BOUNDARY VALUE PROBLEMS FOR ANALYTIC FUNCTIONS IN THE CLASS OF CAUCHY-TYPE INTEGRALS WITH DENSITY IN pot

SAMKO Received 9 July 2004 We study the Riemann boundary value problemΦ+t= GtΦ−t + gt, for analyticfunctions in the class of analytic functions represented by the Cauchy-type integrals w

FUNCTIONS IN THE CLASS OF CAUCHY-TYPE

INTEGRALS WITH DENSITY IN Lp( ·)( Γ)

V KOKILASHVILI, V PAATASHVILI, AND S SAMKO

Received 9 July 2004

We study the Riemann boundary value problemΦ+(t)= G(t)Φ−(t) + g(t), for analyticfunctions in the class of analytic functions represented by the Cauchy-type integrals withdensity in the spacesL p( ·)(Γ) with variable exponent We consider both the case whenthe coefficient G is piecewise continuous and the case when it may be of a more general

nature, admitting its oscillation The explicit formulas for solutions in the variable ponent setting are given The related singular integral equations in the same setting arealso investigated As an application there is derived some extension of the Szeg¨o-Helsontheorem to the case of variable exponents

ex-1 Introduction

LetΓ be an oriented rectifiable closed simple curve in the complex planeC We denote by

D+andD −the bounded and unbounded component ofC \Γ, respectively

The main goal of the paper is to investigate the Riemann problem: find an analyticfunctionΦ on the complex plane cut along Γ whose boundary values satisfy the conjugacycondition

Φ+(t)= G(t)Φ −(t) + g(t), t ∈Γ, (1.1)whereG and g are the given functions onΓ, and Φ+, andΦ−are boundary values ofΦ

onΓ from inside and outside Γ, respectively This problem is also known as the problem

of linear conjugation

We seek the solution of (1.1) in the class of analytic functions represented by theCauchy-type integral with density in the spacesL p( ·)(Γ) with variable exponent assumingthatg belongs to the same class We consider the cases when the coefficient G is contin-

uous or piecewise continuous as well as the case of oscillating coefficient The solvabilityconditions are derived and in all the cases of solvability the explicit formulas are given.The related boundary singular integral equations inL p( ·)(Γ) are treated The solution

of the boundary value problem (BVP) (1.1) allows us to obtain the weight results forCauchy singular integral operator inL p( ·)(Γ)-spaces, among them some extension of thewell-known Helson-Szeg¨o theorem

Copyright©2005 Hindawi Publishing Corporation

Boundary Value Problems 2005:1 (2005) 43–71

DOI: 10.1155/BVP.2005.43

The problem (1.1) is first encountered in Riemann [36] Important results on whichthe posterior solution of problem (1.1) was based, were obtained by Yu Sokhotski, D.Hilbert, I Plemely, and T Carleman The complete solution of the Riemann problem wasfirst given in the works of Gakhov [7,8] and Muskhelishvili [27,28]; we refer also to theworks [14,15,16,17] on investigation of the last decades of the Riemann problem in

L p-spaces (with constantp).

The generalized Lebesgue spaces, that is, Lebesgue spaces with variable exponent, havebeen intensively studied since 1970s One may see an evident rise of interest in thesespaces during the last decade, especially in the last years The interest was aroused, apartfrom mathematical curiosity, by possible applications to models with the so-called non-standard growth in fluid mechanics, elasticity theory, in differential equations (see, e.g.,[5,37] and the references therein)

The development of the operator theory in the spacesL p( ·)encountered essential ficulties from the very beginning For example, the translation operator and the con-volution operators are not in general bounded in these spaces The boundedness of themaximal operator was recently proved by Diening [4] See further results in [2,30] There

dif-is also an evident progress in thdif-is direction for singular operators [5,20]

As is known, for applications to singular integral equations and BVPs the weightedboundedness of singular operators is required The weighted estimates in L p( ·)-spaceswith power weight were proved for the maximal operator on bounded domains in [21]and for singular operators in [20] It is worthwhile mentioning that the Fredholmnesscriteria for singular integral equations with Cauchy kernel were proved in [19] for thespacesL p( ·)and in [12] for such spaces with power weight

2 Preliminaries

Throughout the paper in all statements we suppose thatΓ= { t ∈ C:t = t(s), 0 ≤ s ≤
},with an arc-lengths, is a simple closed rectifiable curve Let a measurable p :Γ→[1,∞).TheL p( ·)-space onΓ may be introduced via the modular

as an example illustrating the theory of modular spaces developed by Nakano

The spacesL p( ·) were studied by Orlicz [31] for the first time in 1931 They are thespecial cases of the Musielak-Orlicz spaces generated by Young functions with parameter,see [25,26,33,34]

However, it was namely the specifics of the spacesL p( ·)which attracted an interest ofmany researchers and allowed to develop a rather rich basic theory of these spaces, thisinterest being also roused by applications in various areas.

Meanwhile, the norm of the type (2.2), as well as a similar norm for the Orlicz spaces isnothing else but the realization of a general norm for “normalizable” topological spacesprovided by the famous Kolmogorov theorem This theorem runs as follows, see [11,Chapter 4] and [22]

Theorem 2.1 (Kolmogorov theorem) A Hausdorff linear topological space X admits a norm if and only if it has a convex bounded neighbourhood of the null-element and in this case Minkowsky functional of this neighbourhood is a norm.

We remind that the Minkowsky functional of a setU ⊂ X is the functional M U(x),

so that the infinum ofI p(f /λ) is nothing else but the Minkowsky functional of f ∈ X =

L p( ·)related to the setU = { f : I p(f ) ≤1}

Therefore, there are many more reasons to call the norm (2.2) the Minkowsky norm.

Kolmogorov-If

1< p =ess infp(t), p =ess supp(t) < ∞, (2.4)then the spaceL p( ·)is reflexive Its associate space coincides, up to equivalence, with thespaceL q( ·), where 1/q(t) + 1/ p(t)=1

In the sequel, byᏼ(Γ), or simply by ᏼ, we denote the class of functions p measurable

with respect to the arc measure and satisfying condition (2.4) Under this condition thespaceL p( ·)coincides with the space

min

1/ p,
1/ p ≤ 1 p( ·)≤max

1/ p,
1/ p (2.8)

Letρ be a measurable, almost everywhere positive function on Γ By L p( ·)

ρ (Γ) we denotethe Banach space of functions f for which

 f  p( ·),ρ =  ρ f  p( ·)< ∞ (2.12)One of the main tools of our investigation is the Cauchy singular integral

᏷p

ρ(Γ)= Φ(z) : Φ(z) =Φ0(z) + const,Φ0∈᏷p

ρ(Γ) . (2.15)

We write᏷p(Γ)=᏷p(Γ) and᏷p(Γ)= ᏷p(Γ) in the case ρ(t)≡1

For a simply connected domainD, bounded by a rectifiable curve Γ, by E δ(D), δ > 0,

we denote the Smirnov class of functionsΦ(z) analytic in D for which

whereΓris the image ofγ r = { z : | z | = r }under conformal mapping ofU = { z : | z | < 1 }

ontoD (When D is an infinite domain, then the conformal mapping means the one

which transforms 0 into infinity.)

A functionΦ∈ E δ(D) possesses almost everywhere angular boundary values onΓ andthe boundary function belongs toL δ(Γ) (see [35, page 205]).

It is known thatE1(D) coincides with the class of analytic functions represented byCauchy integrals Therefore for the functionΦ(z), which is analytic on the plane cutting

along closed curveΓ and belongs to E1(D±),

Φ(z) = KΓ

(see, e.g., [16, page 98])

We make use of the following notations:

As shown in [20] the following statement is true

Proposition 2.2 Let Γ be a Lyapunov curve or a curve of bounded turning (Radon curve) without cusps Assume that p ∈ ᏼ and condition ( 2.10 ) is satisfied Then the weight

3 Some properties of the Cauchy-type integrals with densities inL p( ·)(Γ)

In this section, we present some auxiliary results which provide an extension of knownproperties of the Cauchy singular integrals in the Lebesgue spaces with constantp to the

con-C(Γ) by rational functions, whatsoever Jordan curve Γ we have according to the Walsh

theorem (see, for instance, [40, Chapter II, Theorem 7])

Proposition 3.2 Let Γ be a rectifiable Jordan curve, let p(t) ∈ ᏼ If 1/ρ ∈ L q( ·)(Γ), then

the operator SΓis continuous in measure, that is, for any sequence f n converging in L ρ p( ·)(Γ)

to function f0the sequence SΓf n converges in measure to SΓf0.

The validity of this statement may be obtained by word-for-word repetition from [14,proof of Theorem 2.1, page 21], sinceL ρ p( ·)(Γ)⊂ L1(Γ) according to our assumption

Theorem 3.3 Let Γ be a simple closed rectifiable curve bounding the domains D+and D − The following statements are valid.

(i) Let p and µ belong to ᏼ and let SΓmap L ρ p( ·)(Γ) to Lµ( ·)

ω (Γ) for some weight functions

ρ and ω Then 1/ρ ∈ L q( ·)(Γ) and SΓis bounded from L ρ p( ·)(Γ) into L µ( ·)

ω (Γ).

(ii) Let SΓbe bounded from L p( ρ ·)(Γ) to Lα

ω(Γ), α > 0 Then for arbitrary ϕ∈ L p( ρ ·)(Γ) the

Cauchy-type integral (KΓϕ)(z) belongs to E α(D± ).

(iii) Let p ∈ ᏼ and let SΓbe bounded in L p( ·)(Γ) Then for arbitrary ϕ ∈ L p( ·)(Γ),

(iv) For ρ ∈ W p( ·)(Γ) and ϕ∈ L p( ·)(Γ) the function KΓ(ϕ/ρ) belongs to E1(D± ) Proof (i) Since SΓ is defined for any function in L ρ p( ·)(Γ), we have the embedding

L ρ p( ·)(Γ)⊂ L1(Γ) Then for any ϕ ∈ L p( ·)(Γ) the function ϕ/ρ is integrable on Γ

There-fore, 1/ρ∈ L q( ·)(Γ) According to theProposition 3.2we conclude that for the sequence

of functionsϕ nconverging toϕ in L p( ·)(Γ) the sequence SΓϕ nconverges toSΓϕ in

mea-sure Thus, ifSΓmapsL ρ p( ·)(Γ) into L µ( ·)

ω (Γ), then SΓis a closed operator and by the closedgraph theorem we conclude that it is bounded

(ii) LetSΓbe bounded fromL ρ p( ·)(Γ) into L α(Γ), α > 0 Let ϕ ∈ L ρ p( ·)(Γ) and let ϕ nbe asequence of rational functions (with a unique pole inD+) such thatϕ nconverges toϕ in

L ρ p( ·)(Γ) (seeProposition 3.1) Then for the functionsΦn(z)=(KΓϕ n)(z) we haveΦn(z)∈

L α(D±) andΦ±

n  α ≤ M  ϕ n  p( ·),ρand byProposition 3.2Φ±

nconverges in measure to thefunction±(1/2)ϕ + (1/2)SΓϕ Applying Tumarkin’s Theorem [35, page 269], we concludethatΦ(z) =limn →∞Φn(z) belongs to Eα(D±) In our caseΦ(z) =(KΓϕ)(z).

(iii) From the embeddingL p( ·)(Γ)⊆ L p(Γ) and the boundedness of SΓ in L p( ·)(Γ) itfollows thatSΓ mapsL p( ·)(Γ) into Lp(Γ) Then by (i) SΓ is bounded fromL p( ·)(Γ) into

L p(Γ) In view of (ii), then KΓϕ ∈ E p(D±) for arbitraryϕ ∈ L p( ·)(Γ)

(iv) Sinceρ ∈ W p( ·)(Γ), we have 1/ρ ∈ L q( ·)(Γ) and then SΓ(ϕ/ρ)∈ L1(Γ) for any ϕ ∈

L p( ·)(Γ) The last follows from the equality SΓ(ϕ/ρ)=(1/ρ)(ρSΓ(ϕ/ρ)) Therefore, the eratorSΓ(1/ρ) is defined on Lp( ·)(Γ) and acts into L1(Γ) Then it is continuous in measureand consequently is a closed operator and therefore, it is bounded fromL p( ·)(Γ) to L1(Γ).Applying (ii) whenL α

op-ω(Γ)⊂ L1(Γ) we conclude that KΓ(ϕ/ρ)(z)∈ E1(D±)

Corollary 3.4 IfΓ∈᏾p( ·)and p ∈ ᏼ(Γ), then Γ is a Smirnov curve.

Indeed, sinceΓ∈᏾p( ·) and L p(Γ)⊂ L p( ·)(Γ)⊂ L p(Γ), it follows that SΓ mapsL p(Γ)intoL p(Γ) Then Γ is a Smirnov curve (see [10] and [14, page 22])

Corollary 3.5 LetΓ∈᏾p( ·) and p ∈ ᏼ(Γ) Then for arbitrary bounded function ϕ, it holds that (KΓϕ)(z) ∈β>1 E β(D± ).

Proof Since ϕ ∈α>1 L αp( ·)according to the statement (iii) fromTheorem 3.3we obtainthat (KΓϕ)(z) ∈α>1 E αp(D±) Therefore (KΓϕ)(z) ∈β>p E β(D±), that is, (KΓϕ)(z) ∈



Theorem 3.6 Let p ∈ ᏼ and let SΓ be bounded in the space L p( ·)(Γ) Then SΓ is also bounded in the space L αp( ·)(Γ) for any α > 1 and the inequality

2k+1+ 1sinπ/2k+1

Now we apply the Riesz-type interpolation theorem known for the spacesL p( ·)(see[25, Theorem 14.16]) in the following form: if a linear operator A is bounded in the spaces

L2k p( ·)(Γ) and L2k+1 p( ·)(Γ), then it is also bounded in the space Lαp( ·)(Γ) with α∈[2k, 2k+1 ),

1/α= θ2 − k+ (1− θ)2 − k −1, and

 A  αp( ·)≤  A  θ

2k p( ·) A 1− θ

2k+1 p( ·). (3.11)Then from (3.9) and (3.10) we get

SΓ

αp( ·)≤SΓ

p( ·)

ctg

 π

2k+1

θctg



=ctg

π4α

4 On belongingness of exp(KΓϕ) to the Smirnov classes whenΓ∈᏾p( ·)

Theorem 4.1 Let a closed curveΓ∈᏾p( ·)and p ∈ ᏼ(Γ) Let ϕ be a bounded measurable function on Γ Assume that z0∈ D+ Then

(i) there exists an integer k ≥ 0 such that

Proof We use an idea developed in [18] LetΓr be the image ofγ r = { z : | z | = r },r < 1,

under the conformal mapping ofU = { z : | z | < 1 }ontoD+ We have

According toCorollary 3.5 we haveΦ(z) ∈ E n(D+) for any n ≥1 Then by the knownproperty of the classE p(see [35, Chapter III]), we have

where the series on the right-hand side converges ifc0δ  SΓ p( ·)e < 1.

Thus it was proved thatX(z) ∈ E δ(D+) when

0< δ < δ0= π p

4(1 +
)eMSΓ

p( ·)

In the caseD −and for 1< δ < δ0and arbitraryr we are able to obtain similar estimates by

the same way as in caseD+ As toδ ≤1 it is necessary to consider two cases: 0< r < r0and

r0< r < 1 for some fixed r0 In the last case the appropriate estimates can be proved as inthe caseδ > 1 As to the case 0 < r < r0the needed inequalities are obtained by means ofchoice of numberk > [1/δ].

Now we are able to get a stronger result, namely, thatX(z) ∈ E δ(D+) and (X(z)−1)/( − z0)k ∈ E δ(D−) forδ < 2δ0

From the previous proof it is clear that last series converges whenδ < 2δ0

Now apply Smirnov’s following theorem (see, e.g., [35, Chapter III]): letΦ∈ E γ1(D)

andΦ+∈ L γ2(Γ) where γ2> γ1, thenΦ∈ E γ2(D) According to this statement in our case

we haveX(z) ∈ E δ(D+) and (X(z)−1)/(z− z0)k ∈ E δ(D−) whenδ < 2δ0 withδ0 from(4.12)

By this (i) is proved

Now we prove (ii) For arbitraryε > 0 we can find a H¨older function ψ onΓ such that

ess sup

t ∈Γ

ϕ(t) − ψ(t)< ε. (4.14)

On the other hand, for the H¨older functionψ(t) there exist positive numbers a1and

a2such that 0< a1≤ |exp(KΓψ)(z) | ≤ a2< ∞

Remark 4.2 As it follows from the proof of the final part of previous theorem the number

M in formula (4.12) can be replaced byν(ϕ) =inf ϕ − ψ  C, where the infinum is takenover all rational functionsψ.

5 The problem of linear conjugation with continuous coefficients

In the present paper, we proceed to the solution of problem (1.1) in the class᏷p( ·)

ρ (Γ)under various assumptions with respect to the data

We begin with the case whenp ∈ᏼ, Γ∈᏾p( ·)(Γ), and G is a nonvanishing continuousfunction onΓ The function g is assumed to be in L p( ·)(Γ) We look for a function Φ∈

᏷p( ·)(Γ) whose boundary values Φ±satisfy relation (1.1) almost everywhere onΓ.Letκ=(1/2π)[arg G(t)]Γbe the index ofG onΓ Below we will show that for the aboveformulated problem all the statements for its solvability known for constant p remain

valid in the general case of variable exponent; namely, the following statement is valid

Theorem 5.1 Let p ∈ ᏼ, Γ ∈᏾p( ·), and let g ∈ L p( ·)(Γ) Assume that G∈ C( Γ) and G(t) = 0, t ∈ Γ Then for problem ( 1.1 ) the following statements hold:

(i) forκ≥ 0, problem ( 1.1 ) is unconditionally solvable in the class᏷p( ·)(Γ) and all its solutions are given by

and Qκ−1(z) is an arbitrary polynomial of degreeκ−1(Q−1(z)≡ 0);

(ii) forκ< 0, problem ( 1.1 ) is solvable in this class if and only if

Φ

X

−

+ g

whereX(z) =exp{ KΓ(lnG)(z) } We show thatΦ/ X∈᏷p( ·)(Γ) To this end, we observethatΦ∈ E p( ·)(D±) and 1/X is bounded so thatΦ/ X∈ E p(D±) and therefore

Φ

Φ(z)

X(z) = KΓψ (z), ψ ∈ L p( ·)(Γ) (5.8)Then equality (5.6) yields

that is, the function ψ is a solution of the equation of the type ψ = Kψ in the space

L p( ·)(Γ), where K is a contractive operator Therefore, (5.9) and consequently problem(1.1) have the unique solution in᏷p( ·)(Γ) Basing onTheorem 4.1we construct the solu-tion Let

IfΦ is a solution of problem (1.1), thenΦ∈᏷p(Γ) and therefore, Φ∈ E p(D±) Moreover,

Φ/X ∈ E p − ε(D±) for arbitraryε ∈(0, 1/ p) ThereforeΦ/X ∈᏷p − ε(Γ) So Φ/X ∈᏷1(Γ)

At the same time

is the solution of (1.1) in the class᏷p( ·)(Γ)

Let nowκ> 0 We choose a point z0∈ D+and rewrite (1.1) in the form

Φ+(t)= G1(t)

t − z0 κΦ−(t) + g(t), (5.15)whereG1(t)=(t− z0)−κG(t) is a continuous function with zero index We introduce a

new unknown function

F(z) =





Φ(z), z ∈ D+,

z − z0 κΦ(z), z ∈ D − (5.16)

ForF(z) there exists a polynomial Qκ−1(z) such that

KΓlnG1 (z) . (5.20)Here

(z)= X1(z)KΓ



g

X+ 1

(z) + X1(z)Qκ−1(z)− Qκ−1(z) (5.23)

Then by (5.16) and (5.17) we arrive at formula (5.1)

It can be easily verified that the latter provides the solution of problem (1.1) for anarbitrary polynomialQκ−1(z) which does not depend on the choice of the point z0.Finally, we consider the caseκ< 0 This time the function F given by (5.16) is in

and the conditionΦ(z) =( − z0)−κF ∈ E1(D−) is fulfilled if and only if the conditions

belongs to the class W p( ·)(Γ)

Proof Consider the problem (1.1) in the class᏷p( ·)(Γ) with G(t)=exp (iϕ(t)) and g∈

L p( ·)(Γ) Obviously, G ∈ C( Γ) and indG =0 Consequently, the function

6 The problem of linear conjugation with continuous

coe fficients in the weighted class ᏷p( ·)

ρ

If we assume thatρ ∈ W p( ·)(Γ) and choose the functionG in the proof of Theorem 5.1

such that instead of condition (5.5), the condition

5 The problem of linear conjugation with continuous coefficients

In the present paper, we proceed to the solution of. .. extension of knownproperties of the Cauchy singular integrals in the Lebesgue spaces with constantp to the< /i>

con-C(Γ) by rational functions, whatsoever Jordan curve Γ we have according... show that for the aboveformulated problem all the statements for its solvability known for constant p remain

valid in the general case of variable exponent; namely, the following statement