Adv. Appl. Clifford Algebras 21 (2011), 591–605
© 2011 Springer Basel AG
0188-7009/030591-15
published online January 4, 2011
DOI 10.1007/s00006-010-0272-2

Advances in
Applied Clifford Algebras

Differential Operators Associated to the
Cauchy-Fueter Operator in Quaternion
Algebra
Thanh Van Nguyen
Abstract. This paper deals with the initial value problem of the type
∂w
∂w
= L t, x, w,
∂t
∂xi
w(0, x) = ϕ(x)

(1)
(2)

where t is the time, L is a linear first order operator (matrix-type)
in Quaternionic Analysis and ϕ is a regular function taking values in
the Quaternionic Algebra. The article proves necessary and sufficient
conditions on the coefficients of operator L under which L is associated
to the Cauchy-Fueter operator of Quarternionic Analysis.
This criterion makes it possible to construct the operator L for
which the initial problem (1), (2) is solvable for an arbitrary initial

592

T.V. Nguyen

Adv. Appl. Clifford Algebras

Definition 1. A function f ∈ C 1 (Ω, H) is said to be regular in Ω if f satisfies
Df = 0.

2. Necessary and Sufficient Conditions for Associated Pairs
Suppose that f =

3
j=0

fj ej is a twice continuously differentiable function

with respect to the space-like x0 , x1 , x2 , x3 . Now assume that f is regular.
This means that Df = 0. It is easy to verify that the condition Df = 0 is
equivalent to
3
∂f
Ai
= 0,
∂x
i
i=0
where

⎡

0 0
⎡

0
⎢0
A3 = ⎢
⎣0
1
We define an operator

0
0
1
0

⎤
0 0
0 0⎥
⎥,
0 −1⎦
1 0

⎤
0 −1
−1 0 ⎥
⎥,
0
0⎦
0
0

⎜ ∂xi ⎟
∂f
⎟
=⎜
⎜ ∂f2 ⎟ .
∂xi
⎝ ∂xi ⎠
∂f3
∂xi

as follows,
3

f=

Ai
i=0

∂f
.
∂xi

(3)

It is clear that
⎛Df⎞= 0 if and only if f = 0. Next, we identify the function
f0
⎜f1 ⎟
⎟
f with f := ⎜


593

the operator (in other words, L is associated to the Cauchy-Fueter operator
(j)
of Quaternionic Analysis). Assume that the functions bαβ , cαβ , dα (j, α, β =
0, 1, 2, 3) are continuously differentiable with respect to the space-like variable
x0 , x1 , x2 , x3 and differentiable on t.
Put
(j)

Pj = [pαβ ] = Aj Bj ,
Qij =

(ij)
[qαβ ]
(j)

j = 0, 1, 2, 3

= Ai B j + A j B i ,

Rj = [rαβ ] =

3

Ai
i=0

(5)

(2)
(0)
(3)
(0)
⎪ri0 = ri1
, ri0 = ri2 , ri0 = ri3
⎪
⎪
⎪
⎪
(1)
(0)
(2)
(0)
(3)
(0)
⎪
⎪
r = −ri0 , ri1 = −ri3 , ri1 = ri2
⎪
⎨ i1
(1)
(0)
(2)
(0)
(3)
(0)
ii) ri2)
= ri3 , ri2 = −ri0 , ri2 = −ri1
⎪

⎪
(12)
(1)
(2)
(13)
(1)
(3)
(23)
(2)
(3)
⎪
⎪
⎪qi0 = −pi3 + pi3 , qi0 = pi2 − pi2 , qi0 = −pi1 + pi1
⎪
⎪
⎪
⎪q (01) = −p(0) + p(1) , q (02) = −p(0) + p(2) , q (03) = p(0) − p(3)
⎪
⎪
i1
i0
i0
i1
i3
i3
i1
i2
i2
⎪
⎪

(12)
(1)
(2)
(13)
(1)
(3)
(23)
(2)
(3)
⎪
⎪
qi2 = pi1 − pi1 , qi2 = −pi0 + pi0 , qi2 = −pi3 + pi3
⎪
⎪
⎪
⎪
(01)
(0)
(1)
(02)
(0)
(2)
(03)
(0)
(3)
⎪
qi3 = −pi2 + pi2 , qi3 = pi1 − pi1 , qi3 = −pi0 + pi0
⎪
⎪
⎪

Ai
Bj
+ Cf + K ⎠
∂x
∂x
i
j
i=0
j=0

(Lf ) =

Ai


594

T.V. Nguyen

⎛

3

=

Ai
i=0
3

⎞

3

+

Ai
i=0

3

3

∂2f
∂Bj ∂f
+
Ai
∂xi ∂xj
∂xi ∂xj
i=0 j=0
3

∂C
∂xi

3

Ai C

f+
i=0


+ Aj C
∂xi

∂2f
∂xi ∂xj

3

∂f
+
∂xj

Ai
i=0

(8)

∂C
∂xi

3

Ai

f+
i=0

∂K
.
∂xi

∂2f
∂f
+
Rj
∂xi ∂xj j=0 ∂xj

3

Ai

f+
i=0

∂K
.
∂xi

(9)
⎛

⎞
m0
⎜m1 ⎟
∂2f
∂2f
⎟
M=
Pi 2 +
Qij
=⎜


3

f,

Ai

T =
i=0

∂K
.
∂xi

Then we obtain
l(Lf ) = M + N + S + T.
We get
2
2
2
f0
(0) ∂ f1
(0) ∂ f2
(0) ∂ f3
+
p
+
p
+
p

2
2
2
(2) ∂ f0
(2) ∂ f1
(2) ∂ f2
(2) ∂ f3
+ pi0
+ pi1
+ pi2
+ pi3
2
2
2
∂x2
∂x2
∂x2
∂x22
(0) ∂

mi = pi0

2

(10)


Vol. 21 (2011)

Differential Associated Operators and Their Applications

∂x0 ∂x1
∂x0 ∂x1
∂x0 ∂x1
∂x0 ∂x1
2
2
2
2
(02) ∂ f0
(02) ∂ f1
(02) ∂ f2
(02) ∂ f3
+ qi0
+ qi1
+ qi2
+ qi3
∂x0 ∂x2
∂x0 ∂x2
∂x0 ∂x2
∂x0 ∂x2
2
2
2
2
∂
f
∂
f
∂
f

(12)
(12) ∂ f3
+ qi0
+ qi1
+ qi2
+ qi3
∂x1 ∂x2
∂x1 ∂x2
∂x1 ∂x2
∂x1 ∂x2
2
2
2
2
(13) ∂ f0
(13) ∂ f1
(13) ∂ f2
(13) ∂ f3
+ qi0
+ qi1
+ qi2
+ qi3
∂x1 ∂x3
∂x1 ∂x3
∂x1 ∂x3
∂x1 ∂x3
2
2
2
2

+ ri3
∂x0
∂x0
∂x0
∂x0
(1) ∂f0
(1) ∂f1
(1) ∂f2
(1) ∂f3
+ ri0
+ ri1
+ ri2
+ ri3
∂x1
∂x1
∂x1
∂x1
∂f
∂f
∂f
0
1
2
(2)
(2)
(2)
(2) ∂f3
+ ri0
+ ri1
+ ri2

⎪
⎪
⎪
⎨ ∂f0 +
∂x1
⎪ ∂f0 +
⎪
∂x2
⎪
⎪
⎪ ∂f0
⎩
∂x3 −

∂f1
∂x1
∂f1
∂x0
∂f1
∂x3
∂f1
∂x2

−
−
+
+

∂f2
∂x2

⎧
∂f0
∂f1
∂f2
∂f3
⎪
= ∂x
+ ∂x
+ ∂x
⎪
1
2
3
⎪ ∂x0
⎪
⎪
⎨ ∂f1 = − ∂f0 + ∂f2 − ∂f3
∂x0
∂x1
∂x3
∂x2
∂f2
∂f0
∂f1
∂f3
⎪
=
−
−
+

2
2
f1
f2
f3
= ∂x∂0 ∂x
+ ∂x∂0 ∂x
+ ∂x∂0 ∂x
⎪
⎪
∂x20
1
2
3
⎪
⎪
2
2
2
2
⎪
⎨ ∂ f20 = − ∂ f1 − ∂ f3 + ∂ f2
∂x0 ∂x1
∂x1 ∂x2
∂x1 ∂x3
∂x
2

1


2
2
2
⎩ ∂ f0
f1
f2
f3
= ∂x∂2 ∂x
− ∂x∂1 ∂x
− ∂x∂0 ∂x
.
∂x2
3
3
3

(15)

3

and similar expression for the other

j = 0, 1, 2, 3.

Hence we get 3 remaining systems having the form of (15). Thus, one
has a total of 12 equations. Substituting above 12 equations into (11), and
after a calculation, we obtain
∂ 2 f0
∂ 2 f0
(0)

(23)
−pi2 + pi2 + qi0
+ pi1 − pi1 + qi0
∂x1 ∂x3
∂x2 ∂x3
∂ 2 f1
∂ 2 f1
(0)
(1)
(01)
(0)
(2)
(02)
pi0 − pi0 + qi1
+ pi3 − pi3 + qi1
∂x0 ∂x1
∂x0 ∂x2
2
∂ 2 f1
∂ f1
(0)
(3)
(03)
(1)
(2)
(12)
−pi2 + pi2 + qi1
+ pi2 − pi2 + qi1
∂x0 ∂x3
∂x1 ∂x2

∂ 2 f2
f
2
(0)
(3)
(03)
(1)
(2)
(12)
pi1 − pi1 + qi2
+ −pi1 + pi1 + qi2
∂x0 ∂x3
∂x1 ∂x2
∂ 2 f2
∂ 2 f2
(1)
(3)
(13)
(2)
(3)
(23)
pi0 − pi0 + qi2
+ pi3 − pi3 + qi2
∂x1 ∂x3
∂x2 ∂x3
2
∂ f3
∂ 2 f3
(0)
(1)

(3)
(23)
−pi1 + pi1 + qi3
+ −pi2 + pi2 + qi3
.
∂x1 ∂x3
∂x2 ∂x3
(16)
(0)

(1)

(01)

mi = −pi1 + pi1 + qi0
+
+
+
+
+
+
+
+
+
+
+

Analogously, substituting the relation (14) into (12), one gets
∂f0
(0)


∂f2
(0)
(2) ∂f2
(0)
(3) ∂f2
+ (ri0 + ri2 )
+ (ri1 + ri2 )
∂x1
∂x2
∂x3
(0)
(1) ∂f3
(0)
(2) ∂f3
(0)
(3) ∂f3
+ (ri2 + ri3 )
+ (−ri1 + ri3 )
+ (ri0 + ri3 )
.
∂x1
∂x2
∂x3
(0)

597

(1)


Ai ∂x
= 0. In other words h(α) =
i

3
i=0

ciα ei , α = 0, 1, 2, 3 are regular

functions. Hence S vanished in (10). Now, choose f (3) = x0 + x1 e1 , then (10)
(0)
(1)
leads to N = 0, so ni = 0, i = 0, 1, 2, 3. But in fact ni = ri0 + ri1 .
(1)
(0)
Therefore, we get ri1 = −ri0 .
Note that the equality is the same as the condition 4th of the relation
(i).
By similar method, choose
(4)
f = x1 − x0 e1 , f (5) = x0 e2 + x1 e3 , f (6) = x1 e2 − x0 e3 ,
f (7) = x0 + x2 e2 , f (8) = x0 e1 − x2 e3 , f (9) = x2 − x0 e2 ,
f (10) = x2 e1 + x0 e3 , f (11) = x0 + x3 e3 , f (12) = x0 e1 + x3 e2 ,
f (13) = −x3 e1 + x0 e2 , f (14) = −x3 + x0 e3
and substitute these functions into (10) we obtain N = 0 for all f (i) , i =
4, . . . , 14. From this, we have remaining equalities which are contained in the
condition (ii). Hence N can be omitted in (10).
Now we choose f (15) = (x20 − x21 ) + 2x0 x1 e1 and replace f in (10) by
(15)
f

T.V. Nguyen

Adv. Appl. Clifford Algebras

Note that (18) is the same as the first condition of (iii). Similarly, choose
f (16) = −2x0 x1 + (x20 − x21 )e1

f (17) = (x20 − x21 )e2 + 2x0 x1 e3

f (18) = −2x0 x1 e2 + (x20 − x21 )e3

f (19) = (x20 − x22 ) + 2x0 x2 e2

f (20) = (x20 − x22 )e1 − 2x0 x2 e3

f (21) = −2x0 x2 + (x20 − x22 )e2

f (22) = 2x0 x2 e1 + (x20 − x22 )e3 ,

f (23) = (x21 − x22 ) − 2x1 x2 e3

f (24) = (x21 − x22 )e1 − 2x1 x2 e2

f (25) = 2x1 x2 e1 + (x21 − x22 )e2

f (26) = 2x1 x2 + (x21 − x22 )e3 ,

f (27) = (x20 − x23 ) + 2x0 x3 e3

f (28) = (x20 − x23 )e1 + 2x0 x3 e2 ,


0 ≤ i < j ≤ 3.

So we get
3

Pi

l(Lf ) =
i=0

∂2f
+
∂x2i

3

(Pi − Pj )Aj Ai
0≤i
∂
− P 1 A1
− P 2 A2
− P 3 A3
+ R0 ,
∂x0
∂x1
∂x2
∂x3

and P0 , P1 , P2 , P3 , R0 are given in (5) and (7).
Therefore one gets the following theorem.


Vol. 21 (2011)

Differential Associated Operators and Their Applications

Theorem 2. The operator L is associated to the operator

599

if and only if

lL = V l,
where V = P0

∂
∂
∂

2
1
(0)
(0)
b02 = [c02 x0 − c12 x1 + (γ − c22 )x2 − c32 x3 ] + δ02
2
1
(0)
(0)
b03 = [c03 x0 − c13 x1 − c23 x2 + (γ − c33 )x3 ] + δ03
2
1
(0)
(0)
b10 = [c10 x0 − (γ − c00 )x1 + c30 x2 − c20 x3 ] + δ10
2
1
(0)
(0)
b11 = [−(γ − c11 )x0 + c01 x1 + c31 x2 − c21 x3 ] + δ11
2
1
(0)
(0)
b12 = [c12 x0 + c02 x1 + c32 x2 + (γ − c22 )x3 ] + δ12
2
1
(0)
(0)
b13 = [c13 x0 + c03 x1 − (γ − c33 )x2 − c23 x3 ] + δ13

2
1
(0)
(0)
b32 = [c32 x0 − (γ − c22 )x1 − c12 x2 + c02 x3 ] + δ32
2
1
(0)
(0)
b33 = [−(γ − c33 )x0 + c23 x1 − c13 x2 + c03 x3 ] + δ33 ,
2
where γ,

(0)

δαβ ,

α, β = 0, 1, 2, 3 are arbitrary real-constants.


600

T.V. Nguyen

Adv. Appl. Clifford Algebras

Second, choose
⎧
⎪
⎨B1 = −A1 B0

3
c01 = s0 x2 + γ01
c11 = 3s1 x1 + 3s2 x2 + γ11
c21 = 3s2 x1 − 3s1 x2 + γ21
c31 = s0 x1 + γ31
c02 = −s0 x1 + γ02
c12 = −3s2 x1 + 3s1 x2 + γ12
c22 = 3s1 x1 + 3s2 x2 + γ22
c32 = s0 x2 + γ32
2
c03 = − s0 x0 + s2 x1 − s1 x2 + γ03
3
1
c13 = −2s2 x0 − s0 x1 + γ13
3
1
c23 = 2s1 x0 − s0 x2 + γ23
3
c33 = s1 x1 + s2 x2 + γ33 ,
where s0 , s1 , s2 , γij are arbitrary real-constants.
The elements of the matrix B0 are given by


Vol. 21 (2011)

Differential Associated Operators and Their Applications

601

2


(0)

+ (a4 + γ00 − γ11 )x2 + k31
2
= s2 (x20 + x21 − x22 ) − s0 x0 x1 − 2s1 x1 x2
3
+ (γ02 − γ13 − μ2 + μ4 − μ6 )x0 + (γ30 − γ12 + δ4 )x1

(0)

+ (γ33 − γ22 − c3 + c4 − c6 )x2 + k02
1
= − s0 (x20 + x21 − x22 ) − 2s2 x0 x1 + 2s1 x0 x2
3
+ (γ03 + γ12 + δ3 + δ5 + δ6 )x0 + (γ02 + γ20 + μ5 )x1

b02

b12

(0)

B22

(0)

b32

+ (γ23 + γ32 − b3 + b4 − b5 )x2 + k12


b10 = c2 x0 + b2 x1 + a2 x2 + l10
(1)

b20 = δ1 x0 + μ1 x1 + a3 x2 + l20
(1)

b30 = μ3 x0 + δ2 x1 + a4 x2 + l30
2
(1)
(0)
b01 = −b00 + 2s1 x0 x1 + 2s2 x0 x2 + s0 x1 x2 + (γ11 − γ33 + c3 + c6 )x0
3
+ (γ01 + γ23 − b3 − b5 )x1 + (γ31 − γ20 )x2 + k11 − l10


602

T.V. Nguyen

Adv. Appl. Clifford Algebras

2
(1)
(0)
b11 = −b10 − s1 (x20 − x21 + x22 ) − s0 x0 x2 + 2s2 x1 x2
3
− (γ01 + γ23 − b3 − b5 )x0 + (γ11 − γ33 + c3 + c6 )x1
+ (γ21 + γ30 )x2 − k01 + l00
2

− (γ02 + γ20 − μ2 − μ6 )x0 + (γ12 − γ30 + δ3 + δ6 )x1

(1)

+ (γ22 − γ33 )x2 − (k02 + l03 )
2
(0)
= b23 − s1 (x20 − x21 + x22 ) + s0 x0 x2 + 2s2 x1 x2
3
+ (γ32 + γ10 − b3 − b5 )x0 + (γ22 − γ00 + c3 + c6 )x1

b31

b02

b12

b22

+ (γ03 − γ12 )x2 + k32 + l33
2
(1)
(0)
b32 = b33 − 2s1 x0 x1 − 2s2 x0 x2 + s0 x1 x2 − (γ22 − γ00 + c3 + c6 )x0
3
− (γ10 + γ32 − b3 − b5 )x1 + (γ13 − γ02 )x2 − (k22 + l23 )
(1)

b03 = −2s2 x20 + (γ13 + γ20 − μ4 )x0 − (δ3 + δ4 + δ6 )x1
(1)


603

(2)

b30 = b6 x0 + c6 x1 + δ6 x2 + m30
1
(2)
(0)
b01 = −b03 − s0 (x20 + x21 − x22 ) + 2s2 x0 x1 − 2s1 x0 x2
3
+ (γ21 + γ30 − δ3 − δ6 )x0 − (γ13 + γ31 )x1 + (γ01 + γ10 − b3 − b5 )x2
+ k21 − l13 − m10
2
(2)
(0)
b11 = −b13 − s2 (x20 + x21 − x22 ) − s0 x0 x1 + 2s1 x1 x2
3
− (γ31 − γ20 + μ2 + μ6 )x0 + (γ03 − γ21 )x1 + (γ11 − γ00 + c3 + c6 )x2
− k31 + l03 + l30 + m00
2
(2)
(0)
b21 = −b23 + s1 (x20 + x21 − x22 ) − s0 x0 x2 + 2s2 x1 x2
3
− (γ10 + γ01 − b3 − b5 )x0 + (γ11 − γ33 )x1 + (γ21 + γ30 − δ3 − δ6 )x2
− k01 + l00 − l33 − m30
2
(2)
(0)

3
+ (γ03 + γ12 + δ3 + δ6 )x0 + (γ02 + γ20 )x1 + (γ23 + γ32 − b3 − b5 )x2
+ k12 − m10

(2)
b03

= 2s1 x20 + (b1 − γ10 + γ23 )x0 + (c1 − c3 − c6 )x1
+ (a1 − δ3 − δ6 )x2 + l00 − l33 − m30

(2)

b13 = (c2 + γ00 − γ33 )x0 + (b2 − b3 − b5 )x1
(2)

b23

+ (a2 − μ2 − μ6 )x2 + l10 − l23 − m20
2
= s0 x20 + (δ1 − γ03 − γ30 )x0 + (μ1 − μ2 − μ6 )x1
3
+ (a3 + b3 + b5 )x2 + l20 + l13 + m10


604

T.V. Nguyen

Adv. Appl. Clifford Algebras


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Vol. 21 (2011)

Differential Associated Operators and Their Applications

605

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