Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 947518, 13 pages
doi:10.1155/2010/947518
Research Article
Decomposition of Polyharmonic Functions with
Respect to the Complex Dunkl Laplacian
Guangbin Ren1 and Helmuth R. Malonek2
1
2
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
Departamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro, Portugal
Correspondence should be addressed to Helmuth R. Malonek, hrmalon@ua.pt
Received 28 December 2009; Accepted 26 March 2010
Academic Editor: Yuming Xing
Copyright q 2010 G. Ren and H. R. Malonek. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Let Ω be a G-invariant convex domain in CN including 0, where G is a complex Coxeter group
associated with reduced root system R ⊂ RN . We consider holomorphic functions f defined in Ω
2
which are Dunkl polyharmonic, that is, Δh n f 0 for some integer n. Here Δh N
j1 Dj is the
complex Dunkl Laplacian,
and Dj is the complex Dunkl operator attached to the Coxeter group G,
Dj fz ∂f/∂zj z v∈R κv fz−fσv z/z, vvj , where κv is a multiplicity function on R
and σv is the reflection with respect to the root v. We prove that any complex Dunkl polyharmonic
N 2 n−1
2
function f has a decomposition of the form fz f0 z N
fn−1 z,
n1 zj f1 z · · · n1 zj for all z ∈ Ω, where fj are complex Dunkl harmonic functions, that is, Δh fj 0.
1. Introduction
A fundamental result in the theory of polyharmonic functions is the celebrated Almansi
theorem 1–3, which shows that for any polyharmonic function f of degree n in a starlike
domain D in RN with center 0, that is,
⎞n
N
2
∂
⎠ ,
ΔR n f : ⎝
2
∂x
j1
j
⎛
1.1
f 0,
there exist uniquely harmonic functions f0 , . . . , fn−1 such that
fx f0 x |x|2 f1 x · · · |x|2n−1 fn−1 x,
∀x ∈ D.
1.2
2
Journal of Inequalities and Applications
The Almansi formula is a genuine analogy to the Taylor formula:
ft f0 t
f n 0
f 0
· · · tn
··· .
1!
n!
1.3
Compared with the Taylor formula, the Almansi formula is obtained by the scheme
d
−→ ΔR ,
dt
1.4
and since the constants f n 0/n! are solutions of d/dtf n 0/n! 0, they are replaced
by the solutions of the Laplace equation ΔR fk 0.
In 1, Aronszajn et al. indicated some applications of the Almansi formula in several
complex variables. Its most eminent application is in spherical harmonic function theory 4,
5. The polyharmonic functions have also applications in the theory of elasticity 6, in radar
imaging 7, and in multivariate approximation 8, 9.
The purpose of this article is to extend Almansi’s theorem to the theory of complex
Dunkl harmonics. The theory of Dunkl harmonics developed by Dunkl 10–13 is an
extension of the theory of ordinary harmonics. In 1989, Dunkl 10 constructed for each
Coxeter group a family of commutative differential-difference operators Dj , called Dunkl
operators, which can be considered as perturbations of the usual partial derivatives by
reflection parts. These operators step from the analysis of quantum many body system of
Calogero-Moser-Sutherland type 14 in mathematical physics. They also have roots in the
theory of special functions of several variables. With Dunkl operators in place of the usual
partial derivatives, one can define the Laplacian in the Dunkl setting, which is a parametrized
operator and invariant under reflection groups. These parametrized Laplacian suggests the
theory of Dunkl harmonics. In 15, we obtained the Almansi decomposition for the real
Dunkl operator. Now we continue to consider the Almansi decomposition for the complex
Dunkl operator.
As a direct consequence, we will show that the Almansi Theorem implies the Gauss
decomposition of the homogeneous polynomials into complex Dunkl harmonics.
We need some notations before stating our main result. Let R be a root system in RN
and G the associated Coxeter group. Let κ : R → C be a fixed multiplicity function v → κv
on R. Fix a positive subsystem R of R, and denote γ γκ : v∈R κv . We will always assume
that
Re γκ > −
N
.
2
1.5
Let Dj be the Dunkl operator attached to the Coxeter group G and the multiplicity function
κ, defined by see 16
Dj fz fz − fσv z
∂f
vj ,
κv
z ∂zj
z, v
v∈R
where σv denotes the reflection in the hyperplane orthogonal to v.
1.6
Journal of Inequalities and Applications
3
The Dunkl operators enjoy the regularity property: if f ∈ HΩ, the space of
holomorphic functions in Ω, then Di f ∈ HΩ. This follows immediately from the formula
fz − fσv z
z, v
1
2v
∇ftσv z 1 − tz, 2 dt
|v|
0
1.7
for any f ∈ HΩ and v ∈ R.
The Dunkl Laplacian is defined as
2
,
Δh D12 · · · DN
1.8
more precisely,
Δh fz Δfz 2
v∈R
κv
fz − fσv z 2
∇fz, v
−2
κv
|v| .
z, v
z, v2
v∈R
1.9
Here Δ and ∇ are the complex Laplacian and gradient operator:
∂2
∂2
·
·
·
,
∂z21
∂z2n
∂
∂
,...,
.
∇
∂z1
∂zn
Δ : ΔC 1.10
Throughout this paper we let Ω be a G-invariant convex domain in CN including 0, that is,
GΩ ⊂ Ω, 0 ∈ Ω, and tx 1 − ty ∈ Ω for all t ∈ 0, 1 and x, y ∈ Ω. This class of domain turns
out to be natural for the Almansi decomposition. It is known that Δh is a regular operator in
such a domain. Namely, if f ∈ HΩ, then Δh f ∈ HΩ.
Definition 1.1. A holomorphic function f : Ω → C is Dunkl polyharmonic of degree n if
Δh n f 0. If n 1, it is called Dunkl harmonic function.
Let I be the identity operator. For any s ∈ C with Re s > 0 we define the operator
Is : HΩ → HΩ by
Is fx 1
ftxts−1 dt.
1.11
0
If f is Dunkl harmonic in Ω, then so is Is f. For any j ∈ N, by assumption 1.5 we can
introduce the operator:
Qj 1
IN2j−1/2γκ IN2j−2/2γκ · · · IN/2γκ .
4j j!
1.12
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Journal of Inequalities and Applications
For any z ∈ CN and j ∈ N, we denote
z2 z21 · · · z2N ,
j
j z2j z2 z21 · · · z2N .
1.13
Our main result is the following theorem.
Theorem 1.2. Assume that R is a root system in RN and G its associated complex Coxter group. Let
Ω be a G-invariant convex domain in CN including 0. If f is a Dunkl polyharmonic function in Ω of
degree n, then there exist uniquely Dunkl harmonic functions f0 , . . . , fn−1 such that
fz f0 z z2 f1 z · · · z2n−1 fn−1 z,
∀x ∈ Ω.
1.14
Moreover the Dunkl harmonic functions f0 , . . . , fn−1 are given by the following formulae:
f0 I − z2 Q1 Δh I − z4 Q2 Δ2h · · · I − z2n−1 Qn−1 Δn−1
fz
h
fz
f1 Q1 Δh I − z4 Q2 Δ2h · · · I − z2n−1 Qn−1 Δn−1
h
..
.
1.15
fn−2 Qn−2 Δn−2
I − z2n−1 Qn−1 Δn−1
fz
h
h
fn−1 Qn−1 Δn−1
h fz.
Conversely, the sum in 1.14, with f0 , . . . , fn−1 Dunkl harmonic in Ω, defines a Dunkl
polyharmonic function in Ω of degree n.
Remark 1.3. By the Scheme in 1.4, we know that the formulae of fj above play the role of
Taylor coefficient formulae. These formulae are new even in the classical case κ 0.
2. Preliminaries
Let us recall some notation in the theory of Dunkl harmonics; see 16, 17. Concerning root
system and reflection groups, see 18.
A root system R is a finite set of nonzero vectors in RN such that σv R R and R ∩ Rv {±v} for all v ∈ R.
The positive subsystem R is a subset of R such that R R ∪ −R , where R and
−R are separated by a hyperplane through the origin.
For a nonzero vector v ∈ CN , the reflection σv in the hyperplane orthogonal to v is
defined by
σv z : z − 2
where the symbol z, v N
j1
z, v
|v|2
v,
zj vj and |z|2 z, z.
z ∈ CN ,
2.1
Journal of Inequalities and Applications
5
The Coxeter group G or the finite reflection group generated by the root system R is
the subgroup of the unitary group UN generated by {σu : u ∈ R}.
A multiplicity function κv is a G-invariant complex valued function defined on R, that
is, κv κgv for all g ∈ G.
Notice that Dunkl operators were studied in literature for Re κv ≥ 0.
The Dunkl operator Dj , associated with the Coxeter group G and the multiplicity
function κ, is the first-order differential-difference operator. The remarkable property of
Dunkl operators is that they are commutative:
D i Dj Dj Di .
The Dunkl Laplacian Δh N
j1
2.2
Dj2 can be split into three parts
Δh Δ Gh Dh
2.3
with
Gh fz 2
κv
v∈R
Dh fz −2
κv
∇fz, v
;
z, v
fz − fσv z
v∈R
z, v2
2.4
2
|v| .
When κ 0, the Dunkl Laplacian Δh is just the ordinary complex Laplacian Δ.
Consider the natural action of UN on functions f : CN → C, given by gfz −1
fg z. The Dunkl Laplacian Δh is G-invariant, that is,
g ◦ Δh Δh ◦ g,
∀g ∈ G.
2.5
Example 2.1. Let N be an integer and N ≥ 2. Since we need to consider the sum i /
j and i
runs from 1 to N, this forces N ≥ 2. Take the Coxeter group G SN , which is the symmetric
group in N elements, acting on RN by permuting the standard basis e1 , . . . , eN see 17, page
289. We regard the transposition ij in SN as a reflection σij such that
σij ei − ej − ei − ej .
2.6
Therefore, SN is a finite reflection generated by σij with a root system
R ± ei − ej : 1 ≤ i < j ≤ N .
2.7
As all transpositions are conjugate in SN , the vector space of multiplicity function is one
dimensional. The complex Dunkl operators associated with the multiplicity parameters κ ∈ C
are given by
fz − f σij z
∂f
,
Dj fz z κ
∂zj
zi − zj
i/
j
2.8
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Journal of Inequalities and Applications
where σij acts on CN by interchanging zi and zj ; more precisely, σij z σij z1 , . . . , σij zn with
i, j.
σij zi zj , σij zj zi , and σij zk zk for any k /
In this case, the condition 1.5 of the main theorem reduces to
κ>−
1
.
N−1
2.9
Example 2.2. In the one-dimensional case N 1, the root system R is of type A1 , the reflection
group G Z2 , and the multiplicity function is given by a single parameter κ ∈ C. The Dunkl
operator D : D1 and the Dunkl Laplacian Δh are given, respectively, by
fz − f−z
,
z
fz − f−z
f z
− 2κ
Δh fz f z 2κ
.
z
z2
Dfz f z κ
2.10
If f is an even function, then the third term in the formula of Δh f vanishes, while the
sum of the first two items provides a singular Sturm-Liouville operator.
3. Proof of the Main Theorem
Before proving Theorem 1.2, we need some lemmas.
Denote
Rs sI N
j1
zj
∂
.
∂zj
3.1
We write R instead of R0 when s 0.
Lemma 3.1. If s ∈ C, Re s > 0, and f ∈ HΩ, then
fz Is Rs fz Rs Is fz.
3.2
Proof. For any s ∈ C, Re s > 0 and f ∈ HΩ,
fz 1
0
ds
t ftz dt.
dt
3.3
By direct calculation
N
∂f
ds
s−1
s−1
wi
t ftz st ftz t
tz,
dt
∂wi
i1
3.4
Journal of Inequalities and Applications
7
where wi tzi . Therefore
1
fz N
∂f
sftz wi
tz ts−1 dt;
∂wi
i1
0
fz s
1
ftzts−1 dt N
∂
wi
∂wi
i1
0
1
3.5
ftzts−1 dt.
0
From the above two identities and the definitions of Is and Rs , we have fz Is Rs fz and
fz Rs Is fz.
Lemma 3.2. If f ∈ HΩ, then for any s ∈ C, Re s > 0, and z ∈ Ω
Δh Is fz Is2 Δh fz.
3.6
Proof. By definition, we have for a.e. z ∈ Ω
∇ Is fz , v
Gh Is fz 2
κv
z, v
v∈R
2
κv
v∈R
1 N
1
z, v
0
∂f
tzvi ts dt
∂z
i
i1
1 ∇ftz, v s1
2
κv
t dt
tz, v
0 v∈R
1
3.7
Gh ftzts1 dt
0
Is2 Gh fz.
Similarly
Dh Is fz −2
Is f z − Is f σv z
κv
z, v
v∈R
−2
κv
v∈R
1
0
−2
1
|v|2
z, v
κv
v∈R
Is2 Dh fz.
2
2
|v|2
ftz − ftσv z ts−1 dt
0
ftz − ftσv z ts1 dt
|v|2
2
tz, v
3.8
8
Journal of Inequalities and Applications
It is also easy to see
ΔIs fz Is2 Δfz.
3.9
Since Δh Δ Gh Dh , it follows that Δh Is fz Is2 Δh fz for a.e. z ∈ Ω. From
the regularity property of Dunkl operators, Δh maps C2 Ω into CΩ. By the continuity,
Lemma 3.2 follows.
Lemma 3.3. Let H1 {f ∈ HΩ : Δh f 0}. If s > 0 and Qj as in 1.12, then
Rs H1 H1 ,
Is H1 H1 ,
Qj H1 H1 .
3.10
z ∈ Ω.
3.11
Proof. Note that 3.6 implies
Rs2 Δh fz Δh Rs fz,
Indeed, from Lemma 3.1, Rs2 Δh Rs2 Δh Is Rs Rs2 Is2 Δh Rs Δh Rs . As direct
consequence of 3.6 and 3.11, we find that Is f and Rs f are Dunkl harmonic, whenever
f is Dunkl harmonic. From the definition of Qj , we thus obtain Qj H1 H1 .
Lemma 3.4. Let g ∈ HΩ, j ∈ N, Then for any z ∈ Ω
Δh z2j gx z2j Δh gx 4jz2j−1 RN2j−2/2γ 2jgz.
3.12
Proof. For any f, g ∈ HΩ
Δ fg Δf g 2 ∇f, ∇g f Δg .
3.13
Take fz z2j and apply identities ∂/∂zi z2j 2jzi z2j−1 and Δz2j 2jN 2j −2z2j−1
to yield
Δ z2j g z2j Δg 4jz2j−1 RN2j−2/2 g.
3.14
By our assumption R ⊂ RN . Therefore
v v,
v ∈ R .
3.15
As a result,
z2 σv z2
3.16
Journal of Inequalities and Applications
9
for any z ∈ Ω and v ∈ R . Indeed
2
σv z N
zj − 2
j1
z2 − 4
z, v
|v|2
2
vj
z, vz, v
|v|2
4
z, v2 v, v
3.17
|v|4
z2 .
Then
z2j gz − σv z2j gσv z 2
Dh z2j gz −2
κv
|v|
z, v2
v∈R
3.18
z2j Dh gz.
By definition, we have
Gh
2j ∇ z g ,v
z g 2
κv
z, v
v∈R
2j
2j ∇ z ,v
g
z Gh g 2
κv
z, v
v∈R
2j
3.19
z2j Gh g 4jγz2j−1 gz.
Since Δh Δ Gh Dh , summing up the above identity leads to identity 3.12 for z ∈ Ω.
Lemma 3.5. For any complex Dunkl harmonic function f in Ω,
Δnh z2n Qn fz fz,
z ∈ Ω.
3.20
Proof. From 1.12 and Lemma 3.1, we know that
Qn−1 4n n!RN2n−1/2γ RN2n−2/2γ · · · RN/2γ .
3.21
Denote g Qn f. Then g is Dunkl harmonic in Ω due to 3.10, and
fz 4n n!RN2n−1/2γ RN2n−2/2γ · · · RN/2γ gz.
3.22
We need to show
Δnh z2n gz 4n n!RN2n−1/2γ RN2n−2/2γ · · · RN/2γ gz
for any Dunkl harmonic function g in Ω and n ∈ N.
3.23
10
Journal of Inequalities and Applications
Let g be Dunkl harmonic in Ω and n ∈ N. Then Lemma 3.4 shows
Δh z2n gz 4nz2n−1 RN2n−1/2γ gz.
3.24
We use induction on n to prove 3.23. It is easy to prove when n 1. For the general case,
from 3.24 we have
Δh z2n gz
Δnh z2n gz Δn−1
h
2n−1
.
4nΔn−1
z
R
gz
N2n−1/2γ
h
3.25
Equation 3.23 follows directly from the assumption of induction.
Now we come to the proof of our main theorem.
Proof of Theorem 1.2. Denote Hn {f ∈ HΩ : Δh n f 0}. It is sufficient to show that
Hn Hn−1 Tn−1 H1 ,
n ∈ N,
3.26
where Tn z2n I. Notice that Lemma 3.5 states that
Δnh Tn Qn I.
3.27
We split the proof into two parts.
i Hn ⊃ Hn−1 Tn−1 H1 . Since Hn−1 ⊂ Hn , we need only to show Tn−1 H1 ⊂ Hn . For any
g ∈ H1 , by 3.27 and 3.10 we have
−1
−1
Qn−1
Δnh Tn−1 g Δh Δn−1
T
Q
g Δh Qn−1
g 0.
n−1
n−1
h
3.28
ii Hn ⊂ Hn−1 Tn−1 H1 . For any f ∈ Hn , we have the decomposition
n−1
f
T
Q
f I − Tn−1 Qn−1 Δn−1
Δ
f
.
n−1
n−1
h
h
3.29
We will show that the first summand above is in Hn−1 and the item in the braces of the second
summand is in H1 . This can be verified directly. First,
n−1
n−1
n−1
n−1
Δn−1
I
−
T
f
Δ
Δ
f
Q
Δ
−
Δ
T
Q
n−1
n−1
n−1
n−1
h
h
h
h
Δn−1
h
−
Δn−1
h
3.30
f 0.
Next, since Δn−1
f ∈ H1 and Qn−1 H1 ⊂ H1 , we have Qn−1 Δn−1
f ∈ H1 , as desired.This proves
h
h
that Hn Hn−1 Tn−1 H1 . By induction, we can easily deduce that Hn H1 T1 H1 · · ·Tn−1 H1 .
Journal of Inequalities and Applications
11
Next we prove that for any f ∈ Hn the decomposition
f g Tn−1 fn ,
g ∈ Hn−1 , fn ∈ H1
3.31
on both sides we obtain
is unique. In fact, for such a decomposition, applying Δn−1
h
n−1
n−1
Δn−1
h f Δh g Δh Tn−1 fn
−1
Δn−1
h Tn−1 Qn−1 Qn−1 f1
3.32
−1
Qn−1
fn .
Therefore
fn Qn−1 Δn−1
h f,
3.33
g f − Tn−1 fn I − Tn−1 Qn−1 Δn−1 f.
3.34
so that
Thus the uniqueness follows by induction.
To prove the converse, we see from 3.23 that, for any n ∈ N, Δn1
z2n H1 0.
h
Replacing n by j, we have
Δnh z2j H1 0
3.35
for any n > j.
4. Gauss Decomposition
As a direct consequence of Theorem 1.2, we can get the extended Fischer decomposition
theorem. Let Pm denote the space of homogeneous polynomials of degree m in CN . Notice
that Dj maps Pm into Pm−1 , so that Δh Pm ⊂ Pm−2 . If f ∈ Pm , then
m/21
Δh
fz 0,
4.1
and Is fz 1/m sfz so that
Qj fz dj,m fz,
4.2
−1
4j j!N/2 γ nj and an aa 1 · · · a n − 1. Denote
where dj,n
cj dj,m−2j 4j j!
1
.
N/2 γ m − 2j j
4.3
12
Journal of Inequalities and Applications
Corollary 4.1. Let f be a homogeneous polynomial of degree m in CN . Then there exist uniquely
Dunkl harmonic homogeneous polynomials fj of degree m − 2j such that
fz f0 z z2 f1 z · · · z2m/2 fm/2 z,
∀z ∈ Ω.
4.4
Moreover the Dunkl harmonic functions f0 , . . . , fm/2 are given by the following formulae:
m/2
f0 I − c1 z2 Δh I − c2 z4 Δ2h · · · I − cm/2 z2m/2 Δh
fz
m/2
f1 c1 Δh I − c2 z4 Δ2h · · · I − cm/2 z2m/2 Δh
fz
..
.
4.5
m/2−1
fm/2−1 cm/2−1 Δh
m/2
fm/2 cm/2 Δh
m/2
I − cm/2 z2m/2 Δh
fz
fz.
Proof. Let f ∈ Pm , then f is Dunkl harmonic of degree m/2 1, so that Theorem 1.2 gives
the decomposition of f as in 4.4. It remains to check the formulae of f0 , . . . , fm/2 . We only
consider the formula of f0 , since the others are similar. That is, we need to show
m/2
fz
f0 I − z2 Q1 Δh I − z4 Q2 Δ2h · · · I − z2m/2 Qm/2 Δh
m/2
I − c1 z2 Δh I − c2 z4 Δ2h · · · I − cm/2 z2m/2 Δh
fz.
m/2
Notice that for any f ∈ Pm , Δh
m/2
Qm/2 Δh
4.6
f ∈ Pm−2m/2 ∩ H1 , so that 4.2 implies
m/2
fz cm/2 Δh
fz.
4.7
Therefore
m/2
m/2
z2m/2 Qm/2 Δh
fz cm/2 z2m/2 Δh
fz ∈ Pm ,
4.8
and also
m/2
I − z2m/2 Qm/2 Δh
m/2
fz I − cm/2 z2m/2 Δh
fz ∈ Pm .
4.9
The remaining proof follows by induction.
Acknowledgment
The research is partially supported by the Unidade de Investigação “Matemática e Aplicações” of
University of Aveiro and by the NNSF of China no. 10771201.
Journal of Inequalities and Applications
13
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