Musielak–Orlicz–Sobolev spaces on arbitrary metric space

vol. 56, no. 2 (2016), 169–183

Musielak–Orlicz–Sobolev spaces on arbitrary metric space

Noureddine Aissaoui, Youssef Akdim, and My Cherif Hassib

Summary. In this article we define Musielak–Orlicz–Sobolev spaces on arbitrary metric spaces with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a pointwise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds, and that the Lipschitz functions are dense. We develop a capacity theory based on these spaces.

We study basic properties of capacity and several convergence results. As an application, we prove that each Musielak–Orlicz–Sobolev function has a quasi-continuous representative.

Keywords

Metric measure space;

Musielak–Orlicz–Sobolev spaces;

capacity

MSC 2010

46E35; 31B15; 46E30; 42B25;

28A80

Received: 2015-10-25, Accepted: 2016-07-29

1. Introduction

The theory of Sobolev spaces was originally developed in the domains of R ^N using the notion of distributional derivatives. To generalize this theory to metric spaces, P. Hajlasz showed in [8] that a p-integrable function u, 1 < p < ∞, belongs to W ^{1 , p} ( R ^N ) if and only if there exists a non-negative p-integrable function g such that

∣u(x ) − u( y)∣ ⩽ ∣x − y∣ ∣g(x ) + g( y)∣

Noureddine Aissaoui, Ecole Normale Superieure, B.P 5206, Ben Souda, Fez, Morocco (e-mail: dinoaissaoui@gmail.com)

Youssef Akdim, Faculty Poly-Disciplinary of Taza, Laboratory, LSI, Morocco (e-mail: akdimyoussef@yahoo.fr)

My Cherif Hassib, SMBA University, Faculty of Science and Technique, Fez. Laboratory, LSI, Taza, Morocco (e-mail: cherif_hassib@yahoo.fr)

DOI 10.14708/cm.v56i2.825 © 2016 Polish Mathematical Society

for almost every x , y ∈ R ⁿ . This inequality can be stated also in metric measure spaces if

∣x − y∣ is replaced by the distance between the points x and y. This theory was generalized by N. Aissaoui in [4] to the Orlicz–Sobolev spaces and by Petteri Harjulehto, Peter Hasto and Mikko Pere in [9] to the variable exponent spaces. The objective of this paper is to develop this theory in the setting of Musielak–Orlicz–Sobolev spaces.

The average value of u will be denoted by u A =

1 µ(A) ∫

A ud µ = ∫ −

A ud µ.

The paper is organized as follows. In section 2 we recall some important definitions and results on Musielak–Orlicz spaces L φ (X ); in Section 3, we define the Musielak–Orlicz–

–Sobolev space on a metric space, and we show that: it is a Banach space, the Poincaré inequality holds, and Lipschitz-continuous functions are dense. In Section 4, we develop a capacity theory based on this space; we study basic properties of capacity, including monotonicity and countable subadditivity, as well as several convergence results. As an application we prove that each Musielak–Orlicz–Sobolev function has a quasi-continuous representative.

2. Preliminary

Let (X , Σ, µ) be a σ -finite, complete measure space. A real function φ∶ X × [0, ∞) → [0, ∞) is said to be a Musielak–Orlicz function or a generalized N -function, on (X , Σ, µ) if:

(i) φ(x , ⋅) is an N-function for every x ∈ X [convex, increasing, continuous, φ(x , 0) = 0, φ(x , t) > 0 for every t > 0 and ^{φ(x ,t)} _t → 0 as t → 0, ^{φ(x ,t)} _t → ∞ as t → ∞];

(ii) φ(⋅, t) is a measurable function.

The Musielak–Orlicz function φ is said to satisfy the ∆ ₂ -condition if there exists K ⩾ 2 such that

φ(x , 2t) ⩽ K φ(x , t) for all x ∈ X and t ⩾ 0.

The smallest such K is called the ∆ ₂ -constant of φ. When the last inequality holds only for t ⩾ t ₀ for some t ₀ > 0 then φ is said to satisfy the ∆ 2 -condition near infinity. By L ⁰ (X , µ) we denote the space of all µ-measurable functions on X . We define the functional

ρ φ ( f ) = ∫

X φ( y, ∣ f ( y)∣)d µ( y), f ∈ L ⁰ (X , µ).

The set

K φ (X ) = { f ∈ L ⁰ (X , µ) ∶ ρ φ ( f ) < ∞}

is called the Musielak–Orlicz class.

The Musielak–Orlicz space L φ (X ) is the vector space generated by K φ (X ). We have

L φ (X ) = { f ∈ L ⁰ (X , µ) ∶ ρ φ (

∣ f (x )∣

λ

) < ∞ for some λ > 0}.

K φ (X ) is a convex subset of L φ (X ) .

The equality K φ (Ω) = L φ (Ω) holds if and only if φ satisfies the ∆ 2 -condition for all t or for large t, according to whether X has infinite measure or not.

Let

φ ^∗ (x , s) = sup{st − φ(x , t) ∶ t ⩾ 0}, x ∈ X .

Thus φ ^∗ is the Musielak–Orlicz function complementary to φ in the sense of Young with respect to the variable s.

In the space L φ (X ) we define the following two norms:

∥ f ∥ φ = inf {λ > 0 ∶ ∫

X φ (x ,

∣ f (x )∣

λ

) d µ ⩽ 1}, which is called the Luxemburg norm, and the so-called Orlicz norm by

∣∣∣ f ∣∣∣ φ = sup

∥v ∥

⩽1

∫ X ∣ f (x )v(x )∣d µ

where φ ^∗ is the Musielak–Orlicz function complementary to φ. These two norms are equ- ivalent [11].

We say that a sequence of functions u n ∈ L φ (X ) is modular convergent to u ∈ L φ (X ) if there exists a constant k > 0 such that

n→∞ lim ρ φ ( u n − u

k

) = 0.

If φ satisfies the ∆ 2 -condition; then modular convergence coincides with norm conver- gence.

The space L φ (X ) is reflexive if and only if φ and φ ^∗ satisfy the ∆ ₂ -condition for all t or for large t, according to whether X has infinite measure or not.

2.1. Lemma ([7]). Let φ be a Musielak–Orlicz function and f n , f , g ∈ L ⁰ (X , µ) (i) If f n → f a.e. then ρ φ ( f ) ⩽ lim inf _n→∞ ρ φ ( f n ).

(ii) If ∣ f n ∣ ↗ ∣ f ∣ a.e. then ρ φ ( f ) = lim _n→∞ ρ φ ( f n ).

(iii) If f n → f a.e. ∣ f n ∣ ⩽ ∣g∣ a.e, and ρ φ (λ g) < ∞ for every λ > 0, then f n → f in L φ (X ).

2.2. Theorem ([7]). Let φ ∈ N (X , µ)

(i) ∥ f ∥ φ = ∥∣ f ∣∥ φ for all f ∈ L φ (X ).

(ii) If f ∈ L φ (X ), g ∈ L ⁰ (X , µ) and 0 ⩽ ∣g∣ ⩽ ∣ f ∣ a.e. then g ∈ L φ (X ) and ∥g∥ φ ⩽ ∥ f ∥ φ . (iii) If f n → f a.e. then ∥ f ∥ φ ⩽ lim inf _n→∞ ∥ f n ∥ φ .

(iv) If ∣ f n ∣ ↗ ∣ f ∣ a.e. with f n ∈ L φ (X ) and sup _n ∥ f n ∥ φ < ∞ then f ∈ L φ (X ) and ∥ f n ∥ φ ↗

∥ f ∥ φ .

3. Musielak–Orlicz–Sobolev spaces on arbitrary metric space

Let (X , d ) be a metric space with finite diameter (diam(X) = sup _{x , y∈X} d (x , y) < ∞) and equipped with a finite, positive Borel regular outer measure µ. The triplet(X , d , µ) will be fixed in the sequel and will be denoted by X .

Let u∶ X → [−∞, ∞] be a µ-measurable functions defined on X. We denote by D(u) the set of all µ-mesurable functions g ∶ X → [0, ∞] such that

∣u(x ) − u( y)∣ ⩽ d (x , y)(g(x ) + g( y)) (1) for every x , y ∈ X ∖ F , x ≠ y, with µ(F ) = 0. The set F is called the exceptional set for g, and g is called the Hajlasz gradient of u.

Note that the right-hand side of (1) is always defined for x ≠ y. For the points x , y ∈ X, x ≠ y such that the left-hand side of (1) is undefined, we may assume that the left-hand side is ∞.

3.1. Definition. Let φ be a Musielak–Orlicz function. The Dirichlet–Musielak–Orlicz spa- ce L ¹ _{φ, µ} (X ) is the space of all µ-measurable functions u such that D(u) ∩ L φ (X ) ≠ ∅. This space is equipped with the semi-norm

∥u∥ L

φ , µ

( X ) = inf {∥g∥ φ ∶ g ∈ D(u) ∩ L φ (X )}.

The Musielak–Orlicz–Sobolev space on an arbitrary metric space X is M ¹ _{φ, µ} (X ) = L φ (X )∩

L ¹ _{φ, µ} (X ), equipped with the norm

∥u∥ M

φ , µ

( X ) = ∥u∥ φ + ∥u∥ L

φ , µ

( X ) .

3.2. Lemma. Let φ be a Musielak–Orlicz function such that φ and φ ^∗ satisfy the ∆ ₂ -condition.

For every u ∈ M _{φ, µ} ¹ (X ) there is a unique g ∈ D(u) ∩ L φ (X ) denoted by g u such that

∥g u ∥ φ = ∥u∥ L

φ , µ

( X ) . Proof. Let (g i ) ⊂ D(u) ∩ L φ (X ) be a sequence such that

∥u∥ L

φ , µ

( X ) = lim

i→∞ ∥g i ∥ φ .

Since L φ (X ) is reflexive, there is a subsequence denoted again by (g i ) and a function g u ∈ L φ (X ) such that g i → g u weakly in L φ (X ). By Mazur’s lemma there exist convex combinations f k = ∑

m

i=k α k

g i such that f k → g u strongly in L φ (X ), where α k

⩾ 0 and

∑

m

i=k α k

= 1.

Since every g i ∈ D(u) ∩ L φ (X ), we have

∣u(x ) − u( y)∣ =

i=m

∑

i=k

α k

∣u(x ) − u( y)∣

⩽

i=m

∑

i=k

α k

d (x ; y)(g i (x ) + g i ( y))

= d (x ; y)( f k (x ) + f k ( y))

for µ-almost all x , y ∈ X. Therefore f k ∈ D(u) ∩ L φ (X ) for all k. On the other hand, there exists a subsequence of ( f k ) which converges to g u pointwise µ-almost everywhere in X , thus g u ∈ D(u) ∩ L φ (X ).

Let ε > 0. There is k ∈ N such that

∥g i ∥ φ ⩽ ∥u∥ L

φ , µ

( X ) + ε, i ⩾ k .

For this k we have

∥ f k ∥ φ ⩽

m

∑

i=k

α k

∥g i ∥ φ . Therefore

∥ f k ∥ φ ⩽ ∥u∥ L

φ , µ

( X ) + ε.

Thus

∥g u ∥ φ = ∥u∥ L

φ , µ

( X ) .

Now suppose that there exist two minimal g 1 and g 2 with g 1 ≠ g 2 , so that ∥g 1 − g 2 ∥ φ > 0.

Let G ₁ =

g

∥ g

∥

and G ₂ =

g

∥ g

∥

. By the uniform convexity of the norm we have

∥G ₁ + G ₂ ∥ φ < 2;

then

∥ ¹

2 g 1 + ¹

2 g 2 ∥ _φ < ∥g 1 ∥ φ

which is the contradiction since ( ¹

2 g ₁ + ¹

2 g ₂ ) ∈ D(u) ∩ L φ (X ).

With simple verification, we obtain the following lemma.

3.3. Lemma. Let g 1 ∈ D(u 1 ), g 2 ∈ D(u 2 ) and α , β ∈ R. If g ⩾ ∣α∣g ¹ + ∣β∣g 2 µ-a.e., then g ∈ D(αu 1 + βu 2 ).

3.4. Lemma. Let φ a Musielak–Orlicz function and (u n ) and (g n ) sequences of functions such that

g n ∈ D(u n ), n ∈ N.

If u n → u in L φ (X ) and g n → g in L φ (X ), then g ∈ D(u).

Proof. There are subsequences of (u n ) and (g n ) which we denote again by (u n ) and (g n ) such that u n → u and g n → g µ-a.e. For all n ∈ N, let F ⁿ be the exceptional set for g n , H a set of measure zero such that u n → u and g n → g on H ^c , and F = H ∪ (⋃ n F n ). We have µ(F ) = 0 and

∣u(x ) − u( y)∣ ⩽ d (x , y)∣g(x ) + g( y)∣, x , y ∈ X ∖ F . Therefore g ∈ D(u).

3.5. Theorem. Let φ be a Musielak–Orlicz function. Then (M _{φ, µ} ¹ (X ), ∥ ⋅ ∥ M

φ , µ

) is a Banach

space.

Proof. It is clear that M _{φ, µ} ¹ (X ) is a vector space.

Let (u n ) be a Cauchy sequence in M ¹ _{φ, µ} (X ). Then (u n ) is a Cauchy sequence in L φ (X ), and hence there exists a function u ∈ L φ (X ) such that u n → u in L φ (X ). We can choose a subsequence denoted again (u n ) such that ∥u _n+1 − u n ∥ M

φ , µ

< 2 ⁻ⁿ and u n → u

µ-a.e. in X . There exist non-negative functions g n ∈ L φ (X ) such that ∥g n ∥ φ < 2 ⁻ⁿ and

∣(u _n+1 − u n )(x ) − (u _n+1 − u n )( y)∣ ⩽ d (x , y)(g n (x ) + g n ( y)) µ-a.e. in X . By adding the last inequalities, we have for n > m,

∣(u n − u m )(x ) − (u n − u m )( y)∣ ⩽ d (x , y)(

∞

∑

k=m

g k (x ) +

∞

∑

k=m

g k ( y)) µ-a.e. in X . Letting n → ∞ yields

∣(u − u m )(x ) − (u − u m )( y)∣ ⩽ d (x , y)(

∞

∑

k=m

g k (x ) +

∞

∑

k=m

g k ( y)) µ-a.e. in X . Since

∥

∞

∑

k=m

g k ∥

φ

⩽

∞

∑

k=m

2 ^−k = 2 ^−m+1

the previous inequality implies that (u − u m ) ∈ M _{φ, µ} ¹ (X ) and therefore u ∈ M _{φ, µ} ¹ (X ).

On the other hand, we have

∥u − u m ∥ L

φ , µ

( X ) ⩽ ∥

∞

∑

k=m

g k ∥

φ

⩽ 2 ^−m+1 ,

then u m → u in M ¹ _{φ, µ} (X ).

3.6. Lemma. Let φ be a Musielak–Orlicz function. The function u belongs to M _{φ, µ} ¹ (X ) if and only if u ∈ L φ (X ) and there exist sequences (u n ) ⊂ L φ (X ) and (g n ) ⊂ D(u n ) ∩ L φ (X ) such that u n → u µ-a.e. and g n → g µ-a.e. for some g ∈ L φ (X ).

Proof. The same as in [10, Lemma 2.5].

3.7. Lemma. Let φ be a Musielak–Orlicz function and u 1 , u 1 ∈ M ¹ _{φ, µ} (X ). If g 1 ∈ D(u 1 ) and g 2 ∈ D(u 2 ), then:

(i)

u = max(u 1 , u 2 ) ∈ M ¹ _{φ, µ} (X ) and g = max(g 1 , g 2 ) ∈ D(u) ∩ L φ (X ).

(ii)

v = min(u 1 , u 2 ) ∈ M ¹ _{φ, µ} (X ) and h = max(g 1 , g 2 ) ∈ D(v) ∩ L φ (X ).

Proof. We only prove the case (i), as the proof of (ii) is similar.

Let F ₁ and F ₂ the exceptional sets for g ₁ and g ₂ , respectively. Clearly u, g ∈ L φ (X ). It remains to prove that g ∈ D(u). Let G = {x ∈ X ∖ (F ₁ ∪ F 2 ) ∶ u 1 (x ) ⩾ u 2 (x )}. If x , y ∈ G , then

∣u(x ) − u( y)∣ = ∣u ₁ (x ) − u ₁ ( y)∣ ⩽ d (x , y)(g ₁ (x ) + g ₁ ( y)).

Analogously, for x , y ∈ X ∖ G, we obtain ∣u(x ) − u( y)∣ ⩽ d (x , y)(g ₂ (x ) + g 2 ( y)).

For the remaining cases, let x ∈ G and y ∈ X ∖ G. If u ₁ (x ) ⩾ u ₂ ( y), then

∣u(x ) − u( y)∣ = u ₁ (x ) − u ₂ ( y) ⩽ u ₁ (x ) − u ₁ ( y) ⩽ d (x , y)(g ₁ (x ) + g ₁ ( y)).

If u ₁ (x ) < u ₂ ( y), then

∣u(x ) − u( y)∣ = u ₂ ( y) − u ₁ (x ) ⩽ u ₂ ( y) − u ₂ (x ) ⩽ d (x , y)(g ₂ (x ) + g ₂ ( y)).

The case x ∈ X ∖ G and y ∈ G follows by symmetry and hence

∣u(x ) − u( y)∣ ⩽ d (x , y)(g(x ) + g( y)) for all x , y ∈ X ∖ (F ₁ ∪ F 2 ), with µ(F 1 ∪ F 2 ) = 0.

3.8. Theorem (The Poincare inequality). Let φ be a Musielak–Orlicz function such the function x ↦ 1 belongs to L φ (X ) ∩ L φ

(X ). If u ∈ M _{φ, µ} ¹ (X ), then for every g ∈ D(u) ∩ L φ (X )

∥u − u X ∥ φ ⩽ C(φ) diam(X )∥g∥ φ .

Proof. Integrating the inequality ∣u(x ) − u( y)∣ ⩽ d (x , y)(g(x ) + g( y)) over y, we obtain

∣u(x ) − − ∫

X

u( y)d µ( y)∣ ⩽ ∫ −

X

∣u(x ) − u( y)∣d µ( y) ⩽ diam(X) (g(x ) + ∫ −

X

g( y)d µ( y)) . Applying Hölder’s inequality, we have

∣u(x ) − u X ∣ ⩽ diam(X)(g(x ) + C µ(X)

∥1∥ φ

∥g∥ φ ).

And finally by (ii) of Theorem 2.2,

∥u − u X ∥ φ ⩽ diam(X )(∥g∥ φ + C µ(X)

∥1∥ φ

∥1∥ φ ∥g∥ φ ) ⩽ C(φ) diam(X )∥g∥ φ .

3.9. Theorem. Let φ a Musielak–Orlicz function satisfying the ∆ 2 -condition. For every u ∈ M ¹ _{φ, µ} (X ) and every ε > 0 there exists a Lipschitz function h ∈ M _{φ, µ} ¹ (X ) such that

(i) µ({x ∈ X ∶ u(x ) ≠ h(x )}) ⩽ ε.

(ii) ∥u − h∥ M

φ , µ

( X ) ⩽ ε.

Proof. We fix u ∈ M _{φ, µ} ¹ (X ) and denote by g ∈ L φ (X ) a Hajlasz gradient of u. We write E n = {x ∈ X ∶ ∣u(x )∣ ⩽ n and g(x ) ⩽ n}.

We have u, g ∈ L φ (X ), and φ satisfies the ∆ ₂ -condition, hence u, g ∈ K φ (X ). Furthermore, for all y ∈ E _n ^C

φ( y, n) < φ( y, ∣u( y)∣ + g( y)).

Since lim _n→∞ ^φ(y,n) _n = ∞, we have that for all A > 0 there exists n 0 ∈ N such that for all n ⩾ n ₀ the following inequality holds ^φ(y,n) _n > A. Thus

µ(E _n ^c ) ⩽ C

n ∫ _E

C n

φ( y, n)d µ( y) ⩽ C

n ∫ _X φ( y, ∣u( y)∣ + g( y))d µ( y), so that

n→∞ lim µ(E ^c _n ) = 0.

On the other hand, we have 0 ⩽ χ E

n

φ( y, n) ⩽ χ E

n

φ( y, ∣u( y)∣ + g( y)), hence

n→∞ lim χ E

n

φ( y, n) = 0, and since χ E

n

φ( y, n) < φ( y, ∣u( y)∣ + g( y)), by dominated convergence theorem we have

n→∞ lim ∫

E

n

φ( y, n)d µ( y) = 0.

The restriction u∣ E

is Lipschitz with the constant 2n. We can extend u∣ E

by McShane extention [11] to all of X as a Lipschitz function by defining

¯

u(x ) = inf

y∈E

{u∣ E

( y) + 2n dist(x , y)}.

We put

u n = (sign ¯ u) min (∣ ¯ u∣, n).

It is clear that u n is Lipschitz with constant 2n, u∣ E

= u n ∣ E

, ∣u n ∣ ⩽ n, and for all n ⩾ 1 µ({x ∶ u(x ) ≠ u n (x )}) ⩽ µ(E _n ^c ) → 0 as n → ∞.

On the other hand, we have

∫ X

φ( y, ∣u − u n ∣( y))d µ( y) = ∫

E

n

φ( y, ∣u − u n ∣( y))d µ( y)

⩽ ∫ E

n

φ( y, ∣u∣( y) + ∣u n ∣( y))d µ( y).

Since φ( y, ⋅) is convex and φ satisfies the ∆ 2 -condition, then

∫ X

φ( y, ∣u − u n ∣( y))d µ( y) ⩽ C φ

2 [∫

E

n

φ( y, ∣u∣( y))d µ( y) + ∫

E

n

φ( y, ∣u n ∣( y))d µ( y)], thus

∫ X

φ( y, ∣u − u n ∣( y))d µ( y) ⩽ C φ

2 [∫

E

n

φ( y, ∣u∣( y))d µ( y) + ∫

E

n

φ( y, n)d µ( y)].

Therefore ∫ X φ( y, ∣u − u n ∣( y))d µ( y) → 0 as n → ∞.

Since φ satisfies the ∆ 2 -condition, ∥u − u n ∥ φ → 0 as n → ∞. It remains to estimate the gradient. Let

g n =

⎧ ⎪

⎪

⎨

⎪ ⎪

⎩

0 for x ∈ E n , g(x ) + 3n for x ∈ E _n ^c . It is easy to check that

∣(u − u n )(x ) − (u − u n )( y)∣ ⩽ d (x , y)(g n (x ) + g n ( y)) for µ-almost all x , y ∈ X . We have ∫ X φ( y, g n ( y))d µ( y) = ∫ E

n

φ( y, g n ( y))d µ( y). Since φ( y, ⋅) is convex and φ satisfies the ∆ 2 -condition,

∫ E

n

φ( y, g n ( y))d µ( y) ⩽ C φ

2 [∫

E

n

φ( y, g( y))d µ( y) + ∫

E

n

φ( y, 3n)d µ( y)],

then ∥g n ∥ φ → 0 as n → ∞.

4. Capacity on Musielak–Orlicz–Sobolev space on metric space

4.1. Definition. Let T be a Borel σ -algebra of subsets of X and C∶ T → [0, ∞] a function.

(i) C is called a capacity if the following axioms are satisfied:

a) C(∅) = 0.

b) E ⊂ F ⇒ C(E) ⩽ C(F ) for all E and F in T . c) for all sequences (E n ) ⊂ T

C(⋃

n

E n ) ⩽ ∑

n

C(E n ).

(ii) C is called an outer capacity if for all E ∈ T :

C(E) = inf {C(O) ∶ O ⊃ E , O open}.

(iii) A property that holds true except perhaps on a set of capacity zero is said to be true C-quasi everywhere (abbreviated C-q.e).

(iv) Let f and ( f n ) be real-valued functions finite C-q.e. We say that ( f n ) converges to f in C-capacity if for any ε > 0

n→∞ lim C({x ∶ ∣ f n (x ) − f (x )∣ > ε}) = 0.

(v) Let f and ( f n ) be real-valued functions finite C-q.e. We say that ( f n ) converges to f C-quasi uniformly (abbreviated C-q.u.) if for any ε > 0 there exists X ∈ T such that

C(X) < ε and ( f n ) converges to f uniformly on X ^c .

4.2. Definition. Let φ be a Musielak–Orlicz function. The Sobolev φ-capacity of the set E ⊂ X is defined by

C φ, µ (E) = inf

u∈A

(E )

∥u∥ M

φ , µ

, where

A φ (E) = {u ∈ M ¹ _{φ, µ} (X ) ∶ u ⩾ 1 on an open set containing E and u ⩾ 0}.

If A φ (E) = ∅ we set C φ, µ (E) = ∞. The functions belonging to A φ (E) are called the admissible functions for E.

4.3. Remark. In the defition of C φ, µ (E) we may restrict ourselves to those admissible functions u for which 0 ⩽ u ⩽ 1.

Proof. If A ^′ _φ (E) = {u ∈ A φ (E) ∶ 0 ⩽ u ⩽ 1}, then A ^′ _φ (E) ⊂ A φ (E) implies C _{φ, µ} (E) ⩽ inf

u∈A

(E )

∥u∥ M

φ , µ

.

For the reverse inequality, let ε > 0 and take u ∈ A φ (E) such that ∥u∥ M

φ , µ

< C φ, µ (E) + ε.

Then by Lemma 3.7, we have v = max(0, min(u, 1)) belongs to A ^′ _φ (E). Therefore inf

ω∈A

(E )

∥ω∥ M

φ , µ

⩽ ∥v∥ M

φ , µ

⩽ C _{φ, µ} (E) + ε.

Letting ε → 0, we obtain

inf

ω∈A

(E )

∥ω∥ M

φ , µ

⩽ C φ, µ (E).

This completes the proof.

4.4. Theorem. Let φ be a Musielak–Orlicz function. C φ, µ is an outer capacity.

Proof. It is obvious that C _{φ, µ} (∅) = 0 and C _{φ, µ} (E) ⩽ C _{φ, µ} (F ) if E ⊂ F . Let (E i ) ⊂ X be such that ∑ i C _{φ, µ} (E i ) < ∞. Then C _{φ, µ} (E i ) < ∞ for each i ∈ N.

Therefore for every ε > 0 there exist u i ∈ A φ (E i ) and g u

∈ D(u i ) ∩ L φ (X ) so that

∥u i ∥ φ + ∥g u

∥ φ ⩽ C φ, µ (E i ) + ε2 ^{− i} .

We show that v = sup _i u i is an admissible function for ⋃ ^∞ _i=1 E i and g = sup _i g u

∈ D(v) ∩ L φ (X ). Let v k = max _{1⩽i ⩽k} u i . By Lemma 3.7 the function g v

= max _{1⩽i ⩽k} g u

belongs to D(v k ) ∩ L φ (X ). Since v k → v µ-a.e. and g

k

→ g µ-a.e., Lemma 3.6 gives v ∈ M ¹ _{φ, µ} . Clearly v ⩾ 1 in a neighbourhood of ⋃ ^∞ _i=1 E i . Then

C _{φ, µ} (

∞

⋃

i=1

E i ) ⩽ ∥v∥ M

φ , µ

⩽

∞

∑

i=1

(∥u i ∥ φ + ∥g u

∥ φ ) ⩽

∞

∑

i=1

C _{φ, µ} (E i ) + ε.

The claim follows by letting ε → 0. Hence C φ, µ is a capacity.

It remains to prove that C φ, µ is outer. Indeed,

C _{φ, µ} (E) ⩽ inf {C φ, µ (O) ∶ E ⊂ O , O open}.

For the reverse inequality, let ε > 0 and u ∈ A φ (E) such that ∥u∥ M

φ , µ

⩽ C _{φ, µ} (E) + ε. Since

u ∈ A φ (E), there is an open set O, E ⊂ O, such that u ⩾ 1 on O. Then C _{φ, µ} (O) ⩽ ∥u∥ M

φ , µ

⩽ C φ, µ (E) + ε. The claim follows by letting ε → 0.

4.5. Theorem. If (K n ) is a decreasing sequence of compact sets and K = ⋂ n K n , then

n→∞ lim C _{φ, µ} (K n ) = C _{φ, µ} (K ).

Proof. First, we observe that lim _n→∞ C _{φ, µ} (K n ) ⩾ C φ, µ (K ). Let O be an open set conta- ining K . By the compactness of K , K i ⊂ O for all sufficiently large i. Therefore

n→∞ lim C φ, µ (K n ) ⩽ C φ, µ (O)

and since C φ, µ is an outer capacity, we obtain the claim by taking infimum over all open

sets O containing K .

4.6. Theorem. Let E ⊂ X. If there exists f ∈ M ¹ _{φ, µ} (X ) such that f = ∞ on an open set containing E, then C _{φ, µ} (E) = 0.

Proof. If there exists f ∈ M _{φ, µ} ¹ (X ) such that f = ∞ on an open set O containing E, then f ⩾ α on O for all α > 0. Therefore, C _{φ, µ} (E) ⩽ α ⁻¹ ∥ f ∥ M

φ , µ

for every α > 0. Letting α → ∞,

we obtain C _{φ, µ} (E) = 0.

4.7. Theorem. Consider the following propositions:

(i) f n → f in M ¹ _{φ, µ} (X ) strongly.

(ii) f n → f in C _{φ, µ} -capacity.

(iii) There is a subsequence ( f n

) such that f n

→ f C _{φ, µ} -q.u.

(iv) f n

→ f C φ, µ -q.e.

We have (i) ⇒ (ii) ⇒ (iii) ⇒ (iv).

Proof. Let us show that (i) ⇒ (ii). By Theorem 4.6, we have f and f n are finite C φ, µ -q.e.

for every n. Let ε > 0. Then

C φ, µ ({x ∶ ∣ f n − f ∣(x ) > ε}) ⩽ 1 ε

∥ f − f n ∥ M

φ , µ

.

For (ii) ⇒ (iii), let ε > 0. Then there exists a subsequence ( f n

) of ( f n ) such that C _{φ, µ} ({x ∶ ∣ f n

− f ∣(x ) > 2 ^{− j} }) < ε2 ^{− j} .

We put

E j = {x ∶ ∣ f n

− f ∣(x ) > 2 ^{− j} } and G m = ⋃

j⩾m

E j . We have C φ, µ (G m ) ⩽ ∑ _j⩾m ε2 ^{− j} < ε.

On the other hand for any x ∈ G ^c _m and j ⩾ m

∣ f n

− f ∣(x ) ⩽ 2 ^{− j} . Thus f n

→ f C φ, µ -q.u.

Finally, let us show that (iii) ⇒ (iv). We have that for all j ∈ N there exists X ^j such that C _{φ, µ} (X j ) ⩽ ¹ _j and f n

→ f on X ^c _j . We put X = ⋂ j X j , then C _{φ, µ} (X ) = 0 and f n

→ f on X ^c .

4.8. Theorem. Let φ be a Musielak–Orlicz function such that φ and φ ^∗ satisfy the ∆ ₂ -condition.

If f n , f ∈ L φ (X ) are such that f n → f weakly in L φ (X ) then

lim inf f n ⩽ f ⩽ lim sup f n C _{φ, µ} -q.e.

Proof. The space (L φ (X ), ∥ ⋅ ∥) is reflexive. By the Banach–Saks theorem, there is a subse- quence denoted again ( f n ) such that the sequence g n = _n ¹ ∑

n

i=1 f i converges to f strongly in L φ (X ). By Theorem 4.7, there is a subsequence of (g n ) denoted again (g n ) such that

n→∞ lim g n = f C _{φ, µ} -q.e.

On the other hand,

lim inf f n ⩽ lim

n→∞ g n . Therefore,

lim inf f n ⩽ f C φ, µ -q.e.

For the second inequality, it suffices to replace f n by (− f n ) in the first inequality.

4.9. Theorem. Let φ be a Musielak–Orlicz function such that φ and φ ^∗ satisfy the ∆ 2 -con- dition. Let (O n ) be an increasing sequence of open subsets of X and O = ⋃ n O n , then

n→∞ lim C φ, µ (O n ) = C φ, µ (O).

Proof. We have lim _n→∞ C _{φ, µ} (O n ) ⩽ C _{φ, µ} (O). For the reverse inequality, if

n→∞ lim C _{φ, µ} (O n ) = ∞ there is nothing to show.

Assuming that lim _n→∞ C φ, µ (O n ) < ∞, we have that for any n ∈ N there exists f ⁿ ∈ M ¹ _{φ, µ} (X ) such that

f n ⩾ 1 on O n and ∥ f n ∥ M

φ , µ

( X ) ⩽ C φ, µ (O n ) + 1 n . Then

∥ f n ∥ φ + ∥g n ∥ φ ⩽ C φ, µ (O n ) + 1 n .

Where g n ∈ D( f n ) ∩ L φ (X ). Now ( f n ) and (g n ) are bounded sequences in L φ (X ), hence there exist subsequences, which we denote again by ( f n ) and (g n ), such that f n → f and g n → g weakly in L φ (X ). Thus

∥ f ∥ φ ⩽ lim inf

n ∥ f n ∥ φ and ∥g∥ φ ⩽ lim inf

n ∥g n ∥ φ . Hence

∥ f ∥ M

φ , µ

( X ) ⩽ lim

n→∞ C _{φ, µ} (O n ).

On the other hand, by Theorem 4.8, we have for any n ∈ N

f ⩾ 1 on O n C _{φ, µ} -q.e.

Therefore, f ⩾ 1 on O C φ, µ -q.e. Let Y be a subset of O where f ⩾ 1, then C φ, µ (O) = C _{φ, µ} (Y ). Thus

C φ, µ (O) ⩽ lim

n→∞ C φ, µ (O n ).

4.10. Theorem. Let φ be a Musielak–Orlicz function such that the function x ↦ 1 belongs to L φ

(X ). We have that there exists α > 0 such that for any E ⊂ X

µ(E) ⩽ αC _{φ, µ} (E).

Proof. Let E ⊂ X and f ∈ A φ (E). Then f ⩾ 1 on an open set O containing E and f ⩾ 0.

We have

µ(E) ⩽ µ(O) ⩽ ∫

O f (x )d µ(x ) ⩽ ∫

X

f (x )d µ(x ).

By Hölder’s inequality,

∫ X f (x )d µ(x ) ⩽ 2∥ f ∥ φ ∥1∥ φ

. Therefore

µ(E) ⩽ 2∥1∥ φ

∥ f ∥ M

φ , µ

( X ) . We obtain the claim by taking infimum over f ∈ A(E).

4.11. Corollary. Let φ be a Musielak–Orlicz function such the function x ↦ 1 belongs to L φ

(X ). If ( f n ) is a sequence which converges to f in M ¹ _{φ, µ} (X ), then there exists a subsequ- ence of ( f n ) which converges to f µ-a.e.

Proof. It is an immediate consequence of Theorem 4.7 and Theorem 4.10.

4.12. Theorem. Let φ be a Musielak–Orlicz function satisfying the ∆ 2 -condition and such the function x ↦ 1 belongs to L φ

(X ). For each f ∈ M ¹ _{φ, µ} (X ), there is a C φ, µ -quasi continuous function g ∈ M ¹ _{φ, µ} (X ) such that f = g µ-a.e.

Proof. Let f ∈ M ¹ _{φ, µ} (X ). By Theorem 3.9, there exists a sequence ( f n ) of Lipschitz func- tions such that f n → f in M ¹ _{φ, µ} (X ). By Theorem 4.7, there exists a subsequence of ( f n ) de- noted again by ( f n ) such that f n → f C φ -q.u. The claim now follows by Theorem 4.10.

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