Articulo 12.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 02, (189–197). June 2010

Fischer decomposition by inframonogenic functions

Helmuth R. Malonek1, Dixan Peña Peña2

Department of Mathematics,

Aveiro University,

3810-193 Aveiro, Portugal
1email: hrmalon@ua.pt

2email: dixanpena@ua.pt, dixanpena@gmail.com

and

Frank Sommen

Department of Mathematical Analysis,

Ghent University,

9000 Gent, Belgium

email: fs@cage.ugent.be

ABSTRACT

Let ∂x denote the Dirac operator in R
m. In this paper, we present a refinement of the

biharmonic functions and at the same time an extension of the monogenic functions by

considering the equation ∂xf ∂x = 0. The solutions of this “sandwich” equation, which

we call inframonogenic functions, are used to obtain a new Fischer decomposition for

homogeneous polynomials in Rm.

RESUMEN

Denotemos por ∂x el operador de Dirac en R
m. En este art́ıculo, nosotros presentamos un

refinamiento de las funciones biarmónicas y al mismo tiempo una extensión de las funciones

monogénicas considerando la ecuación ∂xf ∂x = 0. Las soluciones de esta ecuación tipo

“sándwich”, las cuales llamaremos inframonogénicas, son utilizadas para obtener una nueva

descomposición de Fischer para polinomios homogéneos en Rm.



190 Helmuth R. Malonek, Dixan Peña Peña and Frank Sommen CUBO
12, 2 (2010)

Key words and phrases: Inframonogenic functions; Fischer decomposition.

Mathematics Subject Classification: 30G35; 31B30; 35G05.

1 Introduction

Let R0,m be the 2
m-dimensional real Clifford algebra constructed over the orthonormal basis (e1, . . . , em)

of the Euclidean space Rm (see [6]). The multiplication in R0,m is determined by the relations

ej ek + ekej = −2δjk and a general element of R0,m is of the form a =
∑

A aAeA, aA ∈ R, where for

A = {j1, . . . , jk} ⊂ {1, . . . , m}, j1 < · · · < jk, eA = ej1 . . . ejk . For the empty set ∅, we put e∅ = 1, the

latter being the identity element.

Notice that any a ∈ R0,m may also be written as a =
∑m

k=0[a]k where [a]k is the projection of a

on R
(k)
0,m. Here R

(k)
0,m denotes the subspace of k-vectors defined by

R
(k)
0,m =

{

a ∈ R0,m : a =
∑

|A|=k

aAeA, aA ∈ R

}

.

In particular, R
(1)
0,m and R

(0)
0,m ⊕ R

(1)
0,m are called, respectively, the space of vectors and paravectors in

R0,m. Observe that R
m+1 may be naturally identified with R

(0)
0,m ⊕ R

(1)
0,m by associating to any element

(x0, x1, . . . , xm) ∈ R
m+1 the paravector x = x0 + x = x0 +

∑m
j=1 xj ej .

Conjugation in R0,m is given by

a =
∑

A

aAeA, eA = (−1)
|A|(|A|+1)

2 eA.

One easily checks that ab = ba for any a, b ∈ R0,m. Moreover, by means of the conjugation a norm

|a| may be defined for each a ∈ R0,m by putting

|a|2 = [aa]0 =
∑

A

a2A.

The R0,m-valued solutions f (x) of ∂xf (x) = 0, with ∂x =
∑m

j=1 ej ∂xj being the Dirac operator, are

called left monogenic functions (see [4, 8]). The same name is used for null-solutions of the operator

∂x = ∂x0 + ∂x which is also called generalized Cauchy-Riemann operator.

In view of the non-commutativity of R0,m a notion of right monogenicity may be defined in a

similar way by letting act the Dirac operator or the generalized Cauchy-Riemann operator from the

right. Functions that are both left and right monogenic are called two-sided monogenic.

One can also consider the null-solutions of ∂kx and ∂
k
x (k ∈ N) which gives rise to the so-called

k-monogenic functions (see e.g. [2, 3, 15]).

It is worth pointing out that ∂x and ∂x factorize the Laplace operator in the sense that

∆x =
m

∑

j=1

∂2xj = −∂
2
x, ∆x = ∂

2
x0

+ ∆x = ∂x∂x = ∂x∂x.

Let us now introduce the main object of this paper.



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Fischer decomposition by inframonogenic functions 191

Definition 1.1. Let Ω be an open set of Rm (resp. Rm+1). An R0,m-valued function f ∈ C
2(Ω) will

be called an inframonogenic function in Ω if and only if it fulfills in Ω the “sandwich” equation

∂xf ∂x = 0 (resp. ∂xf ∂x = 0).

Here we list some motivations for studying these functions.

1. If a function f is inframonogenic in Ω ⊂ Rm and takes values in R, then f is harmonic in Ω.

2. The left and right monogenic functions are also inframonogenic.

3. If a function f is inframonogenic in Ω ⊂ Rm, then it satisfies in Ω the overdetermined system

∂3xf = 0 = f ∂
3
x. In other words, f is a two-sided 3-monogenic function.

4. Every inframonogenic function f ∈ C4(Ω) is biharmonic, i.e. it satisfies in Ω the equation

∆2xf = 0 (see e.g. [1, 11, 13, 16]).

The aim of this paper is to present some simple facts about the inframonogenic functions (Section 2)

and establish a Fischer decomposition in this setting (Section 3).

2 Inframonogenic functions: simple facts

It is clear that the product of two inframonogenic functions is in general not inframonogenic, even if

one of the factors is a constant.

Proposition 2.1. Assume that f is an inframonogenic function in Ω ⊂ Rm such that ej f (resp. f ej )

is also inframonogenic in Ω for each j = 1, . . . , m. Then f is of the form

f (x) = cx + M (x),

where c is a constant and M a right (resp. left) monogenic function in Ω.

Proof. The proposition easily follows from the equalities

∂x
(

ej f (x)
)

∂x = −2∂xj f (x)∂x − ej
(

∂xf (x)∂x
)

,

∂x
(

f (x)ej
)

∂x = −2∂xj ∂xf (x) −
(

∂xf (x)∂x
)

ej , (1)

j = 1, . . . , m. �

For a vector x and a k-vector Yk, the inner and outer product between x and Yk are defined by

(see [8])

x • Yk =

{

[xYk]k−1 for k ≥ 1

0 for k = 0
and x ∧ Yk = [xYk]k+1 .

In a similar way Yk • x and Yk ∧ x are defined. We thus have that

xYk = x • Yk + x ∧ Yk,

Ykx = Yk • x + Yk ∧ x,



192 Helmuth R. Malonek, Dixan Peña Peña and Frank Sommen CUBO
12, 2 (2010)

where also

x • Yk = (−1)
k−1Yk • x,

x ∧ Yk = (−1)
kYk ∧ x.

Let us now consider a k-vector valued function Fk which is inframonogenic in the open set Ω ⊂ R
m.

This is equivalent to say that Fk satisfies in Ω the system















∂x • (∂x • Fk) = 0

∂x ∧ (∂x • Fk) − ∂x • (∂x ∧ Fk) = 0

∂x ∧ (∂x ∧ Fk) = 0.

In particular, for m = 2 and k = 1, a vector-valued function f = f1e1 + f2e2 is inframonogenic if and

only if
{

∂x1x1 f1 − ∂x2x2 f1 + 2∂x1x2 f2 = 0

∂x1x1 f2 − ∂x2x2 f2 − 2∂x1x2 f1 = 0.

We now try to find particular solutions of the previous system of the form

f1(x1, x2) = α(x1) cos(nx2),

f2(x1, x2) = β(x1) sin(nx2).

It easily follows that α and β must fulfill the system

α′′ + n2α + 2nβ′ = 0

β′′ + n2β + 2nα′ = 0.

Solving this system, we get

f1(x1, x2) =
(

(c1 + c2x1) exp(nx1) + (c3 + c4x1) exp(−nx1)
)

cos(nx2), (2)

f2(x1, x2) =
(

(c3 + c4x1) exp(−nx1) − (c1 + c2x1) exp(nx1)
)

sin(nx2). (3)

Therefore, we can assert that the vector-valued function

f (x1, x2) =
(

(c1 + c2x1) exp(nx1) + (c3 + c4x1) exp(−nx1)
)

cos(nx2)e1

+
(

(c3 + c4x1) exp(−nx1) − (c1 + c2x1) exp(nx1)
)

sin(nx2)e2, cj , n ∈ R,

is inframonogenic in R2. Note that if c1 = c3 and c2 = c4, then

f1(x1, x2) = 2(c1 + c2x1) cosh(nx1) cos(nx2),

f2(x1, x2) = −2(c1 + c2x1) sinh(nx1) sin(nx2).

Since the functions (2) and (3) are harmonic in R2 if and only if c2 = c4 = 0, we can also claim that

not every inframonogenic function is harmonic.

Here is a simple technique for constructing inframonogenic functions from two-sided monogenic

functions.



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Fischer decomposition by inframonogenic functions 193

Proposition 2.2. Let f (x) be a two-sided monogenic function in Ω ⊂ Rm. Then xf (x) and f (x)x

are inframonogenic functions in Ω.

Proof. It is easily seen that

(

xf (x)
)

∂x =

m
∑

j=1

∂xj
(

xf (x)
)

ej = x
(

f (x)∂x
)

+

m
∑

j=1

ej f (x)ej =

m
∑

j=1

ejf (x)ej .

We thus get

∂x
(

xf (x)
)

∂x = −
m

∑

j=1

ej
(

∂xf (x)
)

ej − 2f (x)∂x = 0.

In the same fashion we can prove that f (x)x is inframonogenic. �

We must remark that the functions in the previous proposition are also harmonic. This may be

proved using the following equalities

∆x
(

xf (x)
)

= 2∂xf (x) + x
(

∆xf (x)
)

, (4)

∆x
(

f (x)x
)

= 2f (x)∂x +
(

∆xf (x)
)

x, (5)

and the fact that every monogenic function is harmonic. At this point it is important to notice that

an R0,m-valued harmonic function is in general not inframonogenic. Take for instance h(x)ej , h(x)

being an R-valued harmonic function. If we assume that h(x)ej is also inframonogenic, then from (1)

it may be concluded that ∂xh(x) does not depend on xj . Clearly, this condition is not fulfilled for

every harmonic function.

We can easily characterize the functions that are both harmonic and inframonogenic. Indeed,

suppose that h(x) is a harmonic function in a star-like domain Ω ⊂ Rm. By the Almansi decomposition

(see [12, 15]), we have that h(x) admits a decomposition of the form

h(x) = f1(x) + xf2(x),

where f1(x) and f2(x) are left monogenic functions in Ω. It is easy to check that

∂xh(x) = −mf2(x) − 2Exf2(x),

Ex =
∑m

j=1 xj ∂xj being the Euler operator. Thus h(x) is also inframonogenic in Ω if and only if

mf2(x) + 2Exf2(x) is right monogenic in Ω. In particular, if h(x) is a harmonic and inframonogenic

homogeneous polynomial of degree k, then f1(x) is a left monogenic homogeneous polynomial of degree

k while f2(x) is a two-sided monogenic homogeneous polynomial of degree k − 1.

The following proposition provides alternative characterizations for the case of k-vector valued

functions.

Proposition 2.3. Suppose that Fk is a harmonic (resp. inframonogenic) k-vector valued function in

Ω ⊂ Rm such that 2k 6= m. Then Fk is also inframonogenic (resp. harmonic) if and only if one of

the following assertions is satisfied:

(i) Fk(x)x is left 3-monogenic in Ω;



194 Helmuth R. Malonek, Dixan Peña Peña and Frank Sommen CUBO
12, 2 (2010)

(ii) xFk(x) is right 3-monogenic in Ω;

(iii) xFk(x)x is biharmonic in Ω.

Proof. We first note that

ejeAej =

{

(−1)|A|eA for j ∈ A,

(−1)|A|+1eA for j /∈ A,

which clearly yields
∑m

j=1 ej eAej = (−1)
|A|(2|A|−m)eA. It thus follows that for every k-vector valued

function Fk,
m

∑

j=1

ej Fkej = (−1)
k(2k − m)Fk.

Using the previous equality together with (4) and (5), we obtain

∂x∆x
(

Fk(x)x
)

= 2∂xFk(x)∂x +
(

∂x∆xFk(x)
)

x + (−1)k(2k − m)∆xFk,

∆x
(

xFk(x)
)

∂x = 2∂xFk(x)∂x + x
(

∆xFk(x)∂x
)

+ (−1)k(2k − m)∆xFk,

∆2x
(

xFk(x)x
)

= 4
(

2∂xFk(x)∂x + (−1)
k(2k − m)∆xFk +

(

∂x∆xFk(x)
)

x

+ x
(

∆xFk(x)∂x
)

)

+ x
(

∆2xFk(x)
)

x.

The proof now follows easily. �

Before ending the section, we would like to make two remarks. First, note that if m even, then

a m/2-vector valued function Fm/2(x) is inframonogenic if and only if Fm/2(x) and Fm/2(x)x are left

3-monogenic, or equivalently, Fm/2(x) and xFm/2(x) are right 3-monogenic. Finally, for m odd the

previous proposition remains valid for R0,m-valued functions.

3 Fischer decomposition

The classical Fischer decomposition provides a decomposition of arbitrary homogeneous polynomials

in Rm in terms of harmonic homogeneous polynomials. In this section we will derive a similar decom-

position but in terms of inframonogenic homogeneous polynomials. For other generalizations of the

Fischer decomposition we refer the reader to [5, 7, 8, 9, 10, 12, 14, 17, 18].

Let P(k) (k ∈ N0) denote the set of all R0,m-valued homogeneous polynomials of degree k in R
m.

It contains the important subspace I(k) consisting of all inframonogenic homogeneous polynomials of

degree k.

An an inner product may be defined in P(k) by setting

〈Pk(x), Qk(x)〉k =
[

Pk(∂x) Qk(x)
]

0
, Pk(x), Qk(x) ∈ P(k),

Pk(∂x) is the differential operator obtained by replacing in Pk(x) each variable xj by ∂xj and taking

conjugation.



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Fischer decomposition by inframonogenic functions 195

From the obvious equalities

[eja b]0 = −[aej b]0,

[aej b]0 = −[abej]0, a, b ∈ R0,m,

we easily obtain

〈xPk−1(x), Qk(x)〉k = −
〈

Pk−1(x), ∂xQk(x)
〉

k−1
,

〈Pk−1(x)x, Qk(x)〉k = −
〈

Pk−1(x), Qk(x)∂x
〉

k−1
,

with Pk−1(x) ∈ P(k − 1) and Qk(x) ∈ P(k). Hence for Pk−2(x) ∈ P(k − 2) and Qk(x) ∈ P(k), we

deduce that

〈xPk−2(x)x, Qk(x)〉k =
〈

Pk−2(x), ∂xQk(x)∂x
〉

k−2
. (6)

Theorem 3.1 (Fischer decomposition). For k ≥ 2 the following decomposition holds:

P(k) = I(k) ⊕ xP(k − 2)x.

Moreover, the subspaces I(k) and xP(k − 2)x are orthogonal w.r.t. the inner product 〈 , 〉k.

Proof. The proof of this theorem will be carried out in a similar way to that given in [8] for the

case of monogenic functions.

As P(k) = xP(k − 2)x ⊕ (xP(k − 2)x)
⊥

it is sufficient to show that

I(k) = (xP(k − 2)x)
⊥

.

Take Pk(x) ∈ (xP(k − 2)x)
⊥

. Then for all Qk−2(x) ∈ P(k − 2) it holds

〈

Qk−2(x), ∂xPk(x)∂x
〉

k−2
= 0,

where we have used (6). In particular, for Qk−2(x) = ∂xPk(x)∂x we get that ∂xPk(x)∂x = 0 or

Pk(x) ∈ I(k). Therefore (xP(k − 2)x)
⊥
⊂ I(k).

Conversely, let Pk(x) ∈ I(k). Then for each Qk−2(x) ∈ P(k − 2),

〈xQk−2(x)x, Pk(x)〉k =
〈

Qk−2(x), ∂xPk(x)∂x
〉

k−2
= 0,

whence Pk(x) ∈ (xP(k − 2)x)
⊥

. �

By recursive application of the previous theorem we get:

Corollary 3.1 (Complete Fischer decomposition). If k ≥ 2, then

P(k) =

[k/2]
⊕

s=0

xsI(k − 2s)xs.

Acknowledgement. D. Peña Peña was supported by a Post-Doctoral Grant of Fundação para

a Ciência e a Tecnologia, Portugal (grant number: SFRH/BPD/45260/2008).

Received: March 2009. Revised: May 2009.



196 Helmuth R. Malonek, Dixan Peña Peña and Frank Sommen CUBO
12, 2 (2010)

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