RIEMANN BOUNDARY VALUE PROBLEM FOR HYPERANALYTIC FUNCTIONS

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FOR HYPERANALYTIC FUNCTIONS

RICARDO ABREU BLAYA, JUAN BORY REYES, AND DIXAN PE ˜NA PE ˜NA Received 23 February 2005

We deal with Riemann boundary value problem for hyperanalytic functions. Further- more, necessary and suﬃcient conditions for solvability of the problem are derived. At the end the explicit form of general solution for singular integral equations with a hypercomplex Cauchy kernel in the Douglis sense is established.

1. Introduction

The theory of Riemann boundary value problem for analytic functions of one complex variable and singular integral equations that are equivalent to it has been extensively studied in the literature. For classical books on this topic see [7,12,13] and for an actual overview of them the reader is directed to the monograph by Estrada and Kanwal [6], and the references therein.

In the more recent times several generalizations and extensions of the theory are treated and have led to numerous important results not only for nonsmoothly bounded domain, which differs with the former, but for general assumptions on the data of the problem, such as generalized Hölder coefficients or special subspaces of this space and the desired boundary behavior condition for the solution. During the last decades, the Riemann boundary value problem was studied for generalized analytic functions, as well as for many other linear and nonlinear elliptic systems in the plane [1,2,8,15,16,17].

Letγbe a rectifiable positively oriented Jordan closed curve with diameterdwhich is the boundary of a bounded simply connected domainΩ+in the complex planeCand let Ω−:=C\(Ω+∪γ).

In the Douglis commuting function algebra sense, a continuously diﬀerentiable null solution to the Douglis diﬀerential operator provides us with the class of hyperanalytic functions. Let Ꮽ(Ω±) be the spaces of all continuous functions onΩ±:=Ω±∪γ and hyperanalytic inΩ±.

The classical Riemann boundary value problem for analytic functions consists in find- ing a functionΦ(z) analytic inC\γ, such thatΦhas a finite order at infinity, and satis- fies a prescribed jump condition across the curveγ. The basic boundary condition takes

International Journal of Mathematics and Mathematical Sciences 2005:17 (2005) 2821–2840 DOI:10.1155/IJMMS.2005.2821

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the form

Φ⁺(t)−G(t)Φ⁻(t)=g(t), t∈γ, (1.1) whereG,gare given continuous functions onγ, andΦ⁺(t) andΦ⁻(t) represent the limit values, in a suitable sense, of the desired functionΦat a pointtas this point is approached fromΩ+and fromΩ−, respectively.

The Riemann boundary value problem for analytic functions as well as for hyperanalytic functions in the case where the unknown functions are continuous up to the boundary or their continuity is violated only at a finite number of points are well studied and described in the fundamental monographs [7,8,12,13]. The Riemann problems mentioned above are usually called a continuous or piecewise continuous boundary value problem, respectively.

The present paper is concerned with hyperanalytic Riemann boundary value problem (where instead of analyticity one requires the hyperanalyticity ofΦ) in the continuous case (the solutions including their boundary values onγare continuous). The purpose of the paper is to present an explicit form of the general solution of the problem.

The motivation comes on the one hand from the study of the hyperanalytic Riemann boundary value problem with continuous coeﬃcients [10] and on the other from the necessary and suﬃcient solvability condition, which will be imposed on the layer function of the Cauchy type integral so that this integral provides the solution of the basic jump problem.

The main result is moreover applied to describe the general solution of a singular integral equation with a hypercomplex Cauchy kernel.

2. Preliminaries

For the sake of completeness we recall some basic notions and results in Douglis analysis, that is, a Douglis-algebra-valued function theory which is a generalization of classical complex analysis in the plane. For more details concerning this function theory and its application, we refer the reader to [1,8,16,17].

LetDbe the Douglis algebra generated by the elementsiande. The multiplication in Dis governed by the rules

i²= −1, ie=ei, e^r=0, e⁰=1, (2.1) whereris a positive integer.

An arbitrary elementa∈Dmay be written as a hypercomplex number of the form a=

r−1 k=0

ake^k, (2.2)

where eacha_kis a complex number,a0is called the complex part ofa, andA:=_r₋₁

k=1a_ke^k is the nilpotent part.

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The conjugationa→ainDis defined as a:=

r−1 k=0

¯

ake^k. (2.3)

InDthe algebraic norm ofais defined by

|a|:=

r−1 k=0

a_k. (2.4)

It is easily seen that

|ab| ≤ |a||b|, |a+b| ≤ |a|+|b|, (2.5) for any hypercomplex numbersaandb. The multiplicative inversea⁻¹ofawith complex parta0=0 is given by

a⁻¹or1 a⁼

1 a0

r−1 k=0

(−1)^k A

a0

k

. (2.6)

Conversely, ifa0=0, thenadoes not have a multiplicative inverse andais called nilpotent.

In what follows, we will consider functionsD-valued, which are defined in some subset Ω⊂C.

We say that f =_r₋1

k=0fke^k, where fk are complex-valued functions, belongs to some classical class of functions onΩif each of its components f_kbelongs to that class.

The Douglis operator∂^q_z is given by

∂^q_z:=∂_z+q(z)∂_z, z=x+iy (2.7) hereq(z) is a known nilpotent hypercomplex function and

∂z:=1 2

∂x+i∂y

, ∂z:=1 2

∂x−i∂y

. (2.8)

Definition 2.1. A continuously diﬀerentiable hypercomplex function f is hyperanalytic inΩif∂^q_zf =0 inΩ.

The basic example of a hyperanalytic function is the generating solution of the Douglis operator given by

W(z)=z+

r−1 k=1

Wk(z)e^k, (2.9)

namely,∂^q_zW(z)=0 and its nilpotent part^r_k⁻₌¹1W_ke^kpossess bounded and continuous derivatives up to order two inC.

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Since in the paper we have employed the lettertto denote a generic point in the curve γ, we have decided to denote the generating solution of the Douglis operator withW(z) instead of the standard notationt(z).

The hypercomplex Cauchy kernel, that is, the fundamental solution of the Douglis operator, is given by

2e_z(ζ) := 1 π

∂ζW(ζ)

W(ζ)−W(z), ζ=z. (2.10)

For f,g∈Ꮿ¹(Ω+)∩Ꮿ(Ω+) Green’s identity can be formulated within the framework of hypercomplex function theory in the following way:

γ∂ζW(ζ)f(ζ)nq(ζ)g(ζ)ds=2

Ω+

∂ζW(ζ)

f ∂^q_ζg+g∂^q_ζf

dΩ+, (2.11)

wherenq(ζ) :=n(ζ) +n(ζ)q(ζ),n(ζ) denotes the exterior unit normal vector toγat the pointζ, anddsis an arc length diﬀerential.

Green’s identity leads to the Cauchy-Pompeiu integral representation formula for smooth functions

f(z)=

γe_z(ζ)n_q(ζ)f(ζ)ds−2

Ω+

e_z(ζ)∂^q_ζf(ζ)dΩ+, z∈Ω+, (2.12) while for hyperanalytic functions coincides with Cauchy’s formula.

f(z)=

γez(ζ)nq(ζ)f(ζ)ds, z∈Ω+. (2.13) Definition 2.2. Suppose F is a hyperanalytic function outside of an open ballBR with radiusR >0 and center at the origin and letγ0be any rectifiable positively oriented Jordan closed curve such thatγ0lies inC\B_Rand surroundsB_R. The hypercomplex number

Resζ=∞ F(ζ):= − 1 2π

γ0

W_ζ(ζ)n_q(ζ)F(ζ)ds (2.14) is called the residue ofFat infinity.

Note that, according to (2.11), the integral in the right-hand side does not depend on the choice of the curveγ0.

Theorem2.3. LetF∈Ꮽ(Ω−), then

γez(ζ)nq(ζ)F(ζ)ds=











−Resζ=∞

F(ζ) W(ζ)−W(z)

, z∈Ω+

−F(z)−Resζ=∞

F(ζ) W(ζ)−W(z)

, z∈Ω−.

(2.15)

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Proof. Supposez∈Ω−and letR >0 such thatΩ+⊂B_Randz∈B_R. From Cauchy’s formula we get

γe_z(ζ)n_q(ζ)F(ζ)ds= −F(z) +

∂BR

e_z(ζ)n_q(ζ)F(ζ)ds

= −F(z)−Resζ=∞

F(ζ) W(ζ)−W(z)

.

(2.16)

The casez∈Ω+is similar.

Notationcwill be used for constants which may vary from one occurrence to the next;

in general these constants only depend onq.

Lemma2.4. Letγ(t) := {ζ∈γ:|ζ−t| ≤}, fort∈γand let∈(0,d].

(i)IfF∈Ꮽ(Ω+), then

γ\γ(t)e_t(ζ)n_q(ζ)F(ζ)−F(t)ds≤c max

z∈Ω+,_|z−t|=

F(z)−F(t). (2.17)

(ii)IfF∈Ꮽ(Ω−), then

γ\γ(t)et(ζ)nq(ζ)F(ζ)−F(t)ds+F(t) + Resζ=∞

F(ζ) W(ζ)−W(t)

≤c max

z∈Ω−,_|z−t|=

F(z)−F(t). (2.18)

Proof. The first assertion was already proved in [10]. We consider the second one. Sup- poseB(t) is the open ball with centertand radius, and let

Q:=Ω+∪B(t). (2.19)

Sinceγ_\γ(t)₌∂Q_\(∂Q_∩∂B(t)), we have

γ\γ(t)et(ζ)nq(ζ)F(ζ)−F(t)ds

=

∂Qe_t(ζ)n_q(ζ)F(ζ)₋F(t)ds₋

∂Q∩∂B(t)e_t(ζ)n_q(ζ)F(ζ)₋F(t)ds

=

∂Qe_t(ζ)n_q(ζ)F(ζ)ds−F(t)−

∂Q∩∂B(t)e_t(ζ)n_q(ζ)F(ζ)−F(t)ds.

(2.20)

In view of the previous theorem,

γ\γ(t)et(ζ)nq(ζ)F(ζ)−F(t)ds+F(t) + Resζ=∞

F(ζ) W(ζ)−W(t)

=

∂Q∩∂B(t)e_t(ζ)n_q(ζ)F(ζ)−F(t)ds≤c max

z∈Ω−,_|z−t|=

F(z)−F(t). (2.21)

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For each f ∈Ꮿ(γ) the Cauchy-type integral is given by Cγf(z) :=

γez(ζ)nq(ζ)f(ζ)ds, z /∈γ. (2.22) ClearlyCγf is hyperanalytic inC\γ.

Thus the singular Cauchy integral operator (Hilbert transform) onγreads as Sγf(t) :=2

γet(ζ)nq(ζ)f(ζ)−f(t)ds+f(t), t∈γ, (2.23) where the integral which definesSγf has to be taken in the sense of Cauchy’s principal value, and the function f is such that the integrals

γ(t)et(ζ)nq(ζ)f(ζ)−f(t)ds (2.24) converge uniformly to zero fort∈γas→0.

The space of all continuous functions onγwhich satisfy the above condition will be denoted by᏿(γ). Taking into accountLemma 2.4, we getᏭ(Ω±)⊂᏿(γ).

From now on we always assumeγto be a regular closed curve, that is, the quotient of the length ofγinside any circle to the radius of the circle is less than some fixed constant.

In a previous paper [3] we already studied the problem of establishing necessary and suﬃcient condition for the Cauchy-type integral to be continuously extended ontoγ. The following theorem was proved in [3].

Theorem 2.5. Let f ∈Ꮿ(γ), then Cγf has continuous limit values on γ if and only if f ∈᏿(γ). Moreover,

Ω±limz→t

Cγf(z)=1 2

Sγf(t)±f(t). (2.25)

Therefore, for any f ∈᏿(γ) the functions C^±_γ f(z) :=







Cγf(z), z∈Ω±, 1

2

S_γf(z)±f(z), z∈γ, (2.26)

are continuous onΩ±.

Let᏿^±(γ) be the spaces of all continuous functions f onγwhich have hyperanalytic extensions f^±toΩ±and f⁻(∞)=0.

ByTheorem 2.5we get the splitting of the space᏿(γ) as

᏿(γ)=᏿⁺(γ)⊕᏿⁻(γ), (2.27)

and the corresponding splitting of the functions considered f = f⁺+ f⁻, where f^±∈

᏿^±(γ).

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Now, forf ∈Ꮿ(γ) we consider the modulus of continuity for f as ω_f(ξ) :=ξsup

ρ≥ξ

ρ⁻¹ max

t1,t2∈γ,|t1−t2|≤ρ

ft1

−ft2, ξ >0. (2.28)

Let

I0(γ) :=

f ∈Ꮿ(γ) : _d

0

ω_f(ξ)

ξ dξ <+∞

. (2.29)

For a majorantϕ, that is,ϕis a positive real function on (0,d] such thatϕ(ξ) does not de- crease,ϕ(ξ)/ξdoes not increase, andϕ(ξ)→0 asξ→0, we now introduce the generalized H¨older continuous functions space

Ᏼϕ(γ) :=

f ∈Ꮿ(γ) :ωf(ξ)≤cϕ(ξ),ξ∈(0,d] (2.30) and its subspace

ᐆϕ(γ) :=

f ∈Ᏼϕ(γ) :Θf(ξ)≤cϕ(ξ), ξ∈(0,d], (2.31) where

Θf(ξ) :=ξsup

ρ≥ξ

ρ⁻¹ sup

∈(0,ρ],t∈γ

γ(t)e_t(ζ)n_q(ζ)f(ζ)−f(t)ds. (2.32) One can define a norm inᐆϕ(γ) by

fᐆϕ:=max

t∈γ

f(t)+ sup

ξ∈(0,d]

ωf(ξ) ϕ(ξ) + sup

ξ∈(0,d]

Θf(ξ)

ϕ(ξ) . (2.33)

In the sequel we will make use of the following two technical lemmas, from which other results will follow.

Lemma2.6. Let f ∈᏿(γ), then fort∈γand∈(0,d],

sup

z∈Ω±,|z−t|=

C^±_γ f(z)−

C^±_γ f(t)≤c

ω_f() +Θf() + _d

ω_f(ξ) ξ² dξ

, (2.34)

is true.

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Proof. It is only necessary to distinguish the following two cases.

Case 1. z∈Ω±. Lett_z∈γsuch that|z−t_z| =dist(z,γ)=:ν. Then for|z−t| =, Cγf(z)−

C^±_γ f(t)

=

γν(tz)ez(ζ)nq(ζ)f(ζ)−ftz ds +

γ\γν(tz)

ez(ζ)−etz(ζ)nq(ζ)f(ζ)−ftz ds

−

γν(tz)e_t_z(ζ)n_q(ζ)f(ζ)−ft_zds+1 2

S_γft_z−

S_γf(t)

±1 2

ftz

−f(t)=: 5 k=1

Jk.

(2.35)

Because|z−tz| ≤ |ζ−z|and|ζ−tz| ≤ |ζ−z|+|z−tz| ≤2|ζ−z|, J1+J2≤c

γν(tz)

ω_fζ−t_z

|ζ−z| ds+z−t_z

γ\γν(tz)

ω_fζ−t_z

|ζ−z|ζ−t_zds

≤c

ω_f(ν) +z−t_z

γ\γν(tz)

ω_fζ−t_z ζ−t_z² ds

≤c

ωf(ν) +ν^d

ν

ωf(ξ) ξ² dξ

.

(2.36)

First, we remark that the second term in the last inequality is an almost-increasing function inν. There is no loss of generality in assuming thatν≤≤d/2, then

3 _d

ωf(ξ)

ξ² dξ−ν^d

ν

ωf(ξ) ξ² dξ

=(−ν)^d

ωf(ξ) ξ² dξ+

2

_d

ωf(ξ)

ξ² dξ−ν

ν

ωf(ξ) ξ² dξ

.

(2.37)

Further, it is clear that 2

_d

ωf(ξ)

ξ² dξ≥2ωf() _d

dξ

ξ² ⁼2ωf()d−

d ⁼2ωf()d−

d ^≥ωf(). (2.38) On the other hand,

ν

ω_f(ξ)

ξ² dξ≤νωf()

ν

dξ

ξ² ⁼νωf()⁻ν

ν =ωf()⁻ν

≤ωf(), (2.39)

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and hence

3 _d

ωf(ξ)

ξ² dξ−ν^d

ν

ωf(ξ)

ξ² dξ≥0. (2.40)

It then follows that

J1+J2≤c

ωf() + _d

ωf(ξ) ξ² dξ

. (2.41)

ForJ3we obtain

J3≤Θf(ν)≤Θf(). (2.42)

Since|t_z−t| ≤ |t_z−z|+|z−t| ≤2|z−t| =2, then

J4≤ωS_γf(), J5≤ω_f(). (2.43)

On the other hand, starting from the reasonings in [3, Theorem 2] and using the definition ofΘf, we get

J4≤ωS_γf()≤c

ω_f() +Θf() + _d

ω_f(ξ) ξ² dξ

. (2.44)

Therefore

Cγf(z)−

C^±_γ f(t)≤c

ωf() +Θf() + _d

ωf(ξ) ξ² dξ

. (2.45)

Case 2. z∈γ. If|z−t| =, then C^±_γ f(z)−

C^±_γ f(t)=1 2

S_γf(z)−S_γf(t)±1 2

f(z)−f(t). (2.46)

For that

C^±_γf(z)−

C^±_γf(t)≤ωS_γf() +ωf(), (2.47) hence

(C^±_γ f(z)−

C^±_γ f(t)≤c

ωf() +Θf() + _d

ωf(ξ) ξ² dξ

. (2.48)

The statement of the lemma follows now from the above-considered cases.

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Lemma2.7. Suppose f ∈᏿(γ)and let∈(0,d]. Then ΘS_γf()≤c

ωf() +Θf() + _d

ω_f(ξ) ξ² dξ

. (2.49)

Proof. Fort∈γwe have (Sγf)(t)=2(C⁺_γf)(t)−f(t). Therefore

γ(t)e_t(ζ)n_q(ζ)Sγf(ζ)−

Sγf(t)ds

≤Θf() + 2

γ(t)et(ζ)nq(ζ)C⁺_γf(ζ)−

C⁺_γf(t)ds.

(2.50)

As a consequence ofLemma 2.4we have

γet(ζ)nq(ζ)C⁺_γf(ζ)−

C⁺_γf(t)ds=0, (2.51) for that

γ(t)et(ζ)nq(ζ)C⁺_γf(ζ)−

C⁺_γf(t)ds

=

γ\γ(t)e_t(ζ)n_q(ζ)C⁺_γf(ζ)−

C⁺_γf(t)ds.

(2.52)

Thus, returning to theLemma 2.4, we obtain

γ(t)et(ζ)nq(ζ)C⁺_γf(ζ)−(C⁺_γf(t)ds≤c sup

z∈Ω+,|z−t|=

C⁺_γf(z)−

C⁺_γf(t) (2.53) and the following estimate follows in view of the result ofLemma 2.6:

γ(t)e_t(ζ)n_q(ζ)C⁺_γf(ζ)−

C⁺_γf(t)ds≤c

ω_f() +Θf() + _d

ω_f(ξ) ξ² dξ

. (2.54) Finally

γ(t)e_t(ζ)n_q(ζ)S_γf(ζ)−

S_γf(t)ds≤c

ω_f() +Θf() + _d

ω_f(ξ) ξ² dξ

. (2.55)

This completes the proof.

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The following theorem gives suﬃcient condition for the singular Cauchy integral operator to be bounded onᐆϕ(γ).

Theorem2.8. Supposeϕis a majorant such that for∈(0,d],

_d

ϕ(ξ)

ξ² dξ≤cϕ(). (2.56)

Then the singular Cauchy integral operatorSγis a bounded operator onᐆϕ(γ), that is,

S_γf_ᐆ_ϕ≤cfᐆϕ, (2.57)

for anyf ∈ᐆϕ(γ).

Proof. Suppose f ∈ᐆϕ(γ), then for∈(0,d],

ω_f()≤ fᐆϕϕ(), Θf()≤ fᐆϕϕ(); (2.58) hence

_d

ωf(ξ)

ξ² dξ≤fᐆϕ

_d

ϕ(ξ)

ξ² dξ≤cfᐆϕϕ(). (2.59) Combining the previous estimates withLemma 2.7, we obtain

ΘSγf()≤cfᐆϕϕ(), (2.60) and similarly

ωS_γf()≤cfᐆϕϕ(). (2.61) So we have at onceSγf ∈ᐆϕ(γ). Besides

S_γf(t)≤2Θf(d) +fᐆϕ≤cfᐆϕ, t∈γ, (2.62) thus

Sγf_ᐆ_ϕ≤cfᐆϕ, (2.63)

that is,Sγis a bounded operator onᐆϕ(γ).

Notice that the above theorem is a generalization of an analogous result by Bustamante and Gonz´alez in [11] treated in the framework of complex analysis.

3. Riemann boundary value problem

For the Riemann boundary value problem for hyperanalytic functions, due to the tech- nique of canonical factorization, the interested reader can find a good introduction in [8].

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At the same time early quite more-general results on complete solution of the Riemann problem for hypercomplex functions, in case where the integration curve is rectifiable and the coeﬃcients are assumed to be in special subspace of continuous function space, have been obtained in [10,18].

Systematically the study of the solvability of the Riemann boundary problem for analytic functions is technically involved with the Cauchy-type integral which, by taking boundary values, led to the Sokhotski-Plemelj formulas. In such way the singular Cauchy integral operator appearing in these formulas transforms the Riemann problem to a singular integral equation. A good presentation for excellent examples of such study which is close to the present paper could be [4,5,14].

The main goal of this paper is to develop a theory of the well-posed continuous Rie- mann boundary value problem by assuming that the given continuous coeﬃcients defined onγhave to agree with the desired boundary behavior of the solutions, that is, the solutions including their boundary values onγare continuous functions too.

The results of boundedness of the singular Cauchy integral operator and on the continuous extension of the Cauchy-type integral, as established in the previous section, are now applied to obtain a necessary and suﬃcient condition for the solvability of the Rie- mann problem with coeﬃcientGin the case whereGadmits a canonical factorization

G=H⁺

H⁻, (3.1)

whereH^±∈Ꮽ(Ω±) and the complex partsH₀^±never vanish onΩ±.

Take an arbitrary fixed pointz0inΩ+. We restrict ourselves to the case of a hypercomplex functionGwhich has complex partG0and never vanishes onγ. From this assump- tion the integer

κ:= 1

2π argG0(ζ)_γ (3.2)

has significant importance and is called the index ofGwith respect toγ, also called index of the Riemann boundary value problem. Note that the index of the function (W(ζ)− W(z0))^κwith respect toγisκ, and hence the index of (W(ζ)−W(z0))⁻^κG(ζ) is zero.

We may verify directly that X(z) :=



expΓ(z), z∈Ω+,

W(z)−Wz0

₋κ

expΓ(z), z∈Ω−, (3.3) in which

Γ(z) :=

γez(ζ)nq(ζ) ln W(ζ)−Wz0−κ

G(ζ)ds, z /∈γ, (3.4) is a hyperanalytic function inC\γ.

Remark about the hypercomplex exponential and logarithmic functions can be found for instance in [8,9].

The following theorem gives characterization for hypercomplex function to admit a canonical factorization.

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Theorem3.1. The functionGadmits a canonical factorization if and only ifln[(W(t)− W(z0))⁻^κG(t)]∈᏿(γ).

Proof. AssumeGadmits a canonical factorization, then we deduce ln W(t)−Wz0

₋κ

G(t)v=lnH⁺(t)−ln W(t)−Wz0

κ

H⁻(t), (3.5) fort∈γandH^±satisfying (3.1).

Now the necessity follows fromLemma 2.4and the previous equality.

Conversely, suppose ln[(W(t)−W(z0))⁻^κG(t)]∈᏿(γ). Taking into account (3.5) and Theorem 2.5, it is easy to check thatXgives a canonical factorization ofGonγin the form

G=X⁺/X⁻.

Definition 3.2. p0+p1W(z) +···+ps(W(z))^s,s≥0, is called a hypercomplex polynomial.

Theorem3.3. SupposeGadmits a canonical factorization. For the homogeneous Riemann boundary value problem(g(t)≡0), ifΦ(∞)=0, then it hasκlinearly independent solutions whenκ >0and has only the trivial solution whenκ≤0. The general solution is given by

Φ(z)=X(z)Pκ−1(z), (3.6)

wherePκ−1is an arbitrary hypercomplex polynomial whose degree is not greater thanκ−1 (P_κ₋1≡0whenκ≤0).

Proof. The proof is standard [1,8,9,10] running along similar lines to those in the com-

plex case [4,5,7,12,13] and will not be given here.

Theorem3.4. AssumeGadmits a canonical factorization. The Riemann boundary value problem is solvable if and only ifg/X⁺∈᏿(γ).

Proof. The boundary condition (1.1) may be rewritten as Φ⁺(t)

X⁺(t)⁻ Φ⁻(t) X⁻(t)⁼

g(t)

X⁺(t), t∈γ. (3.7)

The necessary condition is an immediate consequence of (3.7) andLemma 2.4. Let now g/X⁺∈᏿(γ), then obviously the function

X(z)Cγ

g/X⁺(z) (3.8)

is a solution of the Riemann boundary value problem.

We consider nowG∈I0(γ) andg∈᏿(γ). Note that lnW(t)−Wz0

−κ

G(t)∈I0(γ)⊂᏿(γ), (3.9) hence,Gadmits a canonical factorization.

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Taking into account the conditiong∈᏿(γ), it can easily be shown that

γ(t)et(ζ)nq(ζ) g(ζ)

X⁺(ζ)⁻ g(t) X⁺(t)

ds

=

γ(t)et(ζ)nq(ζ) g⁺(ζ)

X⁺(ζ)⁻ g⁺(t) X⁺(t)

ds + 1

G(t)

γ(t)et(ζ)nq(ζ) g⁻(ζ)

X⁻(ζ)⁻ g⁻(t) X⁻(t)

ds +

γ(t)e_t(ζ)n_q(ζ) 1

G(ζ)⁻ 1 G(t)

g⁻(ζ)

X⁻(ζ)ds=:J1+J2+J3, ∈(0,d].

(3.10)

In view of the result ofLemma 2.4,J1,J2converge uniformly to zero fort∈γ, as→0.

Furthermore, as

J3≤c

0

ω_G(ξ)

ξ dξ, (3.11)

we have thatg/X⁺∈᏿(γ).

Taking into account the above andTheorem 3.4, we thus have proved the following theorem.

Theorem3.5. LetG∈I0(γ)andg∈᏿(γ). Under the requirementΦ(∞)=0, the Riemann boundary value problem is solvable whenκ≥0with the general solution

Φ(z)=X(z)Cγ

g/X⁺(z) +X(z)Pκ−1(z). (3.12) HereP_κ₋1(z)is the same as above; when κ <0, it is (uniquely) solvable with the solution (3.12) (Pκ−1(z)≡0whenκ≤0) if and only if the following conditions are satisfied:

γ∂_ζW(ζ)n_q(ζ) g(ζ) X⁺(ζ)

W(ζ)^kds=0, k=0,. . .,−κ−1. (3.13)

4. Singular integral equations in the classᐆϕ(γ)

This section is devoted to the study of the solvability of the singular integral equation of the type

a(t)Υ(t) +b(t)SγΥ(t)=f(t), t∈γ, (4.1) for which we assume thata,b, andf are given hypercomplex functions. For this equation, Υis an unknown hypercomplex function belonging toᐆϕ(γ).

We assume thata,b_∈Ᏼψ(γ), f _∈ᐆϕ(γ), anda²₀(t)₋b²₀(t)₌0 fort_∈γ. Hereϕ,ψare given majorants satisfying the following relations:

_d

ϕ(ξ)

ξ² dξ≤cϕ(),

0

ψ(ξ)

ξ dξ≤cϕ(), ∈(0,d].

(4.2)

(15)

We consider the Riemann boundary value problem associated with the singular integral equation (4.1),

Φ⁺(t)−a(t)−b(t)

a(t) +b(t)Φ⁻(t)= f(t)

a(t) +b(t), t∈γ,Φ(∞)=0, (4.3) which is similar to (1.1). To simplify notation we write

G(t) :=a(t)−b(t)

a(t) +b(t), g(t) := f(t)

a(t) +b(t), t∈γ. (4.4) In view of the indicated relation, the indexκof the function (a0(t)−b0(t))/(a0(t) +b0(t)), and the corresponding functionsΓ(z) andX(z), we arrive at the following assertions.

Theorem4.1. Leta,b∈Ᏼψ(γ)and f ∈ᐆϕ(γ). Then (i)G∈Ᏼψ(γ),

(ii)g,X^±∈ᐆϕ(γ).

Proof. By straightforward computation, the statement (i) can be obtained. On the other hand, sinceψ(ξ)/ξdoes not increase,

ψ()=

0

ψ()

dξ≤

0

ψ(ξ)

ξ dξ≤cϕ(), ∈(0,d], (4.5) we thus have thatg∈Ᏼϕ(γ).

Let now

h(t) :=

a(t) +b(t)⁻¹, t∈γ, (4.6) then

γ(t)e_t(ζ)n_q(ζ)f(ζ)h(ζ)−f(t)h(t)ds

≤h(t)

γ(t)e_t(ζ)n_q(ζ)f(ζ)−f(t)ds +

γ(t)e_t(ζ)n_q(ζ)h(ζ)₋h(t)f(ζ)ds

≤c

Θf() +

0

ωh(ξ) ξ dξ

≤c

Θf() +

0

ψ(ξ) ξ dξ

≤cϕ(), ∈(0,d].

(4.7)

Thereforeg∈ᐆϕ(γ).