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 512141516.

More recently Hermitian Clifford analysis emerged as a refinement of the Euclidean setting for the case of R 2 n . 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 678.

In the papers 10111317, the Hermitian Clifford analysis setting was further refined by considering functions on R 4 n 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 ) circulant matrix approach, similar in spirit to the circulant (2 × 2 ) 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 R4n 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.

Let ( e 1 , … , e m ) be an orthonormal basis of Euclidean space R m and consider the real Clifford algebra R 0 , m constructed over Rm. The non-commutative multiplication in R m is governed by the rules:

e ℓ 2 = − 1 , ℓ = 1 , … , m , e ℓ e k + e k e ℓ = 0 , ℓ ≠ k .

In Rm one can consider the following automorphisms:

(i) the conjugation e ¯ ℓ = − e ℓ and for any a , b ∈ R m , a b ¯ = b ¯ a ¯

(ii) the main involution e ˜ ℓ = − e ℓ and for any a,b∈Rm, a b ˜ = a ˜ b ˜ .

In particular, we consider the skew-field of quaternions H whose elements will be denoted by q = x 0 + i x 1 + j x 2 + k x 3 with i 2 = j 2 = k 2 = − 1 and i j = − j i = k . Clearly, H may be identified with the Clifford algebra R 0 , 2 making the identifications i ↔ e 1 , j ↔ e 2 and k ↔ e 1 e 2 . The automorphisms (i) and (ii) then respectively lead to the H-conjugation

q ¯ = x 0 − i x 1 − j x 2 − k x 3

and to the main H-involution

q γ ≡ q ˜ = x 0 − i x 1 − j x 2 + k x 3 .

However, it is quite natural to introduce two more H-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 .

Definition 1 (see17)

The quaternionic Witt basis of H m = H ⊗ R R m , m = 4 n , is given by { f ℓ , f ℓ α , f ℓ β , f ℓ γ } , ℓ = 1 , … , n , where

f ℓ = e 1 + 4 ( ℓ − 1 ) − i e 2 + 4 ( ℓ − 1 ) − j e 3 + 4 ( ℓ − 1 ) − k e 4 + 4 ( ℓ − 1 ) , f ℓ α = e 1 + 4 ( ℓ − 1 ) − i e 2 + 4 ( ℓ − 1 ) + j e 3 + 4 ( ℓ − 1 ) + k e 4 + 4 ( ℓ − 1 ) , f ℓ β = e 1 + 4 ( ℓ − 1 ) + i e 2 + 4 ( ℓ − 1 ) − j e 3 + 4 ( ℓ − 1 ) + k e 4 + 4 ( ℓ − 1 ) , f ℓ γ = e 1 + 4 ( ℓ − 1 ) + i e 2 + 4 ( ℓ − 1 ) + j e 3 + 4 ( ℓ − 1 ) − k e 4 + 4 ( ℓ − 1 ) .

We will consider the Clifford vectors

X ̲ = X ̲ 0 = ∑ ℓ = 1 n ( e 4 ℓ − 3 x 4 ℓ − 3 + e 4 ℓ − 2 x 4 ℓ − 2 + e 4 ℓ − 1 x 4 ℓ − 1 + e 4 ℓ x 4 ℓ ) , X ̲ 1 = ∑ ℓ = 1 n ( e 4 ℓ − 3 x 4 ℓ − 2 − e 4 ℓ − 2 x 4 ℓ − 3 − e 4 ℓ − 1 x 4 ℓ + e 4 ℓ x 4 ℓ − 1 ) , X ̲ 2 = ∑ ℓ = 1 n ( e 4 ℓ − 3 x 4 ℓ − 1 + e 4 ℓ − 2 x 4 ℓ − e 4 ℓ − 1 x 4 ℓ − 3 − e 4 ℓ x 4 ℓ − 2 ) , X ̲ 3 = ∑ ℓ = 1 n ( e 4 ℓ − 3 x 4 ℓ − e 4 ℓ − 2 x 4 ℓ − 1 + e 4 ℓ − 1 x 4 ℓ − 2 − e 4 ℓ x 4 ℓ − 3 )

for which X ̲ r 2 = − | X ̲ 0 | 2 , while X ̲ r X ̲ s + X ̲ s X ̲ r = 0 , r ≠ s , r , s = 0 , … , 3 . The corresponding Dirac operators are denoted by ∂ X ̲ = ∂ X ̲ 0 , ∂ X ̲ 1 , ∂ X ̲ 2 and ∂ X ̲ 3 . Here we have ∂ X ̲ r 2 = − Δ 4 n , with Δ 4 n the Laplacian in R4n, and ∂ X ̲ r ∂ X ̲ s + ∂ X ̲ s ∂ X ̲ r = 0 , r≠s, r,s=0,…,3. 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

for which Z ̲ 0 Z ̲ 0 † + Z ̲ 1 Z ̲ 1 † + Z ̲ 2 Z ̲ 2 † + Z ̲ 3 Z ̲ 3 † = 16 | X ̲ | 2 , the symbol ^† denoting Hermitian quaternionic conjugation is defined as the composition of H-conjugation and Clifford conjugation in R0,m, i.e. λ † = ∑ A e A ¯ λ A ¯ . 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 )

for which Δ 4 n = 16 ( ∂ Z ̲ 0 ∂ Z ̲ 0 † + ∂ Z ̲ 1 ∂ Z ̲ 1 † + ∂ Z ̲ 2 ∂ Z ̲ 2 † + ∂ Z ̲ 3 ∂ Z ̲ 3 † ) .

Definition 2 (see 17)

Let Ω be an open set in R4n. A continuously differentiable function f : Ω ↦ H 4 n 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 , or, equivalently, the system ∂ X ̲ 0 f = ∂ X ̲ 1 f = ∂ X ̲ 2 f = ∂ X ̲ 3 f = 0 .

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 , r = 0 , … , 3 , 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

with a 4 n the area of the unit sphere S 4 n − 1 in R4n. Explicitly, this means that ∂ X ̲ r E r ( X ̲ ) = δ ( X ̲ ) , r=0,…,3. Next we introduce the Hermitian Cauchy kernels:

E r ( Z ̲ ) = 1 a 4 n Z ̲ r † | Z ̲ | 4 n , r = 0 , … , 3 .

Note that E r is not the fundamental solution of ∂ Z ̲ r . However, the following theorem holds, see 3.

Theorem 1 Introducing the circulant (4×4) 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 )

one obtains that D T E = E D T = δ .

Thus, E is a fundamental solution of D , in a matricial interpretation.

We associate, with functions g 0 , g 1 , g 2 and g 3 defined in Ω ⊂ R 4 n and taking values in H 4 n , the (4×4) 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 ) .

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 ) and of p-integrable (4×4) circulant matrix functions on some suitable subset E of R4n are respectively denoted by C k ( E ) , C 0 , α ( E ) and L p ( E ) . The corresponding spaces of H 4 n -valued functions are denoted by C k ( E ) , C 0 , α ( E ) and L p ( E ) . Moreover, introducing the non-negative function ∥ G ( X ̲ ) ∥ = max r = 0 , 1 , 2 , 3 { | g r ( X ̲ ) | } , the classes C0,α(E) and Lp(E) 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 < + ∞

and

∥ G ∥ p = ( ∫ E ∥ G ( X ̲ ) ∥ p ) 1 p < + ∞ .

Definition 3 The (4×4) circulant matrix function G is called (left) Q-Hermitian monogenic in Ω (or Q-monogenic for short) iff D T G = O in Ω, where O denotes the matrix with zero entries.

Similarly right Q-monogenicity is defined by the system G D T = O . Left and right Q-monogenic matrix functions are called two-sided Q-monogenic. An important special case concerns the diagonal matrix function G 0 , with g 0 = g and g 1 = g 2 = g 3 = 0 . Indeed, G0 is left (respectively right) Q-monogenic iff the function g is left (respectively right) q-monogenic.

Now, let Ω + = Ω be a bounded simply connected domain in R4n with boundary Γ = ∂ Ω , and denote by Ω − the complementary open domain R 4 n ∖ ( Ω ∪ Γ ) . We assume Γ to be a Liapunov surface. The unit normal vector on Γ at X ̲ ∈ Γ is given by

n ̲ 0 ( X ̲ ) = ∑ ℓ = 1 n ( e 4 ℓ − 3 n 4 ℓ − 3 ( X ̲ ) + e 4 ℓ − 2 n 4 ℓ − 2 ( X ̲ ) + e 4 ℓ − 1 n 4 ℓ − 1 ( X ̲ ) + e 4 ℓ n 4 ℓ ( X ̲ ) )

and similarly as above, we also introduce the vectors n ̲ 1 , n ̲ 2 and n ̲ 3 , 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 )

and N 1 , N 2 , N 3 , as well as to the circulant matrix N . 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 − .

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 − ,

where G0 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

for a matrix function G ∈ C ( Γ ) , where Z ̲ and V ̲ denote the Hermitian versions of the Clifford vectors X ̲ and Y ̲ , respectively. C [ G ] is a left Q-monogenic matrix function in R 4 n ∖ Γ , 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

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 ] .

In particular, for the special case of the matrix G0, the action of C 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 ) ) .

In general C [ G 0 ] will not be a diagonal matrix, whence its entries will not be left q-monogenic functions. However C[G0] 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

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 ) .

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 ) (0 < α ≤ 1 ), then the continuous limit values of its Q-Hermitian Cauchy transform C[G] exist and are given by

C ± [ G ] ( U ̲ ) = 1 2 ( H [ G ] ( U ̲ ) ± G ( U ̲ ) ) , U ̲ ∈ Γ .

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 ] ,

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

are Cauchy principal values. H shows the following traditional properties, see 4.

Theorem 5 One has

(i) H is a bounded linear operator on ( C 0 , α ( Γ ) , ∥ • ∥ α ) (0 < α < 1 )

(ii) H is an involution on C 0 , α ( Γ ) (0<α<1).

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

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 ) ) ,

where

g H r , s ( U ̲ ) = 2 ∫ Γ g ( X ̲ ) n ̲ s ( X ̲ ) E r ( X ̲ − U ̲ ) d S ( X ̲ ) , U ̲ ∈ Γ .

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 R4n∖Γ vanishing at infinity and having a prescribed jump G across Γ, i.e.

Ψ + ( U ̲ ) − Ψ − ( U ̲ ) = G ( U ̲ ) , U ̲ ∈ Γ .

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 ) , 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 .

Now consider the important special case of the matrix function G0. 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 ) , the following statements are equivalent:

(i) the jump problem

ψ + ( U ̲ ) − ψ − ( U ̲ ) = g ( U ̲ ) , U ̲ ∈ Γ

is solvable in terms of q-monogenic functions;

(ii) g satisfies the relations (3);

(iii) g satisfies the relations C 0 , 0 g = C 1 , 1 g = C 2 , 2 g = C 3 , 3 g .

Proof (i) → (ii)

Associate to the function g the diagonal matrix function G0. Then G 0 ∈ C 0 , α ( Γ ) , and the jump problem (4) for G0 has the unique solution

Ψ ( Y ̲ ) = C [ G 0 ] ( Y ̲ ) , Y ̲ ∈ R 4 n ∖ Γ .

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

Ψ ( Y ̲ ) = circ ( ψ 0 0 0 )

is another solution of the jump problem (4) for G0, 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 ) )

implying (ii).

(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 , 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

the latter following from the second relation in (3) and the ∂ Y ̲ 3 -monogenicity of C 3 , 1 g . This fact means that C 0 , 0 g − C 3 , 3 g is a ∂ Y ̲ 1 -monogenic function in R4n∖Γ. Moreover, it has a null jump through Γ, whence it vanishes in the whole of R4n. We conclude that C 0 , 0 g = C 3 , 3 g . Similarly, we arrive at C 0 , 0 g = C 1 , 1 g .

(iii) → (i)

It suffices to observe that, under the conditions stated, C 0 , 0 g 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∈C0,α(Γ), the following statements are equivalent:

(i) the jump problem

ψ + ( U ̲ ) − ψ − ( U ̲ ) = g ( U ̲ ) , U ̲ ∈ Γ

is solvable in terms of right q-monogenic functions;

(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 ;

(iii) g satisfies the relations g C 0 , 0 = g C 1 , 1 = g C 2 , 2 = g C 3 , 3 .

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

Theorem 8 Let G∈C0,α(Γ), then the following statements are equivalent:

(i) The Dirichlet problem

D T F = O ( resp.  F D T = O ) , in  Ω , F = G , on  Γ

has a solution.

(ii) H [ G ] = G (resp. [ G ] H = G ).

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

C [ F ] ( Y ̲ ) = F ( Y ̲ ) , Y ̲ ∈ Ω .

Taking limits as Y ̲ → U ̲ ∈ Γ , (ii) follows in view of Theorem 4.

(ii) → (i)

It suffices to observe that, under the condition (ii), F = C [ G ] solves (7). □

Theorem 9 Let g∈C0,α(Γ), then the following statements are equivalent:

(i) The Dirichlet problem

∂ Z ̲ 0 f = ∂ Z ̲ 1 f = ∂ Z ̲ 2 f = ∂ Z ̲ 3 f = 0 , in  Ω , f = g , on  Γ

has a solution.

(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 .

(iii) g satisfies the relations

H 0 , 0 g = H 1 , 1 g = H 2 , 2 g = H 3 , 3 g = g .

Proof (i) → (ii)

From (i) we see that the matrix function

F 0 = circ ( f 0 0 0 )

is a solution of the Dirichlet problem

D T F = O , in  Ω , F = G 0 , on  Γ

whence by Theorem 8 we have that H [ G 0 ] = G 0 . The desired conclusion (ii) then directly follows by comparing the entries in the above equality.

(ii) → (iii)

From the condition H 0 , 0 g = H 2 , 2 g = g it follows that C 0 , 0 ± g = C 2 , 2 ± g . Therefore, as C 0 , 0 g − C 2 , 2 g is harmonic in Ω ± and C 0 , 0 ± g − C 2 , 2 ± g | Γ = 0 , we have C 0 , 0 g = C 2 , 2 g in R4n∖Γ. 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 C0,0g=C1,1g=C2,2g=C3,3g. Consequently, we obtain that H 0 , 0 g = H 1 , 1 g = H 2 , 2 g = H 3 , 3 g = g , as stated in (iii).

(iii) → (i)

The conditions H0,0g=H1,1g=H2,2g=H3,3g=g imply the solvability of the Dirichlet problems

∂ X ̲ r f = 0 , in  Ω , f = g , on  Γ ,

where r=0,…,3. Now, let f 0 , f 1 , f 2 , f 3 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

whence they coincide. The function f = f 0 = f 1 = f 2 = f 3 thus is q-monogenic and constitutes a solution of (8). □

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

Theorem 10 Let g∈C0,α(Γ), then the following statements are equivalent:

(i) The Dirichlet problem

f ∂ Z ̲ 0 = f ∂ Z ̲ 1 = f ∂ Z ̲ 2 = f ∂ Z ̲ 3 = 0 ,   in  Ω , f = g , on  Γ

has a solution.

(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 .

(iii) g satisfies the relations

g H 0 , 0 = g H 1 , 1 = g H 2 , 2 = g H 3 , 3 g = g .

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 ] and [ G | Γ ] H of its trace on the boundary Γ.

Theorem 11 Let G ∈ C 0 , α ( Ω ∪ Γ ) , such that DTG=O in Ω, then the following statements are equivalent:

(i) G is two-sided Q-monogenic in Ω.

(ii) H [ G | Γ ] = [ G | Γ ] H .

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 .

Conversely, suppose that H[G|Γ]=[G|Γ]H. By Theorem 4 and its right-handed version, we conclude that the corresponding left and right Q-Hermitian Cauchy transform of G, C[G] and [ G ] C , have the same boundary values on Γ. This fact, together with their harmonicity, implies that

C [ G ] = [ G ] C .

On the other hand, from the assumed left Q-monogenicity of G we have G = C [ G ] and hence

G = C [ G ] = [ G ] C

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 ) be left q-monogenic in Ω, then the following statements are equivalent:

(i) g is two-sided q-monogenic in Ω.

(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 .

(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 .

Proof (i) ↔ (ii)

From (i) we see that the matrix function G0 corresponding to g is two-sided Q-monogenic in Ω, whence (ii) follows from Theorem 11(ii) applied to G0. Conversely, (ii) can be rewritten in the matricial form H [ G 0 | Γ ] = [ G 0 | Γ ] H , from which (i) follows by observing that the two-sided Q-monogenicity of G0 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. ∂ X ̲ r , r=0,…,3. We may then invoke [2, Theorem 3.2] in order to conclude that H r , r g = g H r , r , r = 0 , … , 3 . Conversely, suppose that (iii) holds. Each of the conditions Hr,rg=gHr,r, r=0,…,3, implies the two-sided monogenicity of g in Ω w.r.t. ∂X̲r, r=0,…,3, see again [2, Theorem 3.2], whence g is two-sided q-monogenic in Ω. □

The authors declare that they have no competing interests.

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