Math. J. Okayama Univ. 53 (2011), 167–172

A CAUCHY-KOWALEVSKI THEOREM FORINFRAMONOGENIC FUNCTIONSHelmuth R. MALONEK, Dixan PEÑA PEÑA and Frank SOMMEN
Abstract. In this paper we prove a Cauchy-Kowalevski theorem forthe functions satisfying the system ∂x f ∂x = 0 (called inframonogenicfunctions).

1. Introduction
Let R0,m be the 2m -dimensional real Clifford algebra constructed over
the orthonormal basis (e1 , . . . , em ) of the Euclidean space Rm (see [3]). The
multiplication in R0,m is determined by the relations ej ek + ek ej = −2δjk ,j, k = 1, . . . , m, where δjk is the Kronecker delta. A general element of R0,m
is of the form
Xa=aA eA , aA ∈ R,A

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
P element.
Notice that any a ∈ R0,m may also be written as a = m
k=0 [a]k where [a]k(k)(k)is the projection of a on R0,m . Here R0,m denotes the subspace of k-vectors
defined byX(k)aA eA , aA ∈ R .
R0,m = a ∈ R0,m : a =|A|=k(0)(1)

Observe that Rm+1 may be naturally identified with R0,m ⊕ R0,m by associatingP
to any element (x0 , x1 , . . . , xm ) ∈ Rm+1 the “paravector” x = x0 + x =x0 + m
j=1 xj ej .
Conjugation in R0,m is given by
X
a=aA eA ,Awhere eA = ejk . . . ej1 , ej = −ej , j = 1, . . . , m. One easily checks thatab = ba for any a, b ∈ R0,m . Moreover, by means of the conjugation a norm
Mathematics Subject Classification. 30G35, 35G10.
Key words and phrases. Inframonogenic functions, Cauchy-Kowalevski theorem.
The second author was supported by a Post-Doctoral Grant of Fundação para a Ciência
e a Tecnologia, Portugal (grant number: SFRH/BPD/45260/2008).
167

168

H. R. MALONEK, D. PEÑA PEÑA AND F. SOMMEN|a| may be defined for each a ∈ R0,m by putting
X|a|2 = [aa]0 =a2A .
APm

Let us denote by ∂x = ∂x0 + ∂x = ∂x0 + j=1 ej ∂xj the generalized CauchyRiemann operator and let Ω be an open set of Rm+1 . According to [11], anR0,m -valued function f ∈ C 2 (Ω) is called an inframonogenic function in Ω
if and only if it fulfills in Ω the “sandwich” equation ∂x f ∂x = 0.
It is obvious that monogenic functions (i.e. null-solutions of ∂x ) are inframonogenic. At this point it is worth mentioning that the monogenic
functions are the central object of study in Clifford analysis (see [2, 4, 5, 7,8, 9, 10, 14]). Furthermore, the concept of monogenicity of a function may
be seen as the higher dimensional counterpart of holomorphy in the complex
plane.
Moreover, asmX∆x =∂x2j = ∂x ∂ x = ∂ x ∂x ,j=0

every inframonogenic function f ∈ C 4 (Ω) satisfies in Ω the biharmonic equation ∆2x f = 0 (see e.g. [1, 6, 12, 15]).
This paper is intended to study the following Cauchy-type problem forthe inframonogenic functions. Given the functions A0 (x) and A1 (x) analytic
in an open and connected set Ω ⊂ Rm , find a function F (x) inframonogenic
e of Ω in Rm+1 which satisfies
in some open neighbourhood Ω(1.1)

F (x)|x0 =0 = A0 (x),(1.2)∂x0 F (x)|x0 =0 = A1 (x).
2. Cauchy-type problem for inframonogenic functions

Consider the formal series(2.1)

F (x) =∞X

xn0 An (x).

n=0

It is clear that F satisfies conditions (1.1) and (1.2). We also see at oncethat∂x An + An ∂x + xn0 ∂x An ∂x .
∂x (xn0 An ) ∂x = n(n − 1)xn−2
An + nxn−100We thus get∂x F ∂x =∞X

n=0xn0(n + 2)(n + 1)An+2 + (n + 1) ∂x An+1 + An+1 ∂x + ∂x An ∂x .

A CAUCHY-KOWALEVSKI THEOREM FOR INFRAMONOGENIC FUNCTIONS 169

From the above it follows that F is inframonogenic if and only if the functions
An satisfy the recurrence relation
1
An+2 = −(n + 1) ∂x An+1 + An+1 ∂x + ∂x An ∂x , n ≥ 0.
(n + 2)(n + 1)
It may be easily proved by induction thatn−2n−1n+1XX(−1)
∂xn−j−1 A0 ∂xj+1 +∂xn−j−1 A1 ∂xj  , n ≥ 2.
(2.2) An =n!
j=0j=0

We now proceed to examine the convergence of the series (2.1) with thefunctions An (n ≥ 2) given by (2.2). Let y be an arbitrary point in Ω. Then
there exist a ball B y, R(y) of radius R(y) centered at y and a positive
constant M (y), such thatn!
∂xn−j As (x)∂xj ≤ M (y) n , x ∈ B y, R(y) , j = 0, . . . , n, s = 0, 1.
R (y)It follows that|An (x)| ≤ M (y)n + R(y) − 1,Rn (y)x ∈ B y, R(y) ,and therefore the series (2.1) converges normally in[e=−R(y), R(y) × B y, R(y) .
Ωy∈Ω

e is a x0 -normal open neighbourhood of Ω in Rm+1 , i.e. for each
Note that Ωe the line segment {x + t : t ∈ R} ∩ Ω̃ is connected and contains onex∈Ωpoint in Ω.
We thus have proved the following.
Theorem 2.1. The function CK[A0 , A1 ] given by(2.3) CK[A0 , A1 ](x) = A0 (x) + x0 A1 (x)n−2∞n−1nXXX(−x0 ) ∂xn−j−1 A0 (x)∂xj+1 +−∂xn−j−1 A1 (x)∂xj n!
n=2

j=0

j=0

is inframonogenic in a x0 -normal open neighbourhood of Ω in Rm+1 andsatisfies conditions (1.1)-(1.2).
It is worth noting that if in particular A1 (x) = −∂x A0 (x), thenCK[A0 , −∂x A0 ](x) =∞X(−x0 )nn=0n!

∂xn A0 (x),170

H. R. MALONEK, D. PEÑA PEÑA AND F. SOMMEN

which is nothing else but the left monogenic extension (or CK-extension)of A0 (x). Similarly, it is easy to see that CK[A0 , −A0 ∂x ](x) yields the rightmonogenic extension of A0 (x) (see [2, 5, 13, 16, 17]).
Let P(k) (k ∈ N0 fixed) denote the set of all R0,m -valued homogeneouspolynomials of degree k in Rm . Let us now take A0 (x) = Pk (x) ∈ P(k) andA1 (x) = Pk−1 (x) ∈ P(k − 1). Clearly,CK[Pk , Pk−1 ](x) = Pk (x) + x0 Pk−1 (x)n−2kn−1nXX (−x0 )X
∂xn−j−1 Pk (x)∂xj+1 +−∂xn−j−1 Pk−1 (x)∂xj  ,n!
n=2

j=0

j=0

since the other terms in the series (2.3) vanish. Moreover, we can also claimthat CK[Pk , Pk−1 ](x) is a homogeneous inframonogenic polynomial of degree
k in Rm+1 .
Conversely, if Pk (x) is a homogeneous inframonogenic polynomial of degree k in Rm+1 , then Pk (x)|x0 =0 ∈ P(k), ∂x0 Pk (x)|x0 =0 ∈ P(k − 1) andobviously CK[Pk |x0 =0 , ∂x0 Pk |x0 =0 ](x) = Pk (x).
Call I(k) the set of all homogeneous inframonogenic polynomials of degree
k in Rm+1 . Then CK[., .] establishes a bijection between P(k) × P(k − 1) andI(k).
It is easy to check thathx, uik
Pk (x) = Pk (∂u ), Pk (x) ∈ P(k),k!
where Pk (∂u ) is the differential operator obtained by replacing in Pk (u)each variable uj by ∂uj . Therefore, in order to characterize I(k), it suffices
to calculate CK hx, uik eA , 0 and CK 0, hx, uik−1 eA with u ∈ Rm .
A simple computation shows that
CK hx, uik eA , 0 (x) = hx, uik eAk  n−2XXk(−x0 )n hx, uik−n −
un−j−1 eA uj+1  ,nn=2j=0
CK 0, hx, uik−1 eA (x) = x0 hx, uik−1 eA
kn−1X1X k−(−x0 )n hx, uik−n un−j−1 eA uj  .
knn=2j=0A CAUCHY-KOWALEVSKI THEOREM FOR INFRAMONOGENIC FUNCTIONS 171

References[1]S. Bock and K. Gürlebeck, On a spatial generalization of the Kolosov-Muskhelishvili
formulae, Math. Methods Appl. Sci. 32 (2009), no. 2, 223–240.
[2] F. Brackx, R. Delanghe and F. Sommen, Clifford analysis, Research Notes in Mathematics, 76, Pitman (Advanced Publishing Program), Boston, MA, 1982.
[3] W. K. Clifford, Applications of Grassmann’s Extensive Algebra, Amer. J. Math. 1(1878), no. 4, 350–358.
[4] J. Cnops and H. Malonek, An introduction to Clifford analysis, Textos de Matemática,Série B, 7, Universidade de Coimbra, Departamento de Matemática, Coimbra, 1995.
[5] R. Delanghe, F. Sommen and V. Souček, Clifford algebra and spinor-valued functions, Mathematics and its Applications, 53, Kluwer Academic Publishers Group,
Dordrecht, 1992.
[6] K. Gürlebeck and U. Kähler, On a boundary value problem of the biharmonic equation,
Math. Methods Appl. Sci. 20 (1997), no. 10, 867–883.
[7] K. Gürlebeck and W. Sprössig, Quaternionic and Clifford calculus for physicists andengineers, Wiley and Sons Publ., 1997.
[8] V. V. Kravchenko, Applied quaternionic analysis, Research and Exposition in Mathematics, 28, Heldermann Verlag, Lemgo, 2003.
[9] V. V. Kravchenko and M. V. Shapiro, Integral representations for spatial models ofmathematical physics, Pitman Research Notes in Mathematics Series, 351, Longman,Harlow, 1996.
[10] H. R. Malonek, Selected topics in hypercomplex function theory, Clifford algebras andpotential theory, 111–150, Univ. Joensuu Dept. Math. Rep. Ser., 7, Univ. Joensuu,Joensuu, 2004.
[11] H. R. Malonek, D. Peña Peña and F. Sommen, Fischer decomposition by inframonogenic functions, to appear in CUBO, A Mathematical Journal.
[12] V. V. Meleshko, Selected topics in the history of the two-dimensional biharmonic
problem, Appl. Mech. Rev. 56 (2003), no. 1, 33–85.
[13] D. Peña Peña, Cauchy-Kowalevski extensions, Fueter’s theorems and boundary valuesof special systems in Clifford analysis, Ph.D. Thesis, Ghent University, Ghent, 2008
(available at http://hdl.handle.net/1854/11636).
[14] J. Ryan, Basic Clifford analysis, Cubo Mat. Educ. 2 (2000), 226–256.
[15] L. Sobrero, Theorie der ebenen Elastizität unter Benutzung eines Systems hyperkomplexer Zahlen, Hamburg. Math. Einzelschriften, Leipzig, 1934.
[16] F. Sommen, Monogenic functions on surfaces, J. Reine Angew. Math. 361 (1985),
145–161.
[17] F. Sommen and B. Jancewicz, Explicit solutions of the inhomogeneous Dirac equation,
J. Anal. Math. 71 (1997), 59–74.

Helmuth R. Malonek
Department of Mathematics
Aveiro University
3810-193 Aveiro
Portugale-mail address: hrmalon@ua.pt

172

H. R. MALONEK, D. PEÑA PEÑA AND F. SOMMEN

Dixan Peña Peña
Department of Mathematics
Aveiro University
3810-193 Aveiro
Portugale-mail address: dixanpena@ua.pt; dixanpena@gmail.com
Frank Sommen
Department of Mathematical Analysis
Ghent University
9000 Gent
Belgiume-mail address: fs@cage.ugent.be(Received May 9, 2009 )