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Boundary value problems for the quaternionic Hermitian system in R 4 n

Ricardo Abreu-Blaya1, Juan Bory-Reyes2, Fred Brackx3, Hennie De Schepper3* and Frank Sommen3

Author Affiliations

1 Facultad de Informática y Matemática, Universidad de Holguín, Holguín, 80100, Cuba

2 Departamento de Matemática, Universidad de Oriente, Santiago de Cuba, 90500, Cuba

3 Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000, Gent, Belgium

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Boundary Value Problems 2012, 2012:74  doi:10.1186/1687-2770-2012-74


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/74


Received:5 January 2012
Accepted:25 May 2012
Published:12 July 2012

© 2012 Abreu-Blaya et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper boundary value problems for quaternionic Hermitian monogenic functions are presented using a circulant matrix approach.

MSC: 30G35.

Keywords:
quaternionic Clifford analysis; Cauchy integral formula

1 Introduction

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, i.e. null solutions of a first-order vector-valued rotation invariant differential operator, called Dirac operator, which factorises the Laplacian; monogenic functions may thus also be seen as a generalisation of holomorphic functions in the complex plane. Its roots go back to the 1930s. For more details on this function theory we refer to the standard references [5,12,14-16].

More recently Hermitian Clifford analysis emerged as a refinement of the Euclidean setting for the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M2">View MathML</a>. Here, Hermitian monogenic functions are considered, i.e. functions taking values either in a complex Clifford algebra or in complex spinor space, and being simultaneous null solutions of two complex Hermitian Dirac operators, which are invariant under the action of the unitary group. For the systematic development of this function theory we refer to [6-8].

In the papers [10,11,13,17], the Hermitian Clifford analysis setting was further refined by considering functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3">View MathML</a> with values in a quaternionic Clifford algebra, being simultaneous null solutions of four mutually related quaternionic Dirac operators, which are invariant under the action of the symplectic group. In [3], Borel-Pompeiu and Cauchy integral formulas are established in this setting, by following a (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M4">View MathML</a>) circulant matrix approach, similar in spirit to the circulant (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M5">View MathML</a>) matrix approach introduced in [9] within the complex Hermitian Clifford case. Subsequently, in [4] a quaternionic Hermitian Cauchy integral is introduced, as well as its boundary limit values, leading to the definition of a matrix quaternionic Hermitian Hilbert transform. These operators provide a useful tool for studying boundary value problems for the quaternionic Hermitian system. This is precisely the main objective of the present paper. The main problems that we address are the problem of finding a quaternionic Hermitian monogenic function with a given jump over a given surface of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3">View MathML</a> as well as problems of Dirichlet type for the quaternionic Hermitian system. Finally, we also prove an equivalence between both-sided quaternionic Hermitian monogenicity and a certain integral conservation law.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M7">View MathML</a> be an orthonormal basis of Euclidean space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M8">View MathML</a> and consider the real Clifford algebra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M9">View MathML</a> constructed over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M8">View MathML</a>. The non-commutative multiplication in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M11">View MathML</a> is governed by the rules:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M12">View MathML</a>

In <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M11">View MathML</a> one can consider the following automorphisms:

(i) the conjugation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M14">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M16">View MathML</a>

(ii) the main involution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M17">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M19">View MathML</a>.

In particular, we consider the skew-field of quaternions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20">View MathML</a> whose elements will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M21">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M23">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20">View MathML</a> may be identified with the Clifford algebra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M25">View MathML</a> making the identifications <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M28">View MathML</a>. The automorphisms (i) and (ii) then respectively lead to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20">View MathML</a>-conjugation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M30">View MathML</a>

and to the main <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20">View MathML</a>-involution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M32">View MathML</a>

However, it is quite natural to introduce two more <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20">View MathML</a>-involutions defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M34">View MathML</a>

Definition 1 (see[17])

The quaternionic Witt basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M36">View MathML</a>, is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M38">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M39">View MathML</a>

We will consider the Clifford vectors

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M40">View MathML</a>

for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M41">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M44">View MathML</a>. The corresponding Dirac operators are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M48">View MathML</a>. Here we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M49">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M50">View MathML</a> the Laplacian in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M44">View MathML</a>. Next, the quaternionic Hermitian variables are introduced:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M55">View MathML</a>

for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M56">View MathML</a>, the symbol denoting Hermitian quaternionic conjugation is defined as the composition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M20">View MathML</a>-conjugation and Clifford conjugation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M9">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M59">View MathML</a>. The Hermitian Dirac operators are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M60">View MathML</a>

for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M61">View MathML</a>.

Definition 2 (see [17])

Let Ω be an open set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3">View MathML</a>. A continuously differentiable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M63">View MathML</a> is said to be (left) q-Hermitian monogenic in Ω (or q-monogenic for short) iff it satisfies in Ω the system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M64">View MathML</a>, or, equivalently, the system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M65">View MathML</a>.

Similarly right q-monogenicity is defined. Left and right q-monogenic functions are called two-sided q-monogenic. A q-monogenic function in Ω is monogenic, and thus harmonic in Ω. Note that Definition 2 was proven in [10] to be equivalent to the system introduced in [13] by group invariance considerations.

The fundamental solutions of the Dirac operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67">View MathML</a>, i.e. the Euclidean Cauchy kernels, are respectively given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M68">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M69">View MathML</a> the area of the unit sphere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M70">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3">View MathML</a>. Explicitly, this means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67">View MathML</a>. Next we introduce the Hermitian Cauchy kernels:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M74">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M75">View MathML</a> is not the fundamental solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M76">View MathML</a>. However, the following theorem holds, see [3].

Theorem 1Introducing the circulant (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M4">View MathML</a>) matrices

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M78">View MathML</a>

one obtains that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M79">View MathML</a>.

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M80">View MathML</a> is a fundamental solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M81">View MathML</a>, in a matricial interpretation.

We associate, with functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M85">View MathML</a> defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M86">View MathML</a> and taking values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M87">View MathML</a>, the (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M4">View MathML</a>) circulant matrix function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M89">View MathML</a>

(1)

We say that G belongs to some class of functions if all its entries belong to that class. In particular, the spaces of k-times continuously differentiable, of α-Hölder continuous (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M90">View MathML</a>) and of p-integrable (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M4">View MathML</a>) circulant matrix functions on some suitable subset E of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3">View MathML</a> are respectively denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M95">View MathML</a>. The corresponding spaces of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M96">View MathML</a>-valued functions are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M99">View MathML</a>. Moreover, introducing the non-negative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M100">View MathML</a>, the classes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M95">View MathML</a> may also be defined by means of the respective traditional conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M103">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M104">View MathML</a>

Definition 3 The (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M4">View MathML</a>) circulant matrix function G is called (left) Q-Hermitian monogenic in Ω (or Q-monogenic for short) iff <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M106">View MathML</a> in Ω, where O denotes the matrix with zero entries.

Similarly right Q-monogenicity is defined by the system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M107">View MathML</a>. Left and right Q-monogenic matrix functions are called two-sided Q-monogenic. An important special case concerns the diagonal matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M110">View MathML</a>. Indeed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a> is left (respectively right) Q-monogenic iff the function g is left (respectively right) q-monogenic.

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M112">View MathML</a> be a bounded simply connected domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3">View MathML</a> with boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M114">View MathML</a>, and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M115">View MathML</a> the complementary open domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M116">View MathML</a>. We assume Γ to be a Liapunov surface. The unit normal vector on Γ at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M117">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M118">View MathML</a>

and similarly as above, we also introduce the vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M120">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M121">View MathML</a>, giving rise in the usual way (up to a constant factor) to their Hermitian counterparts

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M122">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M125">View MathML</a>, as well as to the circulant matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M126">View MathML</a>. Then, in [3], the following Cauchy integral formulae were proven for Q-monogenic matrix functions and for q-monogenic functions, respectively.

Theorem 2 (Q-Hermitian Cauchy integral formula)

If the matrix functionG, (1), isQ-monogenic in Ω then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M127">View MathML</a>

Theorem 3 (q-Hermitian Cauchy integral formula)

If the functiongisq-monogenic in Ω then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M128">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a>is the corresponding diagonal matrix.

Next, in [4] a Q-Hermitian Cauchy transform was introduced, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M130">View MathML</a>

(2)

for a matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M131">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M133">View MathML</a> denote the Hermitian versions of the Clifford vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M134">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M135">View MathML</a>, respectively. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M136">View MathML</a> is a left Q-monogenic matrix function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M137">View MathML</a>, vanishing at infinity; in terms of the Euclidean Cauchy type integrals

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M138">View MathML</a>

it reads as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M139">View MathML</a>

In particular, for the special case of the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a>, the action of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M141">View MathML</a> is reduced to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M142">View MathML</a>

In general <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M143">View MathML</a> will not be a diagonal matrix, whence its entries will not be left q-monogenic functions. However <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M143">View MathML</a> does become diagonal if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M145">View MathML</a>

(3)

in which case we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M146">View MathML</a>

The following Plemelj-Sokhotski formula, proven in [4], then asserts the existence of the continuous boundary limits of the Q-Hermitian Cauchy transform.

Theorem 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M147">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M148">View MathML</a>), then the continuous limit values of itsQ-Hermitian Cauchy transform<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M136">View MathML</a>exist and are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M150">View MathML</a>

Here we have introduced the matrix Q-Hermitian Hilbert operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M151">View MathML</a>

where the singular integrals

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M152">View MathML</a>

are Cauchy principal values. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M153">View MathML</a> shows the following traditional properties, see [4].

Theorem 5One has

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M153">View MathML</a>is a bounded linear operator on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M155">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M156">View MathML</a>)

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M153">View MathML</a>is an involution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M158">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M156">View MathML</a>).

Similar results may be obtained for right-hand versions of the Q-Hermitian Cauchy and Hilbert transforms by means of the alternative definitions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M160">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M161">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M162">View MathML</a>

3 Boundary value problems for Q-monogenic functions

In this section we study the so-called jump problem (reconstruction problem) for Q-monogenic functions; that is, we will investigate the problem of reconstructing a Q-monogenic matrix function Ψ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M137">View MathML</a> vanishing at infinity and having a prescribed jump G across Γ, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M164">View MathML</a>

(4)

First, it should be noted that if this problem has a solution, then it necessarily is unique. This assertion can be easily proven using the Painlevé and Liouville theorems in the Clifford analysis setting, see [1]. Next, under the condition that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M165">View MathML</a>, Theorem 4 ensures the solvability of the jump problem (4), its unique solution being given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M166">View MathML</a>

Now consider the important special case of the matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a>. The reconstruction problem (4) then is strongly related to the jump problem for the involved q-monogenic function, as addressed in the following theorem.

Theorem 6For a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M168">View MathML</a>, the following statements are equivalent:

(i) the jump problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M169">View MathML</a>

(5)

is solvable in terms ofq-monogenic functions;

(ii) gsatisfies the relations (3);

(iii) gsatisfies the relations<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M170">View MathML</a>.

Proof (i) → (ii)

Associate to the function g the diagonal matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M172">View MathML</a>, and the jump problem (4) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a> has the unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M174">View MathML</a>

Let ψ be a solution of (5), then the circulant matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M175">View MathML</a>

is another solution of the jump problem (4) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a>, whence the uniqueness yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M177">View MathML</a>

implying (ii).

(ii) → (iii)

From the third relation in (3), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M178">View MathML</a>, and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M179">View MathML</a>

the latter following from the second relation in (3) and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M180">View MathML</a>-monogenicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M181">View MathML</a>. This fact means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M182">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M183">View MathML</a>-monogenic function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M137">View MathML</a>. Moreover, it has a null jump through Γ, whence it vanishes in the whole of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M3">View MathML</a>. We conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M186">View MathML</a>. Similarly, we arrive at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M187">View MathML</a>.

(iii) → (i)

It suffices to observe that, under the conditions stated, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M188">View MathML</a> is q-monogenic, whence it solves the jump problem (5). □

For right q-monogenic functions the following analogue is obtained.

Theorem 7For a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M168">View MathML</a>, the following statements are equivalent:

(i) the jump problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M190">View MathML</a>

(6)

is solvable in terms of rightq-monogenic functions;

(ii) gsatisfies the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M191">View MathML</a>

(iii) gsatisfies the relations<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M192">View MathML</a>.

The next result deals with the Dirichlet boundary value problem for Q-monogenic functions.

Theorem 8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M147">View MathML</a>, then the following statements are equivalent:

(i) The Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M194">View MathML</a>

(7)

has a solution.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M195">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M196">View MathML</a>).

Proof We give the proof for the left-sided version of the theorem, the right-sided one being completely similar.

(i) → (ii)

Let F be a solution of the Dirichlet problem (7). Then, by the Q-Hermitian Cauchy formula, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M197">View MathML</a>

Taking limits as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M198">View MathML</a>, (ii) follows in view of Theorem 4.

(ii) → (i)

It suffices to observe that, under the condition (ii), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M199">View MathML</a> solves (7). □

Theorem 9Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M168">View MathML</a>, then the following statements are equivalent:

(i) The Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M201">View MathML</a>

(8)

has a solution.

(ii) gsatisfies the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M202">View MathML</a>

(iii) gsatisfies the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M203">View MathML</a>

Proof (i) → (ii)

From (i) we see that the matrix function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M204">View MathML</a>

is a solution of the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M205">View MathML</a>

whence by Theorem 8 we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M206">View MathML</a>. The desired conclusion (ii) then directly follows by comparing the entries in the above equality.

(ii) → (iii)

From the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M207">View MathML</a> it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M208">View MathML</a>. Therefore, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M209">View MathML</a> is harmonic in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M211">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M212">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M137">View MathML</a>. Using the remaining conditions in (ii) and following a similar reasoning as above, we obtain that g satisfies the relations (3) and hence by Theorem 6 we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M170">View MathML</a>. Consequently, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M215">View MathML</a>, as stated in (iii).

(iii) → (i)

The conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M215">View MathML</a> imply the solvability of the Dirichlet problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M217">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67">View MathML</a>. Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M222">View MathML</a> be the respective solutions of (9), then these functions all are solutions of the classical Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M223">View MathML</a>

whence they coincide. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M224">View MathML</a> thus is q-monogenic and constitutes a solution of (8). □

For right q-monogenic functions the following analogue is obtained.

Theorem 10Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M168">View MathML</a>, then the following statements are equivalent:

(i) The Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M226">View MathML</a>

(10)

has a solution.

(ii) gsatisfies the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M227">View MathML</a>

(iii) gsatisfies the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M228">View MathML</a>

We now turn our attention towards establishing a connection between the two-sided Q-monogenicity of a matrix function G and the matrix Hilbert transforms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M229">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M230">View MathML</a> of its trace on the boundary Γ.

Theorem 11Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M231">View MathML</a>, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M106">View MathML</a>in Ω, then the following statements are equivalent:

(i) Gis two-sidedQ-monogenic in Ω.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M233">View MathML</a>.

Proof Assume that, next to its already assumed left Q-monogenicity, G also is right Q-monogenic in Ω. Then by Theorem 8 it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M234">View MathML</a>

Conversely, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M233">View MathML</a>. By Theorem 4 and its right-handed version, we conclude that the corresponding left and right Q-Hermitian Cauchy transform of G, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M237">View MathML</a>, have the same boundary values on Γ. This fact, together with their harmonicity, implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M238">View MathML</a>

On the other hand, from the assumed left Q-monogenicity of G we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M239">View MathML</a> and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M240">View MathML</a>

which clearly forces G to be two-sided Q-monogenic. □

The following result illustrates the utility of the above theorem when considering q-monogenic functions.

Theorem 12Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M241">View MathML</a>be leftq-monogenic in Ω, then the following statements are equivalent:

(i) gis two-sidedq-monogenic in Ω.

(ii) gsatisfies the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M242">View MathML</a>

(iii) gsatisfies the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M243">View MathML</a>

Proof (i) ↔ (ii)

From (i) we see that the matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a> corresponding to g is two-sided Q-monogenic in Ω, whence (ii) follows from Theorem 11(ii) applied to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a>. Conversely, (ii) can be rewritten in the matricial form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M246">View MathML</a>, from which (i) follows by observing that the two-sided Q-monogenicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M108">View MathML</a> implied by Theorem 11 is equivalent to the q-monogenicity of g.

(i) ↔ (iii)

It follows from (i) that g is two-sided monogenic w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67">View MathML</a>. We may then invoke [[2], Theorem 3.2] in order to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M251">View MathML</a>. Conversely, suppose that (iii) holds. Each of the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67">View MathML</a>, implies the two-sided monogenicity of g in Ω w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/74/mathml/M67">View MathML</a>, see again [[2], Theorem 3.2], whence g is two-sided q-monogenic in Ω. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have worked jointly on the manuscript, which is the result of an intensive collaboration. All authors read and approved the final manuscript.

Acknowledgement

Ricardo Abreu-Blaya and Juan Bory-Reyes wish to thank all members of the Department of Mathematical Analysis of Ghent University, where the paper was written, for the invitation and hospitality. They were supported respectively by the Research Council of Ghent University and by the Research Foundation - Flanders (FWO, project 31506208).

References

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