeystheglobal△ -condition,abbreviatedbyF ∈▽ . NotethatF ∈▽
2 2 2
ifandonlyifthereexistsaconstantc>1suchthatF (x)≤(2c)−1F (cx)forallx∈ [18,
R
Theorem1.1.2,p.3].
Example2.13. For p∈(1,¥ )thefunctionF : → definedbyF (x):=|x|p/pisan
p + p
N -functionandF ∈△ ∩▽ . Moreover, theRcompRlementaryN -functionto F is F
p 2 2 p q
whereq∈(1,¥ )isgivenby1/p+1/q=1. qiscalledtheconjugateindexto p.
Definition2.14(Orlicz-Space). LetF beanN -functionand(X,S ,m )beameasurespace.
ThentheOrliczspaceLF (X)=LF (X,S ,m )isgivenby
LF (X):= u∈L0(X):r F (u/a )<¥ forsomea >0
where r F isthe Luxemburg(cid:8)functionalgivenby r F (u):= XF (u)dm .(cid:9)The spaceLF (X)
endowed with the Luxemburg norm k·kF is a BanachspacRe [17, Theorem 3.3.10, p.67].
4 MARKUSBIEGERT
If in addition F ∈D ∩▽ , then LF (X) is reflexive [17, Theorem 4.2.10, p.112]. For
2 2
reflexivity/p-convexityofMusielak-Orliczspacessee[12].Moreover,wehavethatkfgk ≤
L1
2kfkF kgkY whereF andY arecomplementaryN -functions.
Example2.15(Orlicz-Sobolevspaces). ForanN -functionF andanopensetW ⊂ N we
letW1,F (W )⊂LF (W )⊂L1 (W )bethefirstorderOrlicz-SobolevspaceconsistingRofall
loc
functionsuintheOrliczspaceLF (W )whosedistributionalderivativesoforderonebelong
toLF (W ). ThentheOrlicz-SobolevspaceW1,F (W )isaBanachspacefortheLuxemburg
normk·k1,F associatedtotheLuxemburgfunctional
r 1,F ,W (u):=Z F (u)+F (|(cid:209) u|)dl
X
IfinadditionF ∈△ ∩▽ ,thenW1,F (W )isareflexiveLuxemburgspaceoftypeL :
2 2
• For property (L1) see Rao and Ren [17, Theorem 3.4.12, p.52 and Corollary
3.4.15];
• Forproperty(L2)seeRaoandRen[17,Corollary5.3.4(iii),p.174];
• Forproperty(L3)seeRaoandRen[18,Theorem1.2.7(iii),p.16];
• Forreflexivity: Firstcheckthatk·kgivenbykuk:= (cid:229) kDa ukF isequivalentto
|a |≤1
k·k1,F andthenidentify(W1,F (W ),k·k)withaclosedsubspaceof(LF (W ))N+1.
LetV :=W˜1,F (W ) be the closure ofCc(W )∩W1,F (W ) in W1,F (W ). Then (V,r 1,F ,W ) is
ofclass ¡ (W ) where the tms X :=W is as in Example 2.6(2). Moreover, if W ⊂W are
1 2
non-emptyopensetin N,thenW˜1,F (W )(cid:22)W˜1,F (W ).
1 2
R
Definition 2.16 (V-Admissibility). Let (V,r ) be of class ¡ . We call a continuous and
bijectivefunctiony : → V-admissibleif
+ +
R R
(A1) ForallC>0thereexistsK(C)>0suchthaty (Ca)≤K(C)y (a)foralla>0;
(A2) Forallu,v∈V itholdstruethat(y ◦r )(u∨v)≤(y ◦r )(u)+(y ◦r )(v).
WewillcallaV-admissiblefunctiony stronglyV-admissibleifforallu,v∈V
(y ◦r )(u∨v)+(y ◦r )(u∧v)≤(y ◦r )(u)+(y ◦r )(v)
Example 2.17. With theassumptionsandnotationsofExample2.10andy (x):=xp we
getthaty isstronglyW˜1,p(W )-admissible. Infact, let u ,u ∈W˜1,p(W ). Byconsidering
1 2
thedisjointsetsD :={x∈W :u (x)<u (x)},D :={x∈W :u (x)>u (x)}andD :=
1 1 2 2 1 2 3
{x∈W :u (x)=u (x)}wegetfromStampacchia’sLemma
1 2
ku ∨u kp = |u |p+|(cid:209) u |p+ |u |p+|(cid:209) u |p+ |u |p+|(cid:209) u |p
1 2 W1,p(W ) Z 2 2 Z 1 1 Z 1 1
D1 D2 D3
ku ∧u kp = |u |p+|(cid:209) u |p+ |u |p+|(cid:209) u |p+ |u |p+|(cid:209) u |p.
1 2 W1,p(W ) Z 1 1 Z 2 2 Z 2 2
D1 D2 D3
Fromthiswededucethat
y (ku1∨u2kW1,p(W ))+y (ku1∧u2kW1,p(W ))=y (ku1kW1,p(W ))+y (ku2kW1,p(W )).
Example2.18. WiththeassumptionsandnotationsfromExample2.15andy (x):=xwe
getthaty isstronglyW˜1,F (W )-admissible. Infact,letu ,u ∈W˜1,F (W ). Byconsidering
1 2
thedisjointsetsD :={x∈W :u (x)<u (x)},D :={x∈W :u (x)>u (x)}andD :=
1 1 2 2 1 2 3
{x∈W :u (x)=u (x)}wegetfromStampacchia’sLemma
1 2
r 1,F (u1∨u2)=Z F (u2)+F (|(cid:209) u2|)+Z F (u1)+F (|(cid:209) u1|)+Z F (u1)+F (|(cid:209) u1|)
D1 D2 D3
ONACAPACITYFORMODULARSPACES 5
r 1,F (u1∧u2)=Z F (u1)+F (|(cid:209) u1|)+Z F (u2)+F (|(cid:209) u2|)+Z F (u2)+F (|(cid:209) u2|).
D1 D2 D3
Fromthiswededucethatr 1,F ,W (u1∨u2)+r 1,F ,W (u1∧u2)=r 1,F ,W (u1)+r 1,F ,W (u2).
Definition 2.19 (Cutoff-Property). Let (V,r ) be of class ¡ , K ⊂X compactandU ⊂X
opencontainingK. Afunctionh ∈V∩C (U)iscalleda(K,U)-cutofffunctionif
c
(K1) h ≡1onKand0≤h ≤1onX;
(K2) ThemappingV →V,u7→h uiswell-definedandcontinuous;
(K3) ThereexistsaconstantC>0suchthatr (h u)≤C·r (u)forallu∈V.
WesaythataspaceV ofclass¡ satisfiesthecutoff-propertyifforeverycompactsetKand
foreveryopensetU containingK thereexistsa(K,U)-cutofffunction.
Example2.20. LetW ⊂ N,X:=W ,UanopensetinthetmsX,andK⊂Ucompact.Then
there exists an open setRO in N such thatU =O∩X and a test functionj ∈D( N)=
C¥ ( N) such that j ≡1 onRK, 0≤j ≤1 on N and supp(j )⊂ O. We remarRk that
c
if uR∈W1,1(W ) and j ∈W1,¥ (W ), then uj ∈WR1,1(W ) and D (uj )=j D u+uD j in
loc loc j j j
D(W )′. Thisshowthat|j u|≤kj kL¥ (W )|u|and
|Dj(j u)|≤kj kL¥ (W ) Dju +kDjj kL¥ (W )|u|⇒|(cid:209) (j u)|≤C1(|(cid:209) u|+|u|)
(cid:12) (cid:12)
forsomeconstantC =C (j ,(cid:12)N)>(cid:12)0. Hence,usingthatF ∈△ ,weget
1 1 2
F (j u)+F (|(cid:209) j u|)dx≤C F (u)+F (|(cid:209) u|).
ZW 2ZW
This shows that for F ∈△ ∩▽ we get thatW˜1,F (W )∈¡ (W ) has the cutoff-property.
2 2
Similarly,wegetthatfor p∈(1,¥ )theSobolevW1,p(W )∈¡ (W )hasthecutoff-property.
2.2. The¡ -Capacity.
Definition2.21(ChoquetCapacity). Let(X,t )beatopologicalspaceandletK denote
thecollectionofallcompactsetsinX. Thenamapping CfromthepowersetP(X)ofX
into[−¥ ,¥ ]iscalledaChoquetcapacityonthepavedspace(X,K )if
(C1) IfA ⊂A ⊂X,then C(A )≤ C(A );
1 2 1 2
(C2) If(A ) ⊂X isincreasingandA= A ,thenlim C(A )= C(A);
n n n n n n
(C3) K ∈K isdecreasingandK= KS,thenlim C(K )= C(K).
n n n n n
WewillcallaChoquetcapacity CnormeTd,if C(0/)=0.
Definition2.22. LetV beofclass¡ andy beaV-admissiblefunction.Thenwedefinethe
¡ -capacityCapy ,V ofanarbitrarysetA⊂X by
Capy ,V(A):=inf{(y ◦r )(u):u∈YV(A)}
whereY (A):={u∈V :∃OopeninX,A⊂O,u≥1m -a.e.onO}. WhenF ∈△ ∩▽ is
V 2 2
anN -function,thenwecallthecapacityCapF ,W :=Capy ,V therelativeF -capacitywhere
y (x):=xandV :=W˜1,F (W ).Whenp∈(1,¥ )thenwecallthecapacityCapp,W :=Capy ,V
therelativep-capacitywherey (x):=|x|pandV :=W˜1,p(W ).NotethatwhenW = N,then
R
Capp:=Capp,W =Capp, N istheclassical p-capacity.
R
3. PROPERTIES OF THE¡ -CAPACITY
3.1. ElementaryProperties. Inthissubsectionweassumethat(V,r )isofclass¡ andy
isV-admissible.
Lemma3.1. ForeveryopensetO⊂X thesetY (O)isconvexand(weakly)closedinV.
V
6 MARKUSBIEGERT
Proof. Since Y (O) = {u∈V :u≥1m -a.e.onO} convexity is clear. That Y (O) is
V V
closedfollowsfromproperty(V3). (cid:3)
Proposition 3.2. Let O⊂X be an open set with Capy ,V(O)<¥ . Then there exists a
functionu∈YV(O) such thatCapy ,V(O)=y (r (u)), u=1 m -a.e. on O and 0≤u≤1
m -a.e.onX. Ifinadditionr isstrictlyconvex,thenuisunique.
Proof. Letun∈V besuchthaty (r (un))→Capy ,V(O)=:y (c). Then(un)nisabounded
sequenceinV [property(L2)]. SinceV isreflexive,bypossiblypassingtoasubsequence,
wemay assumethat(u ) convergesweaklyto a functionv∈V. UsingLemma3.1and
n n
Remark 2.2 we get that v∈YV(O) and Capy ,V(O)=y (r (v)). Now let u:=(v∧1)+.
Then u∈YV(O) and r (u)≤r (v) and hence r (u)=r (v) and therefore Capy ,V(O)=
y (r (u)). Ifr isstrictlyconvexweget, usingthaty isstrictlyincreasing,uniquenessof
theminimizeru. (cid:3)
Theorem3.3. The¡ -capacityCapy ,V isanormedChoquetcapacityonX andforevery
A⊂X wehavethat
(1) Capy ,V(A)=inf{Capy ,V(O):O⊂X openandA⊂O}.
Proof. Equation(1),Capy ,V(0/)=0andA⊂B⊂X ⇒Capy ,V(A)≤Capy ,V(B)aredirect
consequencesofDefinition2.22.
Nowlet(K ) beadecreasingsequenceofcompactsubsetsinX anddenotebyK the
n n
intersectionofallK . IfO⊂X isopenandcontainsK,thenthereexistsn ∈ suchthat
n 0
N
Kn⊂Oforalln≥n0[property(M1)].HenceCapy ,V(K)≤limnCapy ,V(Kn)≤Capy ,V(O).
TakingtheinfimumoverallopensetsOinX containingK wegetfromEquation(1)that
Capp,W (K)=limnCapp,W (Kn).
Nowlet(A ) beanincreasingsequenceofsubsetsofX anddenotebyAtheunionofall
n n
An. Lets:=limnCapy ,V(An)≤Capy ,V(A)∈[0,¥ ]. Togettheconverseinequalitywemay
assumethats<¥ weandletun∈YV(An)besuchthat(y ◦r )(un)≤Capy ,V(An)+1/n.
Therefore(u ) is a boundedsequence in the reflexiveBanach spaceV and hence has a
n n
weaklyconvergentsubsequence.Letu∈V denotetheweaklimitofthissubsequence.By
Mazur’slemmathereisasequence(v ) consistingofconvexcombinationsoftheu with
j j n
n≥ jwhichconvergesstronglytou. Bytheconvexityofr wegetthat
r (v )≤supr (u )≤y −1(s)+1/j.
j n
n≥j
Sinceu ≥1m -a.e.onU foranopensetU containingA wegetthatthereexistsanopen
n n n n
setW (the finite intersectionofU with j≥n)containingA such thatv ≥1 m -a.e.on
n j n n
W . Since(v ) convergestouwemayassume,bypossiblypassingtoasubsequence,that
n j j
k|v −v |k ≤2−j. Let
j+1 j V
¥ k−1
w :=v +(cid:229) |v −v|≥v +(cid:229) (v −v)=v fork≥ j.
j j i+1 i j i+1 i k
i=j i=j
Thenw ∈V andw ≥1m -a.e.onO wheretheopensetO isgivenbyO := ¥ W ⊃A.
j j j j j i=j i
Sincew →uandv →uinV wegetusingproperty(L1) S
j j
Capy ,V(A)≤limy (r (wj))=y (r (u))=limy (r (vj))≤s=limCapy ,V(An).
j j n
(cid:3)
Lemma 3.4. If A⊂X is Capy ,V-polar (Definition 3.21), that is, Capy ,V(A)=0, then
m (A)=0.
ONACAPACITYFORMODULARSPACES 7
Proof. Letu ∈Y (A) be suchthatr (u )→0. Thenbyproperty(L3)u →0 inV. By
n V n n
possiblypassingtoasubsequence[property(V3)]wemayassumethatu →0m -a.e.Since
n
u ≥1m -a.e.onAwegetthatm (A)=0. (cid:3)
n
Proposition3.5. AssumethatV hasthecutoff-property.ThenforeverycompactsetK⊂X
wehavethat
Capy ,V(K) = inf{y (r (u)):u∈V∩Cc(X),u≥1onK}
= inf{y (r (u)):u∈V∩C(X),u≥1onK}.
Proof. WithoutlossofgeneralitywemayassumethatY (K)6=0/.Letu∈Y (K)befixed.
V V
ThenthereexistsanopensetU containingK suchthatv:=(u∧1)+=1m -a.e.onU. Let
h bea(K,U)-cutofffunctionandlet(v ) beasequenceinV∩C (X)whichconvergesto
n n c
v inV. Thenu :=h +(1−h )v+ convergesinV to h +(1−h )v=h v+(1−h )v=v
n n
[properties(K2)+(V6)].Usingthatu ∈V∩C (X),u ≥1onKandr (v)≤r (u)[property
n c n
(V5)]wegetthat
Capy ,V(K) ≥ inf{y (r (u)):u∈V∩Cc(X),u≥1onK}
≥ inf{y (r (u)):u∈V∩C(X),u≥1onK}.
Fortheconverseinequalitywefixafunctionu∈V∩C(X)suchthatu≥1onK.Thenu :=
n
(1+1/n)u∈YV(K)andhenceCapy ,V(K)≤y (r (un))→y (r (u))[property(L1)]. (cid:3)
Theorem3.6. Ify isstronglyV-admissible,thenthe¡ -capacityisstronglysubadditive,
thatis,forallM ,M ⊂X
1 2
Capy ,V(M1∪M2)+Capy ,V(M1∩M2)≤Capy ,V(M1)+Capy ,V(M2)
Proof. Letu ∈Y (M )for j=1,2andletu:=u ∨u ,v:=u ∧u . Thenwehavethat
j V j 1 2 1 2
u∈Y (M ∪M ),v∈Y (M ∩M )and
V 1 2 V 1 2
Capy ,V(M1∪M2)+Capy ,V(M1∩M2) ≤ y (r (u1∨u2))+y (r (u1∧u2))
≤ y (r (u ))+y (r (u )).
1 2
Takingtheinfimumoverallu ∈Y (M )theclaimfollows. (cid:3)
j V 1
Theorem3.7. The¡ -capacityissubadditive,thatis,forallM ,M ⊂X
1 2
Capy ,V(M1∪M2)≤Capy ,V(M1)+Capy ,V(M2)
Proof. TheprooffollowsthelinesintheproofofTheorem3.6. Notethatthe¡ -capacity
Capy ,V isdefinedonlyforV-admissibley andthattheV-admissibilityofy wasassumed
atthebeginningofthissection. (cid:3)
Theorem3.8. The¡ -capacityCapy ,V iscountablysubadditive,thatis,forallAk⊂X
(cid:229)
Capy ,V(cid:16)[k∈NAk(cid:17)≤k∈ Capy ,V(Ak).
N
Proof. Let B be the unionof A with 1≤k≤n andlet A be the unionof all A . From
n k k
Theorem3.7wegetbyinductionthatforalln∈NCapy ,V(Bn)≤(cid:229) nk=1Capy ,V(Ak). Using
thatCapy ,V isaChoquetcapacity[property(C2)]weget
n
(cid:229) (cid:229)
Capy ,V(A)=limCapy ,V(Bn)≤lim Capy ,V(Ak)= Capy ,V(Ak).
n n
k=1 k∈
N
(cid:3)
8 MARKUSBIEGERT
3.2. Relations between ¡ -Capacities. In this subsection we assume that for j = 1,2
the tms X =(X ,t ,S ,m ) is of type Q , X (cid:22)X , (V ,r ) is of class ¡ (X ), y is V -
j j j j j 1 2 j j j j j
admissible.Moreover,for1≤i,j≤2welety : → bethebijectiveandcontinuous
i,j + +
R R
functiongivenbyy :=y ◦y −1.
i,j i j
Definition3.9((V ,V )-extensionproperty). WewillsaythatasetU⊂X hasthe(V ,V )-
1 2 1 1 2
extensionpropertyif
(X1) m (N)=0⇒m (N)=0forallS -measurablesetsN⊂U;
1 2 2
(X2) There exists a constant c>0 and a mapping E :V0(U)→V where V0(U) is
1 2 1
givenby
V0(U):={u∈V :u=0m -a.e.onX \U},
1 1 1 1
suchthatEu=um -a.e.onX andr (Eu)≤cr (u).
1 1 2 1
WesaythatasetU inX hasthes (V ,V )-extensionpropertyifU isthecountableunion
1 1 2
ofopensetsinX satisfyingthe(V ,V )-extensionproperty.
1 1 2
Example3.10. LetW ⊂W benon-emptyopensetsin N andU ⊂⊂W (thismeansU
1 2 1
isopen,U iscompactandU ⊂W ). R
1
(1) If 1<q≤ p<¥ , thenU has the (W˜1,p(W ),W˜1,q(W ))-extension property and
1 2
henceW hasthes (W˜1,p(W ),W˜1,q(W ))-extensionproperty.
1 1 2
(2) IfF ∈△ ∩▽ isanN -function,thenUhasthe(W˜1,F (W ),W˜1,F (W ))-extension
2 2 1 2
propertyandhenceW hasthes (W˜1,F (W ),W˜1,F (W ))-extensionproperty.
1 1 2
Lemma3.11. AssumethatU isopeninX andhasthe(V ,V )-extensionproperty,K⊂U
1 1 2
iscompactandV hasthecutoffproperty. ThenthereexistsaconstantC=C(K)>0such
1
thatforallA⊂K
Capy 2,V2(A)≤y 2,1(C·Capy 1,V1(A)).
Inparticular,wheny :=y =y ,thenthereexistsaconstantC>0suchthat
1 2
Capy ,V2(A)≤C·Capy ,V1(A).
Proof. Let h ∈V ∩C (U) be a (K,U)-cutoff function and u∈Y (A) be fixed. Then
1 c V1
h u∈V0(U),v:=E(h u)∈Y (A)andhence[properties(K3)+(X2)]
1 V2
Capy 2,V2(A) ≤ (y 2◦r 2)(v)≤y 2(c1r 1(h u))≤y 2(c2r 1(u))
= y (y (c r (u)))≤y (C·(y ◦r )(u)).
2,1 1 2 1 2,1 1 1
Takingtheinfimumoverallu∈Yy ,V1(A)wegettheclaim. (cid:3)
Example3.12. LetW ⊂W benon-emptyopensetsin N,1<q≤p<¥ andletK⊂W
1 2 1
R
becompact.ThenthereexistsaconstantC>0suchthatforallA⊂K
Capq,W 2(A)≤C·Capp,W 1(A)q/p.
Infact,letU⊂⊂W besuchthatK⊂U.ThenbyExample3.10(2)wegetthatallassump-
1
tionsfromLemma3.11aresatisfied.
Lemma3.13. IfV (cid:22)V thenthereexistsaC>0suchthatforallA⊂X
1 2 1
(2) Capy 1,V1(A)≤y 1,2(C·Capy 2,V2(A)).
Proof. Let u ∈ Y (A). Then u| ∈ Y (A) and r (u| ) ≤cr (u) by Definition 2.9.
V2 X1 V1 1 X1 2
Hence
Capy 1,V1(A) ≤ (y 1◦r 1)(u|X1)≤y 1(c·r 2(u))
= y (y (c·r (u)))≤y (C·(y ◦r )(u)).
1,2 2 2 1,2 2 2
ONACAPACITYFORMODULARSPACES 9
Takingtheinfimumoverallu∈Y (A)wegettheclaim. (cid:3)
V2
Proposition 3.14. Assume thatU is an open set in X and has the s (V ,V )-extension
1 1 2
property,V hasthecutoffpropertyandV (cid:22)V . ThenforeveryA⊂U wehavethat
1 1 2
(3) Capy 1,V1(A)=0 ⇐⇒ Capy 2,V2(A)=0.
Proof. FromEquation(2)wegetthatCapy 2,V2(A)=0impliesthatCapy 1,V1(A)=0.Hence
toprove(3)itremainstoprovetheconverseimplication.ForthisletU ⊂X beopensets
m 1
withthe(V ,V )-extensionpropertysuchthatU = U andletK ⊂U becompact
1 2 m m m,n m
setssuchthatUm= nKm,n andassumethatCapy 1,VS1(A)=0. ThenbyLemma3.11there
existconstantsC Ssuchthat
n,m
Capy 2,V2(Km,n∩A)≤y 2,1(Cm,n·Capy 1,V1(Km,n∩A))=0.
UsingthatCapy 2,V2 isa countablysubadditive[Theorem3.8]andthat n,mKn,m =U we
getthat S
(cid:229)
Capy 2,V2(A)≤ Capy 2,V2(Kn,m∩A)=0.
m,n
(cid:3)
Corollary3.15. LetW ⊂W benon-emptyopensetsin N, p∈(1,¥ )andF ∈△ ∩▽
1 2 2 2
beanN -function.ThenforallsetsinA⊂W wehavethRat
1
Capp,W 1(A)=0⇔Capp,W 2(A)=0 and CapF ,W 1(A)=0⇔CapF ,W 2(A)=0.
Remark3.16. IngeneraltheassertionofCorollary3.15(andhenceofProposition3.14)
is untrue for A⊂W . To see this we let N ≥2 and W :=(0,1)N\ [2−2j,21−2j]×
1 j∈
[2−j,1]×(0,1)N−2. ThentheopensetW ⊂ N isboundedandconnecStedNandA:={0}×
[0,1]n−1 ⊂¶ W is such that the (N−1)-dimRensional Hausdorff measure HN−1(A)=1.
Thenforevery p∈(1,¥ )wehavethatCapp(A)>0andCapp,W (A)=0.
Definition3.17(ContinuousExtensionProperty). WewillsaythatV hasthecontinuous
1
V -extension property if there exists a (possibly non-linear)mapping E :V →V and a
2 1 2
constantC=C(E)suchthatE(V ∩C(X ))⊂V ∩C(X )andforallu ∈V itholdstrue
1 1 2 2 1 1
thatr (E(u ))≤Cr (u )andEu =u m -a.e.onX .
2 1 1 1 1 1 1 1
Definition 3.18. Let W ⊂W be non-empty open sets in N and p∈(1,¥ ). Then we
1 2
say that W1,p(W ) has the W1,p(W )-extension property ifRthe restriction W1,p(W )→
1 2 2
Wthe1n,p(wWe1s)a,yub7→rieufl|yW 1thisatsuW rjehcatisveth.eIfWW 12,p=-eRxNtenasnidonW p1rhoapserthtye.W1,p(W 2)-extensionproperty,
1
ThefollowingisanimmediateconsequenceofShvartsman[19]andHajłaszandKoskela
andTuominen[8].
Theorem3.19. Let p∈(1,¥ ) andW ⊂ N be aW1,p-extension domain. Then W has
1 1
thecontinuousW1,p(W )-extensionproperRtyforeveryopensetW ⊂ N containingW .
2 2 1
R
Proof. SinceW isa(1,p)-extensiondomain,wegetfrom[8,Theorem2andLemma2.1]
that there exists a constant d W > 0 such that l (B(x,r)∩W )≥ d W rN for all 0 < r ≤ 1
and l (¶ W )=0. For a measurable set A⊂ we let M1,p(A) be the Sobolev-typespace
introducedbyHajłaszconsistingofthosefunRctionu∈Lp(A)withgeneralizedgradientin
Lp(A). Itfollowsfrom[19,Theorem1.3]thatM1,p( N)| =M1,p(W )andthatthereexists
W
alinearcontinuousextensionoperatorE :M1,p(W )R→M1,p( N). UsingthatM1,p( N)=
W1,p( N)assetswithequivalentnorms,wegetthatM1,p(W )R=W1,p(W )areequalRassets
R
10 MARKUSBIEGERT
withequivalentnormsandhencetheextensionoperatorE constructedforM1,p(W )isalso
alinearcontinuousextensionoperatorfromW1,p(W )intoW1,p( N). Itislefttothereader
to verify that the extension operator E constructed by ShvartsRman [19, Equation (1.5)]
mapsW1,p(W )∩C(W )intoW1,p( N)∩C( N). (cid:3)
R R
Theorem 3.20. If V (cid:22)V , V has the continuousV -extension property and the cutoff
1 2 1 2
property,thenthereexistconstantsC ,C >0suchthatforeverysetA⊂X
1 2 1
Capy 2,V2(A)≤y 2,1[C1·Capy 1,V1(A)]≤y 2,1[C1·y 1,2(C2Capy 2,V2(A))].
Proof. Let K ⊂X be a compact set. By Proposition 3.5 there exist u ∈V ∩C (X )
1 n 1 c 1
suchthatun≥1 onK andy 1(r 1(un))→Capy 1,V1(K). LetE :V1→V2 be a continuous
extensionoperatoranddefinev :=Eu . Thenv ∈V ∩C(X )andv ≥1onK. Henceby
n n n 2 2 n
Proposition3.5wegetthat
Capy 2,V2(K) ≤ y 2(r 2(vn))≤y 2(C(E)r 1(un))=y 2,1(y 1(C(E)r 1(un)))
≤ y 2,1(C(E,y 1)y 1(r 1(un)))→y 2,1(C(E,y 1)Capy 1,V1(K)).
LetW be an openset in X . Thenthere exists an increasingsequence (K ) of compact
1 n n
setssuchthat nKn=W [property(M4)]. UsingthatCapy 1,V1 andCapy 2,V2 areChoquet
capacities[proSperty(C2)]wegetthat
Capy 2,V2(W) = linmCapy 2,V2(Kn)≤linmy 2,1(C(E,y 1)Capy 1,V1(Kn))
= y 2,1(C(E,y )Capy ,V1(W)).
NowletA⊂X bearbitrary.ThenbyTheorem3.3
1
Capy 2,V2(A) = inf{Capy 2,V2(O):OisopeninX2 andA⊂O}
= inf{Capy 2,V2(O∩X1):OisopeninX2andA⊂O}
= inf{Capy 2,V2(W):W isopeninX1andA⊂W}
≤ y 2,1[C(E,y 1)inf{Capy 1,V1(W):W isopeninX1andA⊂W}]
= y 2,1[C(E,y )Capy ,V1(A)].
TheremaininginequalityfollowsfromLemma3.13. (cid:3)
3.3. Quasicontinuity and Polar Sets. In this subsection we assume that the tms X =
(X,t ,S ,m )isoftypeQ ,(V,r )isofclass¡ (X)andy isV-admissible.Thepurposeofthis
subsectionistoproveexistenceanduniquenessofCapy ,V-quasicontinuousrepresentatives
onX.
Definition 3.21. A set P⊂X is said to be Capy ,V-polar if Capy ,V(P)=0. A pointwise
definedfunctionuonD⊂X iscalledCapp,W -quasicontinuousonDifforeache >0there
existsanopensetOinX withCapy ,V(O)<e suchthaturestrictedtoD\Oiscontinuous.
We say that a property holds Capy ,V-quasi everywhere (briefly Capy ,V-q.e.) if it holds
exceptforaCapy ,V-polarset.
Lemma3.22. Ifu∈V andu ∈V∩C (X)aresuchthat
k c
¥
(cid:229) y (2k+2ku−u k +2k+1ku−u k )<¥ ,
k+1 V k V
k=1
then the pointwise limit u˜:=limkuk exists Capy ,V-quasi everywhere on X, u˜:X → is
Capy ,V-quasicontinuousandu˜=um -almosteverywhereonX. R