**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 hyper- complex 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 stud- ied 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 H¨older 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].

Let*γ*be a rectifiable positively oriented Jordan closed curve with diameter*d*which 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 that*Φhas a finite order at infinity, and satis-
fies a prescribed jump condition across the curve*γ. The basic boundary condition takes*

Copyright©2005 Hindawi Publishing Corporation

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

the form

Φ^{+}(t)*−**G(t)*Φ* ^{−}*(t)

*=*

*g(t),*

*t*

*∈*

*γ,*(1.1) where

*G,g*are given continuous functions on

*γ, and*Φ

^{+}(t) andΦ

*(t) represent the limit values, in a suitable sense, of the desired functionΦat a point*

^{−}*t*as this point is approached fromΩ+and fromΩ

*−*, respectively.

The Riemann boundary value problem for analytic functions as well as for hyper- analytic 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 stud- ied 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 on*γ*are 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 elements*i*and*e. The multiplication in*
Dis governed by the rules

*i*^{2}*= −*1, *ie**=**ei,* *e*^{r}*=*0, *e*^{0}*=*1, (2.1)
where*r*is a positive integer.

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

*r**−*1
*k**=*0

*a**k**e** ^{k}*, (2.2)

where each*a** _{k}*is a complex number,

*a*0is called the complex part of

*a, andA*:

*=*

_{r}

_{−}_{1}

*k**=*1*a*_{k}*e** ^{k}*
is the nilpotent part.

The conjugation*a**→**a*inDis defined as
*a*:*=*

*r**−*1
*k**=*0

¯

*a**k**e*^{k}*.* (2.3)

InDthe algebraic norm of*a*is 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 numbers*a*and*b. The multiplicative inversea*^{−}^{1}of*a*with complex
part*a*0*=*0 is given by

*a*^{−}^{1}or1
*a*^{=}

1
*a*0

*r**−*1
*k**=*0

(*−*1)^{k}*A*

*a*0

*k*

*.* (2.6)

Conversely, if*a*0*=*0, then*a*does not have a multiplicative inverse and*a*is called nilpo-
tent.

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

We say that *f* *=*_{r}* _{−}*1

*k**=*0*f**k**e** ^{k}*, where

*f*

*k*are complex-valued functions, belongs to some classical class of functions onΩif each of its components

*f*

*belongs to that class.*

_{k}The Douglis operator*∂*^{q}* _{z}* is given by

*∂*^{q}* _{z}*:

*=*

*∂*

*+*

_{z}*q(z)∂*

*,*

_{z}*z*

*=*

*x*+

*iy*(2.7) here

*q(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}_{z}*f* *=*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

*W**k*(z)e* ^{k}*, (2.9)

namely,*∂*^{q}_{z}*W(z)**=*0 and its nilpotent part^{}^{r}_{k}^{−}_{=}^{1}1*W*_{k}*e** ^{k}*possess bounded and continuous
derivatives up to order two inC.

Since in the paper we have employed the letter*t*to 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 notation*t(z).*

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

2e* _{z}*(ζ) :

*=*1

*π*

*∂**ζ**W(ζ)*

*W(ζ)**−**W(z)*, *ζ**=**z.* (2.10)

For *f*,g*∈*Ꮿ^{1}(Ω+)*∩*Ꮿ(Ω+) Green’s identity can be formulated within the framework of
hypercomplex function theory in the following way:

*γ**∂**ζ**W(ζ)f*(ζ)n*q*(ζ)g(ζ)ds*=*2

Ω+

*∂**ζ**W(ζ)*

*f ∂*^{q}_{ζ}*g*+*g∂*^{q}_{ζ}*f*

*dΩ*+, (2.11)

where*n**q*(ζ) :*=**n(ζ) +n(ζ)q(ζ*),*n(ζ) denotes the exterior unit normal vector toγ*at the
point*ζ, andds*is an arc length diﬀerential.

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

*f*(z)*=*

*γ**e** _{z}*(ζ)n

*(ζ)*

_{q}*f*(ζ)ds

*−*2

Ω+

*e** _{z}*(ζ)∂

^{q}

_{ζ}*f*(ζ)dΩ+,

*z*

*∈*Ω+, (2.12) while for hyperanalytic functions coincides with Cauchy’s formula.

*f*(z)*=*

*γ**e**z*(ζ)n*q*(ζ)*f*(ζ)ds, *z**∈*Ω+*.* (2.13)
*Definition 2.2.* Suppose *F* is a hyperanalytic function outside of an open ball*B**R* with
radius*R >*0 and center at the origin and let*γ*0be any rectifiable positively oriented Jordan
closed curve such that*γ*0lies inC*\**B** _{R}*and surrounds

*B*

*. The hypercomplex number*

_{R}Res*ζ**=∞* *F(ζ)*^{}:*= −* 1
2π

*γ*0

*W** _{ζ}*(ζ)n

*(ζ)F(ζ)ds (2.14) is called the residue of*

_{q}*F*at infinity.

Note that, according to (2.11), the integral in the right-hand side does not depend on
the choice of the curve*γ*0.

Theorem2.3. *LetF**∈*Ꮽ(Ω*−*), then

*γ**e**z*(ζ)n*q*(ζ)F(ζ)ds*=*

*−*Res*ζ**=∞*

*F(ζ)*
*W(ζ*)*−**W(z)*

, *z**∈*Ω+

*−**F(z)**−*Res*ζ**=∞*

*F(ζ)*
*W(ζ)**−**W(z)*

, *z**∈*Ω*−**.*

(2.15)

*Proof.* Suppose*z**∈*Ω*−*and let*R >*0 such thatΩ+*⊂**B** _{R}*and

*z*

*∈*

*B*

*. From Cauchy’s for- mula we get*

_{R}

*γ**e** _{z}*(ζ)n

*(ζ)F(ζ)ds*

_{q}*= −*

*F(z) +*

*∂B**R*

*e** _{z}*(ζ)n

*(ζ)F(ζ)ds*

_{q}*= −**F(z)**−*Res*ζ**=∞*

*F(ζ)*
*W(ζ)**−**W(z)*

*.*

(2.16)

The case*z**∈*Ω+is similar.

Notation*c*will be used for constants which may vary from one occurrence to the next;

in general these constants only depend on*q.*

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

*(ζ)*

_{q}^{}

*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)*e**t*(ζ)n*q*(ζ)^{}*F(ζ)**−**F(t)*^{}*ds*+*F(t) + Res**ζ**=∞*

*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-
pose*B** _{}*(t) is the open ball with center

*t*and radius, and let

*Q*:*=*Ω+*∪**B** _{}*(t). (2.19)

Since*γ*_{\}*γ** _{}*(t)

_{=}*∂Q*

*(∂Q*

_{\}

_{∩}*∂B*

*(t)), we have*

_{}*γ**\**γ*(t)*e**t*(ζ)n*q*(ζ)^{}*F(ζ)**−**F(t)*^{}*ds*

*=*

*∂Q**e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

*F(ζ)*

_{−}*F(t)*

^{}

*ds*

_{−}*∂Q**∩**∂B*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

*F*(ζ)

_{−}*F(t)*

^{}

*ds*

*=*

*∂Q**e** _{t}*(ζ)n

*(ζ)F(ζ)ds*

_{q}*−*

*F*(t)

*−*

*∂Q**∩**∂B*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

*F(ζ)*

*−*

*F(t)*

^{}

*ds.*

(2.20)

In view of the previous theorem,

*γ**\**γ*(t)*e**t*(ζ)n*q*(ζ)^{}*F(ζ*)*−**F(t)*^{}*ds*+*F(t) + Res**ζ**=∞*

*F(ζ)*
*W(ζ)**−**W(t)*

*=*

*∂Q**∩**∂B*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

*F(ζ)*

*−*

*F(t)*

^{}

*ds*

^{}

_{}

*≤*

*c*max

*z**∈*Ω*−*,_{|}*z**−**t**|=*

*F(z)**−**F(t)*^{}*.*
(2.21)

For each *f* *∈*Ꮿ(γ) the Cauchy-type integral is given by
**C***γ**f*^{}(z) :*=*

*γ**e**z*(ζ)n*q*(ζ)*f*(ζ)ds, *z /**∈**γ.* (2.22)
Clearly**C***γ**f* is hyperanalytic inC*\**γ.*

Thus the singular Cauchy integral operator (Hilbert transform) on*γ*reads as
**S***γ**f*^{}(t) :*=*2

*γ**e**t*(ζ)n*q*(ζ)^{}*f*(ζ)*−**f*(t)^{}*ds*+*f*(t), *t**∈**γ,* (2.23)
where the integral which defines**S***γ**f* has to be taken in the sense of Cauchy’s principal
value, and the function *f* is such that the integrals

*γ*(t)*e**t*(ζ)n*q*(ζ)^{}*f*(ζ)*−**f*(t)^{}*ds* (2.24)
converge uniformly to zero for*t**∈**γ*as*→*0.

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

From now on we always assume*γ*to be a regular closed curve, that is, the quotient of
the length of*γ*inside 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 onto*γ. The*
following theorem was proved in [3].

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

Ω*±*lim*z**→**t*

**C***γ**f*^{}(z)*=*1
2

**S***γ**f*^{}(t)*±**f*(t)^{}*.* (2.25)

Therefore, for any *f* *∈*(γ) the functions
**C**^{±}_{γ}*f*^{}(z) :*=*

**C***γ**f*^{}(z), *z**∈*Ω*±*,
1

2

**S**_{γ}*f*^{}(z)*±**f*(z)^{}, *z**∈**γ,* (2.26)

are continuous onΩ*±*.

Let* ^{±}*(γ) be the spaces of all continuous functions

*f*on

*γ*which 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).

Now, for*f* *∈*Ꮿ(γ) we consider the modulus of continuity for *f* as
*ω** _{f}*(ξ) :

*=*

*ξ*sup

*ρ**≥**ξ*

*ρ*^{−}^{1} max

*t*1,t2*∈**γ,**|**t*1*−**t*2*|≤**ρ*

*f*^{}*t*1

*−**f*^{}*t*2, *ξ >*0. (2.28)

Let

*I*0(γ) :*=*

*f* *∈*Ꮿ(γ) :
_{d}

0

*ω** _{f}*(ξ)

*ξ* *dξ <*+*∞*

*.* (2.29)

For a majorant*ϕ, that is,ϕ*is a positive real function on (0,d] such that*ϕ(ξ) does not de-*
crease,*ϕ(ξ)/ξ*does not increase, and*ϕ(ξ*)*→*0 as*ξ**→*0, we now introduce the generalized
H¨older continuous functions space

Ᏼ*ϕ*(γ) :*=*

*f* *∈*Ꮿ(γ) :*ω**f*(ξ)*≤**cϕ(ξ),ξ**∈*(0,d]^{} (2.30)
and its subspace

ᐆ*ϕ*(γ) :*=*

*f* *∈*Ᏼ*ϕ*(γ) :Θ*f*(ξ)*≤**cϕ(ξ),* *ξ**∈*(0,d]^{}, (2.31)
where

Θ*f*(ξ) :*=**ξ*sup

*ρ**≥**ξ*

*ρ*^{−}^{1} sup

*∈*(0,ρ],t*∈**γ*

*γ*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

*f*(ζ)

*−*

*f*(t)

^{}

*ds*

^{}

_{}

*.*(2.32) One can define a norm inᐆ

*ϕ*(γ) by

*f*ᐆ*ϕ*:*=*max

*t**∈**γ*

*f*(t)^{}+ sup

*ξ**∈*(0,d]

*ω**f*(ξ)
*ϕ(ξ)* + sup

*ξ**∈*(0,d]

Θ*f*(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 for*t**∈**γand**∈*(0,d],

sup

*z**∈*Ω*±*,*|**z**−**t**|=*

**C**^{±}_{γ}*f*^{}(z)*−*

**C**^{±}_{γ}*f*^{}(t)^{}*≤**c*

*ω** _{f}*() +Θ

*f*() +

_{d}

*ω** _{f}*(ξ)

*ξ*

^{2}

*dξ*

, (2.34)

*is true.*

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

*Case 1.* *z**∈*Ω*±*. Let*t*_{z}*∈**γ*such that*|**z**−**t*_{z}*| =*dist(z,γ)*=*:*ν*. Then for*|**z**−**t**| =*,
**C***γ**f*^{}(z)*−*

**C**^{±}_{γ}*f*^{}(t)

*=*

*γ**ν*(t*z*)*e**z*(ζ)n*q*(ζ)^{}*f*(ζ)*−**f*^{}*t**z*
*ds*
+

*γ**\**γ**ν*(t*z*)

*e**z*(ζ)*−**e**t**z*(ζ)^{}*n**q*(ζ)^{}*f*(ζ)*−**f*^{}*t**z*
*ds*

*−*

*γ**ν*(t*z*)*e*_{t}* _{z}*(ζ)n

*(ζ)*

_{q}^{}

*f*(ζ)

*−*

*f*

^{}

*t*

_{z}^{}

*ds*+1 2

**S**_{γ}*f*^{}*t*_{z}^{}*−*

**S**_{γ}*f*^{}(t)^{}

*±*1
2

*f*^{}*t**z*

*−**f*(t)^{}*=*:
5
*k**=*1

*J**k**.*

(2.35)

Because*|**z**−**t**z**| ≤ |**ζ**−**z**|*and*|**ζ**−**t**z**| ≤ |**ζ**−**z**|*+*|**z**−**t**z**| ≤*2*|**ζ**−**z**|*,
*J*1+^{}*J*2*≤**c*

*γ**ν*(t*z*)

*ω*_{f}^{}*ζ**−**t*_{z}^{}

*|**ζ**−**z**|* *ds*+^{}*z**−**t*_{z}^{}

*γ**\**γ**ν*(t*z*)

*ω*_{f}^{}*ζ**−**t*_{z}^{}

*|**ζ**−**z**|**ζ**−**t*_{z}^{}*ds*

*≤**c*

*ω** _{f}*(ν) +

^{}

*z*

*−*

*t*

_{z}^{}

*γ**\**γ**ν*(t*z*)

*ω*_{f}^{}*ζ**−**t*_{z}^{}
*ζ**−**t*_{z}^{}^{2} *ds*

*≤**c*

*ω**f*(ν) +*ν*^{}^{d}

*ν*

*ω**f*(ξ)
*ξ*^{2} *dξ*

*.*

(2.36)

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

3
_{d}

*ω**f*(ξ)

*ξ*^{2} *dξ**−**ν*^{}^{d}

*ν*

*ω**f*(ξ)
*ξ*^{2} *dξ*

*=*(*−**ν)*^{}^{d}

*ω**f*(ξ)
*ξ*^{2} *dξ*+

2

_{d}

*ω**f*(ξ)

*ξ*^{2} *dξ**−**ν*^{}^{}

*ν*

*ω**f*(ξ)
*ξ*^{2} *dξ*

*.*

(2.37)

Further, it is clear that 2

_{d}

*ω**f*(ξ)

*ξ*^{2} *dξ**≥*2*ω**f*()
_{d}

*dξ*

*ξ*^{2} * ^{=}*2

*ω*

*f*()

*d*

*−*

*d* * ^{=}*2ω

*f*()

*d*

*−*

*d* ^{≥}*ω**f*(). (2.38)
On the other hand,

*ν*^{}^{}

*ν*

*ω** _{f}*(ξ)

*ξ*^{2} *dξ**≤**νω**f*()
_{}

*ν*

*dξ*

*ξ*^{2} ^{=}*νω**f*()^{}^{−}*ν*

*ν* *=**ω**f*()^{}^{−}*ν*

*≤**ω**f*(), (2.39)

and hence

3
_{d}

*ω**f*(ξ)

*ξ*^{2} *dξ**−**ν*^{}^{d}

*ν*

*ω**f*(ξ)

*ξ*^{2} *dξ**≥*0. (2.40)

It then follows that

*J*1+^{}*J*2*≤**c*

*ω**f*() +
_{d}

*ω**f*(ξ)
*ξ*^{2} *dξ*

*.* (2.41)

For*J*3we obtain

*J*3*≤*Θ*f*(ν)*≤*Θ*f*(). (2.42)

Since*|**t*_{z}*−**t**| ≤ |**t*_{z}*−**z**|*+*|**z**−**t**| ≤*2*|**z**−**t**| =*2, then

*J*4*≤**ω***S**_{γ}*f*(), ^{}*J*5*≤**ω** _{f}*(). (2.43)

On the other hand, starting from the reasonings in [3, Theorem 2] and using the defini-
tion ofΘ*f*, we get

*J*4*≤**ω***S**_{γ}*f*()*≤**c*

*ω** _{f}*() +Θ

*f*() +

_{d}

*ω** _{f}*(ξ)

*ξ*

^{2}

*dξ*

*.* (2.44)

Therefore

**C***γ**f*^{}(z)*−*

**C**^{±}_{γ}*f*^{}(t)^{}*≤**c*

*ω**f*() +Θ*f*() +
_{d}

*ω**f*(ξ)
*ξ*^{2} *dξ*

*.* (2.45)

*Case 2.* *z**∈**γ. If**|**z**−**t**| =*, then
**C**^{±}_{γ}*f*^{}(z)*−*

**C**^{±}_{γ}*f*^{}(t)*=*1
2

**S**_{γ}*f*(z)*−***S**_{γ}*f*(t)^{}*±*1
2

*f*(z)*−**f*(t)^{}*.* (2.46)

For that

**C**^{±}_{γ}*f*^{}(z)*−*

**C**^{±}_{γ}*f*^{}(t)^{}*≤**ω***S**_{γ}*f*() +*ω**f*(), (2.47)
hence

(C^{±}_{γ}*f*^{}(z)*−*

**C**^{±}_{γ}*f*^{}(t)^{}*≤**c*

*ω**f*() +Θ*f*() +
_{d}

*ω**f*(ξ)
*ξ*^{2} *dξ*

*.* (2.48)

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

Lemma2.7. *Suppose* *f* *∈*(γ)*and let**∈*(0,d]. Then
Θ**S**_{γ}*f*()*≤**c*

*ω**f*() +Θ*f*() +
_{d}

*ω** _{f}*(ξ)

*ξ*

^{2}

*dξ*

*.* (2.49)

*Proof.* For*t**∈**γ*we have (S*γ**f*)(t)*=*2(C^{+}_{γ}*f*)(t)*−**f*(t). Therefore

*γ*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

**S**

*γ*

*f*

^{}(ζ)

*−*

**S***γ**f*^{}(t)^{}*ds*^{}_{}

*≤*Θ*f*() + 2^{}_{}

*γ*(t)*e**t*(ζ)n*q*(ζ)^{}**C**^{+}_{γ}*f*^{}(ζ)*−*

**C**^{+}_{γ}*f*^{}(t)^{}*ds*^{}_{}*.*

(2.50)

As a consequence ofLemma 2.4we have

*γ**e**t*(ζ)n*q*(ζ)^{}**C**^{+}_{γ}*f*^{}(ζ)*−*

**C**^{+}_{γ}*f*^{}(t)^{}*ds**=*0, (2.51)
for that

*γ*(t)*e**t*(ζ)n*q*(ζ)^{}**C**^{+}_{γ}*f*^{}(ζ)*−*

**C**^{+}_{γ}*f*^{}(t)^{}*ds*^{}_{}

*=*

*γ**\**γ*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

**C**

^{+}

_{γ}*f*

^{}(ζ)

*−*

**C**^{+}_{γ}*f*^{}(t)^{}*ds*^{}_{}*.*

(2.52)

Thus, returning to theLemma 2.4, we obtain

*γ*(t)*e**t*(ζ)n*q*(ζ)^{}**C**^{+}_{γ}*f*^{}(ζ)*−*(C^{+}_{γ}*f*^{}(t)^{}*ds*^{}_{}*≤**c* sup

*z**∈*Ω+,*|**z**−**t**|=*

**C**^{+}_{γ}*f*^{}(z)*−*

**C**^{+}_{γ}*f*^{}(t)^{}
(2.53)
and the following estimate follows in view of the result ofLemma 2.6:

*γ*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

**C**

^{+}

_{γ}*f*

^{}(ζ)

*−*

**C**^{+}_{γ}*f*^{}(t)^{}*ds*^{}_{}*≤**c*

*ω** _{f}*() +Θ

*f*() +

_{d}

*ω** _{f}*(ξ)

*ξ*

^{2}

*dξ*

*.*
(2.54)
Finally

*γ*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

**S**

_{γ}*f*

^{}(ζ)

*−*

**S**_{γ}*f*^{}(t)^{}*ds*^{}_{}*≤**c*

*ω** _{f}*() +Θ

*f*() +

_{d}

*ω** _{f}*(ξ)

*ξ*

^{2}

*dξ*

*.*
(2.55)

This completes the proof.

The following theorem gives suﬃcient condition for the singular Cauchy integral op-
erator to be bounded onᐆ*ϕ*(γ).

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

_{d}

*ϕ(ξ)*

*ξ*^{2} *dξ**≤**cϕ(*). (2.56)

*Then the singular Cauchy integral operator***S***γ**is a bounded operator on*ᐆ*ϕ*(γ), that is,

**S**_{γ}*f*^{}_{ᐆ}_{ϕ}*≤**c**f*ᐆ*ϕ*, (2.57)

*for anyf* *∈*ᐆ*ϕ*(γ).

*Proof.* Suppose *f* *∈*ᐆ*ϕ*(γ), then for*∈*(0,d],

*ω** _{f}*()

*≤*

*f*ᐆ

*ϕ*

*ϕ(*), Θ

*f*()

*≤*

*f*ᐆ

*ϕ*

*ϕ(*); (2.58) hence

_{d}

*ω**f*(ξ)

*ξ*^{2} *dξ**≤**f*ᐆ*ϕ*

_{d}

*ϕ(ξ*)

*ξ*^{2} *dξ**≤**c**f*ᐆ*ϕ**ϕ(*). (2.59)
Combining the previous estimates withLemma 2.7, we obtain

Θ**S***γ**f*()*≤**c**f*ᐆ*ϕ**ϕ(*), (2.60)
and similarly

*ω***S**_{γ}*f*()*≤**c**f*ᐆ*ϕ**ϕ(*). (2.61)
So we have at once**S***γ**f* *∈*ᐆ*ϕ*(γ). Besides

**S**_{γ}*f*(t)^{}*≤*2Θ*f*(d) +*f*ᐆ*ϕ**≤**c**f*ᐆ*ϕ*, *t**∈**γ,* (2.62)
thus

**S***γ**f*^{}_{ᐆ}_{ϕ}*≤**c**f*ᐆ*ϕ*, (2.63)

that is,**S***γ*is a bounded operator onᐆ*ϕ*(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].

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 an- alytic 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 sin- gular 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 de-
fined on*γ*have to agree with the desired boundary behavior of the solutions, that is, the
solutions including their boundary values on*γ*are continuous functions too.

The results of boundedness of the singular Cauchy integral operator and on the con-
tinuous 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 coeﬃcient*G*in the case where*G*admits a canonical factorization

*G**=**H*^{+}

*H** ^{−}*, (3.1)

where*H*^{±}*∈*Ꮽ(Ω*±*) and the complex parts*H*_{0}* ^{±}*never vanish onΩ

*±*.

Take an arbitrary fixed point*z*0inΩ+. We restrict ourselves to the case of a hypercom-
plex function*G*which has complex part*G*0and never vanishes on*γ. From this assump-*
tion the integer

*κ*:*=* 1

2π arg*G*0(ζ)^{}* _{γ}* (3.2)

has significant importance and is called the index of*G*with respect to*γ, also called index*
of the Riemann boundary value problem. Note that the index of the function (W(ζ)*−*
*W(z*0))* ^{κ}*with respect to

*γ*is

*κ, and hence the index of (W(ζ)*

*−*

*W(z*0))

^{−}

^{κ}*G(ζ) is zero.*

We may verify directly that
*X(z) :**=*

expΓ(z), *z**∈*Ω+,

*W(z)**−**W*^{}*z*0

_{−}*κ*

expΓ(z), *z**∈*Ω*−*, (3.3)
in which

Γ(z) :*=*

*γ**e**z*(ζ)n*q*(ζ) ln *W(ζ)**−**W*^{}*z*0*−**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.

Theorem3.1. *The functionGadmits a canonical factorization if and only if*ln[(W(t)*−*
*W(z*0))^{−}^{κ}*G(t)]**∈*(γ).

*Proof.* Assume*G*admits a canonical factorization, then we deduce
ln *W(t)**−**W*^{}*z*0

_{−}*κ*

*G(t)v*^{}*=*ln*H*^{+}(t)*−*ln *W(t)**−**W*^{}*z*0

*κ*

*H** ^{−}*(t)

^{}, (3.5) for

*t*

*∈*

*γ*and

*H*

*satisfying (3.1).*

^{±}Now the necessity follows fromLemma 2.4and the previous equality.

Conversely, suppose ln[(W(t)*−**W(z*0))^{−}^{κ}*G(t)]**∈*(γ). Taking into account (3.5) and
Theorem 2.5, it is easy to check that*X*gives a canonical factorization of*G*on*γ*in the form

*G**=**X*^{+}*/X** ^{−}*.

*Definition 3.2.* *p*0+*p*1*W(z) +**···*+*p**s*(W(z))* ^{s}*,

*s*

*≥*0, is called a hypercomplex polyno- mial.

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

Φ(z)*=**X(z)P**κ**−*1(z), (3.6)

*whereP**κ**−*1*is an arbitrary hypercomplex polynomial whose degree is not greater thanκ**−*1
*(P*_{κ}* _{−}*1

*≡*0

*whenκ*

*≤*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)*^{}**C***γ*

*g/X*^{+}^{}(z) (3.8)

is a solution of the Riemann boundary value problem.

We consider now*G**∈**I*0(γ) and*g**∈*(γ). Note that
ln^{}*W(t)**−**W*^{}*z*0

*−**κ*

*G(t)*^{}*∈**I*0(γ)*⊂*(γ), (3.9)
hence,*G*admits a canonical factorization.

Taking into account the condition*g**∈*(γ), it can easily be shown that

*γ*(t)*e**t*(ζ)n*q*(ζ)
*g*(ζ)

*X*^{+}(ζ)^{−}*g*(t)
*X*^{+}(t)

*ds*

*=*

*γ*(t)*e**t*(ζ)n*q*(ζ)
*g*^{+}(ζ)

*X*^{+}(ζ)^{−}*g*^{+}(t)
*X*^{+}(t)

*ds*
+ 1

*G(t)*

*γ*(t)*e**t*(ζ)n*q*(ζ)
*g** ^{−}*(ζ)

*X** ^{−}*(ζ)

^{−}*g*

*(t)*

^{−}*X*

*(t)*

^{−}*ds*
+

*γ*(t)*e** _{t}*(ζ)n

*(ζ) 1*

_{q}*G(ζ)** ^{−}*
1

*G(t)*

*g** ^{−}*(ζ)

*X** ^{−}*(ζ)

*ds*

*=*:

*J*1+

*J*2+

*J*3,

*∈*(0,d].

(3.10)

In view of the result ofLemma 2.4,*J*1,*J*2converge uniformly to zero for*t**∈**γ, as**→*0.

Furthermore, as

*J*3*≤**c*
_{}

0

*ω** _{G}*(ξ)

*ξ* *dξ,* (3.11)

we have that*g/X*^{+}*∈*(γ).

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

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

Φ(z)*=**X(z)*^{}**C***γ*

*g/X*^{+}^{}(z) +*X(z)P**κ**−*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) (P*

*κ*

*−*1(z)

*≡*0

*whenκ*

*≤*0) if and only if the following conditions are satisfied:

*γ**∂*_{ζ}*W(ζ)n** _{q}*(ζ)

*g(ζ)*

*X*

^{+}(ζ)

*W(ζ)*^{}^{k}*ds**=*0, *k**=*0,*. . .,**−**κ**−*1. (3.13)

**4. Singular integral equations in the class**ᐆ*ϕ*(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)*^{}**S***γ*Υ^{}(t)*=**f*(t), *t**∈**γ,* (4.1)
for which we assume that*a,b, andf* are given hypercomplex functions. For this equation,
Υis an unknown hypercomplex function belonging toᐆ*ϕ*(γ).

We assume that*a,b** _{∈}*Ᏼ

*ψ*(γ),

*f*

*ᐆ*

_{∈}*ϕ*(γ), and

*a*

^{2}

_{0}(t)

_{−}*b*

^{2}

_{0}(t)

*0 for*

_{=}*t*

_{∈}*γ. Hereϕ,ψ*are given majorants satisfying the following relations:

_{d}

*ϕ(ξ)*

*ξ*^{2} *dξ**≤**cϕ(*),
_{}

0

*ψ(ξ)*

*ξ* *dξ**≤**cϕ(*), *∈*(0,d].

(4.2)

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 index*κ*of the function (a0(t)*−**b*0(t))/(a0(t) +*b*0(t)),
and the corresponding functionsΓ(z) and*X(z), we arrive at the following assertions.*

Theorem4.1. *Leta,b**∈*Ᏼ*ψ*(γ)*and* *f* *∈*ᐆ*ϕ*(γ). Then
(i)*G**∈*Ᏼ*ψ*(γ),

(ii)*g,X*^{±}*∈*ᐆ*ϕ*(γ).

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

*ψ(*)*=*
_{}

0

*ψ(*)

*dξ**≤*

_{}

0

*ψ(ξ*)

*ξ* *dξ**≤**cϕ(*), *∈*(0,d], (4.5)
we thus have that*g**∈*Ᏼ*ϕ*(γ).

Let now

*h(t) :**=*

*a(t) +b(t)*^{}^{−}^{1}, *t**∈**γ,* (4.6)
then

*γ*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

*f*(ζ)h(ζ)

*−*

*f*(t)h(t)

^{}

*ds*

^{}

*≤**h(t)*^{}^{}_{}

*γ*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

*f*(ζ)

*−*

*f*(t)

^{}

*ds*

^{}

_{}+

^{}

_{}

*γ*(t)*e** _{t}*(ζ)n

*(ζ)*

_{q}^{}

*h(ζ)*

_{−}*h(t)*

^{}

*f*(ζ)ds

^{}

_{}

*≤**c*

Θ*f*() +
_{}

0

*ω**h*(ξ)
*ξ* *dξ*

*≤**c*

Θ*f*() +
_{}

0

*ψ(ξ)*
*ξ* *dξ*

*≤**cϕ(*), *∈*(0,d].

(4.7)

Therefore*g**∈*ᐆ*ϕ*(γ).