Open Access

Boundary value problems for the quaternionic Hermitian system in R 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq1_HTML.gif

  • Ricardo Abreu-Blaya1,
  • Juan Bory-Reyes2,
  • Fred Brackx3,
  • Hennie De Schepper3Email author and
  • Frank Sommen3
Boundary Value Problems20122012:74

DOI: 10.1186/1687-2770-2012-74

Received: 5 January 2012

Accepted: 25 May 2012

Published: 12 July 2012

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, 1416].

More recently Hermitian Clifford analysis emerged as a refinement of the Euclidean setting for the case of R 2 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq2_HTML.gif. 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 [68].

In the papers [10, 11, 13, 17], the Hermitian Clifford analysis setting was further refined by considering functions on R 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq3_HTML.gif 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 ( 4 × 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq4_HTML.gif) circulant matrix approach, similar in spirit to the circulant ( 2 × 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq5_HTML.gif) 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 R 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq3_HTML.gif 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 ( e 1 , , e m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq6_HTML.gif be an orthonormal basis of Euclidean space R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq7_HTML.gif and consider the real Clifford algebra R 0 , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq8_HTML.gif constructed over R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq7_HTML.gif. The non-commutative multiplication in R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq9_HTML.gif is governed by the rules:
e 2 = 1 , = 1 , , m , e e k + e k e = 0 , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equa_HTML.gif
In R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq9_HTML.gif one can consider the following automorphisms:
  1. (i)

    the conjugation e ¯ = e https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq10_HTML.gif and for any a , b R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq11_HTML.gif, a b ¯ = b ¯ a ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq12_HTML.gif

     
  2. (ii)

    the main involution e ˜ = e https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq13_HTML.gif and for any a , b R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq11_HTML.gif, a b ˜ = a ˜ b ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq14_HTML.gif.

     
In particular, we consider the skew-field of quaternions H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq15_HTML.gif whose elements will be denoted by q = x 0 + i x 1 + j x 2 + k x 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq16_HTML.gif with i 2 = j 2 = k 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq17_HTML.gif and i j = j i = k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq18_HTML.gif. Clearly, H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq15_HTML.gif may be identified with the Clifford algebra R 0 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq19_HTML.gif making the identifications i e 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq20_HTML.gif, j e 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq21_HTML.gif and k e 1 e 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq22_HTML.gif. The automorphisms (i) and (ii) then respectively lead to the H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq15_HTML.gif-conjugation
q ¯ = x 0 i x 1 j x 2 k x 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equb_HTML.gif
and to the main H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq15_HTML.gif-involution
q γ q ˜ = x 0 i x 1 j x 2 + k x 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equc_HTML.gif
However, it is quite natural to introduce two more H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq15_HTML.gif-involutions defined by
q α = x 0 + i x 1 j x 2 k x 3 , q β = x 0 i x 1 + j x 2 k x 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equd_HTML.gif

Definition 1 (see[17])

The quaternionic Witt basis of H m = H R R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq23_HTML.gif, m = 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq24_HTML.gif, is given by { f , f α , f β , f γ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq25_HTML.gif, = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq26_HTML.gif, where
f = e 1 + 4 ( 1 ) i e 2 + 4 ( 1 ) j e 3 + 4 ( 1 ) k e 4 + 4 ( 1 ) , f α = e 1 + 4 ( 1 ) i e 2 + 4 ( 1 ) + j e 3 + 4 ( 1 ) + k e 4 + 4 ( 1 ) , f β = e 1 + 4 ( 1 ) + i e 2 + 4 ( 1 ) j e 3 + 4 ( 1 ) + k e 4 + 4 ( 1 ) , f γ = e 1 + 4 ( 1 ) + i e 2 + 4 ( 1 ) + j e 3 + 4 ( 1 ) k e 4 + 4 ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Eque_HTML.gif
We will consider the Clifford vectors
X ̲ = X ̲ 0 = = 1 n ( e 4 3 x 4 3 + e 4 2 x 4 2 + e 4 1 x 4 1 + e 4 x 4 ) , X ̲ 1 = = 1 n ( e 4 3 x 4 2 e 4 2 x 4 3 e 4 1 x 4 + e 4 x 4 1 ) , X ̲ 2 = = 1 n ( e 4 3 x 4 1 + e 4 2 x 4 e 4 1 x 4 3 e 4 x 4 2 ) , X ̲ 3 = = 1 n ( e 4 3 x 4 e 4 2 x 4 1 + e 4 1 x 4 2 e 4 x 4 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equf_HTML.gif
for which X ̲ r 2 = | X ̲ 0 | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq27_HTML.gif, while X ̲ r X ̲ s + X ̲ s X ̲ r = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq28_HTML.gif, r s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq29_HTML.gif, r , s = 0 , , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq30_HTML.gif. The corresponding Dirac operators are denoted by X ̲ = X ̲ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq31_HTML.gif, X ̲ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq32_HTML.gif, X ̲ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq33_HTML.gif and X ̲ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq34_HTML.gif. Here we have X ̲ r 2 = Δ 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq35_HTML.gif, with Δ 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq36_HTML.gif the Laplacian in R 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq3_HTML.gif, and X ̲ r X ̲ s + X ̲ s X ̲ r = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq37_HTML.gif, r s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq29_HTML.gif, r , s = 0 , , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq30_HTML.gif. Next, the quaternionic Hermitian variables are introduced:
Z ̲ = Z ̲ 0 = X ̲ 0 + i X ̲ 1 + j X ̲ 2 + k X ̲ 3 , Z ̲ 1 = X ̲ 0 + i X ̲ 1 j X ̲ 2 k X ̲ 3 , Z ̲ 2 = X ̲ 0 i X ̲ 1 + j X ̲ 2 k X ̲ 3 , Z ̲ 3 = X ̲ 0 i X ̲ 1 j X ̲ 2 + k X ̲ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equg_HTML.gif
for which Z ̲ 0 Z ̲ 0 + Z ̲ 1 Z ̲ 1 + Z ̲ 2 Z ̲ 2 + Z ̲ 3 Z ̲ 3 = 16 | X ̲ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq38_HTML.gif, the symbol denoting Hermitian quaternionic conjugation is defined as the composition of H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq15_HTML.gif-conjugation and Clifford conjugation in R 0 , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq8_HTML.gif, i.e. λ = A e A ¯ λ A ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq39_HTML.gif. The Hermitian Dirac operators are
Z ̲ 0 = 1 16 ( X ̲ 0 + i X ̲ 1 + j X ̲ 2 + k X ̲ 3 ) , Z ̲ 1 = 1 16 ( X ̲ 0 + i X ̲ 1 j X ̲ 2 k X ̲ 3 ) , Z ̲ 2 = 1 16 ( X ̲ 0 i X ̲ 1 + j X ̲ 2 k X ̲ 3 ) , Z ̲ 3 = 1 16 ( X ̲ 0 i X ̲ 1 j X ̲ 2 + k X ̲ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equh_HTML.gif

for which Δ 4 n = 16 ( Z ̲ 0 Z ̲ 0 + Z ̲ 1 Z ̲ 1 + Z ̲ 2 Z ̲ 2 + Z ̲ 3 Z ̲ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq40_HTML.gif.

Definition 2 (see [17])

Let Ω be an open set in R 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq3_HTML.gif. A continuously differentiable function f : Ω H 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq41_HTML.gif is said to be (left) q-Hermitian monogenic in Ω (or q-monogenic for short) iff it satisfies in Ω the system Z ̲ 0 f = Z ̲ 1 f = Z ̲ 2 f = Z ̲ 3 f = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq42_HTML.gif, or, equivalently, the system X ̲ 0 f = X ̲ 1 f = X ̲ 2 f = X ̲ 3 f = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq43_HTML.gif.

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 X ̲ r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq44_HTML.gif, r = 0 , , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq45_HTML.gif, i.e. the Euclidean Cauchy kernels, are respectively given by
E r ( X ̲ ) = 1 a 4 n X ̲ r | X ̲ | 4 n , r = 0 , , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equi_HTML.gif
with a 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq46_HTML.gif the area of the unit sphere S 4 n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq47_HTML.gif in R 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq3_HTML.gif. Explicitly, this means that X ̲ r E r ( X ̲ ) = δ ( X ̲ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq48_HTML.gif, r = 0 , , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq45_HTML.gif. Next we introduce the Hermitian Cauchy kernels:
E r ( Z ̲ ) = 1 a 4 n Z ̲ r | Z ̲ | 4 n , r = 0 , , 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equj_HTML.gif

Note that E r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq49_HTML.gif is not the fundamental solution of Z ̲ r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq50_HTML.gif. However, the following theorem holds, see [3].

Theorem 1 Introducing the circulant ( 4 × 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq4_HTML.gif) matrices
D = ( Z ̲ 0 Z ̲ 3 Z ̲ 2 Z ̲ 1 Z ̲ 1 Z ̲ 0 Z ̲ 3 Z ̲ 2 Z ̲ 2 Z ̲ 1 Z ̲ 0 Z ̲ 3 Z ̲ 3 Z ̲ 2 Z ̲ 1 Z ̲ 0 ) , E = ( E 0 E 3 E 2 E 1 E 1 E 0 E 3 E 2 E 2 E 1 E 0 E 3 E 3 E 2 E 1 E 0 ) , δ = ( δ 0 0 0 0 δ 0 0 0 0 δ 0 0 0 0 δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equk_HTML.gif

one obtains that D T E = E D T = δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq51_HTML.gif.

Thus, E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq52_HTML.gif is a fundamental solution of D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq53_HTML.gif, in a matricial interpretation.

We associate, with functions g 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq54_HTML.gif, g 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq55_HTML.gif, g 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq56_HTML.gif and g 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq57_HTML.gif defined in Ω R 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq58_HTML.gif and taking values in H 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq59_HTML.gif, the ( 4 × 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq4_HTML.gif) circulant matrix function
G = ( g 0 g 3 g 2 g 1 g 1 g 0 g 3 g 2 g 2 g 1 g 0 g 3 g 3 g 2 g 1 g 0 ) circ ( g 0 g 1 g 2 g 3 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equ1_HTML.gif
(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 ( 0 < α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq60_HTML.gif) and of p-integrable ( 4 × 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq4_HTML.gif) circulant matrix functions on some suitable subset E of R 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq3_HTML.gif are respectively denoted by C k ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq61_HTML.gif, C 0 , α ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq62_HTML.gif and L p ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq63_HTML.gif. The corresponding spaces of H 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq64_HTML.gif-valued functions are denoted by C k ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq65_HTML.gif, C 0 , α ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq66_HTML.gif and L p ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq67_HTML.gif. Moreover, introducing the non-negative function G ( X ̲ ) = max r = 0 , 1 , 2 , 3 { | g r ( X ̲ ) | } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq68_HTML.gif, the classes C 0 , α ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq62_HTML.gif and L p ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq63_HTML.gif may also be defined by means of the respective traditional conditions
G α = max X ̲ E G ( X ̲ ) + sup X ̲ , Y ̲ E , X ̲ Y ̲ G ( X ̲ ) G ( Y ̲ ) | X ̲ Y ̲ | α < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equl_HTML.gif
and
G p = ( E G ( X ̲ ) p ) 1 p < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equm_HTML.gif

Definition 3 The ( 4 × 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq4_HTML.gif) circulant matrix function G is called (left) Q-Hermitian monogenic in Ω (or Q-monogenic for short) iff D T G = O https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq69_HTML.gif in Ω, where O denotes the matrix with zero entries.

Similarly right Q-monogenicity is defined by the system G D T = O https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq70_HTML.gif. Left and right Q-monogenic matrix functions are called two-sided Q-monogenic. An important special case concerns the diagonal matrix function G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif, with g 0 = g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq72_HTML.gif and g 1 = g 2 = g 3 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq73_HTML.gif. Indeed, G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif is left (respectively right) Q-monogenic iff the function g is left (respectively right) q-monogenic.

Now, let Ω + = Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq74_HTML.gif be a bounded simply connected domain in R 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq3_HTML.gif with boundary Γ = Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq75_HTML.gif, and denote by Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq76_HTML.gif the complementary open domain R 4 n ( Ω Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq77_HTML.gif. We assume Γ to be a Liapunov surface. The unit normal vector on Γ at X ̲ Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq78_HTML.gif is given by
n ̲ 0 ( X ̲ ) = = 1 n ( e 4 3 n 4 3 ( X ̲ ) + e 4 2 n 4 2 ( X ̲ ) + e 4 1 n 4 1 ( X ̲ ) + e 4 n 4 ( X ̲ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equn_HTML.gif
and similarly as above, we also introduce the vectors n ̲ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq79_HTML.gif, n ̲ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq80_HTML.gif and n ̲ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq81_HTML.gif, giving rise in the usual way (up to a constant factor) to their Hermitian counterparts
N 0 = 1 16 ( n ̲ 0 + i n ̲ 1 + j n ̲ 2 + k n ̲ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equo_HTML.gif

and N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq82_HTML.gif, N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq83_HTML.gif, N 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq84_HTML.gif, as well as to the circulant matrix N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq85_HTML.gif. 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 function G, (1), is Q-monogenic in Ω then
Γ E ( Z ̲ V ̲ ) N T ( Z ̲ ) G ( X ̲ ) d S ( X ̲ ) = { G ( Y ̲ ) , Y ̲ Γ + , O , Y ̲ Γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equp_HTML.gif

Theorem 3 (q-Hermitian Cauchy integral formula)

If the function g is q-monogenic in Ω then
Γ E ( Z ̲ V ̲ ) N T ( Z ̲ ) G 0 ( X ̲ ) d S ( X ̲ ) = { G 0 ( Y ̲ ) , Y ̲ Γ + , O , Y ̲ Γ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equq_HTML.gif

where G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif is the corresponding diagonal matrix.

Next, in [4] a Q-Hermitian Cauchy transform was introduced, given by
C [ G ] ( Y ̲ ) = Γ E ( Z ̲ V ̲ ) N T ( Z ̲ ) G ( X ̲ ) d S ( X ̲ ) , Y ̲ Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equ2_HTML.gif
(2)
for a matrix function G C ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq86_HTML.gif, where Z ̲ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq87_HTML.gif and V ̲ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq88_HTML.gif denote the Hermitian versions of the Clifford vectors X ̲ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq89_HTML.gif and Y ̲ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq90_HTML.gif, respectively. C [ G ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq91_HTML.gif is a left Q-monogenic matrix function in R 4 n Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq92_HTML.gif, vanishing at infinity; in terms of the Euclidean Cauchy type integrals
C r , s g ( Y ̲ ) : = Γ E r ( X ̲ Y ̲ ) n ̲ s ( X ̲ ) g ( X ̲ ) d S ( X ̲ ) , Y ̲ Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equr_HTML.gif
it reads as
C [ G ] = 1 4 circ ( C 0 , 0 + C 1 , 1 + C 2 , 2 + C 3 , 3 C 0 , 0 C 2 , 2 + j ( C 1 , 3 + C 3 , 1 ) C 0 , 0 C 1 , 1 + C 2 , 2 C 3 , 3 C 0 , 0 C 2 , 2 j ( C 1 , 3 + C 3 , 1 ) ) [ G ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equs_HTML.gif
In particular, for the special case of the matrix G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif, the action of C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq93_HTML.gif is reduced to
C [ G 0 ] = 1 4 circ ( C 0 , 0 g + C 1 , 1 g + C 2 , 2 g + C 3 , 3 g C 0 , 0 g C 2 , 2 g + j ( C 1 , 3 g + C 3 , 1 g ) C 0 , 0 g C 1 , 1 g + C 2 , 2 g C 3 , 3 g C 0 , 0 g C 2 , 2 g j ( C 1 , 3 g + C 3 , 1 g ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equt_HTML.gif
In general C [ G 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq94_HTML.gif will not be a diagonal matrix, whence its entries will not be left q-monogenic functions. However C [ G 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq94_HTML.gif does become diagonal if and only if
C 0 , 0 g = C 2 , 2 g , C 1 , 3 g = C 3 , 1 g , 2 C 0 , 0 g = C 1 , 1 g + C 3 , 3 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equ3_HTML.gif
(3)
in which case we obtain
C [ G 0 ] = circ ( C 0 , 0 g 0 0 0 ) = circ ( C 2 , 2 g 0 0 0 ) = 1 2 circ ( C 1 , 1 g + C 3 , 3 g 0 0 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equu_HTML.gif

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

Theorem 4 Let G C 0 , α ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq95_HTML.gif ( 0 < α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq96_HTML.gif), then the continuous limit values of its Q-Hermitian Cauchy transform C [ G ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq91_HTML.gif exist and are given by
C ± [ G ] ( U ̲ ) = 1 2 ( H [ G ] ( U ̲ ) ± G ( U ̲ ) ) , U ̲ Γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equv_HTML.gif
Here we have introduced the matrix Q-Hermitian Hilbert operator
H [ G ] = 1 4 circ ( H 0 , 0 + H 1 , 1 + H 2 , 2 + H 3 , 3 H 0 , 0 H 2 , 2 + j ( H 1 , 3 + H 3 , 1 ) H 0 , 0 H 1 , 1 + H 2 , 2 H 3 , 3 H 0 , 0 H 2 , 2 j ( H 1 , 3 + H 3 , 1 ) ) [ G ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equw_HTML.gif
where the singular integrals
H r , s g ( U ̲ ) = 2 Γ E r ( X ̲ U ̲ ) n ̲ s ( X ̲ ) g ( X ̲ ) d S ( X ̲ ) , U ̲ Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equx_HTML.gif

are Cauchy principal values. H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq97_HTML.gif shows the following traditional properties, see [4].

Theorem 5 One has
  1. (i)

    H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq97_HTML.gif is a bounded linear operator on ( C 0 , α ( Γ ) , α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq98_HTML.gif ( 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq99_HTML.gif)

     
  2. (ii)

    H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq97_HTML.gif is an involution on C 0 , α ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq100_HTML.gif ( 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq99_HTML.gif).

     
Similar results may be obtained for right-hand versions of the Q-Hermitian Cauchy and Hilbert transforms by means of the alternative definitions
[ G ] C ( Y ̲ ) = Γ G ( X ̲ ) N T ( Z ̲ ) E ( Z ̲ V ̲ ) d S ( X ̲ ) , Y ̲ Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equy_HTML.gif
and
[ G ] H = [ G ] 1 4 circ ( H 0 , 0 + H 1 , 1 + H 2 , 2 + H 3 , 3 H 0 , 0 H 2 , 2 + j ( H 1 , 3 + H 3 , 1 ) H 0 , 0 H 1 , 1 + H 2 , 2 H 3 , 3 H 0 , 0 H 2 , 2 j ( H 1 , 3 + H 3 , 1 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equz_HTML.gif
where
g H r , s ( U ̲ ) = 2 Γ g ( X ̲ ) n ̲ s ( X ̲ ) E r ( X ̲ U ̲ ) d S ( X ̲ ) , U ̲ Γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equaa_HTML.gif

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 R 4 n Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq92_HTML.gif vanishing at infinity and having a prescribed jump G across Γ, i.e.
Ψ + ( U ̲ ) Ψ ( U ̲ ) = G ( U ̲ ) , U ̲ Γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equ4_HTML.gif
(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 G C 0 , α ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq101_HTML.gif, Theorem 4 ensures the solvability of the jump problem (4), its unique solution being given by
Ψ ( Y ̲ ) = C [ G ] ( Y ̲ ) , Y ̲ R 4 n Γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equab_HTML.gif

Now consider the important special case of the matrix function G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif. 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 6 For a function g C 0 , α ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq102_HTML.gif, the following statements are equivalent:
  1. (i)
    the jump problem
    ψ + ( U ̲ ) ψ ( U ̲ ) = g ( U ̲ ) , U ̲ Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equ5_HTML.gif
    (5)
     
is solvable in terms of q-monogenic functions;
  1. (ii)

    g satisfies the relations (3);

     
  2. (iii)

    g satisfies the relations C 0 , 0 g = C 1 , 1 g = C 2 , 2 g = C 3 , 3 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq103_HTML.gif.

     

Proof (i) → (ii)

Associate to the function g the diagonal matrix function G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif. Then G 0 C 0 , α ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq104_HTML.gif, and the jump problem (4) for G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif has the unique solution
Ψ ( Y ̲ ) = C [ G 0 ] ( Y ̲ ) , Y ̲ R 4 n Γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equac_HTML.gif
Let ψ be a solution of (5), then the circulant matrix
Ψ ( Y ̲ ) = circ ( ψ 0 0 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equad_HTML.gif
is another solution of the jump problem (4) for G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif, whence the uniqueness yields
circ ( ψ 0 0 0 ) = 1 4 circ ( C 0 , 0 g + C 1 , 1 g + C 2 , 2 g + C 3 , 3 g C 0 , 0 g C 2 , 2 g + j ( C 1 , 3 g + C 3 , 1 g ) C 0 , 0 g C 1 , 1 g + C 2 , 2 g C 3 , 3 g C 0 , 0 g C 2 , 2 g j ( C 1 , 3 g + C 3 , 1 g ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equae_HTML.gif
implying (ii).
  1. (ii)

    → (iii)

     
From the third relation in (3), we have 2 Y ̲ 1 C 0 , 0 g = Y ̲ 1 C 3 , 3 g + Y ̲ 1 C 1 , 1 g = Y ̲ 1 C 3 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq105_HTML.gif, and hence
2 Y ̲ 1 C 0 , 0 g = Γ ( Y ̲ 1 E 3 ( X ̲ Y ̲ ) ) n ̲ 3 ( X ̲ ) g ( X ̲ ) d S ( X ̲ ) = Γ ( Y ̲ 3 E 1 ( X ̲ Y ̲ ) ) n ̲ 3 ( X ̲ ) g ( X ̲ ) d S ( X ̲ ) = Y ̲ 3 C 1 , 3 g = Y ̲ 3 C 3 , 1 g = 0 , Y ̲ Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equaf_HTML.gif
the latter following from the second relation in (3) and the Y ̲ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq106_HTML.gif-monogenicity of C 3 , 1 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq107_HTML.gif. This fact means that C 0 , 0 g C 3 , 3 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq108_HTML.gif is a Y ̲ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq109_HTML.gif-monogenic function in R 4 n Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq92_HTML.gif. Moreover, it has a null jump through Γ, whence it vanishes in the whole of R 4 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq3_HTML.gif. We conclude that C 0 , 0 g = C 3 , 3 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq110_HTML.gif. Similarly, we arrive at C 0 , 0 g = C 1 , 1 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq111_HTML.gif.
  1. (iii)

    → (i)

     

It suffices to observe that, under the conditions stated, C 0 , 0 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq112_HTML.gif is q-monogenic, whence it solves the jump problem (5). □

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

Theorem 7 For a function g C 0 , α ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq102_HTML.gif, the following statements are equivalent:
  1. (i)
    the jump problem
    ψ + ( U ̲ ) ψ ( U ̲ ) = g ( U ̲ ) , U ̲ Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equ6_HTML.gif
    (6)
     
is solvable in terms of right q-monogenic functions;
  1. (ii)
    g satisfies the relations
    g C 0 , 0 = g C 2 , 2 , g C 1 , 3 = g C 3 , 1 , 2 g C 0 , 0 = g C 1 , 1 + g C 3 , 3 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equag_HTML.gif
     
  2. (iii)

    g satisfies the relations g C 0 , 0 = g C 1 , 1 = g C 2 , 2 = g C 3 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq113_HTML.gif.

     

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

Theorem 8 Let G C 0 , α ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq95_HTML.gif, then the following statements are equivalent:
  1. (i)
    The Dirichlet problem
    D T F = O ( resp.  F D T = O ) , in  Ω , F = G , on  Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equ7_HTML.gif
    (7)
     
has a solution.
  1. (ii)

    H [ G ] = G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq114_HTML.gif (resp. [ G ] H = G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq115_HTML.gif).

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

    → (ii)

     
Let F be a solution of the Dirichlet problem (7). Then, by the Q-Hermitian Cauchy formula, we have
C [ F ] ( Y ̲ ) = F ( Y ̲ ) , Y ̲ Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equah_HTML.gif
Taking limits as Y ̲ U ̲ Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq116_HTML.gif, (ii) follows in view of Theorem 4.
  1. (ii)

    → (i)

     

It suffices to observe that, under the condition (ii), F = C [ G ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq117_HTML.gif solves (7). □

Theorem 9 Let g C 0 , α ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq102_HTML.gif, then the following statements are equivalent:
  1. (i)
    The Dirichlet problem
    Z ̲ 0 f = Z ̲ 1 f = Z ̲ 2 f = Z ̲ 3 f = 0 , in  Ω , f = g , on  Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equ8_HTML.gif
    (8)
     
has a solution.
  1. (ii)
    g satisfies the relations
    H 0 , 0 g = H 2 , 2 g = g , H 1 , 3 g = H 3 , 1 g , H 1 , 1 g + H 3 , 3 g = 2 g . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equai_HTML.gif
     
  2. (iii)
    g satisfies the relations
    H 0 , 0 g = H 1 , 1 g = H 2 , 2 g = H 3 , 3 g = g . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equaj_HTML.gif
     

Proof (i) → (ii)

From (i) we see that the matrix function
F 0 = circ ( f 0 0 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equak_HTML.gif
is a solution of the Dirichlet problem
D T F = O , in  Ω , F = G 0 , on  Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equal_HTML.gif
whence by Theorem 8 we have that H [ G 0 ] = G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq118_HTML.gif. The desired conclusion (ii) then directly follows by comparing the entries in the above equality.
  1. (ii)

    → (iii)

     
From the condition H 0 , 0 g = H 2 , 2 g = g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq119_HTML.gif it follows that C 0 , 0 ± g = C 2 , 2 ± g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq120_HTML.gif. Therefore, as C 0 , 0 g C 2 , 2 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq121_HTML.gif is harmonic in Ω ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq122_HTML.gif and C 0 , 0 ± g C 2 , 2 ± g | Γ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq123_HTML.gif, we have C 0 , 0 g = C 2 , 2 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq124_HTML.gif in R 4 n Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq92_HTML.gif. 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 C 0 , 0 g = C 1 , 1 g = C 2 , 2 g = C 3 , 3 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq103_HTML.gif. Consequently, we obtain that H 0 , 0 g = H 1 , 1 g = H 2 , 2 g = H 3 , 3 g = g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq125_HTML.gif, as stated in (iii).
  1. (iii)

    → (i)

     
The conditions H 0 , 0 g = H 1 , 1 g = H 2 , 2 g = H 3 , 3 g = g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq125_HTML.gif imply the solvability of the Dirichlet problems
X ̲ r f = 0 , in  Ω , f = g , on  Γ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equ9_HTML.gif
(9)
where r = 0 , , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq45_HTML.gif. Now, let f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq126_HTML.gif, f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq127_HTML.gif, f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq128_HTML.gif, f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq129_HTML.gif be the respective solutions of (9), then these functions all are solutions of the classical Dirichlet problem
Δ 4 n f = 0 , in  Ω , f = g , on  Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equam_HTML.gif

whence they coincide. The function f = f 0 = f 1 = f 2 = f 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq130_HTML.gif thus is q-monogenic and constitutes a solution of (8). □

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

Theorem 10 Let g C 0 , α ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq102_HTML.gif, then the following statements are equivalent:
  1. (i)
    The Dirichlet problem
    f Z ̲ 0 = f Z ̲ 1 = f Z ̲ 2 = f Z ̲ 3 = 0 , in  Ω , f = g , on  Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equ10_HTML.gif
    (10)
     
has a solution.
  1. (ii)
    g satisfies the relations
    g H 0 , 0 = g H 2 , 2 = g , g H 1 , 3 = g H 3 , 1 , g H 1 , 1 + g H 3 , 3 = 2 g . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equan_HTML.gif
     
  2. (iii)
    g satisfies the relations
    g H 0 , 0 = g H 1 , 1 = g H 2 , 2 = g H 3 , 3 g = g . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equao_HTML.gif
     

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 H [ G | Γ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq131_HTML.gif and [ G | Γ ] H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq132_HTML.gif of its trace on the boundary Γ.

Theorem 11 Let G C 0 , α ( Ω Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq133_HTML.gif, such that D T G = O https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq69_HTML.gif in Ω, then the following statements are equivalent:
  1. (i)

    G is two-sided Q-monogenic in Ω.

     
  2. (ii)

    H [ G | Γ ] = [ G | Γ ] H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq134_HTML.gif.

     
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
H [ G | Γ ] = G | Γ = [ G | Γ ] H . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equap_HTML.gif
Conversely, suppose that H [ G | Γ ] = [ G | Γ ] H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq134_HTML.gif. By Theorem 4 and its right-handed version, we conclude that the corresponding left and right Q-Hermitian Cauchy transform of G, C [ G ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq91_HTML.gif and [ G ] C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq135_HTML.gif, have the same boundary values on Γ. This fact, together with their harmonicity, implies that
C [ G ] = [ G ] C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equaq_HTML.gif
On the other hand, from the assumed left Q-monogenicity of G we have G = C [ G ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq136_HTML.gif and hence
G = C [ G ] = [ G ] C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equar_HTML.gif

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 12 Let g C 0 , α ( Ω Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq137_HTML.gif be left q-monogenic in Ω, then the following statements are equivalent:
  1. (i)

    g is two-sided q-monogenic in Ω.

     
  2. (ii)
    g satisfies the relations
    H 0 , 0 g = g H 0 , 0 , H 2 , 2 g = g H 2 , 2 , H 1 , 3 g + H 3 , 1 g = g H 1 , 3 + g H 3 , 1 , H 1 , 1 g + H 3 , 3 g = g H 1 , 1 + g H 3 , 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equas_HTML.gif
     
  3. (iii)
    g satisfies the relations
    H 0 , 0 g = g H 0 , 0 , H 1 , 1 g = g H 1 , 1 , H 2 , 2 g = g H 2 , 2 , H 3 , 3 g = g H 3 , 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_Equat_HTML.gif
     

Proof (i) ↔ (ii)

From (i) we see that the matrix function G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif corresponding to g is two-sided Q-monogenic in Ω, whence (ii) follows from Theorem 11(ii) applied to G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif. Conversely, (ii) can be rewritten in the matricial form H [ G 0 | Γ ] = [ G 0 | Γ ] H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq138_HTML.gif, from which (i) follows by observing that the two-sided Q-monogenicity of G 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq71_HTML.gif implied by Theorem 11 is equivalent to the q-monogenicity of g.
  1. (i)

    ↔ (iii)

     

It follows from (i) that g is two-sided monogenic w.r.t. X ̲ r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq139_HTML.gif, r = 0 , , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq45_HTML.gif. We may then invoke [[2], Theorem 3.2] in order to conclude that H r , r g = g H r , r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq140_HTML.gif, r = 0 , , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq141_HTML.gif. Conversely, suppose that (iii) holds. Each of the conditions H r , r g = g H r , r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq140_HTML.gif, r = 0 , , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq45_HTML.gif, implies the two-sided monogenicity of g in Ω w.r.t. X ̲ r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq139_HTML.gif, r = 0 , , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-74/MediaObjects/13661_2012_Article_195_IEq45_HTML.gif, see again [[2], Theorem 3.2], whence g is two-sided q-monogenic in Ω. □

Declarations

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).

Authors’ Affiliations

(1)
Facultad de Informática y Matemática, Universidad de Holguín
(2)
Departamento de Matemática, Universidad de Oriente
(3)
Department of Mathematical Analysis, Faculty of Engineering, Ghent University

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© Abreu-Blaya et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. 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.