**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 space*N*_{Φ}^{1} is a Banach space.

The relationship between *N*_{Φ}^{1} and*M*_{Φ}^{1} (the first extension defined in A¨ıssaoui (2002))
is studied. We also explore and compare diﬀerent definitions of capacities and give a
criterion under which*N*_{Φ}^{1} is strictly smaller than the Orlicz space**L**_{Φ}.

**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 by*M*_{Φ}^{1}(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 space*N*_{Φ}^{1}(X) is a Banach space for anyᏺ-functionΦand the space*M*^{1}_{Φ}(X) con-
tinuously embeds on*N*_{Φ}^{1}(X) whenΦsatisfies the∆2condition. We know that Lipschitz
functions are dense in*M*_{Φ}^{1}(X) forΦverifying the∆2 condition. To expect the same re-
sult with the vaster space*N*_{Φ}^{1}(X), we must add some assumptions, as in the Sobolev case,
on the metric space*X, namely,X*must be doubling and support a (1,Φ)-Poincar´e in-
equality, andΦverifies the∆* ^{}*condition. Remark that whenΦ(x)

*=*(1/ p)x

*(p >1), we rediscover the same result in the setting of Sobolev spaces. On the other hand, whenΩis a domain inR*

^{p}*, we give a new characterization of the Orlicz-Sobolev space*

^{N}*W*

^{1}

**L**Φ(Ω), and we show that

*N*

_{Φ}

^{1}(Ω)

*=*

*W*

^{1}

**L**

_{Φ}(Ω) whenΦsatisfies the∆2condition. Hence, for re- flexive Orlicz spaces

**L**

_{Φ}(R

*), we get*

^{N}*N*

_{Φ}

^{1}(R

*)*

^{N}*=*

*M*

^{1}

_{Φ}(R

*)*

^{N}*=*

*W*

^{1}

**L**

_{Φ}(R

*), since we know that*

^{N}*M*

_{Φ}

^{1}(R

*)*

^{N}*=*

*W*

^{1}

**L**Φ(R

*). See [4, Theorem 3.3]. We also study the mean equivalent class with respect toΦ(MEC*

^{N}_{Φ}) criterion under which

*N*

_{Φ}

^{1}(X) is strictly included in the Orlicz space

**L**

_{Φ}(X) and we compare between natural capacities defined on

*N*

_{Φ}

^{1}(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 *N*_{Φ}^{1}(X). Section 4 deals with comparison between *N*_{Φ}^{1}(X) and
*M*_{Φ}^{1}(X) and with some properties of*N*_{Φ}^{1}(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 for*t >*0, lim_{t}* _{→}*0(Φ(t)/t)

*=*0, and lim

_{t}*+*

_{→}*∞*(Φ(t)/t)

*=*+

*∞*. We have the representationΦ(t)

*=*

_{|}

_{t}

_{|}0 *ϕ(x)d*L(x), where*ϕ*:R^{+}*→*R^{+}is nondecreasing,
right continuous, with*ϕ(0)**=*0,*ϕ(t)>*0 for*t >*0, lim*t**→*0^{+}*ϕ(t)**=*0, and lim*t**→*+*∞**ϕ(t)**=*
+*∞*. HereLstands for the Lebesgue measure. We put in the sequel, as usual,*dx**=**d*L(x).

Theᏺ-function Φ^{∗}*conjugate* toΦis defined by Φ* ^{∗}*(t)

*=*

_{|}

_{t}

_{|}0 *ϕ** ^{∗}*(x)dx, where

*ϕ*

*is given by*

^{∗}*ϕ*

*(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 :

*X*Φ^{}*f*(x)^{}*dµ(x)<**∞*

*.* (2.1)

We define the*Orlicz space***L**_{Φ,µ}(X) by
**L**_{Φ},µ(X)*=*

*f* :*X**−→*Rmeasurable :

*X*Φ^{}*α f*(x)^{}*dµ(x)<**∞*for some*α >*0

*.* (2.2)
The Orlicz space**L**_{Φ},µ(X) is a Banach space with the following norm, called the*Lux-*
*emburg norm:*

*|**f**|*_{Φ},µ,X*=*inf

*r >*0 :

*X*Φ^{}*f*(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* *∈***L**_{Φ,µ}(X) and *g**∈*
**L**_{Φ}* ^{∗}*,µ(X), then

*f g*

*∈*

**L**

^{1}and

*X**|**f g**|**dµ**≤*2*|**f**|*Φ,µ,X*· |**g**|*Φ* ^{∗}*,µ,X

*.*(2.4) LetΦbe anᏺ-function. We say thatΦ

*verifies the*∆2

*condition*if there is a constant

*C >*0 such thatΦ(2t)

*≤*

*C*Φ(t) for all

*t*

*≥*0.

The∆2condition forΦcan be formulated in the following equivalent way: for every
*C >*0, there exists*C*^{}*>*0 such thatΦ(Ct)*≤**C** ^{}*Φ(t) for all

*t*

*≥*0.

We have alwaysᏸΦ,µ(X)*⊂***L**Φ,µ(X). The equalityᏸΦ,µ(X)*=***L**Φ,µ(X) occurs ifΦveri-
fies the∆2condition.

We know that**L**_{Φ},µ(X) is reflexive ifΦandΦ* ^{∗}*verify the∆2condition.

Note that ifΦverifies the∆2condition, then^{}Φ(*f** _{i}*(x))dµ

*→*0 as

*i*

*→ ∞*if and only if

*|**f*_{i}*|*Φ,µ,X*→*0 as*i**→ ∞*.

Recall that anᏺ-functionΦsatisfies the∆* ^{}*condition if there is a positive constant

*C*such that for all

*x,y*

*≥*0,Φ(xy)

*≤*

*C*Φ(x)Φ(y). See [16,21]. If anᏺ-functionΦsatisfies the∆

*condition, then it satisfies also the∆2condition.*

^{}LetΩbe an open set inR* ^{N}*, let

**C**

*(Ω) be the space of functions which, together with all their partial derivatives of any order, are continuous onΩ, and let*

^{∞}**C**

^{∞}_{0}(R

*)*

^{N}*=*

**C**

^{∞}_{0}stand for all functions in

**C**

*(R*

^{∞}*) which have compact support inR*

^{N}*. The space*

^{N}**C**

*(Ω) stands for the space of functions having all derivatives of order*

^{k}*≤*

*k*continuous onΩ, and

**C(Ω)**is the space of continuous functions onΩ.

The (weak) partial derivative of *f* of order*|**β**|*is denoted by
*D*^{β}*f* *=* *∂*^{|}^{β}^{|}

*∂x*_{1}^{β}^{1}*·**∂x*^{β}_{2}^{2}*· ··· ·**∂x*_{N}^{β}^{N}*f .* (2.5)
LetΦbe anᏺ-function and*m**∈*N. We say that a function *f* :R^{N}*→*Rhas a distribu-
tional (weak partial) derivative of order*m, denoted byD*^{β}*f*,*|**β**| =**m, if*

*f D*^{β}*θ dx**=*(*−*1)^{|}^{β}^{|}*D*^{β}*f*^{}*θ dx,* *∀**θ**∈***C*** ^{∞}*0

*.*(2.6) LetΩbe an open set inR

*and denote*

^{N}**L**

_{Φ,L}(Ω) by

**L**

_{Φ}(Ω). The

*Orlicz-Sobolev space*

*W*

^{m}**L**

_{Φ}(Ω) is the space of real functions

*f*such that

*f*and its distributional derivatives up to the order

*m*are in

**L**

_{Φ}(Ω).

The space*W*^{m}**L**_{Φ}(Ω) is a Banach space equipped with the norm

*|**f**|**m,Φ,Ω**=*

0_{≤|}*β**|≤**m*

*D*^{β}*f*^{}_{Φ}, *f* *∈**W*^{m}**L**Φ(Ω), (2.7)

where*|**D*^{β}*f**|*Φ*= |**D*^{β}*f**|*Φ,L,Ω.

Recall that if Φ verifies the ∆2 condition, then **C*** ^{∞}*(Ω)

*∩*

*W*

^{m}**L**

_{Φ}(Ω) is dense in

*W*

^{m}**L**

_{Φ}(Ω), and

**C**

^{∞}_{0}(R

*) is dense in*

^{N}*W*

^{m}**L**

_{Φ}(R

*).*

^{N}For more details on the theory of Orlicz spaces, see [1,16,17,18,21].

In this paper, the letter*C*will denote various constants which may diﬀer from one
formula to the next one even within a single string of estimates.

**3. The Orlicz-Sobolev space***N*_{Φ}^{1}(X)

**3.1.**Φ-modulus in metric spaces. Let (X,d,µ) be a metric, Borel measure space, such
that*µ*is positive and finite on balls in*X.*

If*I*is an interval inR, a path in*X*is a continuous map*γ*:*I**→**X. By abuse of language,*
the image*γ(I)**=*:*|**γ**|*is also called a path. If*I**=*[a,b] is a closed interval, then the length
of a path*γ*:*I**→**X*is

*l(γ)**=*length(γ)*=*sup

*n*
*i**=*1

*γ*^{}*t**i+1*

*−**γ*^{}*t**i*, (3.1)

where the supremum is taken over all finite sequences*a**=**t*1*≤**t*2*≤ ··· ≤**t*_{n}*≤**t*_{n+1}*=**b. If*
*I*is not closed, we set

*l(γ)**=*supl^{}*γ**|**J*

, (3.2)

where the supremum is taken over all closed subintervals*J*of*I. A path is said to be recti-*
fiable if its length is a finite number. A path*γ*:*I**→**X*is locally rectifiable if its restriction
to each closed subinterval of*I*is rectifiable.

For any rectifiable path*γ, there are its associated length functions**γ*:*I**→*[0,l(γ)] and
a unique 1-Lipschitz continuous map*γ** _{s}*: [0,l(γ)]

*→*

*X*such that

*γ*

*=*

*γ*

_{s}*◦*

*s*

*. The path*

_{γ}*γ*

*is the arc-length parametrization of*

_{s}*γ.*

Let*γ* be a rectifiable path in *X. The line integral overγ* of each nonnegative Borel
function*ρ*:*X** _{→}*[0,

*∞*] is

*γ**ρ ds**=*
_{l(γ)}

0 *ρ**◦**γ**s*(t)dt. (3.3)

If the path*γ*is only locally rectifiable, we set

*γ**ρ ds**=*sup

*γ*^{}*ρ ds,* (3.4)

where the supremum is taken over all rectifiable subpaths*γ** ^{}*of

*γ. See [14] for more de-*tails.

Denote byΓrectthe collection of all nonconstant compact (i.e.,*I*is compact) rectifiable
paths in*X. IfA*is a subset of*X, then*Γ*A*is the family of all paths inΓrectthat intersect the
set*A, and*Γ^{+}* _{A}*is the family of all paths

*γ*inΓrectsuch 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 Mod_{Φ}(Γ), is defined as

*ρ**∈*infᏲ(Γ)*|**ρ*_{|}_{Φ}, (3.5)

whereᏲ(Γ) is the set of all nonnegative Borel functions*ρ* such that^{}_{γ}*ρ ds**≥*1 for all
rectifiable paths*γ*inΓ. Such functions*ρ*used 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. *Let*Φ*be an*ᏺ-function and letΓ*be a collection of paths inX. Then the*
Φ-modulus of the familyΓ*is an outer measure on*Γ. That is,

(1) Mod_{Φ}(*∅*)*=*0,

(2) Mod_{Φ}(Γ1)*≤*Mod_{Φ}(Γ2)*if* Γ1*⊂*Γ2*,*
(3) Mod_{Φ}(^{}^{∞}_{i}* _{=}*1Γ

*i*)

*≤*

_{∞}*i**=*1Mod_{Φ}(Γ*i*).

*Proof.* Assertions (1) and (2) are obvious. We prove (3). We may assume that ModΦ(Γ*i*)<

*∞*for all*i. Forε >*0, there is*ρ*_{i}*∈*Ᏺ(Γ*i*) such that
*ρ**i*

Φ*≤*Mod_{Φ}^{}Γ*i*

+*ε2*^{−}^{i}*.* (3.6)

Set*ρ**=*sup_{i}*ρ**i*andΓ*=*_{∞}

*i**=*1Γ*i*. Since*ρ**≥**ρ**i*for all*i,ρ**∈*Ᏺ(Γ). Thus Mod_{Φ}(Γ)*≤ |**ρ**|*Φ.
By [5, Lemma 2],*|**ρ**|*Φ*≤*_{∞}

*i**=*1*|**ρ**i**|*Φ. Hence,
ModΦ(Γ)*≤* ^{∞}

*i**=*1

ModΦ(Γ*i*) +*ε.* (3.7)

Since*ε*is arbitrary, (3) is proved.

A property relevant to paths in*X*is 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 *γ** _{xy}* to be the
subpath

*γ*

*|*[t

*x*,t

*y*], where the two distinct numbers

*t*

*x*and

*t*

*y*are chosen from the domain of

*γ*such that

*γ(t*

*x*)

*=*

*x*and

*γ(t*

*y*)

*=*

*y. The subpathγ*

*xy*is not a well-defined notion as there can be more than one choice of the related numbers

*t*

*and*

_{x}*t*

*. Because of this ambiguity, any property that is required for one choice of the subpath*

_{y}*γ*

*xy*is also required for all such choices of subpaths.

*Definition 3.3.* LetΦbe anᏺ-function and let*l(γ) denote the length ofγ. A functionu*
is said to be absolutely continuous onΦ-almost every curve (ACCΦ) if*u**◦**γ*is absolutely
continuous on [0,l(γ)] forΦ-almost every rectifiable arc-length parametrized path*γ*in
*X. IfX*is a domain inR* ^{N}*, a function

*u*is 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 ACL

_{Φ}if its directional derivatives are in

**L**

_{Φ}.

*Definition 3.4.* Let*u*be a real-valued function on a metric space*X. A nonnegative Borel-*
measurable function*ρ*is said to be an upper gradient of*u*if for all compact rectifiable
paths*γ, the following inequality holds:*

*u(x)**−**u(y)*^{}*≤*

*γ**ρ ds,* (3.8)

where*x*and*y*are the end points of the path.

*Definition 3.5.* LetΦbe anᏺ-function and let*u*be an arbitrary real-valued function on
*X. Letρ*be a nonnegative Borel function on*X. If there exists a family*Γ*⊂*Γrectsuch that
Mod_{Φ}(Γ)*=*0 and the inequality (3.8) is true for all paths*γ*inΓrect*\*Γ, then*ρ*is said to
be aΦ-weak upper gradient of*u. If inequality (3.8) holds true for*Φ-modulus almost all
paths in a set*B**⊂**X, thenρ*is said to be aΦ-weak upper gradient of*u*on*B.*

Lemma3.6. *Let*Φ*be an*ᏺ-function and letΓ*be a collection of paths inX. Then*ModΦ(Γ)*=*
0*if and only if there is a nonnegative Borel functionρonXsuch thatρ**∈***L**_{Φ}*and for all paths*
*γ**∈*Γ,

*γ**ρ ds**= ∞**.* (3.9)

*Proof.* Suppose that Mod_{Φ}(Γ)*=*0. Then if*n**∈*N* ^{∗}*, there exists a nonnegative Borel func-
tion

*ρ*

*n*on

*X*such that

*ρ*

*n*

*∈*

**L**

_{Φ}and

*|*

*ρ*

*n*

*|*Φ

*≤*2

^{−}*. The function*

^{n}*ρ*

*=*

_{∞}*n**=*1*ρ**n*is a nonneg-
ative Borel function on*X*and, by [5, Lemma 2],*|**ρ**|*Φ*≤*_{∞}

*n**=*1*|**ρ*_{n}*|*Φ, which implies
that*ρ**∈***L**_{Φ}. It is evident that^{}_{γ}*ρ ds**= ∞*for all paths*γ**∈*Γ.

Assume that there is a nonnegative Borel function*ρ*on*X*such that*ρ**∈***L**_{Φ}and for all
paths*γ**∈*Γ,^{}_{γ}*ρ ds**= ∞*. Then for each*n, the function 2*^{−}^{n}*ρ*is 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
Mod_{Φ}(Γ^{+}*E*)*=*0.

*Proof.* Since*∞**χ**E*is an admissible function for calculating Mod_{Φ}(Γ^{+}*E*), the corollary fol-

lows byLemma 3.6.

**3.2. The Orlicz-Sobolev space***N*_{Φ}^{1}(X)

*Definition 3.8.* LetΦbe anᏺ-function and let the set*N*^{}_{Φ}^{1}(X,d,µ) be the collection of all
real-valued function*u*on*X*such that*u**∈***L**_{Φ}and*u*has aΦ-weak upper gradient in**L**_{Φ}.

We remark that^{}*N*_{Φ}^{1} is a vector space, since if*α,β**∈*Rand*u,v**∈**N*_{Φ}^{1} with respect to
Φ-weak upper gradients*ρ*and*σ, then**|**α**|**ρ*+*|**β**|**σ*is aΦ-weak upper gradient of*αρ*+*βσ.*

If*u**∈**N*_{Φ}^{1}, we set

*|**u**|*_{}* _{N}*1

Φ*= |**u**|*Φ+ inf

*ρ* *|**ρ**|*Φ, (3.10)

where the infimum is taken over allΦ-weak upper gradient*ρ*of*u*such that*ρ*_{∈}**L**Φ.
If*u,v**∈**N*_{Φ}^{1}, let*uv*if*|**u**−**v**|**N*_{Φ}^{1} *=*0. It can be easily seen thatis an equivalence
relation, partitioning^{}*N*_{Φ}^{1} 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 by*N*_{Φ}^{1}(X), is defined to be the space*N*^{}_{Φ}^{1}(X,d,µ)with the norm*|**u**|*_{N}_{Φ}^{1}:*=*

*|**u**|**N*_{Φ}^{1}.

If*u,v**∈**N*_{Φ}^{1}, then it is easily verified that the functions*|**u**|*, min*{**u,v**}*, max*{**u,v**} ∈**N*_{Φ}^{1},
that if*λ**≥*0, then min*{**u,λ**} ∈**N*_{Φ}^{1}, and that if*λ**≤*0, then max*{**u,λ**} ∈**N*_{Φ}^{1}. Thus*N*_{Φ}^{1}(X)
enjoys all the lattice properties in classical first-order Orlicz-Sobolev spaces.

Lemma3.10. *Let*Φ*be an*ᏺ-function and*u**∈**N*_{Φ}^{1}*. Thenuis*ACCΦ*.*

*Proof.* By hypothesis,*u**∈***L**Φand*u*has aΦ-weak upper gradient*ρ**∈***L**Φ. LetΓbe the
collection of all paths inΓrectfor which inequality (3.8) does not hold. Then Mod_{Φ}(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,

Mod_{Φ}^{}Γ1

*≤*Mod_{Φ}(Γ)*=*0. (3.11)

LetΓ2be the collection of all paths*γ*inΓrectsuch that^{}_{γ}*ρ ds**= ∞*. Since*ρ**∈***L**_{Φ}, then
ModΦ(Γ2)*=*0. Thus ModΦ(Γ1*∪*Γ2)*=*0. If*γ*is a path inΓrect*\*(Γ1*∪*Γ2),*γ*has no subpath
inΓ1, and hence for all*x,y**∈ |**γ**|*,

*u(x)**−**u(y)*^{}*≤*

*γ**xy*

*ρ ds <**∞**.* (3.12)

Therefore,*u*is absolutely continuous on each path*γ*inΓrect*\*(Γ1*∪*Γ2). The proof is

complete.

Lemma 3.11. *Let* Φ*be an* ᏺ-function and let *u**∈**N*_{Φ}^{1} *be such that* *|**u**|*Φ*=*0. Then
ModΦ(Γ)*=*0, where

Γ*=*

*γ**∈*Γrect:*u(x)**=*0*for somex**∈ |**γ**|*

*.* (3.13)

*Proof.* Since*|**u**|*Φ*=*0, the set*S**= {**x**∈**X*:*u(x)**=*0*}*has measure zero. Hence,Γ*=*Γ*S*

and

Γ*=*Γ^{+}*S**∪*
Γ*S**\*Γ^{+}*S*

*.* (3.14)

The subfamilyΓ^{+}*S* can be disregarded since

Mod_{Φ}^{}Γ^{+}_{S}^{}*≤ |∞ ·**χ**S**|*Φ*=*0, (3.15)
where*χ**S*is the characteristic function of the set*S. The pathsγ**∈*Γ*S**\*Γ^{+}*S* intersect*S*only
on a set of linear measure zero, and hence, with respect to the linear measure almost
everywhere on*γ, the functionu*takes on the value of zero. By the fact that*γ*also intersect
*S, therefore,u* is not absolutely continuous on*γ. By*Lemma 3.10, Mod_{Φ}(Γ*S**\*Γ^{+}*S*)*=*0.

Thus Mod_{Φ}(Γ)*=*0 and the proof is complete.

We deduce from the previous lemma the following corollary.

Corollary3.12. *Let*Φ*be an*ᏺ-function. If*u,v**∈**N*_{Φ}^{1} *are such that**|**u**−**v**|*Φ*=*0, then
*uandvbelong to the same equivalent class inN*_{Φ}^{1}(X).

In the sequel, we will not distinguish between the functions in*N*^{}_{Φ}^{1} and their equiva-
lence classes in*N*_{Φ}^{1}.

Lemma3.13. *Let*Φ*be an*ᏺ-function. If(ρ* _{i}*)

*i*

*∈*N

^{∗}*is a sequence of Borel functions in*

**L**Φ

*such*

*that*lim

_{i}

_{→∞}*|*

*ρ*

_{i}*|*Φ

*=*0, then there exist a subsequence(ρ

_{i}*)*

_{k}

_{k}

_{∈}_{N}

^{∗}*and a family*Γ

*⊂*Γrect

*such*

*that*Mod

_{Φ}(Γ)

*=*0

*and for all pathsγ*

*∈*Γrect

*\*Γ,

*k*lim*→∞*

*γ**ρ**i**k**ds**=*0. (3.16)

*Proof.* Let (ρ*i**k*)*k**∈*N* ^{∗}* be a subsequence of the sequence (ρ

*i*)

*i*

*∈*N

*such that*

^{∗}*|*

*ρ*

*i*

*k*

*|*Φ

*≤*2

^{−}*. Then*

^{k}Γ*=*

*γ**∈*Γrect: lim sup

*k**→∞*

*γ**ρ**i**k**ds >*0

*=*

*n**∈*N

*γ**∈*Γrect: lim sup

*k**→∞*

*γ**ρ*_{i}_{k}*ds**≥*1
*n*

*=*

*n**∈*N

*γ**∈*Γrect: for infinitely many*k,*

*γ**ρ*_{i}_{k}*ds**≥* 1
2n

*.*

(3.17)

Hence, it suﬃces to show that for each*n**∈*N, the family of paths
Γ*n**=*

*γ**∈*Γrect: for infinitely many*k,*

*γ**ρ*_{i}_{k}*ds**≥* 1
2n

(3.18)
is such that Mod_{Φ}(Γ*n*)*=*0. For this goal, let*ρ**=*_{∞}

*k**=*1*ρ*_{i}* _{k}*. Then by [5, Lemma 2],

*ρ*

*∈*

**L**

_{Φ}. For all

*γ*

*∈*Γ

*n*,

*γ**ρ ds**≥*

*∞*
*k**=*1

*γ**ρ**i**k**ds**= ∞**.* (3.19)

Hence, Mod_{Φ}(Γ*n*)*=*0. The proof is complete.

**3.3. The capacity***C*_{Φ}

*Definition 3.14.* LetΦbe anᏺ-function. For a set*E**⊂**X, defineC*_{Φ}(E) by

*C*_{Φ}(E)*=*inf^{}*|**u**|**N*_{Φ}^{1} :*u**∈*Ꮾ(E)^{}, (3.20)
whereᏮ(E)*= {**u**∈**N*_{Φ}^{1}:*u**|**E**≥*1*}*.

IfᏮ(E)*= ∅*, we set *C*_{Φ}(E)*= ∞*. Functions belonging toᏮ(E) are called admissible
functions for*E.*

We define a *capacity*as an increasing positive set function*C* given on a*σ*-additive
class of setsΓ, which contains compact sets and such that*C(**∅*)*=*0 and *C(*^{}_{i}_{≥}_{1}*X** _{i}*)

*≤*

*i**≥*1*C(X**i*) for*X**i**∈*Γ,*i**=*1, 2,. . . .

The set function*C*is called outer capacity if for every*X**∈*Γ,

*C(X)**=*inf^{}*C(O) :O*open, *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. *Let*Φ*be an*ᏺ*-function. The set functionC*_{Φ}*is an outer capacity.*

Lemma3.16. *Let*Φ*be an*ᏺ*-function and let*(u* _{i}*)

_{i}*be a Cauchy sequence inN*

_{Φ}

^{1}(X). Then

*there are a functionuinN*

_{Φ}

^{1}(X)

*and a subsequence*(u

*i*

*k*)

*k*

*such that*(u

*i*

*k*)

*k*

*converges touin*

**L**

_{Φ}

*and pointwiseµ-almost everywhere.*

*Proof.* Since (u*i*)*i* is a Cauchy sequence in *N*_{Φ}^{1}(X), it is also a Cauchy sequence in**L**_{Φ}.
By passing to a subsequence if necessary, there is a function*v**∈***L**_{Φ}to which the subse-
quence converges both pointwise*µ-almost everywhere and in***L**_{Φ}. We choose a further
subsequence, also denoted by (u*n*)*n*for simplicity in notation, such that

*u*_{i}*−**v*^{}_{Φ}*≤*2^{−}* ^{i}*, (3.22)

*u**i**−→**v* pointwise*µ-a.e.,* (3.23)

*g**i+1,i*

Φ*≤*2^{−}* ^{i}*, (3.24)

where*g** _{i,j}*is an upper gradient of

*u*

_{i}*−*

*u*

*. If*

_{j}*g*1is an upper gradient of

*u*1such that

*g*1

*∈*

**L**Φ, then

*u*2

*=*

*u*1+ (u2

*−*

*u*1) has an upper gradient

*g*2

*=*

*g*1+

*g*1,2.

In general,*u**i**=**u*1+^{}^{i}_{k}^{−}_{=}^{1}_{1}(u*k+1**−**u**k*) has an upper gradient*g**i**=**g*1+^{}^{i}_{k}^{−}_{=}^{1}_{1}*g**k+1,k*such
that*g**i**∈***L**_{Φ}.

For*j < i,*
*g**i**−**g**j*

Φ*≤*

*i**−*1

*k**=**j*

*g**k+1,k*

Φ*≤*

*i**−*1

*k**=**j*

2^{−}^{k}*≤*2^{−}^{j+1}*−→*0 as *j**−→ ∞**.* (3.25)
Hence, (g*i*)*i*is a Cauchy sequence in**L**_{Φ}, which implies that it converges in**L**_{Φ}-norm
to a nonnegative Borel function*g*. Let*u*be a function defined by

*u(x)**=*1
2

lim sup

*i**→∞* *u**i*(x) + lim inf

*i**→∞* *u**i*(x)

(3.26)
whenever the definition makes sense. By (3.23), we get*u(x)**=**v(x)µ-almost everywhere,*
and hence,*u**∈***L**_{Φ}. Set*T**= {**x*: lim sup_{i}_{→∞}*|**u**i*(x)*| = ∞}*. The function*u*is well defined
outside of*T. To prove thatu**∈**N*_{Φ}^{1}, byLemma 3.10, we must show that*u*is well defined
on almost all paths. To this end, we must prove that Mod_{Φ}(Γ*T*)*=*0.

LetΓ1be the collection of all paths*γ**∈*Γrectsuch that either^{}_{γ}*g ds**= ∞*or lim*i**→∞*

*γ**g**i**ds*

*=*

*γ**g ds. By*Lemma 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, Mod_{Φ}(Γ^{+}* _{T}*)

*=*0. Therefore, Mod

_{Φ}(Γ1

*∪*Γ

^{+}

*T*)

*=*0. Let

*γ*

*∈*Γrect

*\*(Γ1

*∪*Γ

^{+}

*T*). Then, since

*γ /*

*∈*Γ

^{+}

*T*, there exists a point

*y*

*∈ |*

*γ*

*|*such that

*y*

*∈*

*T*. Since

*g*

*is an upper gradient of*

_{i}*u*

*, for any point*

_{i}*x*

*∈ |*

*γ*

*|*, we get

*u**i*(x)^{}*−**u**i*(y)^{}*≤**u**i*(x)*−**u**i*(y)^{}*≤*

*γ**g**i**ds.* (3.27)

Hence,*|**u** _{i}*(x)

*| ≤ |*

*u*

*(y)*

_{i}*|*+

^{}

_{γ}*g*

_{i}*ds. Sinceγ /*

*∈*Γ1, we deduce that lim sup

*i**→∞*

*u** _{i}*(x)

^{}

*≤*lim sup

*i**→∞* *|**u** _{i}*(y)

*|*+

*γ**g ds <**∞*, (3.28)
and hence*x /**∈**T*. ThusΓ*T**⊂*Γ1*∪*Γ^{+}*T*. This implies that Mod_{Φ}(Γ*T*)*=*0.

On the other hand, if*γ**∈*Γrect*\*(Γ1*∪*Γ^{+}*T*), denoting*x*and*y*as the end points of*γ*and
noting by the above argument that*x,y /**∈**T*, we get

*u(x)**−**u(y)*^{}*=*1
2

lim sup

*i**→∞* *u**i*(x)*−*lim inf

*i**→∞* *u**i*(y) + lim inf

*i**→∞* *u**i*(x)*−*lim sup

*i**→∞* *u**i*(y)^{}_{}

*≤*lim sup

*i**→∞*

*u**i*(x)*−**u**i*(y)^{}

*≤*lim

*i**→∞*

*γ**g**i**ds**=*

*γ**g ds.*

(3.29)

This means that *g* is a weak upper gradient of*u, and hence,u**∈**N*_{Φ}^{1}. The proof is

complete.

Lemma3.17. *Let*Φ*be an*ᏺ-function. If*E**⊂**Xis such thatC*_{Φ}(E)*=*0, thenMod_{Φ}(Γ*E*)*=*0.

*Proof.* Since *C*_{Φ}(E)*=*0, for each *i**∈*N* ^{∗}*, there exists a function

*u*

*i*

*∈*

*N*

_{Φ}

^{1}such that

*|**u**i**|**N*_{Φ}^{1}*≤*2^{−}* ^{i}*with

*u*

*i*

*|*

*E*

*≥*1. Pose

*v*

*n*

*=*

_{n}*i**=*1*|**u**i**|*. Then for each*n,v**n**∈**N*_{Φ}^{1} and
*v**n**−**v**m*

*N*_{Φ}^{1}*≤*

*n*
*i**=**m+1*

*u**i*

*N*_{Φ}^{1} *≤*2^{−}^{m}*−→*0 as*m**−→ ∞**.* (3.30)
Hence, the sequence (v*n*)*n* is a Cauchy sequence in*N*_{Φ}^{1}. By Lemma 3.16, there is a
function*v**∈***L**_{Φ}such that*|**v*_{n}*−**v**|*Φ*→*0. By the construction used inLemma 3.16and
since the sequence (v*n*(x))*n*is increasing outside of a set*T*such that Mod_{Φ}(Γ*T*)*=*0, we
get

*v(x)**=*lim

*n**→∞**v**n*(x) (3.31)

with*v(x)<**∞*.

If*E**\**T**= ∅*, then for arbitrary large*n,*
*v**|**E**\**T**≥**v*_{n}*|**E**\**T**=*

*n*
*i**=*1

*u*_{i}^{}_{E}_{\}_{T}*≥**n.* (3.32)

Hence,*v**|**E**\**T**= ∞*, which is not possible because*x /**∈**T*. Therefore,*E**\**T**= ∅*, and
hence,Γ*E**⊂*Γ*T*. Thus Mod_{Φ}(Γ*E*)*=*0. The proof is complete.

Corollary3.18. *Let*Φ*be an*ᏺ-function and let*Ebe a subset ofXsuch thatC*_{Φ}(E)*=*0. If
*u**∈**N*_{Φ}^{1}(X*\**E), then there is an extension ofutoEthat is inN*_{Φ}^{1}(X). Any two such extensions
*ofuto all ofXare in the same equivalence class ofN*_{Φ}^{1}(X).

Theorem3.19. *For any*ᏺ-functionΦ,*N*_{Φ}^{1}(X)*is a Banach space.*

*Proof.* Let (u* _{i}*)

*i*

*∈*N

*be a Cauchy sequence in*

^{∗}*N*

_{Φ}

^{1}(X). It suﬃces to show that some subse- quence is a convergent sequence in

*N*

_{Φ}

^{1}(X). By passing to a subsequence if necessary, we can assume that

*u**k**−**u**k+1*

Φ*≤*2^{−}^{2k} (3.33)

and that

*g**i+1,i*

Φ*≤*2^{−}* ^{i}*, (3.34)

where*g**i,j*is an upper gradient of*u**i**−**u**j*chosen to satisfy the above inequality.

Let

*E**k**=*

*x**∈**X*:^{}*u**k*(x)*−**u**k+1*(x)^{}*≥*2^{−}^{k}^{}*.* (3.35)
Then 2^{k}*|**u**k**−**u**k+1**| ∈**N*_{Φ}^{1}(X) and 2^{k}*|**u**k**−**u**k+1**||**E**k**≥*1. Hence, by (3.33),

*C*_{Φ}^{}*E**k*

*≤*2^{k}^{}*u**k**−**u**k+1*

Φ*≤*2^{−}^{k}*.* (3.36)

Let*F**j**= ∪*^{∞}_{k}_{=}_{j}*E**k*and*F**= ∩**j**∈*N*F**j*. Then

*C*_{Φ}^{}*F*_{j}^{}*≤* ^{∞}

*k**=**j*

*C*_{Φ}^{}*E*_{k}^{}*≤*2^{−}^{j+1}*.* (3.37)

This implies that*C*_{Φ}(F)*=*0.

For*x**∈**X**\**F, there is* *j**∈*Nsuch that for all*k**∈*Nand*k**≥**j,x /**∈**E** _{k}*. Hence, for all

*k*

*∈*Nand

*k*

*≥*

*j,*

*|*

*u*

*k*(x)

*−*

*u*

*k+1*(x)

*|*

*<*2

^{−}*. Therefore, whenever*

^{k}*l*

*≥*

*k*

*≥*

*j, we get*

*u**k*(x)*−**u**l*(x)^{}*≤*2^{−}^{k+1}*.* (3.38)

Thus the sequence (u*i*(x))*i**∈*N* ^{∗}* is a Cauchy sequence inR, and therefore is convergent
to a finite number. For

*x*

*∈*

*X*

*\*

*F, we let*

*u(x)**=*lim

*i**→∞**u**i*(x). (3.39)

For*x**∈**X**\**F, we have*

*u(x)**−**u** _{k}*(x)

*=*

^{∞}*n**=**k*

*u** _{n+1}*(x)

*−*

*u*

*(x)*

_{n}^{}

*.*(3.40)

ByLemma 3.17, ModΦ(Γ*F*)*=*0, and for each path*γ**∈*Γrect*\*Γ*F*, for all points*x**∈ |**γ**|*,
(3.40) holds. Thus^{}^{∞}_{n}_{=}_{k}*g** _{n+1,n}*is a weak upper gradient of

*u*

*−*

*u*

*. Therefore,*

_{k}*u**−**u**k*

*N*_{Φ}^{1}*≤**u**−**u**k*

Φ+

*∞*
*n**=**k*

*g**n+1,n*

Φ

*≤**u**−**u**k*

Φ+

*∞*
*n**=**k*

2^{−}^{n}

*≤**u**−**u*_{k}^{}_{Φ}+ 2^{−}^{k+1}*−→*0 as*k**−→ ∞**.*

(3.41)

This means that the subsequence converges in the norm of*N*_{Φ}^{1}(X) to*u. The proof is*

complete.

In particular, we have shown that if*j**∈*N, there is a set*F** _{j}*such that

*C*

_{Φ}(F

*)*

_{j}*≤*2

^{−}*and the chosen subsequence converges uniformly outside of*

^{j+1}*F*

*j*. Thus we have the following corollary.

Corollary3.20. *For any*ᏺ-function Φ, any Cauchy sequence (u*i*)*i**∈*N^{∗}*⊂**N*_{Φ}^{1}(X)*has a*
*subsequence that converges pointwise outside a set of*Φ-capacity zero. Furthermore, the sub-
*sequence can be chosen so that there exist sets of arbitrarily small*Φ*-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 to**L*** ^{p}*Lebesgue spaces. We omit these proofs.

Lemma3.21. *Let* Φ*be an*ᏺ-function. Let *u*1 *andu*2 *be* ACC_{Φ} *functions on* *X* *with*Φ-
*weak upper gradientsg*1*andg*2*, respectively. Letube another*ACC_{Φ}*function inXsuch that*
*there is an open setO**⊂**Xverifyingu**=**u*1*onOandu**=**u*2*onX**\**O. Theng*1*χ**O*+*g*2*and*
*g*1+*g*2*χ**X**\**O**are*Φ-weak upper gradients of*u.*

Remark that if we are in the hypotheses of the previous lemma and if*g*2*≥**g*1almost
everywhere on*O, theng*2is aΦ-weak upper gradient of*u; and ifg*2*≤**g*1almost every-
where on*X**\**O, theng*1is aΦ-weak upper gradient of*u.*

Lemma3.22. *Let*Φ*be an*ᏺ*-function and letube an*ACC_{Φ}*function onXsuch thatu**=*0
*µ-almost everywhere onX**\**O, whereOis an open set inX. Ifgis a*Φ-weak upper gradient
*ofu, thengχ*_{O}*is also a*Φ-weak upper gradient of*u.*

Lemma3.23. *Let*Φ*be an*ᏺ-function and let*ube an*ACCΦ*function onX. Ifg*,h*∈***L**Φ

*are two*Φ*-weak upper gradients ofuandFis a closed subset ofX, then the functionv**=*
*gχ**F*+*hχ**X**\**F**is also a*Φ-weak upper gradient of*u.*

**3.4. A characterization of***N*_{Φ}^{1}(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 space*N*_{Φ}^{1}(X).