TO METRIC SPACES
NOUREDDINE A¨ISSAOUI Received 12 May 2003
We propose another extension of Orlicz-Sobolev spaces to metric spaces based on the concepts of theΦ-modulus andΦ-capacity. The resulting spaceNΦ1 is a Banach space.
The relationship between NΦ1 andMΦ1 (the first extension defined in A¨ıssaoui (2002)) is studied. We also explore and compare different definitions of capacities and give a criterion under whichNΦ1 is strictly smaller than the Orlicz spaceLΦ.
1. Introduction
In [22], Shanmugalingam studies extensively an extension of Sobolev spaces on metric spaces different from the approach of Hajłasz in [12]. In particular, he gives a comparison between the obtained two spaces. See also [6,9,13,22] for further developments of these two theories.
Since a first extension of Orlicz-Sobolev spaces on metric spaces, denoted byMΦ1(X), following Hajłasz’ method, was studied in [4], it is natural to examine Shanmugalingam’s definition based on the notions of modulus of paths families and on the capacity. The resulting spaceNΦ1(X) is a Banach space for anyᏺ-functionΦand the spaceM1Φ(X) con- tinuously embeds onNΦ1(X) whenΦsatisfies the∆2condition. We know that Lipschitz functions are dense inMΦ1(X) forΦverifying the∆2 condition. To expect the same re- sult with the vaster spaceNΦ1(X), we must add some assumptions, as in the Sobolev case, on the metric spaceX, namely,Xmust be doubling and support a (1,Φ)-Poincar´e in- equality, andΦverifies the∆condition. Remark that whenΦ(x)=(1/ p)xp(p >1), we rediscover the same result in the setting of Sobolev spaces. On the other hand, whenΩis a domain inRN, we give a new characterization of the Orlicz-Sobolev spaceW1LΦ(Ω), and we show thatNΦ1(Ω)=W1LΦ(Ω) whenΦsatisfies the∆2condition. Hence, for re- flexive Orlicz spacesLΦ(RN), we getNΦ1(RN)=M1Φ(RN)=W1LΦ(RN), since we know thatMΦ1(RN)=W1LΦ(RN). See [4, Theorem 3.3]. We also study the mean equivalent class with respect toΦ(MECΦ) criterion under whichNΦ1(X) is strictly included in the Orlicz spaceLΦ(X) and we compare between natural capacities defined onNΦ1(X). We expect that other developments will be done in forthcoming papers.
Copyright©2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:1 (2004) 1–26
2000 Mathematics Subject Classification: 46E35, 31B15, 28A80 URL:http://dx.doi.org/10.1155/S1085337504309012
We organize this paper as follows. InSection 2, we list the required prerequisites from the Orlicz theory. Section 3 is reserved to the study of Φ-modulus, the capacity, and Orlicz-Sobolev space NΦ1(X). Section 4 deals with comparison between NΦ1(X) and MΦ1(X) and with some properties ofNΦ1(X). InSection 5, we study the MECΦcriterion and we compare between some capacities.
2. Preliminaries
Anᏺ-functionis a continuous convex and even functionΦdefined onR, verifyingΦ(t)>
0 fort >0, limt→0(Φ(t)/t)=0, and limt→+∞(Φ(t)/t)=+∞. We have the representationΦ(t)=|t|
0 ϕ(x)dL(x), whereϕ:R+→R+is nondecreasing, right continuous, withϕ(0)=0,ϕ(t)>0 fort >0, limt→0+ϕ(t)=0, and limt→+∞ϕ(t)= +∞. HereLstands for the Lebesgue measure. We put in the sequel, as usual,dx=dL(x).
Theᏺ-function Φ∗ conjugate toΦis defined by Φ∗(t)=|t|
0 ϕ∗(x)dx, whereϕ∗ is given byϕ∗(s)=sup{t:ϕ(t)≤s}.
Let (X,Γ,µ) be a measure space and Φan ᏺ-function. The Orlicz class ᏸΦ,µ(X) is defined by
ᏸΦ,µ(X)=
f :X−→Rmeasurable :
XΦf(x)dµ(x)<∞
. (2.1)
We define theOrlicz spaceLΦ,µ(X) by LΦ,µ(X)=
f :X−→Rmeasurable :
XΦα f(x)dµ(x)<∞for someα >0
. (2.2) The Orlicz spaceLΦ,µ(X) is a Banach space with the following norm, called theLux- emburg norm:
|f|Φ,µ,X=inf
r >0 :
XΦf(x) r
dµ(x)≤1
. (2.3)
If there is no confusion, we set|f|Φ= |f|Φ,µ,X.
The H¨older inequality extends to Orlicz spaces as follows: if f ∈LΦ,µ(X) and g∈ LΦ∗,µ(X), then f g∈L1and
X|f g|dµ≤2|f|Φ,µ,X· |g|Φ∗,µ,X. (2.4) LetΦbe anᏺ-function. We say thatΦverifies the∆2 conditionif there is a constant C >0 such thatΦ(2t)≤CΦ(t) for allt≥0.
The∆2condition forΦcan be formulated in the following equivalent way: for every C >0, there existsC>0 such thatΦ(Ct)≤CΦ(t) for allt≥0.
We have alwaysᏸΦ,µ(X)⊂LΦ,µ(X). The equalityᏸΦ,µ(X)=LΦ,µ(X) occurs ifΦveri- fies the∆2condition.
We know thatLΦ,µ(X) is reflexive ifΦandΦ∗verify the∆2condition.
Note that ifΦverifies the∆2condition, thenΦ(fi(x))dµ→0 asi→ ∞if and only if
|fi|Φ,µ,X→0 asi→ ∞.
Recall that anᏺ-functionΦsatisfies the∆condition if there is a positive constantC such that for allx,y≥0,Φ(xy)≤CΦ(x)Φ(y). See [16,21]. If anᏺ-functionΦsatisfies the∆condition, then it satisfies also the∆2condition.
LetΩbe an open set inRN, letC∞(Ω) be the space of functions which, together with all their partial derivatives of any order, are continuous onΩ, and letC∞0(RN)=C∞0 stand for all functions inC∞(RN) which have compact support inRN. The spaceCk(Ω) stands for the space of functions having all derivatives of order≤kcontinuous onΩ, andC(Ω) is the space of continuous functions onΩ.
The (weak) partial derivative of f of order|β|is denoted by Dβf = ∂|β|
∂x1β1·∂xβ22· ··· ·∂xNβN f . (2.5) LetΦbe anᏺ-function andm∈N. We say that a function f :RN→Rhas a distribu- tional (weak partial) derivative of orderm, denoted byDβf,|β| =m, if
f Dβθ dx=(−1)|β| Dβfθ dx, ∀θ∈C∞0. (2.6) LetΩbe an open set inRN and denoteLΦ,L(Ω) byLΦ(Ω). TheOrlicz-Sobolev space WmLΦ(Ω) is the space of real functions f such that f and its distributional derivatives up to the ordermare inLΦ(Ω).
The spaceWmLΦ(Ω) is a Banach space equipped with the norm
|f|m,Φ,Ω=
0≤|β|≤m
DβfΦ, f ∈WmLΦ(Ω), (2.7)
where|Dβf|Φ= |Dβf|Φ,L,Ω.
Recall that if Φ verifies the ∆2 condition, then C∞(Ω)∩WmLΦ(Ω) is dense in WmLΦ(Ω), andC∞0(RN) is dense inWmLΦ(RN).
For more details on the theory of Orlicz spaces, see [1,16,17,18,21].
In this paper, the letterCwill denote various constants which may differ from one formula to the next one even within a single string of estimates.
3. The Orlicz-Sobolev spaceNΦ1(X)
3.1.Φ-modulus in metric spaces. Let (X,d,µ) be a metric, Borel measure space, such thatµis positive and finite on balls inX.
IfIis an interval inR, a path inXis a continuous mapγ:I→X. By abuse of language, the imageγ(I)=:|γ|is also called a path. IfI=[a,b] is a closed interval, then the length of a pathγ:I→Xis
l(γ)=length(γ)=sup
n i=1
γti+1
−γti, (3.1)
where the supremum is taken over all finite sequencesa=t1≤t2≤ ··· ≤tn≤tn+1=b. If Iis not closed, we set
l(γ)=suplγ|J
, (3.2)
where the supremum is taken over all closed subintervalsJofI. A path is said to be recti- fiable if its length is a finite number. A pathγ:I→Xis locally rectifiable if its restriction to each closed subinterval ofIis rectifiable.
For any rectifiable pathγ, there are its associated length functionsγ:I→[0,l(γ)] and a unique 1-Lipschitz continuous mapγs: [0,l(γ)]→Xsuch thatγ=γs◦sγ. The pathγs is the arc-length parametrization ofγ.
Letγ be a rectifiable path in X. The line integral overγ of each nonnegative Borel functionρ:X→[0,∞] is
γρ ds= l(γ)
0 ρ◦γs(t)dt. (3.3)
If the pathγis only locally rectifiable, we set
γρ ds=sup
γρ ds, (3.4)
where the supremum is taken over all rectifiable subpathsγofγ. See [14] for more de- tails.
Denote byΓrectthe collection of all nonconstant compact (i.e.,Iis compact) rectifiable paths inX. IfAis a subset ofX, thenΓAis the family of all paths inΓrectthat intersect the setA, andΓ+Ais the family of all pathsγinΓrectsuch that the Hausdorffone-dimensional measureᏴ1(|γ| ∩A) is positive.
Definition 3.1. Let Φ be an ᏺ-function and Γbe a collection of paths in X. The Φ- modulus of the familyΓ, denoted by ModΦ(Γ), is defined as
ρ∈infᏲ(Γ)|ρ|Φ, (3.5)
whereᏲ(Γ) is the set of all nonnegative Borel functionsρ such thatγρ ds≥1 for all rectifiable pathsγinΓ. Such functionsρused to define theΦ-modulus ofΓare said to be admissible for the familyΓ.
FromDefinition 3.1, theΦ-modulus of the family of all nonrectifiable paths is 0.
We have the following important proposition.
Proposition3.2. LetΦbe anᏺ-function and letΓbe a collection of paths inX. Then the Φ-modulus of the familyΓis an outer measure onΓ. That is,
(1) ModΦ(∅)=0,
(2) ModΦ(Γ1)≤ModΦ(Γ2)if Γ1⊂Γ2, (3) ModΦ(∞i=1Γi)≤∞
i=1ModΦ(Γi).
Proof. Assertions (1) and (2) are obvious. We prove (3). We may assume that ModΦ(Γi)<
∞for alli. Forε >0, there isρi∈Ᏺ(Γi) such that ρi
Φ≤ModΦΓi
+ε2−i. (3.6)
Setρ=supiρiandΓ=∞
i=1Γi. Sinceρ≥ρifor alli,ρ∈Ᏺ(Γ). Thus ModΦ(Γ)≤ |ρ|Φ. By [5, Lemma 2],|ρ|Φ≤∞
i=1|ρi|Φ. Hence, ModΦ(Γ)≤ ∞
i=1
ModΦ(Γi) +ε. (3.7)
Sinceεis arbitrary, (3) is proved.
A property relevant to paths inXis said to hold forΦ-almost all pathsif the family of rectifiable compact paths on which that property does not hold hasΦ-modulus zero.
For any path γ∈Γrect and for distinct points x and y in |γ|, denote γxy to be the subpathγ|[tx,ty], where the two distinct numberstxandtyare chosen from the domain of γsuch thatγ(tx)=xandγ(ty)=y. The subpathγxyis not a well-defined notion as there can be more than one choice of the related numberstxandty. Because of this ambiguity, any property that is required for one choice of the subpathγxyis also required for all such choices of subpaths.
Definition 3.3. LetΦbe anᏺ-function and letl(γ) denote the length ofγ. A functionu is said to be absolutely continuous onΦ-almost every curve (ACCΦ) ifu◦γis absolutely continuous on [0,l(γ)] forΦ-almost every rectifiable arc-length parametrized pathγin X. IfXis a domain inRN, a functionuis said to have the absolute continuity on almost every line (ACL) property if on almost every line parallel to the coordinate axes with respect to the Hausdorff(N−1)-measure, the function is absolutely continuous. An ACL function therefore has directional derivatives almost everywhere. An ACL function is said to have the property ACLΦif its directional derivatives are inLΦ.
Definition 3.4. Letube a real-valued function on a metric spaceX. A nonnegative Borel- measurable functionρis said to be an upper gradient ofuif for all compact rectifiable pathsγ, the following inequality holds:
u(x)−u(y)≤
γρ ds, (3.8)
wherexandyare the end points of the path.
Definition 3.5. LetΦbe anᏺ-function and letube an arbitrary real-valued function on X. Letρbe a nonnegative Borel function onX. If there exists a familyΓ⊂Γrectsuch that ModΦ(Γ)=0 and the inequality (3.8) is true for all pathsγinΓrect\Γ, thenρis said to be aΦ-weak upper gradient ofu. If inequality (3.8) holds true forΦ-modulus almost all paths in a setB⊂X, thenρis said to be aΦ-weak upper gradient ofuonB.
Lemma3.6. LetΦbe anᏺ-function and letΓbe a collection of paths inX. ThenModΦ(Γ)= 0if and only if there is a nonnegative Borel functionρonXsuch thatρ∈LΦand for all paths γ∈Γ,
γρ ds= ∞. (3.9)
Proof. Suppose that ModΦ(Γ)=0. Then ifn∈N∗, there exists a nonnegative Borel func- tionρnonXsuch thatρn∈LΦand|ρn|Φ≤2−n. The functionρ=∞
n=1ρnis a nonneg- ative Borel function onXand, by [5, Lemma 2],|ρ|Φ≤∞
n=1|ρn|Φ, which implies thatρ∈LΦ. It is evident thatγρ ds= ∞for all pathsγ∈Γ.
Assume that there is a nonnegative Borel functionρonXsuch thatρ∈LΦand for all pathsγ∈Γ,γρ ds= ∞. Then for eachn, the function 2−nρis admissible for calculating theΦ-modulus of the familyΓ. This implies that ModΦ(Γ)=0. The proof is complete.
Corollary 3.7. Let Φ be an ᏺ-function and let E⊂X be such that µ(E)=0. Then ModΦ(Γ+E)=0.
Proof. Since∞χEis an admissible function for calculating ModΦ(Γ+E), the corollary fol-
lows byLemma 3.6.
3.2. The Orlicz-Sobolev spaceNΦ1(X)
Definition 3.8. LetΦbe anᏺ-function and let the setNΦ1(X,d,µ) be the collection of all real-valued functionuonXsuch thatu∈LΦanduhas aΦ-weak upper gradient inLΦ.
We remark thatNΦ1 is a vector space, since ifα,β∈Randu,v∈NΦ1 with respect to Φ-weak upper gradientsρandσ, then|α|ρ+|β|σis aΦ-weak upper gradient ofαρ+βσ.
Ifu∈NΦ1, we set
|u|N1
Φ= |u|Φ+ inf
ρ |ρ|Φ, (3.10)
where the infimum is taken over allΦ-weak upper gradientρofusuch thatρ∈LΦ. Ifu,v∈NΦ1, letuvif|u−v|NΦ1 =0. It can be easily seen thatis an equivalence relation, partitioningNΦ1 into equivalence classes, which is a normed vector space under the norm defined by (3.10).
Definition 3.9. LetΦbe anᏺ-function. The Orlicz-Sobolev space corresponding toΦ, denoted byNΦ1(X), is defined to be the spaceNΦ1(X,d,µ)with the norm|u|NΦ1:=
|u|NΦ1.
Ifu,v∈NΦ1, then it is easily verified that the functions|u|, min{u,v}, max{u,v} ∈NΦ1, that ifλ≥0, then min{u,λ} ∈NΦ1, and that ifλ≤0, then max{u,λ} ∈NΦ1. ThusNΦ1(X) enjoys all the lattice properties in classical first-order Orlicz-Sobolev spaces.
Lemma3.10. LetΦbe anᏺ-function andu∈NΦ1. ThenuisACCΦ.
Proof. By hypothesis,u∈LΦanduhas aΦ-weak upper gradientρ∈LΦ. LetΓbe the collection of all paths inΓrectfor which inequality (3.8) does not hold. Then ModΦ(Γ)=0.
LetΓ1be the collection of all paths inΓrectthat have a subpath inΓ. Then any admissible function used to estimate theΦ-modulus ofΓis an admissible function forΓ1. Hence,
ModΦΓ1
≤ModΦ(Γ)=0. (3.11)
LetΓ2be the collection of all pathsγinΓrectsuch thatγρ ds= ∞. Sinceρ∈LΦ, then ModΦ(Γ2)=0. Thus ModΦ(Γ1∪Γ2)=0. Ifγis a path inΓrect\(Γ1∪Γ2),γhas no subpath inΓ1, and hence for allx,y∈ |γ|,
u(x)−u(y)≤
γxy
ρ ds <∞. (3.12)
Therefore,uis absolutely continuous on each pathγinΓrect\(Γ1∪Γ2). The proof is
complete.
Lemma 3.11. Let Φbe an ᏺ-function and let u∈NΦ1 be such that |u|Φ=0. Then ModΦ(Γ)=0, where
Γ=
γ∈Γrect:u(x)=0for somex∈ |γ|
. (3.13)
Proof. Since|u|Φ=0, the setS= {x∈X:u(x)=0}has measure zero. Hence,Γ=ΓS
and
Γ=Γ+S∪ ΓS\Γ+S
. (3.14)
The subfamilyΓ+S can be disregarded since
ModΦΓ+S≤ |∞ ·χS|Φ=0, (3.15) whereχSis the characteristic function of the setS. The pathsγ∈ΓS\Γ+S intersectSonly on a set of linear measure zero, and hence, with respect to the linear measure almost everywhere onγ, the functionutakes on the value of zero. By the fact thatγalso intersect S, therefore,u is not absolutely continuous onγ. ByLemma 3.10, ModΦ(ΓS\Γ+S)=0.
Thus ModΦ(Γ)=0 and the proof is complete.
We deduce from the previous lemma the following corollary.
Corollary3.12. LetΦbe anᏺ-function. Ifu,v∈NΦ1 are such that|u−v|Φ=0, then uandvbelong to the same equivalent class inNΦ1(X).
In the sequel, we will not distinguish between the functions inNΦ1 and their equiva- lence classes inNΦ1.
Lemma3.13. LetΦbe anᏺ-function. If(ρi)i∈N∗is a sequence of Borel functions inLΦsuch thatlimi→∞|ρi|Φ=0, then there exist a subsequence(ρik)k∈N∗and a familyΓ⊂Γrectsuch thatModΦ(Γ)=0and for all pathsγ∈Γrect\Γ,
klim→∞
γρikds=0. (3.16)
Proof. Let (ρik)k∈N∗ be a subsequence of the sequence (ρi)i∈N∗ such that|ρik|Φ≤2−k. Then
Γ=
γ∈Γrect: lim sup
k→∞
γρikds >0
=
n∈N
γ∈Γrect: lim sup
k→∞
γρikds≥1 n
=
n∈N
γ∈Γrect: for infinitely manyk,
γρikds≥ 1 2n
.
(3.17)
Hence, it suffices to show that for eachn∈N, the family of paths Γn=
γ∈Γrect: for infinitely manyk,
γρikds≥ 1 2n
(3.18) is such that ModΦ(Γn)=0. For this goal, letρ=∞
k=1ρik. Then by [5, Lemma 2],ρ∈LΦ. For allγ∈Γn,
γρ ds≥
∞ k=1
γρikds= ∞. (3.19)
Hence, ModΦ(Γn)=0. The proof is complete.
3.3. The capacityCΦ
Definition 3.14. LetΦbe anᏺ-function. For a setE⊂X, defineCΦ(E) by
CΦ(E)=inf|u|NΦ1 :u∈Ꮾ(E), (3.20) whereᏮ(E)= {u∈NΦ1:u|E≥1}.
IfᏮ(E)= ∅, we set CΦ(E)= ∞. Functions belonging toᏮ(E) are called admissible functions forE.
We define a capacityas an increasing positive set functionC given on aσ-additive class of setsΓ, which contains compact sets and such thatC(∅)=0 and C(i≥1Xi)≤
i≥1C(Xi) forXi∈Γ,i=1, 2,. . . .
The set functionCis called outer capacity if for everyX∈Γ,
C(X)=infC(O) :Oopen, X⊂O. (3.21)
We omit the proof of the following lemma, since it is an easy adaptation of the one [4, Theorem 4.3].
Lemma3.15. LetΦbe anᏺ-function. The set functionCΦis an outer capacity.
Lemma3.16. LetΦbe anᏺ-function and let(ui)ibe a Cauchy sequence inNΦ1(X). Then there are a functionuinNΦ1(X)and a subsequence(uik)ksuch that(uik)kconverges touin LΦand pointwiseµ-almost everywhere.
Proof. Since (ui)i is a Cauchy sequence in NΦ1(X), it is also a Cauchy sequence inLΦ. By passing to a subsequence if necessary, there is a functionv∈LΦto which the subse- quence converges both pointwiseµ-almost everywhere and inLΦ. We choose a further subsequence, also denoted by (un)nfor simplicity in notation, such that
ui−vΦ≤2−i, (3.22)
ui−→v pointwiseµ-a.e., (3.23)
gi+1,i
Φ≤2−i, (3.24)
wheregi,jis an upper gradient ofui−uj. Ifg1is an upper gradient ofu1such thatg1∈LΦ, thenu2=u1+ (u2−u1) has an upper gradientg2=g1+g1,2.
In general,ui=u1+ik−=11(uk+1−uk) has an upper gradientgi=g1+ik−=11gk+1,ksuch thatgi∈LΦ.
Forj < i, gi−gj
Φ≤
i−1
k=j
gk+1,k
Φ≤
i−1
k=j
2−k≤2−j+1−→0 as j−→ ∞. (3.25) Hence, (gi)iis a Cauchy sequence inLΦ, which implies that it converges inLΦ-norm to a nonnegative Borel functiong. Letube a function defined by
u(x)=1 2
lim sup
i→∞ ui(x) + lim inf
i→∞ ui(x)
(3.26) whenever the definition makes sense. By (3.23), we getu(x)=v(x)µ-almost everywhere, and hence,u∈LΦ. SetT= {x: lim supi→∞|ui(x)| = ∞}. The functionuis well defined outside ofT. To prove thatu∈NΦ1, byLemma 3.10, we must show thatuis well defined on almost all paths. To this end, we must prove that ModΦ(ΓT)=0.
LetΓ1be the collection of all pathsγ∈Γrectsuch that eitherγg ds= ∞or limi→∞
γgids
=
γg ds. ByLemma 3.13, ModΦ(Γ1)=0. On the other hand, recall thatΓ+T= {γ∈Γrect: Ᏼ1(|γ| ∩T)>0}. By (3.23), µ(T)=0. Hence, ModΦ(Γ+T)=0. Therefore, ModΦ(Γ1∪ Γ+T)=0. Letγ∈Γrect\(Γ1∪Γ+T). Then, sinceγ /∈Γ+T, there exists a point y∈ |γ|such thaty∈T. Sincegiis an upper gradient ofui, for any pointx∈ |γ|, we get
ui(x)−ui(y)≤ui(x)−ui(y)≤
γgids. (3.27)
Hence,|ui(x)| ≤ |ui(y)|+γgids. Sinceγ /∈Γ1, we deduce that lim sup
i→∞
ui(x)≤lim sup
i→∞ |ui(y)|+
γg ds <∞, (3.28) and hencex /∈T. ThusΓT⊂Γ1∪Γ+T. This implies that ModΦ(ΓT)=0.
On the other hand, ifγ∈Γrect\(Γ1∪Γ+T), denotingxandyas the end points ofγand noting by the above argument thatx,y /∈T, we get
u(x)−u(y)=1 2
lim sup
i→∞ ui(x)−lim inf
i→∞ ui(y) + lim inf
i→∞ ui(x)−lim sup
i→∞ ui(y)
≤lim sup
i→∞
ui(x)−ui(y)
≤lim
i→∞
γgids=
γg ds.
(3.29)
This means that g is a weak upper gradient ofu, and hence,u∈NΦ1. The proof is
complete.
Lemma3.17. LetΦbe anᏺ-function. IfE⊂Xis such thatCΦ(E)=0, thenModΦ(ΓE)=0.
Proof. Since CΦ(E)=0, for each i∈N∗, there exists a function ui∈NΦ1 such that
|ui|NΦ1≤2−iwithui|E≥1. Posevn=n
i=1|ui|. Then for eachn,vn∈NΦ1 and vn−vm
NΦ1≤
n i=m+1
ui
NΦ1 ≤2−m−→0 asm−→ ∞. (3.30) Hence, the sequence (vn)n is a Cauchy sequence inNΦ1. By Lemma 3.16, there is a functionv∈LΦsuch that|vn−v|Φ→0. By the construction used inLemma 3.16and since the sequence (vn(x))nis increasing outside of a setTsuch that ModΦ(ΓT)=0, we get
v(x)=lim
n→∞vn(x) (3.31)
withv(x)<∞.
IfE\T= ∅, then for arbitrary largen, v|E\T≥vn|E\T=
n i=1
uiE\T ≥n. (3.32)
Hence,v|E\T= ∞, which is not possible becausex /∈T. Therefore,E\T= ∅, and hence,ΓE⊂ΓT. Thus ModΦ(ΓE)=0. The proof is complete.
Corollary3.18. LetΦbe anᏺ-function and letEbe a subset ofXsuch thatCΦ(E)=0. If u∈NΦ1(X\E), then there is an extension ofutoEthat is inNΦ1(X). Any two such extensions ofuto all ofXare in the same equivalence class ofNΦ1(X).
Theorem3.19. For anyᏺ-functionΦ,NΦ1(X)is a Banach space.
Proof. Let (ui)i∈N∗ be a Cauchy sequence inNΦ1(X). It suffices to show that some subse- quence is a convergent sequence inNΦ1(X). By passing to a subsequence if necessary, we can assume that
uk−uk+1
Φ≤2−2k (3.33)
and that
gi+1,i
Φ≤2−i, (3.34)
wheregi,jis an upper gradient ofui−ujchosen to satisfy the above inequality.
Let
Ek=
x∈X:uk(x)−uk+1(x)≥2−k. (3.35) Then 2k|uk−uk+1| ∈NΦ1(X) and 2k|uk−uk+1||Ek≥1. Hence, by (3.33),
CΦEk
≤2kuk−uk+1
Φ≤2−k. (3.36)
LetFj= ∪∞k=jEkandF= ∩j∈NFj. Then
CΦFj≤ ∞
k=j
CΦEk≤2−j+1. (3.37)
This implies thatCΦ(F)=0.
Forx∈X\F, there is j∈Nsuch that for allk∈Nandk≥j,x /∈Ek. Hence, for all k∈Nandk≥j,|uk(x)−uk+1(x)|<2−k. Therefore, wheneverl≥k≥j, we get
uk(x)−ul(x)≤2−k+1. (3.38)
Thus the sequence (ui(x))i∈N∗ is a Cauchy sequence inR, and therefore is convergent to a finite number. Forx∈X\F, we let
u(x)=lim
i→∞ui(x). (3.39)
Forx∈X\F, we have
u(x)−uk(x)= ∞
n=k
un+1(x)−un(x). (3.40)
ByLemma 3.17, ModΦ(ΓF)=0, and for each pathγ∈Γrect\ΓF, for all pointsx∈ |γ|, (3.40) holds. Thus∞n=kgn+1,nis a weak upper gradient ofu−uk. Therefore,
u−uk
NΦ1≤u−uk
Φ+
∞ n=k
gn+1,n
Φ
≤u−uk
Φ+
∞ n=k
2−n
≤u−ukΦ+ 2−k+1−→0 ask−→ ∞.
(3.41)
This means that the subsequence converges in the norm ofNΦ1(X) tou. The proof is
complete.
In particular, we have shown that ifj∈N, there is a setFjsuch thatCΦ(Fj)≤2−j+1and the chosen subsequence converges uniformly outside ofFj. Thus we have the following corollary.
Corollary3.20. For anyᏺ-function Φ, any Cauchy sequence (ui)i∈N∗ ⊂NΦ1(X)has a subsequence that converges pointwise outside a set ofΦ-capacity zero. Furthermore, the sub- sequence can be chosen so that there exist sets of arbitrarily smallΦ-capacity such that the subsequence converges uniformly in the complement of each of these sets.
The proofs of the following three lemmas are an easy adaptation of those in [22, Lem- mas 2.1.5, 2.1.7, and 2.1.8] relative toLpLebesgue spaces. We omit these proofs.
Lemma3.21. Let Φbe anᏺ-function. Let u1 andu2 be ACCΦ functions on X withΦ- weak upper gradientsg1andg2, respectively. Letube anotherACCΦfunction inXsuch that there is an open setO⊂Xverifyingu=u1onOandu=u2onX\O. Theng1χO+g2and g1+g2χX\OareΦ-weak upper gradients ofu.
Remark that if we are in the hypotheses of the previous lemma and ifg2≥g1almost everywhere onO, theng2is aΦ-weak upper gradient ofu; and ifg2≤g1almost every- where onX\O, theng1is aΦ-weak upper gradient ofu.
Lemma3.22. LetΦbe anᏺ-function and letube anACCΦfunction onXsuch thatu=0 µ-almost everywhere onX\O, whereOis an open set inX. Ifgis aΦ-weak upper gradient ofu, thengχOis also aΦ-weak upper gradient ofu.
Lemma3.23. LetΦbe anᏺ-function and letube anACCΦfunction onX. Ifg,h∈LΦ
are twoΦ-weak upper gradients ofuandFis a closed subset ofX, then the functionv= gχF+hχX\Fis also aΦ-weak upper gradient ofu.
3.4. A characterization ofNΦ1(X). Next, we define another characterization of Orlicz- Sobolev spaces on metric measure spaces using only upper gradients and bypassing the notions of moduli of path families and weak upper gradients. We show inTheorem 3.27 that this characterization gives the same spaceNΦ1(X).
