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SOUTHWEST JOURNAL

OF

PURE AND APPLIED MATHEMATICS

EDITOR IN CHIEF: IOANNIS ARGYROS MANAGING EDITOR: MOHAMMAD TABATABAI

TECHNICAL EDITOR: GARY HUCKABAY EDITORIAL BOARD

I. ARGYROS E. C ˇATINAS¸

D. CHEN S¸ COBZAS¸

J. EZQUERRO J. GUTIERREZ

M. HERNADEZ D. JANKOVIC

T. McKELLIPS C. MUST ˇATA

I. P ˇAV ˇALOIU F. SZIDAROVSZKY

M. TABATABAI B. VASIGH

G. ANASTASSIOU

CAMERON UNIVERSITY

DEPARTMENT OF MATHEMATICAL SCIENCES LAWTON, OK 73505, U.S.A

Southwest Journal of

Pure and Applied Mathematics Editor in Chief: Ioannis Argyros Managing Editor: Mohammad Tabatabai

Technical Editor: Gary Huckabay Editorial Board

G. Anastassiou ganastss@memphis.edu I. Argyros ioannisa@cameron.edu E. Cˇatina¸s ecatinas@ictp-acad.ubbcluj.ro

D. Chen dchen@uafsysb.uark.edu

ˇ

S. Cobza¸s scobzas@math.ubbcluj.ro J. Ezquerro jezquer@siur.unirioja.es J. Gutierrez jmguti@siur.unirioja.es M. Hernadez mahernan@dmc.unirioja.es D. Jankovic draganj@cameron.edu T. McKellips terralm@cameron.edu

C. Mustˇata cmustata@ictp-acad.ubbcluj.ro I. Pˇavˇaloiu pavaloiu@ictp-acad.ubbcluj.ro F. Szidarovszky szidar@sie.arizona.edu M. Tabatabai mohammadt@cameron.edu B. Vasigh basighb@cts.db.erau.edu

Editorial Office Publication Program

SWJPAM 2 issues per year

Cameron University July, December Department of Mathematical Sciences

Lawton, OK 73505, U.S.A.

Information for Authors

Scope. SWJPAM is an electronic journal devoted to all aspects of Pure and Applied mathematics, and related topics. Authoritative expository and survey articles on subjects of special interest are also welcomed.

Since SWJPAM aims to serve not only the specialists, but also the user of math-ematics (promoting, in such a way, a strong interaction between them), particu-lar emphasis is placed on applications. Thus, SWJPAM publishes papers which, although not directly contributing to mathematics, contain, however, significant results (from any field) obtained by making an essential and well stressed use of the methods and results of mathematics.

OF

PURE AND APPLIED MATHEMATICS ISSUE 1, JULY – DECEMBER 2004

CONTENTS

J. M. Guti´errez, M. A. Hern´andez On the approximate solution of some Fredholm integral 1–9 and M. A. Salanova equations by Newton’s method

Noureddine A¨ıssaoui Orlicz-Sobolev Spaces with Zero Boundary Values on 10–32 Metric Spaces

Anthony Gamst An Interacting Particles Process for Burgers Equation 33–47 on the Circle

Ioannis K. Argyros An Iterative Method for Computing Zeros of Operators 47–53 Satisfying Autonomous Differential Equations

Michael Deutch Proving Matrix Equations 54–56

Y. Abu Muhanna and Yusuf Abu Absolutely Continuous Measures and Compact 57–67 Muhanna Composition Operator On Spaces of Cauchy Transforms

Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html

ISBN 1083-0464

Issue 1 July 2004, pp. 1 – 9

Submitted: October 21, 2003. Published: July 1, 2004

ON THE APPROXIMATE SOLUTION OF SOME FREDHOLM INTEGRAL EQUATIONS BY NEWTON’S

METHOD

J. M. GUTI´ERREZ, M. A. HERN ´ANDEZ AND M. A. SALANOVA Abstract. _{The aim of this paper is to apply Newton’s method to} solve a kind of nonlinear integral equations of Fredholm type. The study follows two directions: firstly we give a theoretical result on existence and uniqueness of solution. Secondly we illustrate with an example the technique for constructing the functional sequence that approaches the solution.

A.M.S. (MOS) Subject Classification Codes. 45B05,47H15, 65J15

Key Words and Phrases. Fredholm integral equations, Newton’s method.

1. Introduction

In this paper we give an existence and uniqueness of solution result for a nonlinear integral equation of Fredholm type:

(1) φ(x) =f(x) +λ

Z b a

K(x, t)φ(t)pdt, x∈[a, b], p≥2,

where λ is a real number, the kernel K(x, t) is a continuous function in [a, b]_×[a, b] andf(x) is a given continuous function defined in [a, b].

University of La Rioja, Department of Mathematics and Computation E-mail: jmguti@dmc.unirioja.es,mahernan@dmc.unirioja.es

C/ Luis de Ulloa s/n. 26006 Logro˜no, Spain. masalano@dmc.unirioja.es,

Supported in part by a grant of the Spanish Ministry of Science and Technology (ref. BFM2002-00222) and a grant of the University of La Rioja (ref. API-02-98). Copyright c2004 by Cameron University

There exist various results about Fredholm integral equations of sec-ond kind

φ(x) =f(x) +λ

Z b a

K(x, t, φ(t))dt, x∈[a, b]

when the kernel K(x, t, φ(t)) is linear in φ or it is of Lipschitz type in the third component. These two points have been considered, for instance, in [7] or [3] respectively. However the above equation (1) does not satisfy either of these two conditions.

In [3] we can also find a particular case of (1), for f(x) = 0 and

K(x, t) a degenerate kernel. In this paper we study the general case. The technique will consist in writing equation (1) in the form:

(2) F(φ) = 0,

where F : Ω_⊆X _→Y is a nonlinear operator defined by

F(φ)(x) =φ(x)−f(x)−λ

Z b a

K(x, t)φ(t)pdt, p≥2,

and X = Y = C([a, b]) is the space of continuous functions on the interval [a, b], equipped with the max-norm

kφ_k= max

x∈[0,1]|φ(x)|, φ∈X.

In addition, Ω = X if p ∈ N, p ≥ 2, and when it will be necessary, Ω = C+([a, b]) = {φ ∈ C([a, b]);φ(t) > 0, t ∈ [a, b]} for p ∈ R, with

p >2.

The aim of this paper is to apply Newton’s method to equation (2) in order to obtain a result on the existence and unicity of solution for such equation. This idea has been considered previously in different situations [1], [2], [4], [6].

At it is well known, Newton’s iteration is defined by (3) φn+1 =φn−ΓnF(φn), n ≥0,

where Γn is the inverse of the linear operatorFφ′n. Notice that for each

φ_∈Ω, the first derivative F′

φ is a linear operator defined from X to Y by the following formula:

(4)

F_φ′[ψ](x) =ψ(x)₋λp

Z b a

K(x, t)φ(t)p−1ψ(t)dt, x_∈[a, b], ψ _∈X.

In the second section we establish two main theorems, one about the existence of solution for (2) and other about the unicity of solution for the same equation. In the third section we illustrate these theoretical

results with an example. For this particular case, we construct some iterates of Newton’s sequence.

2. The main result

Let us denote N = max x∈[a,b]

Z b a

|K(x, t)|dt. Let φ0 be a function in Ω such that Γ0 = [Fφ′0]

−1 _{exists and kΓ}

0F(φ0)k ≤ η. We consider the following auxiliary scalar function

(5) f(t) = 2(η−t) +M(kφ0k+t)p−2[(p−1)ηt−2(η−t)(kφ0k+t)], where, M = _|λ_|pN. Let us note that if p _∈ N, with p _≥ 2, f(t) is a polynomial of degree p₋2. Firstly, we establish the following two technical lemmas:

Lemma 2.1.

_{Let us assume that the equation f(t) = 0 has at least} a positive real solution and let us denote by R the smaller one. Then we have the following relations:

i) η < R.

ii) a=M(_kφ0k+R)p−1 <1. iii) If we denoteb = (p−1)η

2(kφ0k+R)

andh(t) = 1

1−t, then, abh(a)<

1.

iv) R= η

1₋abh(a).

Proof: First, notice that iv) follows from the relation f(R) = 0. So, as R > 0, we deduce that abh(a) < 1, and iii) holds. Moreover, 1>1₋abh(a)>0, then 1< 1

1₋abh(a), so η < R, and i)also holds. To prove ii), we consider the relation f(R) = 0 that can be written in the form:

2(η₋R)1₋M(_kφ0k+R)p−1=−Mη(p−1)R(kφ0k+R)p−2 <0. As η₋R <0, 1₋M(_kφ0k+R)p−1 = 1−a >0, and therefore a <1.

Let us denote B(φ0, R) = {φ ∈ X; kφ−φ0k < R} and B(φ0, R) =

{φ_∈X; _kφ₋φ0k ≤R}.

Lemma 2.2.

_{If B(φ0, R)⊆Ω, the following conditions hold} i) For allφ ∈B(φ0, R) there exists [Fφ′]−1 and k[Fφ′]−1k ≤h(a).

ii) If φn, φn−1 ∈B(φ0, R), then

kF(φn)k ≤

(p₋1)a

2(kφ0k+R)k

φn−φn−1k2.

Proof: To prove i)we apply the Banach lemma on invertible oper-ators [5]. Taking into account

(I₋F_φ′)ψ(x) =λp

Z b a

K(x, t)φ(t)p−1ψ(t)dt,

then

kI−F_φ′k ≤ |λ|pNkφkp−1 _≤M(kφ

0k+R)p−1 =a <1, therefore, there exists [F′

φ]−1 and k[Fφ′]−1k ≤ 1

1₋a =h(a).

To prove ii), using Taylor’s formula, we have

F(φn)(x) = Z 1

0 [F_φ′_n

−1+s(φn−φn−1)−F ′

φn−1](φn−φn−1)(x)ds

=₋λp

Z 1 0

Z b a

K(x, t)ρn(s, t)p−1−φn−1(t)p−1(φn(t)−φn−1(t))dt ds,

−λp

Z 1 0

Z b a

K(x, t) "_p−2

X j=0

ρn(s, t)p−2−jφn−1(t)j

#

[φn(t)−φn−1(t)]2s dt ds,

where ρn(s, t) = φn−1(t) +s(φn −φn−1) and we have considered the equality

xp−1₋yp−1 = p−2 X j=0

xp−2−jyj

!

(x₋y), x, y _∈R.

As φn−1, φn ∈ B(φ0, R), for each s ∈ [0,1], ρn(s,·) ∈ B(φ0, R), then

kρn(s,·)k ≤ kφ0k+R. Consequently

kF(φn)k ≤ |

λ|pN

2

p−2 X j=0

(_kφ0k+R))p−2−jkφn−1kj !

kφn−φn−1k2

≤ |λ_|p(p−1)N

2 [kφ0k+R] p−2

kφn−φn−1k2 =

(p₋1)a

2(_kφ0k+R)k

φn−φn−1k2, and the proof is complete.

Next, we give the following results on existence and uniqueness of solutions for the equation (2). Besides, we obtain that the sequence given by Newton’s method has R-order two.

Theorem 2.3.

_{Let us assume that equationf(t) = 0, withf defined} in (5) has at least a positive solution and let R be the smaller one. If

B(φ0, R)⊆Ω, then there exists at least a solutionφ∗ of (2) inB(φ0, R). In addition, the Newton’s sequence (3) converges to φ∗ _{with at least} R-order two.

Proof: Firstly, as_kφ1−φ0k ≤η < R, we have φ1 ∈B(φ0, R). Then, Γ1 exists and _kΓ1_{k ≤}h(a). In addition,

kF(φ1)k ≤ (p−1)a

2(_kφ0k+R)k

φ1−φ0k2 =abη and therefore

kφ2−φ1k ≤abh(a)η. Then, applying iv) from Lemma 2.1,

kφ2−φ0k ≤ kφ2−φ1k+kφ1−φ0k ≤(1−(abh(a))2)R < R, and we have thatx2 ∈B(φ0, R). By induction is easy to prove that (6) kφn−φn−1k ≤(abh(a))2

n−1 −1_kφ

1−φ0k.

In addition, taking into account Bernoulli’s inequality, we also have:

kφn−φ0k ≤ n−1 X j=0

(abh(a))2j−1 !

kφ1−φ0k<

∞

X j=0

(abh(a))2j−1 !

η

<

∞

X j=0

(abh(a))j !

η=R

Consequently, φn∈B(φ0, R) for all n≥0.

Next, we prove that {φn} is a Cauchy sequence. From (6), Lemma 2.1 and Bernouilli’s inequality, we deduce

kφn+m−φnk ≤ kφn+m−φn+m−1k+kφn+m−1−φn+m−2k+· · ·+kφn−φn−1k

≤h(abh(a))2n+m−1−1+ (abh(a))2n+m−2−1+· · ·+ (abh(a))2n−1ikφ1−φ0k

≤(abh(a))2n−1h(abh(a))2n(2m−1−1)+ (abh(a))2n(2m−2−1)+_{· · ·}+ (abh(a))2n + 1iη <(abh(a))2n−1(abh(a))2n(m−1)+ (abh(a))2n(m−2)+· · ·+ (abh(a))2n + 1η

= (abh(a))2n−11−(abh(a)) 2n_m 1₋(abh(a))2n η.

But this last quantity goes to zero when n → ∞. Let φ∗ = lim n→∞φn,

then, by letting m_{→ ∞}, we have

kφ∗−φnk ≤(abh(a))2

n₋₁ η

1₋(abh(a))2n =

η

(1₋(abh(a))2n

)(abh(a))(abh(a)) 2n

≤ η

(1−(abh(a)))(abh(a))(abh(a)) 2n

=Cγ2n

with C >0 and γ =abh(a)<1. This inequality guarantees that {φn} has at least R-order of convergence two [8].

Finally, for n= 0, we obtain

kφ∗_−φ

0k<

η

1−abh(a) =R then, φ∗ _∈B(φ

0, R). Moreover, as

kF(φn)k ≤ 1

2M(p−1)(kφ0k+R) p−2_kφ

n−φn−1k2,

when n_{→ ∞} we obtainF(φ∗_{) = 0, and φ}∗ _{is a solution of F(x) = 0.}

Now we give a uniqueness result:

Theorem 2.4.

_{Let kΓ0k ≤ β, then the solution of (2) is unique in}

B(φ0, R)TΩ, with R is the bigger positive solution of the equation

(7) Mβ(p−1)

2 (2kφ0k+R+x)

p−2_{(R+x) = 1.} Proof: To show the uniqueness, we suppose thatγ∗ _∈B(φ

0, R)TΩ is another solution of (2). Then

0 = Γ0F(γ∗)₋Γ0F(φ∗) = Z 1

0

Γ0F_φ′∗_+s(γ∗_−φ∗₎ds(γ

∗

−φ∗).

We are going to prove that A−1 _{exists, where A is a linear operator} defined by

A= Z 1

0

Γ0Fφ′∗_+s(γ∗_−φ∗₎ds,

then γ∗ _=φ∗_{. For this, notice that for each ψ ∈ X and x ∈ [a, b], we}

have

(A₋I)(ψ)(x) = Z 1

0

Γ0[F_φ′∗_+s(γ∗_−φ∗₎−F_φ′₀]ψ(x)ds,

=₋λp

Z 1 0

Γ0 Z b

a

K(x, t)ρ∗(s, t)p−1₋φ0(t)p−1

=₋λp

Z 1 0

Γ0 Z b

a

K(x, t) "_p−2

X j=0

ρ∗(s, t)p−2−jφ0(t)j #

(ρ∗(s, t)₋φ0(t))ψ(t)dt ds, where ρ∗_{(s, t) =φ}∗_{(t) +s(γ}∗_(t)−φ∗_(t)).

Taking into account that

|ρ∗(s, t)−φ0(t)| ≤ kφ∗−φ0+s(γ∗−φ∗)k ≤(1−s)kφ∗−φ0k+skγ∗−φ0k<(1−s)R+sR, we obtain

k(A−I)ψk ≤ |λ|pNkΓ0k

"Z 1 0

p−2 X j=0

kρ∗(s,·)kp−2−j_kφ 0kj

!

((1−s)R+sR)ds

#

kψk.

Therefore, as

kρ∗(s,·)k ≤(1−s)kφ∗k+skγ∗k ≤(1−s)(kφ0k+R)+s(kφ0k+R)≤2kφ0k+R+R, we have, from (7),

kA₋I_{k ≤} kΓ0kM

2 (R+R) "_p−2

X j=0

kφ0k 2kφ0k+R+R

j#

(2_kφ0k+R+R)p−2

< Mβ

2 (R+R)(p−1)(2kφ0k+R+R)

p−2 _{= 1.} So, the operator

Z 1 0

F′(φ∗ +t(γ∗ ₋φ∗))dt has an inverse and conse-quently, γ∗ _=φ∗_{.Then, the proof is complete.}

3. An example

To illustrate the above theoretical results, we consider the following example

(8) φ(x) = sin(πx) + 1 5

Z 1 0

cos(πx) sin(πt)φ(t)3dt, x_∈[0,1].

Let X = C[0,1] be the space of continuous functions defined on the interval [0,1], with the max-norm and let F :X _→X be the operator given by

(9)

F(φ)(x) =φ(x)₋sin(πx)₋ 1 5

Z 1 0

cos(πx) sin(πt)φ((t)3dt, x_∈[0,1].

By differentiating (9) we have: (10) F_φ′[u](x) =u(x)₋3

5cos(πx) Z 1

0

With the notation of section 2,

λ= 1

5, N = maxx∈[0,1] Z 1

0 |

sin(πt)|dt= 1 and M =|λ|pN = 3 5. We take as starting-point φ0(x) = sin(πx), then we obtain from (10)

F′

φ0[u](x) =u(x)−

3

5cos(πx) Z 1

0

sin3_(πt)u(t)dt If F′

φ[u](x) = ω(x), then [Fφ′]−1[ω](x) = u(x) and u(x) = ω(x) + 3

5cos(πx)Ju, where

Ju = Z 1

0

sin(πt)φ(t)2u(t)dt.

Therefore the inverse of F′

φ0 is given by

[F_φ′0]

−1_{[ω](x) =ω(x) +} 3 5

R1 0 sin

3_(πt)w(t)dt 1₋ 3

5 R1

0 cos(πt) sin

3_(πt)dtcos(πx). Then

kΓ0k ≤ kI+ 4

5πcos(πx)k ≤1.25468· · ·=β,

and kF(φ0)k ≤ 3

40 = 0.075. Consequently kΓ0F(φ0)k ≤ 0.094098· · ·=

η.

The equation f(t) = 0, with f given by (5) is now 1.2t3+ 2.4t2₋0.912918t+ 0.0752789 = 0.

This equation has two positive solutions. The smaller one is R = 0.129115. . .. Then, by Theorem 2.3, we know there exists a solution of (8) in B(φ0, R). To obtain the uniqueness domain we consider the equation (7) whose positive solution is the uniqueness ratio. In this case, the solution is unique in B(φ0,0.396793. . .).

Finally, we are going to deal with the computational aspects to solve (8) applying Newton’s method (3). To calculate the iterations

φn+1(x) =φn(x)−[Fφ′n]

−1_[F(φ

n)](x) with the functionφ0(x) as starting-point, we proceed in the following way:

(1) First we compute the integrals

An= Z 1

0

sin(πt)φn(t)3dt; Bn= Z 1

0

sin(πt)2φn(t)2dt;

Cn= Z 1

0

(2) Next we define

φn+1(x) = sin(πx) + 1 5

−2An+ 3Bn 1₋ 3

5Cn

cos(πx).

So we obtain the following approximations

φ0(x) = sinπx,

φ1(x) = sinπx+ 0.075 cosπx,

φ2(x) = sinπx+ 0.07542667509481667 cosπx, φ3(x) = sinπx+ 0.07542668890493719 cosπx, φ4(x) = sinπx+ 0.07542668890493714 cosπx, φ5(x) = sinπx+ 0.07542668890493713 cosπx,

As we can see, in this case Newton’s method converges to the solution

φ∗(x) = sinπx+ 20−

√

391

3 cosπx.

References

[1] I. K. Argyros and F. Szidarovszky,Theory and applications of iteration methods,

CRC-Press Inc., Boca Raton, Florida, USA, 1993.

[2] I. K. Argyros,Advances in the Efficiency of Computational Methods and Appli-cations,World Scientific, Singapore, 2000.

[3] H. T. Davis,Introduction to nonlinear differential and integral equations,Dover, Nueva York, 1962.

[4] J. M. Guti´errez and M. A. Hern´andez, “An application of Newton’s method to differential and integral equations”,ANZIAM J.,42, no. 3, (2001), 372–386. [5] L.V. Kantorovich and G.P. Akilov,Functional Analysis, Pergamon, 1982. [6] R. H. Moore, “Newton’s method and variations”, in Nonlinear integral

equa-tions, Proceedings of an advanced seminar conducted by the Mathematics Re-search Center, United States Army. Editor: P. M. Anselone, Universidad de Wisconsin (Madison), (1963), 65–98.

[7] D. Porter and D. S. G. Stirling,Integral equations, a practical treatment, from spectral theory to applications, Cambridge University Press, Cambridge, 1990. [8] F. A. Potra and V.Pt´ak,Nondiscrete Induction and Iterative Processes,Pitman,

Internet: http://rattler.cameron.edu/swjpam.html ISBN 1083-0464

Issue 1 July 2004, pp. 10 – 32

Submitted: September 10, 2003. Published: July 1, 2004

ORLICZ-SOBOLEV SPACES WITH ZERO BOUNDARY VALUES ON METRIC SPACES

NOUREDDINE A¨ISSAOUI

Abstract. _{In this paper we study two approaches for the} defi-nition of the first order Orlicz-Sobolev spaces with zero boundary values on arbitrary metric spaces. The first generalization, de-noted byMΦ1,0(E), where E is a subset of the metric space X, is

defined by the mean of the notion of the trace and is a Banach space when the N-function satisfies the ∆2condition. We give also

some properties of these spaces. The second, following another def-inition of Orlicz-Sobolev spaces on metric spaces, leads us to three definitions that coincide for a large class of metric spaces and N-functions. These spaces are Banach spaces for any N-function. A.M.S. (MOS) Subject Classification Codes.46E35, 31B15, 28A80.

Key Words and Phrases. Orlicz spaces, Orlicz-Sobolev spaces, modulus of a family of paths, capacities.

1. Introduction

This paper treats definitions and study of the first order Orlicz-Sobolev spaces with zero boundary values on metric spaces. Since we have introduce two definitions of Orlicz-Sobolev spaces on metric spaces, we are leading to examine two approaches.

The first approach follows the one given in the paper [7] relative to Sobolev spaces. This generalization, denoted by M_Φ1,0(E), where E is a subset of the metric space X, is defined as Orlicz-Sobolev functions onX, whose trace on X_\E vanishes.

Ecole Normale Sup´erieure, B.P 5206 Ben Souda, F`es, Maroc n.aissaoui@caramail.com

Copyright c2004 by Cameron University

This is a Banach space when the N-function satisfies the ∆2 con-dition. For the definition of the trace of Orlicz-Sobolev functions we need the notion of Φ-capacity on metric spaces developed in [2]. We show that sets of Φ-capacity zero are removable in the Orlicz-Sobolev spaces with zero boundary values. We give some results closely re-lated to questions of approximation of Orlicz-Sobolev functions with zero boundary values by compactly supported functions. The approx-imation is not valid on general sets. As in Sobolev case, we study the approximation on open sets. Hence we give sufficient conditions, based on Hardy type inequalities, for an Orlicz-Sobolev function to be approximated by Lipschitz functions vanishing outside an open set.

The second approach follows the one given in the paper [13] relative to Sobolev spaces; see also [12]. We need the rudiments developed in [3]. Hence we consider the set of Lipschitz functions on X vanishing on X \ E, and close that set under an appropriate norm. Another definition is to consider the space of Orlicz-Sobolev functions on X

vanishing Φ-q.e. in X_\E. A third space is obtained by considering the closure of the set of compactly supported Lipschitz functions with support in E. These spaces are Banach for any N-function and are, in general, different. For a large class of metric spaces and a broad family of N-functions, we show that these spaces coincide.

2. Preliminaries

An N-function is a continuous convex and even function Φ de-fined on R, verifying Φ(t) > 0 for t > 0, limt→0t−1Φ(t) = 0 and limt→∞t−1Φ(t) = +∞.

We have the representation Φ(t) =

|t|

R 0

ϕ(x)dL_{(x), where ϕ : R}+ _→

R+ is non-decreasing, right continuous, with ϕ(0) = 0, ϕ(t) > 0 for

t >0, limt→0+ϕ(t) = 0 and lim_t→∞ϕ(t) = +∞. Here L stands for the

Lebesgue measure. We put in the sequel, as usually, dx= dL_(x).

The N-function Φ∗ _{conjugate to Φ is defined by Φ}∗_{(t) =} |t|

R 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 Φ an N-function. The Orlicz class LΦ,µ(X) is defined by

LΦ,µ(X) =

f :X _→Rmeasurable :R_XΦ(f(x))dµ(x)<_∞ .

LΦ,µ(X) =

f :X _→Rmeasurable :R_XΦ(αf(x))dµ(x)<_∞ for some α >0 .

The Orlicz spaceLΦ,µ(X) is a Banach space with the following norm, called the Luxemburg norm,

|||f_|||Φ,µ,X = inf n

r >0 :R_X Φf(_rx)dµ(x)_≤1o.

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∈L1 and

R

X |f g|dµ≤2|||f|||Φ,µ,X. |||g|||Φ∗,µ,X.

Let Φ be anN-function. We say that Φverifies the ∆2 condition if there is a constant C >0 such that Φ(2t)_≤CΦ(t) for all t _≥0.

The ∆2 condition for Φ can be formulated in the following equivalent way: for everyC >0 there existsC′ _{>0 such that Φ(Ct)≤C}′_{Φ(t) for}

allt _≥0.

We have always _LΦ,µ(X) ⊂ LΦ,µ(X). The equality LΦ,µ(X) = LΦ,µ(X) occurs if Φ verifies the ∆2 condition.

We know that LΦ,µ(X) is reflexive if Φ and Φ∗ verify the ∆2 condi-tion.

Note that if Φ verifies the ∆2 condition, then R Φ(fi(x))dµ → 0 as

i→ ∞ if and only if |||fi|||Φ,µ,X →0 as i→ ∞.

Recall that an N-function Φ satisfies the ∆′ _{condition if there is a}

positive constant C such that for all x, y ≥ 0, Φ(xy) ≤ CΦ(x)Φ(y). See [9] and [12]. If an N-function Φ satisfies the ∆′ _{condition, then it}

satisfies also the ∆2 condition.

Let Ω be an open set inRN,C∞_{(Ω) be the space of functions which,}

together with all their partial derivatives of any order, are continuous on Ω, and C∞

0 (RN) = C∞0 stands for all functions in C∞(RN) which have compact support in RN. The space Ck_{(Ω) stands for the space} of functions having all derivatives of order _≤ k continuous on Ω, and C(Ω) is the space of continuous functions on Ω.

The (weak) partial derivative off of order _|β_|is denoted by

Dβ_{f =} ∂

|β|

∂xβ1

1 .∂x β2

2 ...∂x βN N

f.

Let Φ be an _N-function and m _∈ N. We say that a function f :

RN _→ R has a distributional (weak partial) derivative of order m, denoted Dβ_{f,|β|=m, if}

R

Let Ω be an open set in RN and denote LΦ,L(Ω) by LΦ(Ω). The Orlicz-Sobolev space Wm_{LΦ(Ω) is the space of real functions f, such} thatf and its distributional derivatives up to the orderm, are inLΦ(Ω).

The space Wm_{LΦ(Ω) is a Banach space equipped with the norm} |||f_|||m,Φ,Ω = P

0≤|β|≤m|||

Dβ_{f|||Φ, f ∈W}m_LΦ(Ω), where _|||Dβ_f|||

Φ =|||Dβf|||Φ,L_,Ω.

Recall that if Φ verifies the ∆2 condition, then C∞_(Ω)∩Wm_LΦ(Ω) is dense in Wm_{LΦ(Ω), and C}∞

0 (RN) is dense in WmLΦ(RN).

For more details on the theory of Orlicz spaces, see [1, 8, 9, 10, 11].

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. Orlicz-Sobolev space with zero boundary values M_Φ1,0(E)

3.1. The Orlicz-Sobolev space M1

Φ(X). We begin by recalling the definition of the space M1

Φ(X).

Let u : X → [−∞, +∞] be a µ-measurable function defined on X. We denote by D(u) the set of all µ-measurable functions g : X →

[0, +_∞] such that

(3.1) |u(x)−u(y)| ≤d(x, y)(g(x) +g(y))

for every x, y ∈X \F, x=6 y, with µ(F) = 0. The set F is called the exceptional set for g.

Note that the right hand side of (3.1) is always defined for x 6= y. For the points x, y ∈ X, x 6=y such that the left hand side of (3.1) is undefined we may assume that the left hand side is +∞.

Let Φ be an _N-function. The Dirichlet-Orlicz space L1

Φ(X) is the space of all µ-measurable functions u such that D(u)∩LΦ(X) 6= ∅. This space is equipped with the seminorm

(3.2) _|||u_|||L1

Φ(X)= inf{|||g|||Φ :g ∈D(u)∩LΦ(X)}.

The Orlicz-Sobolev space M1

Φ(X) is defined by MΦ1(X) = LΦ(X)∩ L1

Φ(X) equipped with the norm (3.3) _|||u_|||M1

Φ(X) =|||u|||Φ+|||u|||L 1 Φ(X).

We define a capacity as an increasing positive set function C given on a σ-additive class of sets Γ, which contains compact sets and such that C(∅) = 0 and C(S

i≥1

Xi)≤P i≥1

C(Xi) for Xi∈ Γ, i= 1,2, ... .

C is called outer capacity if for every X ∈Γ,

C(X) = inf_{C(O) :O open, X _⊂O_}.

Let C be a capacity. If a statement holds except on a set E where

C(E) = 0, then we say that the statement holds C-quasieverywhere (abbreviatedC-q.e.). A functionu:X →[−∞,∞] isC-quasicontinuous in X if for every ε > 0 there is a set E such that C(E) < ε and the restriction of u to X\E is continuous. When C is an outer capacity, we may assume thatE is open.

Recall the following definition in [2]

Definition 1. LetΦbe an_N-function. For a setE _⊂X, define CΦ(E) by

CΦ(E) = inf{|||u|||M1

Φ(X) :u∈B(E)}, where B(E) ={u∈M1

Φ(X) :u≥1 on a neighborhood of E}. If B(E) = ∅, we set CΦ,µ(E) = ∞.

Functions belonging to B(E) are called admissible functions for E. In the definition ofCΦ(E), we can restrict ourselves to those admis-sible functions u such that 0 ≤ u ≤ 1. On the other hand, CΦ is an outer capacity.

Let Φ be anN-function satisfying the ∆2 condition, then by [2 The-orem 3.10] the set

Lip1

Φ(X) ={u∈MΦ1(X) :u is Lipschitz in X} is a dense subspace of M1

Φ(X). Recall the following result in [2, Theo-rem 4.10]

Theorem 1. Let Φ be an _N-function satisfying the ∆2 condition and

u _∈ M1

Φ(X). Then there is a function v ∈ MΦ1(X) such that u = v

µ-a.e. and v is CΦ-quasicontinuous in X.

The function v is called a CΦ-quasicontinuous representative of u. Recall also the following theorem, see [6]

Theorem 2. Let C be an outer capacity on X and µ be a nonnega-tive, monotone set function on X such that the following compatibility condition is satisfied: If G is open and µ(E) = 0, then

C(G) = C(G_\E).

Let f and g be C-quasicontinuous on X such that

Then f =g C-quasi everywhere on X.

It is easily verified that the capacity CΦ satisfies the compatibility condition. Thus from Theorem 2, we get the following corollary.

Corollary 1. LetΦbe an_N-function. Ifuandv areCΦ-quasicontinuous on an open set O and if u=v µ-a.e. in O, then u=v CΦ-q.e. in O.

Corollary 1 make it possible to define the trace of an Orlicz-Sobolev function to an arbitrary set.

Definition 2. Let Φ be an _N-function, u _∈ M1

Φ(X) and E be such that CΦ(E) > 0. The trace of u to E is the restriction to E of any

CΦ-quasicontinuous representative of u.

Remark 1. LetΦbe anN-function. Ifuandv areCΦ-quasicontinuous and u_≤ v µ-a.e. in an open set O, then max(u₋v,0) = 0 µ-a.e. in

O and max(u− v,0) is CΦ-quasicontinuous. Hence by Corollary 1, max(u−v,0) = 0 CΦ-q.e. in O, and consequently u≤v CΦ-q.e. in O. Now we give a characterization of the capacity CΦ in terms of qua-sicontinuous functions. We begin by a definition

Definition 3. Let Φ be an _N-function. For a set E _⊂ X, define

DΦ(E) by

DΦ(E) = inf_{|||u_|||M1

Φ(X) :u∈ B(E)}, where

B(E) ={u∈M_Φ1(X) :u is CΦ-quasicontinuous and u≥1 CΦ-q.e. in E}. If _B(E) =_∅, we set DΦ(E) =_∞.

Theorem 3. Let Φ be an N-function and E a subset in X. Then

CΦ(E) =DΦ(E). Proof. Let u ∈ M1

Φ(X) be such that u ≥ 1 on an open neighborhood

O of E. Then, by Remark 1, the CΦ-quasicontinuous representative v

of usatisfiesv ≥1CΦ-q.e. onO, and hencev ≥1CΦ-q.e. onE. Thus

DΦ(E)≤CΦ(E).

For the reverse inequality, let v _{∈ B}(E). By truncation we may assume that 0 _≤ v _≤ 1. Let ε be such that 0 < ε < 1 and choose an open set V such that CΦ(V) < ε with v = 1 on E \V and v

_X\V is continuous. We can find, by topology, an open set U _⊂ X such that

{x_∈X :v(x)>1₋ε_{} \}V = U _\V. We have E _\V _⊂ U _\V. We choose u _∈ B(V) such that _|||u_|||M1

definew= ₁₋v_ε+u. Then w≥1µ-a.e. in (U\V)∪V =U∪V, which is an open neighbourhood ofE. Hence w_∈B(E). This implies that

CΦ(E) ≤ |||w|||M1 Φ(X) ≤

1

1₋ε|||v|||M1

Φ(X)+|||u|||M 1 Φ(X)

≤ 1

1₋ε|||v|||M1

Φ(X)+ε.

We get the desired inequality sinceε and v are arbitrary. The proof is complete.

We give a sharpening of [2, Theorem 4.8].

Theorem 4. Let Φ be an _N-function and (ui)_i be a sequence of CΦ -quasicontinuous functions inM1

Φ(X)such that(ui)_iconverges inMΦ1(X) to a CΦ-quasicontinuous function u. Then there is a subsequence of (ui)_i which converges to u CΦ-q.e. in X.

Proof. There is a subsequence of (ui)i, which we denote again by (ui)i, such that

(3.4)

∞

X i=1

2i_|||ui−u|||M1

Φ(X)<∞.

We setEi ={x∈X :|ui(x)−u(x)|>2−i} fori= 1,2, ..., andFj = ∞

S i=j

Ei. Then 2i|ui−u| ∈ B(Ei) and by Theorem 3 we obtainCΦ(Ei)≤ 2i_|||u

i−u|||M1

Φ(X). By subadditivity we get

CΦ(Fj)≤ ∞

X i=j

CΦ(Ei)≤ ∞

X i=j

2i|||ui−u|||_M1 Φ(X).

Hence

CΦ(

∞

\ j=1

Fj)≤ lim

j→∞CΦ(Fj) = 0.

Thus ui →u pointwise in X\ ∞

T j=1

Fj and the proof is complete.

3.2. The Orlicz-Sobolev space with zero boundary valuesM_Φ1,0(E). Definition 4. LetΦbe an_N-function andE a subspace ofX. We say that u belongs to the Orlicz-Sobolev space with zero boundary values, and denote u ∈ M_Φ1,0(E), if there is a CΦ-quasicontinuous function e

u_∈M1

Φ(X) such that ue=u µ-a.e. in E and eu= 0 CΦ-q.e. in X\E. In other words,u belongs toM_Φ1,0(E) if there is_eu∈M1

Φ(X)as above such that the trace of u_evanishes CΦ-q.e. in X\E.

The space M_Φ1,0(E) is equipped with the norm |||u|||MΦ1,0(E)=|||eu|||M

1 Φ(X).

Recall that CΦ(E) = 0 implies that µ(E) = 0 for every E ⊂ X; see [2]. It follows that the norm does not depend on the choice of the quasicontinuous representative.

Theorem 5. Let Φ be an N-function satisfying the ∆2 condition and

E a subspace of X. Then M_Φ1,0(E) is a Banach space.

Proof. Let (ui)_i be a Cauchy sequence in M_Φ1,0(E). Then for every ui, there is a CΦ-quasicontinuous function u_ei ∈ MΦ1(X) such that uei =ui

µ-a.e. inE and u_ei = 0 CΦ-q.e. in X\E. By [2, Theorem 3.6] MΦ1(X) is complete. Hence there is u _∈ M1

Φ(X) such that uei → u in MΦ1(X) as i → ∞. Let u_e be a CΦ-quasicontinuous representative of u given by Theorem 1. By Theorem 4 there is a subsequence (u_ei)i such that

e

ui → u Ce Φ-q.e. in X as i → ∞. This implies that ue = 0 CΦ-q.e. in

X\E and hence u∈M_Φ1,0(E). The proof is complete.

Moreover the spaceM_Φ1,0(E) has the following lattice properties. The proof is easily verified.

Lemma 1. Let Φ be an _N-function and let E be a subset in X. If

u, v ∈M_Φ1,0(E), then the following claims are true.

1) If α_≥0, then min(u, α)_∈M_Φ1,0(E) and _|||min(u, α)_|||M1,0 Φ (E) ≤ |||u_|||M1,0

Φ (E).

2) Ifα_≤0, thenmax(u, α)_∈M_Φ1,0(E)and_|||max(u, α)_|||M1,0 Φ (E) ≤ |||u|||MΦ1,0(E).

3) _|u_{| ∈}M_Φ1,0(E) and _||||u_||||M1,0

Φ (E)≤ |||u|||M 1,0 Φ (E).

4) min(u, v)_∈M_Φ1,0(E) and max(u, v)_∈M_Φ1,0(E).

Theorem 6. Let Φ be an N-function satisfying the ∆2 condition and

E a µ-measurable subset in X. If u _∈ M_Φ1,0(E) and v _∈ M1

Φ(X) are such that |v| ≤u µ-a.e. in E, then v ∈M_Φ1,0(E).

Proof. Let w be the zero extension of v to X_\E and let _eu_∈ M1 Φ(X) be aCΦ-quasicontinuous function such thatu_e=u µ-a.e. inE and that e

u= 0CΦ-q.e. inX_\E. Letg1 ∈D(eu)∩LΦ(X) andg2 ∈D(v)∩LΦ(X). Define the function g3 by

g3(x) =

max(g1(x), g2(x)),

g1(x),

x∈E x∈X\E.

Then it is easy to verify thatg3 ∈D(w)∩LΦ(X). Hencew∈MΦ1(X). Letw_e∈M1

Φ(X) be a CΦ-quasicontinuous function such thatwe=w µ -a.e. in X given by Theorem 1. Then |we| ≤eu µ-a.e. inX. By Remark 1 we get |we| ≤ u Ce Φ-q.e. in X and consequently w_e = 0 CΦ-q.e. in

X\E. This shows that v ∈M_Φ1,0(E). The proof is complete. The following lemma is easy to verify.

Lemma 2. Let Φbe an _N-function and let E be a subset in X. Ifu_∈ M_Φ1,0(E) and v _∈M1

Φ(X) are bounded functions, then uv∈M 1,0 Φ (E). We show in the next theorem that the sets of capacity zero are re-movable in the Orlicz-Sobolev spaces with zero boundary values. Theorem 7. Let Φ be an _N-function and let E be a subset inX. Let

F _⊂E be such that CΦ(F) = 0. Then M_Φ1,0(E) =M_Φ1,0(E_\F).

Proof. It is evident that M_Φ1,0(E \ F) ⊂ M_Φ1,0(E). For the reverse inclusion, letu_∈M_Φ1,0(E), then there is aCΦ-quasicontinuous function e

u _∈ M1

Φ(X) such that ue = u µ-a.e. in E and that eu = 0 CΦ-q.e. in X \ E. Since CΦ(F) = 0, we get that _eu = 0 CΦ-q.e. in X \

(E _\F). This implies that u|E\F ∈ M_Φ1,0(E\F). Moreover we have

u|E\F

MΦ1,0(E\F)=|||u|||M 1,0

Φ (E). The proof is complete.

As in the Sobolev case, we have the following remark.

Remark 2. 1) If CΦ(∂F) = 0, then M_Φ1,0(int E) =M_Φ1,0(E). 2) We have the equivalence: M_Φ1,0(X_\F) =M_Φ1,0(X) =M1

Φ(X) if and only if CΦ(F) = 0.

The converse of Theorem 7 is not true in general. In fact it suffices to take Φ(t) = 1_ptp _{(p >1) and consider the example in [7].}

Nevertheless the converse of Theorem 7 holds for open sets.

Theorem 8. Let Φ be an _N-function and suppose that µ is finite in bounded sets and that O is an open set. Then M_Φ1,0(O) =M_Φ1,0(O_\F) if and only if CΦ(F ∩O) = 0.

Proof. We must show only the necessity. We can assume that F _⊂O. Letx0 ∈Oand fori∈N∗, poseOi =B(x0, i)∩{x∈O : dist(x, X \O)>1/i}. We define fori∈N∗,ui:X →Rbyui(x) = max(0,1−dist(x, F∩Oi)). Then ui ∈MΦ1(X), ui is continuous, ui = 1 in F ∩Oi and 0≤ ui ≤1. For i ∈ N∗, define vi : Oi → R by vi(x) = dist(x, X \ Oi). Then

vi ∈ MΦ1,0(Oi) ⊂ MΦ1,0(O). By Lemma 2 we have, for every i ∈ N∗,

uivi ∈ MΦ1,0(O) =M 1,0

Φ (O\F). If w is a CΦ-quasicontinuous function such that w = uivi µ-a.e. in O \F, then w = uivi µ-a.e. in O since

µ(F) = 0. By Corollary 1 we get w = uivi CΦ-q.e. in O. In partic-ular w = uivi > 0 CΦ-q.e. in F ∩Oi. Since uivi ∈ MΦ1,0(O \F) we may define w = 0 CΦ-q.e. in X \(O \F). Hence w = 0 CΦ-q.e. in

F∩Oi. This is possible only ifCΦ(F∩Oi) = 0 for everyi∈N∗. Hence

CΦ(F)≤ P∞ i=1

CΦ(F ∩Oi) = 0. The proof is complete.

3.3. Some relations between H_Φ1,0(E) and M_Φ1,0(E). We would de-scribe the Orlicz-Sobolev space with zero boundary values on E _⊂ X

as the completion of the set Lip1_Φ,0(E) defined by

Lip1_Φ,0(E) ={u∈M1

Φ(X) :u is Lipschitz in X and u= 0 inX\E} in the norm defined by (3.3). SinceM1

Φ(X) is complete, this completion is the closure of Lip1_Φ,0(E) in M1

Φ(X). We denote this completion by

H_Φ1,0(E).

Let Φ be an N-function satisfying the ∆2 condition and E a sub-space of X. By [2, Theorem 3.10] we have H_Φ1,0(X) = M_Φ1,0(X). Since Lip1_Φ,0(E) ⊂ M_Φ1,0(E) and M_Φ1,0(E) is complete, then H_Φ1,0(E) ⊂

M_Φ1,0(E). When Φ(t) = 1 pt

p _{(p > 1), simple examples show that the} equality is not true in general; see [7]. Hence for the study of the equal-ity, we restrict ourselves to open sets as in the Sobolev case. We begin by a sufficient condition.

Theorem 9. Let Φ be an _N-function satisfying the ∆2 condition, O

an open subspace of X and suppose that u _∈ M1

Φ(O). Let v be the function defined on O by v(x) = u(x)

dist(x, X _\O). If v ∈ LΦ(O), then

u_∈H_Φ1,0(O).

Proof. Letg _∈D(u)_∩LΦ(O) and define the function g by

g(x) = max(g(x), v(x)) if x∈O g(x) = 0 if x∈X\O.

Then g _∈ LΦ(X). Define the function u as the zero extension of u

toX\O. For µ-a.e. x, y ∈O orx, y ∈X\O, we have

|u(x)₋u(y)_{| ≤}d(x, y)(g(x) +g(y)). Forµ-a.e. x_∈O and y_∈X_\O, we get

|u(x)₋u(y)_|=_|u(x)_{| ≤}d(x, y) |u(x)|

dist(x, X \O) ≤d(x, y)(g(x) +g(y)). Thus g _∈D(u)_∩LΦ(X) which implies thatu_∈M1

Φ(O). Hence (3.5) |u(x)−u(y)| ≤d(x, y)(g(x) +g(y))

for every x, y ∈X\F withµ(F) = 0. Fori∈N∗, set

(3.6) Fi ={x∈O\F :|u(x)| ≤i, g(x)≤i} ∪X\O.

From (3.5) we see that u|Fi is 2i-Lipschitz and by the McShane ex-tension

ui(x) = inf{u(y) + 2id(x, y) :y∈Fi}

we extend it to a 2i-Lipschitz function on X. We truncate ui at the level i and set ui(x) = min(max(ui(x),−i), i). Then ui is such that ui is 2i-Lipschitz function in X, _|ui| ≤ i in X and ui = u in Fi and, in particular, ui = 0 in X\O. We show that ui ∈ MΦ1(X). Define the function gi by

gi(x) = g(x), if x∈Fi,

gi(x) = 2i, if x∈X\Fi. We begin by showing that

(3.7) |ui(x)−ui(y)| ≤d(x, y)(gi(x) +gi(y)),

for x, y _∈X_\F. If x, y _∈Fi, then (3.7) is evident. Fory∈X\Fi, we have

|ui(x)−ui(y)| ≤ 2id(x, y)≤d(x, y)(gi(x) +gi(y)), if x∈X\Fi, |ui(x)−ui(y)| ≤ 2id(x, y)≤d(x, y)(g(x) + 2i), if x∈X\Fi.

This implies that (3.7) is true and thus gi ∈D(ui). Now we have |||gi|||Φ ≤ |||gi|||Φ,Fi+ 2i|||1|||Φ,X\Fi

≤ |||g|||Φ,Fi+

2i

Φ−1₍ 1 µ(X\Fi))

<∞, and

|||ui|||Φ ≤ |||u|||Φ,Fi+ 2i|||1|||Φ,X\Fi

≤ |||u|||Φ,Fi+

2i

Φ−1₍ 1 µ(X\Fi))

<∞. Hence ui ∈MΦ1(X). It follows that ui ∈Lip1Φ,0(O).

It remains to prove that ui →u inMΦ1(X). By (3.6) we have

µ(X_\Fi)≤µ({x∈X :|u(x)| > i}) +µ({x∈X : g(x)> i}). Since u_∈LΦ(X) and Φ satisfies the ∆2 condition, we get

Z

{x∈X:|u(x)|>i}

Φ(u(x))dµ(x)_≥Φ(i)µ_{x_∈X :_|u(x)_|> i_}, which implies that Φ(i)µ{x∈X :|u(x)|> i} →0 as i→ ∞.

By the same argument we deduce that Φ(i)µ{x∈X :g(x)> i} →0 asi→ ∞.

Thus

(3.8) Φ(i)µ(X\Fi)→0 as i→ ∞.

Using the convexity of Φ and the fact that Φ satisfies the ∆2 condi-tion, we get

Z X

Φ(u₋ui)dµ ≤ Z

X\Fi

Φ(_|u_|+_|ui|)dµ

≤ C

2[ Z

X\Fi

Φ◦ |u|dµ+ Φ(i)µ(X\Fi)]→0 as i→ ∞. On the other hand, for each i∈N∗ we define the function hi by

hi(x) = g(x) + 3i, if x∈X\Fi,

hi(x) = 0, if x∈Fi.

We claim that hi ∈D(u−ui)∩LΦ(X). In fact, the only nontrivial case is x_∈Fi and y∈X\Fi; but then

|(u₋ui)(x)−(u−ui)(y)| ≤ d(x, y)(g(x) +g(y) + 2i) ≤ d(x, y)(g(y) + 3i). By the convexity of Φ and by the ∆2 condition we have Z

X

Φ◦hidµ ≤ Z

X\Fi

Φ◦(g+ 3i)dµ

≤ C[ Z

X\Fi

Φ_◦gdµ+ Φ(i)µ(X_\Fi)]→0 as i→ ∞. This implies that |||hi|||Φ → 0 as i → ∞ since Φ verifies the ∆2 condition.

Now

|||u₋ui|||_L1

Φ(X) ≤ |||hi|||Φ →0 as i→ ∞.

Thus u∈H_Φ1,0(O). The proof is complete.

Definition 5. A locally finite Borel measure µ is doubling if there is a positive constant C such that for every x∈X and r >0,

µ(B(x,2r))≤Cµ(B(x, r)).

Definition 6. A nonempty set E _⊂X is uniformlyµ-thick if there are constants C >0 and 0< r0 ≤1 such that

µ(B(x, r)_∩E)_≥Cµ(B(x, r)), for every x∈E, and 0< r < r0.

Now we give a Hardy type inequality in the context of Orlicz-Sobolev spaces.

Theorem 10. Let Φ be an N-function such that Φ∗ _{satisfies the ∆2} condition and suppose that µ is doubling. Let O ⊂ X be an open set such that X\O is uniformly µ-thick. Then there is a constant C > 0 such that for every u∈M_Φ1,0(O),

|||v_|||Φ,O ≤C|||u|||MΦ1,0(O),

where v is the function defined on O by v(x) = u(x)

dist(x, X _\O). The constant C is independent of u.

Proof. Let u ∈ M_Φ1,0(O) and u_e ∈ M1

Φ(O) be Φ-quasicontinuous such thatu=_eu µ-a.e. inOandu_e= 0 Φ-q.e. inX_\O. Letg _∈D(u_e)_∩LΦ(X) and set O′ _{= {x∈O : dist(x, X \O)< r}

0}. For x ∈ O′, we choose

x0 ∈ X \O such that rx =dist(x, X \O) = d(x, x0). Recall that the Hardy-Littlewood maximal function of a locally µ-integrable function

f is defined by

Mf(x) = sup r>0

1

µ(B(x, r)) Z

B(x,r)

f(y)dµ(y).

Using the uniform µ-thickness and the doubling condition, we get 1

µ(B(x0, rx)\O) Z

B(x0,rx)\O

g(y)dµ(y) ≤ C

µ(B(x0, rx)) Z

B(x0,rx)

g(y)dµ(y)

≤ _µ(B(x,C_2r x))

Z B(x,2rx)

g(y)dµ(y)

≤ C_Mg(x). On the other hand, for µ-a.e. x∈O′ _{there is y∈B(x}

0, rx)\O such that

|u(x)_{| ≤} d(x, y)(g(x) + 1

µ(B(x0, rx)\O) Z

B(x0,rx)\O

g(y)dµ(y))

≤ Crx(g(x) +Mg(x)) ≤ Cdist(x, X\O)Mg(x).

By [5], M is a bounded operator from LΦ(X) to itself since Φ∗

satisfies the ∆2 condition. Hence

|||v_|||Φ,O′ ≤C|||Mg|||_Φ ≤C|||g|||Φ.

On O\O′ _{we have}

Thus

|||v_|||Φ,O ≤C(|||eu|||Φ+|||g|||Φ).

By taking the infimum over allg ∈D(u_e)∩LΦ(X), we get the desired result.

By Theorem 9 and Theorem 10 we obtain the following corollaries Corollary 2. Let Φ be an N-function such that Φ and Φ∗ _{satisfy the}

∆2 condition and suppose that µ is doubling. Let O ⊂ X be an open set such that X\O is uniformly µ-thick. Then M_Φ1,0(O) =H_Φ1,0(O). Corollary 3. Let Φ be an _N-function such that Φ and Φ∗ _{satisfy the}

∆2 condition and suppose that µ is doubling. Let O _⊂ X be an open set such that X _\O is uniformly µ-thick and let (ui)i ⊂ M_Φ1,0(O) be a bounded sequence in M_Φ1,0(O). If ui →u µ-a.e., then u∈MΦ1,0(O).

In the hypotheses of Corollary 3 we get M_Φ1,0(O) =H_Φ1,0(O). Hence the following property (P) is satisfied for sets E whose complement is

µ-thick:

(_P) Let (ui)i be a bounded sequence in HΦ1,0(E). If ui →u µ-a.e., then

u_∈H_Φ1,0(E). Remark 3. IfM1

Φ(X) is reflexive, then by Mazur’s lemma closed con-vex sets are weakly closed. Hence every open subset O of X satisfies property (P). But in general we do not know whether the space M1

Φ(X) is reflexive or not.

Recall that a space X is proper if bounded closed sets in X are compact.

Theorem 11. LetΦ be an_N-function satisfying the ∆2 condition and suppose that X is proper. LetO be an open set inX satisfying property (P). Then M_Φ1,0(O) =H_Φ1,0(O).

Proof. It suffices to prove that M_Φ1,0(O) ⊂ H_Φ1,0(O). Let u ∈ M_Φ1,0(O) be a Φ-quasicontinuous function from M1

Φ(X) such that u = 0 Φ-q.e. on X \O. By using the property (P), we deduce, by truncating and considering the positive and the negative parts separately, that we can assume that uis bounded and non-negative. If x0 ∈O is a fixed point, define the sequence (ηi)i by

ηi(x) =   

1 if d(x0, x)≤i−1,

i₋d(x0, x) if i−1< d(x0, x)< i 0 if d(x0, x)≥i.

If we define the sequence (vi)i by vi =uηi, then since vi →u µ-a.e. in X and |||vi|||M1

Φ(X) ≤ 2|||u|||M 1

Φ(X), by the property (P) it clearly

suffices to show that vi ∈HΦ1,0(O). Remark that

|vi(x)−vi(y)| ≤ |u(x)−u(y)|+|ηi(x)−ηi(y)| ≤ d(x, y)(g(x) +g(y) +u(x). Hence vi ∈MΦ1(X).

Now fixiand set v =vi. Since v vanishes outside a bounded set, we can find a bounded open subsetU ⊂Osuch thatv = 0 Φ-q.e. inX\U. We choose a sequence (wj)⊂MΦ1(X) of quasicontinuous functions such that 0_≤wj ≤1,wj = 1 on an open setOj, with|||wj|||M1

Φ(X) →0, and

so that the restrictions v|X\Oj are continuous andv = 0 inX\(U∪Oj). The sequence (sj)j, defined by sj = (1−wj) max(v−1_j,0), is bounded in M1

Φ(X), and passing if necessary to a subsequence, sj → v µ-a.e. Since v|X\Oj is continuous, we get

{x_∈X :sj(x)6= 0} ⊂

x_∈X :v(x)_≥ 1

j

\Oj ⊂U.

This means that_{x_∈X :sj(x)6= 0}is a compact subset ofO, whence by Theorem 9, sj ∈ HΦ1,0(O). The property (P) implies v ∈ H

1,0 Φ (O) and the proof is complete.

Corollary 4. Let Φ be an _N-function satisfying the ∆2 condition and suppose that X is proper. Let O be an open set in X and suppose that

M1

Φ(X) is reflexive. Then M 1,0

Φ (O) =H 1,0 Φ (O).

Proof. By Remark 3, O satisfies property (P), and Theorem 11 gives the result.

4. Orlicz-Sobolev space with zero boundary values N_Φ1,0(E) 4.1. The Orlicz-Sobolev space N1

Φ(X). We recall the definition of the space N1

Φ(X).

Let (X, d, µ) be a metric, Borel measure space, such thatµis positive and finite on balls in X.

IfI is an interval in R, 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 P i=1|

γ(ti+1)−γ(ti)|,

where the supremum is taken over all finite sequences a = t1 ≤ t2 ≤

...≤ tn ≤ tn+1 =b. If I is not closed, we set l(γ) = supl(γ|J), where the supremum is taken over all closed sub-intervals J of I. A path is

said to be rectifiable if its length is a finite number. A path γ :I →X

is locally rectifiable if its restriction to each closed sub-interval of I is 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)]→X such that γ =γs◦sγ. The pathγsis the arc length parametrization of

γ.

Let γ be a rectifiable path in X. The line integral over γ of each non-negative Borel functionρ:X →[0,∞] is R_γρds =R₀l(γ)ρ◦γs(t)dt. If the path γ is only locally rectifiable, we set R_γρds= supR_γ′ρds,

where the supremum is taken over all rectifiable sub-pathsγ′ _{of γ. See}

[5] for more details.

Denote by Γrect the collection of all non-constant compact (that is,

I is compact) rectifiable paths in X.

Definition 7. Let Φ be an _N-function and Γ be a collection of paths in X. The Φ-modulus of the familyΓ, denoted ModΦ(Γ), is defined as

inf

ρ∈F(Γ)|||ρ|||Φ,

where _F(Γ) is the set of all non-negative Borel functions ρ such that R

γρds ≥ 1 for all rectifiable paths γ in Γ. Such functions ρ used to define the Φ-modulus of Γ are said to be admissible for the family Γ.

From the above definition the Φ-modulus of the family of all non-rectifiable paths is 0.

A property relevant to paths in X is said to hold for Φ-almost all paths if the family of rectifiable compact paths on which that property does not hold has Φ-modulus zero.

Definition 8. Let u be a real-valued function on a metric space X. A non-negative Borel-measurable function ρ is said to be an upper gra-dient of u if for all compact rectifiable paths γ the following inequality holds

(4.1) _|u(x)₋u(y)_{| ≤} Z

γ

ρds,

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

Definition 9. Let Φ be an N-function and let u be an arbitrary real-valued function on X. Let ρ be a non-negative Borel function on X. If there exists a family Γ ⊂ Γrect such that ModΦ(Γ) = 0 and the inequality (4.1) is true for all paths γ in Γrect \Γ, then ρ is said to

be a Φ-weak upper gradient of u. If inequality (4.1) holds true for Φ -modulus almost all paths in a set B ⊂X, then ρis said to be a Φ-weak upper gradient of u on B.

Definition 10. Let Φ be an _N-function and let the set Nf1

Φ(X, d, µ) be the collection of all real-valued function u on X such that u_∈ LΦ and

u have a Φ-weak upper gradient in LΦ. Ifu ∈Nf1

Φ, we set (4.2) |||u|||g_N1

Φ

=|||u|||Φ+ inf

ρ |||ρ|||Φ,

where the infimum is taken over allΦ-weak upper gradient,ρ, of usuch that ρ_∈LΦ.

Definition 11. LetΦbe anN-function. The Orlicz-Sobolev space cor-responding toΦ, denotedN1

Φ(X), is defined to be the spaceNfΦ1(X, d, µ)∽, with norm _|||u_|||N1

Φ :=|||u|||gN_Φ1.

For more details and developments, see [3].

4.2. The Orlicz-Sobolev space with zero boundary valuesN_Φ1,0(E). Definition 12. Let Φ be an N-function. For a set E ⊂ X define

CapΦ(E) by

CapΦ(E) = inf n

|||u_|||N1

Φ :u∈ D(E)

o , where _D(E) =_{u_∈N1

Φ :u|E ≥1}.

If _D(E) = _∅, we set CapΦ(E) = _∞. Functions belonging to _D(E) are called admissible functions forE.

Definition 13. Let Φ be an _N-function and E a subset of X. We define Ng_Φ1,0(E) as the set of all functions u :E _→ [_−∞,_∞] for which there exists a functionu_e∈Nf1

Φ(E)such thateu=u µ-a.e. in E andue= 0 CapΦ-q.e. in X\E; which means CapΦ({x∈X\E :ue(x)6= 0}) = 0.

Let u, v _∈ Ng_Φ1,0(E). We say that u _∼ v if u = v µ-a.e. in E. The relation∼is an equivalence relation and we setN_Φ1,0(E) = Ng_Φ1,0(E)_∽. We equip this space with the norm |||u|||NΦ1,0(E) :=|||u|||N

1 Φ(X).

It is easy to see that for every setA _⊂X,µ(A)_≤CapΦ(A). On the other hand, by [3, Corollary 2] if _euand u_e′ _{both correspond tou in the}

above definition, then |||ue−eu′_||| N1

Φ(X) = 0. This means that the norm

Definition 14. Let Φ be an N-function and E a subset of X. We set

Lip1_Φ,0_,N(E) = u_∈N_Φ1(X) :u is Lipschitz in X and u= 0 in X_\E ,

and

Lip1_Φ,_,C0 (E) =u∈Lip1_Φ,_,N0 (E) :u has compact support .

We let H_Φ1,_,N0 (E) be the closure of Lip1_Φ,_,N0 (E) in the norm of N1 Φ(X), and H_Φ1,_,C0 (E) be the closure of Lip1_Φ,_,C0 (E) in the norm of N1

Φ(X). By definition H_Φ1,_,N0 (E) and H_Φ1,_,C0 (E) are Banach spaces. We prove that N_Φ1,0(E) is also a Banach space.

Theorem 12. Let Φ be an _N-function and E a subset of X. Then

N_Φ1,0(E) is a Banach space.

Proof. Let (ui)i be a Cauchy sequence inNΦ1,0(E). Then there is a cor-responding Cauchy sequence (u_ei)i in NΦ1(X), where uei is the function corresponding to ui as in the definition of N_Φ1,0(E). Since NΦ1(X) is a Banach space, see [3, Theorem 1], there is a function _eu _∈ N1

Φ(X), and a subsequence, also denoted (u_ei)i for simplicity, so that as in the proof of [3, Theorem 1], _eui → eu pointwise outside a set T with

CapΦ(T) = 0, and also in the norm of N1

Φ(X). For every i, set

Ai = {x∈X\E :eui(x)6= 0}. Then CapΦ(∪iAi) = 0. Moreover, on (X\E)\(∪iAi∪T), we have ue(x) = lim

i→∞uei(x) = 0.

Since CapΦ(_∪iAi ∪T) = 0, the function u = eu|E is in NΦ1,0(E). On the other hand we have

|||u₋ui|||_N1,0

Φ (E) =|||eu−uei|||N 1

Φ(X) →0 as i→ ∞.

Thus N_Φ1,0(E) is a Banach space and the proof is complete.

Proposition 1. Let Φ be an N-function and E a subset of X. Then the space H_Φ1,_,N0 (E) embeds isometrically into N_Φ1,0(E), and the space

H_Φ1,_,C0 (E) embeds isometrically into H_Φ1,_,N0 (E).

Proof. Let u _∈ H_Φ1,_,N0 (E). Then there is a sequence (ui)i ⊂ NΦ1(X) of Lipschitz functions such that ui →u in NΦ1(X) and for each integer i,

ui|X\E = 0. Considering if necessary a subsequence of (ui)i, we proceed as in the proof of [3, Theorem 1], we can consider the function_eudefined outside a set S with CapΦ(S) = 0, by u_e = 1

2(lim sup i

ui + lim inf i ui). Then _eu ∈ N1

Φ(X) and u|E = eu|E µ-a.e and ue|(X\E)\S = 0. Hence

u|E ∈NΦ1,0(E), with the two norms equal. SinceH 1,0

Φ,C(E)⊂Lip 1,0 Φ,N(E), it is easy to see thatH_Φ1,_,C0 (E) embeds isometrically intoH_Φ1,_,N0 (E). The proof is complete.

When Φ(t) = 1_ptp_{, there are examples of spaces X and E ⊂ X for} which N_Φ1,0(E),H_Φ1,_,N0 (E) andH_Φ1,_,C0 (E) are different. See [13]. We give, in the sequel, sufficient conditions under which these three spaces agree. We begin by a definition and some lemmas.

Definition 15. LetΦbe anN-function. The spaceXis said to support a (1,Φ)-Poincar´e inequality if there is a constant C > 0 such that for all balls B _⊂ X, and all pairs of functions u and ρ, whenever ρ is an upper gradient of u on B and u is integrable on B, the following inequality holds

1

µ(B) Z

B|

u−uE| ≤Cdiam(B)|||g|||_LΦ(B)Φ −1₍ 1

µ(B)).

Lemma 3. Let Φ be an _N-function and Y a metric measure space with a Borel measure µ that is finite on bounded sets. Let u_∈ N1

Φ(Y) be non-negative and define the sequence (ui)i by ui = min(u, i), i∈ N. Then (ui)i converges to u in the norm of NΦ1(Y).

Proof. Set Ei = {x∈Y :u(x)> i}. If µ(Ei) = 0, then ui = u µ-a.e. and since ui ∈NΦ1(Y), by [3, Corollary 2] theNΦ1(Y) norm ofu−ui is zero for sufficiently large i. Now, suppose that µ(Ei) > 0. Since µ is finite on bounded sets, it is an outer measure. Hence there is an open set Oi such thatEi ⊂Oi and µ(Oi)≤µ(Ei) + 2−i.

We have 1

i |||u|||LΦ(Ei) ≥ |||1|||LΦ(Ei) =

1 Φ−1₍ 1

µ(Ei)) .

Since Φ−1_{is continuous, increasing and verifies Φ(x)→ ∞asx→ ∞,} we get

1 Φ−1₍ 1

µ(Oi)−2−i)

≤ 1_i |||u_{|||LΦ →}0 as i_{→ ∞}, and

µ(Oi)→0 as i→ ∞.

Note that u=ui onY \Oi. Thus u−ui has 2gχOi as a weak upper gradient whenever g is an upper gradient of u and hence of ui as well; see [3, Lemma 9]. Thus ui →u in NΦ1(Y). The proof is complete. Remark 4. By [3, Corollary 7], and in conditions of this corollary, if

u _∈ N1

Φ(X), then for each positive integer i, there is a wi ∈ NΦ1(X) such that 0 _≤ wi ≤ 1, |||wi|||N1

Φ(X) ≤ 2

−i_{, and w}

i|Fi = 1, with Fi an open subset of X such that u is continuous on X_\Fi.

We define, as in the proof of Theorem 11, fori∈N∗, the function ti by

ti = (1−wi) max(u−1_i,0).

Lemma 4. Let Φ be an _N-function satisfying the ∆′ _{condition. Let}

X be a proper doubling space supporting a (1,Φ)-Poincar´e inequality, and let u _∈ N1

Φ(X) be such that 0 ≤ u ≤ M, where M is a constant. Suppose that the set A={x∈X :u(x)6= 0} is a bounded subset ofX. Then ti →u in NΦ1(X).

Proof. Set Ei =

x∈X :u(x)< 1_i . By [3, Corollary 7] and by the choice of Fi, there is an open set Ui such that Ei \ Fi = Ui \ Fi. Pose Vi = Ui ∪Fi and remark that wi|Fi = 1 and u|Ei <

1

i. Then {x∈X :ti(x)6= 0} ⊂A\Vi ⊂A. If we setvi =u−ti, then 0≤vi ≤M since 0 ≤ ti ≤ u. We can easily verify that ti = (1−wi)(u−1/i) on

A\Vi, and ti = 0 onVi. Therefore

(4.3) vi =wiu+ (1−wi)i onA\Vi, and

(4.4) vi =u on Vi.

Letx, y ∈X. Then

|wi(x)u(x)−wi(y)u(y)| ≤ |wi(x)u(x)−wi(x)u(y)|+|wi(x)u(y)−wi(y)u(y)| ≤ wi(x)|u(x)−u(y)|+M|wi(x)−wi(y)|.

Letρi be an upper gradient ofwi such that|||ρi|||_LΦ ≤2

−i+1 _{and let}

ρ be an upper gradient ofu belonging toLΦ. If γ is a path connecting two points x, y _∈X, then

|wi(x)u(x)−wi(y)u(y)| ≤ wi(x) Z

γ

ρds+M

Z γ

ρids. Hence, if z _{∈ |}γ_|, then

|wi(x)u(x)−wi(y)u(y)| ≤ |wi(x)u(x)−wi(z)u(z)|+|wi(z)u(z)−wi(y)u(y)| ≤ wi(z)

Z γxz

ρds+M

Z γxz

ρids+wi(z) Z

γzy

ρds+M

Z γzy

ρids

≤ wi(z) Z

γ

ρds+M

Z γ

ρids,

whereγxzandγzyare such that the concatenation of these two segments gives the original pathγ back again. Therefore

|wi(x)u(x)−wi(y)u(y)| ≤ Z

γ

inf

z∈|γ|wi(z)ρ+Mρi

where

|

z

|

<

1.

Notice that all functions are positive and harmonic in

D

and that the radial limits of

wn(z) and

w(z) are

wn(t) and

w(t)

respectively.

Then, for

|

z

| ≤

ρ <

1,

|

gx

(z)

₋

gy

(z)

_{| ≤}

1

1

₋

ρ

gx

e

it

₋

gy

e

it

_L1

Then the continuity condition implies that

gx

(z) is uniformly

contin-uous in

x

for all

_|

z

_{| ≤}

ρ.

Hence, weak star convergence, implies that

wn(z)

→

w(z) uniformly on

|

z

| ≤

ρ

and consequently the

conver-gence is locally uniformly on

D

.

In addition, we have

_k

w

n

(ρe

it

)

kL

1

→

k

w(ρe

it

₎

_kL

1

.

Hence we conclude that

wn(ρe

it

)

−

w(ρe

it

)

_L1

−→

0 as

n

→ ∞

.

Now using Fatou’s Lemma we conclude that

wn(e

it

)

−

w(e

it

)

_L1

−→

0.

Lemma 3.

Let

gx

(e

it

₎

_{be a nonnegative}

_L

1

_{continuous function of}

_x

such that

k

gxk

L1

≤

a <

∞

and

gx(e

it

)

defines a bounded operator on

H

1

0

.

If

f(z) =

Z

₁

(1

₋

xz)

α

dµ(x)

, let

L

be the operator given by

L

[f

(z)] =

ZZ

gx

(e

it

)

(1

−

e

−it

_z)

α

dtdµ(x)

then

L

is compact operator on

Fα, α

_≥

1.

Proof.

First note that the condition that

gx(e

it

) defines a bounded

op-erator on

H

1

0

implies that the

L

operator is a well defined function on

F

α. Let

{

fn(z)

}

be a bounded sequence in

Fα

and let

{

µn}

be the

corresponding norm bounded sequence of measures in

M

.

Since every

norm bounded sequence of measures in

M

has a weak star

conver-gent subsequence, let

{

µn}

be such subsequence that is convergent to

µ

∈

M

.

We want to show that

{

L(fn)

}

has a convergent subsequence

in

Fα.

First, let us assume that

dµ

n

(x)

>>

0 for all

n,

and let

w

n

(t) =

R

g

x

(e

it

)

dµ

n

(x) and

w(t) =

R

g

x

(e

it

)

dµ

(x)

,then we know from the

previous lemma that

wn(t), w

(t)

∈

L

1

_{for all}

_n,

_and

_wn(t)

_→

_{w(t) in}

{

µn}

is weak star convergent to

µ,

then

L(fn(z)) =

ZZ

gx

(e

it

₎

_d(t)

(1

₋

e

−it

_z)

α

dµn

(x) =

Z

wn(t)

(1

₋

e

−it

_z)

α

dt

L(f

(z)) =

ZZ

gx

(e

it

₎

_d(t)

(1

₋

e

−it

_z)

α

dµ

(x) =

Z

w(t)

(1

₋

e

−it

_z)

α

dt

Furthermore because

wn(t) is nonnegative then

k

L(fn)

k

Fα

=

k

wnk

L1

k

L(f

)

_kFα

=

_k

w

_kL

1

Now since

_k

w

n

−

w

kL

1

→

0 then

k

L(f

_n

)

−

L(f

)

kFα

→

0 which shows

that

_{

L(fn)

_}

has convergent subsequence in

Fα

and thus

L

is a compact

operator for the case where

µ

is a positive measure.

In the case where

µ

is complex measure we write

dµn

(x) = (dµ

1

n

(x)

−

dµ

2

n

(x)) +

i(dµ

3n

(x)

−

dµ

4n

(x)),

where each

dµ

j

n

(x)

>>

0 and define

w

jn

(t) =

R

gx

(e

it

₎

_dµ

j

n

(x) then

wn(t) =

R

gx

(e

it

₎

_dµn

_{(x) = (w}

1

n

(t)

−

w

n2

(t)) +

i

(w

n3

(t)

−

w

n4

(t))

.

Using an argument similar to the one above we get that

w

j

n

(t), w

j

(t)

∈

L

1

_,

_and

_k

_w

j

n

−

w

j

kL

1

−→

0.

Consequently,

k

wn

−

w

kL

1

−→

0,

where

w(t) = (w

1

_(t)

₋

_w

2

_{(t)) +}

_i

_(w

3

_(t)

₋

_w

4

_{(t)) =}

R

_gx

_(e

it

₎

_dµ

_(x)

_.

Hence,

k

L(fn)

−

L(f

)

kFα

≤ k

wn

−

w

kL

1

−→

0.

Finally, we conclude that the operator is compact.

The following is the converse of Theorem 1.

Theorem 2.

For a holomorphic self-map

ϕ

of the unit disc

D

,

if

1

(1

−

xϕ(z))

α

=

Z

_gx

_(e

it

₎

(1

−

e

−it

_z)

α

dt

where

g

x

∈

L

1

,

nonnegative,

k

g

xkL1

≤

a <

∞

for all

x

∈

T

and

g

_x

is an

L

1

_{continuous function of}

_{x ,}

_then

_Cϕ

_{is compact on}

_Fα.

Proof.

We want to show that

C

ϕ

is compact on

F

α

. Let

f(z)

∈

F

α

then

there exists a measure

µ

in

M

such that for every

z

in

D

f

(z) =

Z

₁

(1

−

xz)

α

dµ(x)

Using the assumption of the theorem we get that

(f

◦

ϕ)(z) =

Z

1

(1

₋

xϕ(z))

α

dµn

(x) =

ZZ

gx

(e

it

₎

(1

₋

e

−it

_z)

α

dtdµn

(x)

which by the previous lemma was shown to be compact on

Fα.

Now we give some examples:

Corollary 2.

Let

ϕ

∈

H(

D

),

with

k

ϕ

k

∞

<

1.

Then

Cϕ

is compact on

Fα, α

≥

1.

Proof.

(Cϕ

◦

K

α x

)(z) =

1

(1

₋

xϕ(z))

α

∈

H

∞

_∩

_Fα

_⊂

_Fαa

_{and is}

subordi-nate to

1

(1

₋

z)

α

,

hence

(Cϕ

_◦

K

_xα

)(z) =

Z

K

_xα

(z)gx

e

it

dt

with

gx

(e

it

₎

_≥

_{0 and since 1 = (Cϕ}

_◦

_K

α

x

)(0) =

Z

gx

(e

it

₎

_dt

_{we get that}

k

gx

(e

it

₎

_k

1

= 1.

Remark 1.

In fact one can show that

Cϕ,

as in the above corollary, is

compact from

F

α

, α

≥

1

into

F

1

.

In other words a contraction.

Corollary 3.

If

Cϕ

is compact on

Fα, α

_≥

1

and

lim

r→1

ϕ(re

iθ

₎

_{= 1}

then

_ϕ

′

_(e

1

iθ

₎

= 0

.

Proof.

If

Cϕ

is compact then

(Cϕ

◦

K

_xα

)(z) =

Z

K

_xα

(z)gx

e

it

dt

Hence, if

z

=

e

iθ

_and

_ϕ(e

iθ

_{) =}

_x

_then

lim

r→1

(e

iθ

₋

_re

iθ

₎

α

(1

−

xϕ(re

iθ

₎₎

α

= 0.

Corollary 4.

If

Cϕ

∈

K(Fα, Fα)

for

α

≥

1,

then

Cϕ

is contraction.

4.

Miscellaneous Results

We first start by giving another characterization of compactness on

Fα.

Lemma 4.

Let

ϕ

∈

C(Fα, Fα), α >

0

then

ϕ

∈

K(Fα, Fα)

if and only if

for any bounded sequence

(fn)

in

Fα

with

fn

→

0

uniformly on compact

subsets of

D

as

n

_{→ ∞}

,

_k

C

ϕ

(f

n

)

k

_Fα

→

0

as

n

→ ∞

.

Proof.

Suppose

Cϕ

_∈

K(Fα, Fα) and let (fn) be a bounded sequence

(fn) in

Fα

with lim

n→∞

fn

→

0 uniformly on compact subsets of

D

.

If

the conclusion is false then there exists an

ǫ >

0 and a subsequence

n

1

< n

2

< n

3

<

· · ·

such that

Since (fn) is bounded and

Cϕ

is compact, one can find a another

sub-sequence

nj

1

< nj

2

< nj

3

<

· · ·

and

f

in

Fα

such that

lim

k→∞

Cϕ(fn

_jk

)

−

f

Fα

= 0

Since point functional evaluation are continuous in

Fα

then for any

z

∈

D

there exist

A >

0 such that

C

ϕ

(f

n_jk

)

−

f

(z)

_≤

A

C

ϕ

(f

n_jk

)

−

f

Fα

→

0 as

k

→ ∞

Hence

lim

k→∞

h

Cϕ(fn

_jk

)

₋

f

i

_→

0

uniformly on compact subsets of

D

.

Moreover since

f

n_jk

→

0 uniformly

on compact subsets of

D

,

then

f

= 0 i.e.

Cϕ(fn

_jk

)

_→

0 on compact

subsets of

Fα.

Hence

lim

k→∞

C

ϕ

(f

n_jk

)

Fα

= 0

which contradicts our assumption. Thus we must have

lim

n→∞

k

Cϕ(fn)

kFα

= 0.

Conversely, let (fn) be a bounded sequence in the closed unit ball of

Fα.

We want to show that

Cϕ(fn) has a norm convergent subsequence.

The closed unit ball of

Fα

is compact subset of

Fα

in the topology

of uniform convergence on compact subsets of

D

.

Therefore there is a

subsequence (f

nk

) such that

f

nk

→

f

uniformly on compact subsets of

D.

Hence by hypothesis

k

Cϕ(fnk

)

₋

Cϕ(f)

_k

_Fα

_→

0 as

k

_{→ ∞}

which completes the proof.

Proposition 2.

If

Cϕ

∈

C(Fα, Fα)

then

Cϕ

∈

K(Fα, Fβ

)

for all

β >

α >

0.

Proof.

Let (fn) be a bounded sequence in the closed unit ball of

Fα.

Then (fn◦

ϕ) is bounded in

Fα

and since the inclusion map

i

:

Fα

→

Fβa

is compact, (f

n

◦

ϕ) has a convergent subsequence in

F

β

.

Proposition 3.

C

ϕ

(f) = (f

◦

ϕ)

is compact on

F

α

if and only if the

operator

ϕ

′

_{Cϕ(g) =}

_ϕ

′

_(g

_◦

_ϕ)

_{is compact on}

_Fα

Proof.

Suppose that

Cϕ(f

) = (f

◦

ϕ) is compact on

Fα.

It is known

from [6] that

ϕ

′

_{Cϕ(g) =}

_ϕ

′

_(g

_◦

_{ϕ) is bounded on}

_Fα

+1

.

Let (gn) be a

bounded sequence in

Fα

+1

with

gn

→

0 uniformly on compact subsets

of

D

as

n

→ ∞

.

We want to show that lim

n→∞

k

ϕ

′

_(gn

_◦

_ϕ)

_kFα

+1

= 0 .

Let (f

n

) be the sequence defined by

f

n

(z) =

R

z

0

g

n

(w)dw.

Then

f

n

∈

Fα

and

k

fnk

Fα

≤

2

α

k

gnk

Fα+1

,

thus (fn) is a bounded sequence in

Fα.

Furthermore, using the Lebesgue dominated convergence theorem we

get that

fn

→

0 uniformly on compact subsets of

D

.

Thus

k

ϕ

′

(g

n

◦

ϕ)

kFα

+1

=

k

ϕ

′

_(f

′

n

◦

ϕ)

kFα

+1

=

_k

(fn

_◦

ϕ)

′

_kFα

₊₁

≤

α

k

(fn

◦

ϕ)

kFα

→

0 as

n

→ ∞

which shows that

ϕ

′

_{Cϕ(g) =}

_ϕ

′

_(g

_◦

_{ϕ) is compact on}

_Fα

+1

.

Conversely, assume that

ϕ

′

_{Cϕ(g) =}

_ϕ

′

_(g

_◦

_{ϕ) is compact on}

_Fα

+1

.

Then

in particular

ϕ

′

_Cϕ(f

′

_{) =}

_ϕ

′

_(f

′

_◦

_{ϕ) = (f}

_◦

_ϕ)

′

_{is a compact for every}

_f

_∈

Fα.

Now since

_k

(f

_◦

ϕ)

_kFα

_≤

_α2

_k

(f

_◦

ϕ)

′

_kFα

+1

.

Let (fn) be a bounded

sequence in

Fα

with

fn

_→

0 uniformly on compact subsets of

D

as

n

_→

∞

.

We want to show that lim

n→∞

k

(fn

◦

ϕ)

kFα

= 0.

Since any bounded

sequence of

Fα

is also a bounded sequence of

Fα

+1

,

then

k

(fn

◦

ϕ)

kFα

≤

2

α

k

(fn

◦

ϕ)

′

kFα

+1

→

0 as

n

→ ∞

and the proof is complete.

References

[1] Joseph A. Cima and Alec Matheson, Cauchy transforms and composition op-erators, Illinois J. Math. Vol. 42, No.1, (1998), 58-69.

[2] J. Garnett, Bounded Analytic Functions, Academic Press, New York 1980. [3] D. J. Hallenbeck and K. Samotij, On Cauchy integrals of

logarithmic potentials and their multipliers, J. Math. Anal. Appl., 174 (1993), 614-634.

[4] D. J. Hallenbeck, T. H. MacGregor and K. Samotij, Fractional Cauchy trans-forms, inner functions and multipliers, Proc. London Math. Soc. (3) 72 (1996) 157-187.

[5] R. A. Hibschweiler and T. H. MacGregor, Closure properties of families of Cauchy-Stieltjes transforms, Proc. Amer. Math. Soc. Vol. 105, No. 3 (1989), 615-621.

[6] R. A. Hibschweiler, Composition Operators on Spaces of Cauchy transforms, Contemporary Mathematics Vol. 213, (1998), 57-63.

[7] R. A. Hibschweiler and E. Nordgren, Cauchy transforms of measures and weighted shift operators on the disc algebra, Rock. Mt. J. Vol(26), 2, (1996), 627-654.

[8] T. H. MacGregor, Analytic and univalent functions with integral representa-tions involving complex measures, Indiana Univ. J. 36, (1987), 109-130 [9] S. V. Hruscev and S. A. Vinogradov, Inner functions and multipliers of Cauchy

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