Decomposition of Polyharmonic Functions with Respect to the Complex Dunkl Laplacian

Abstract

Let be a -invariant convex domain in including , where is a complex Coxeter group associated with reduced root system . We consider holomorphic functions defined in which are Dunkl polyharmonic, that is, for some integer . Here is the complex Dunkl Laplacian, and is the complex Dunkl operator attached to the Coxeter group , where is a multiplicity function on and is the reflection with respect to the root . We prove that any complex Dunkl polyharmonic function has a decomposition of the form , for all , where are complex Dunkl harmonic functions, that is, .

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 of degree in a starlike domain in with center , that is,

(1.1)

there exist uniquely harmonic functions such that

(1.2)

The Almansi formula is a genuine analogy to the Taylor formula:

(1.3)

Compared with the Taylor formula, the Almansi formula is obtained by the scheme

(1.4)

and since the constants are solutions of , they are replaced by the solutions of the Laplace equation .

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 , 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 be a root system in and the associated Coxeter group. Let be a fixed multiplicity function on . Fix a positive subsystem of , and denote . We will always assume that

(1.5)

Let be the Dunkl operator attached to the Coxeter group and the multiplicity function , defined by (see [16])

(1.6)

where denotes the reflection in the hyperplane orthogonal to .

The Dunkl operators enjoy the regularity property: if , the space of holomorphic functions in , then . This follows immediately from the formula

(1.7)

for any and .

The Dunkl Laplacian is defined as

(1.8)

more precisely,

(1.9)

Here and are the complex Laplacian and gradient operator:

(1.10)

Throughout this paper we let be a -invariant convex domain in including , that is, , , and for all and . This class of domain turns out to be natural for the Almansi decomposition. It is known that is a regular operator in such a domain. Namely, if , then .

Definition 1.1.

A holomorphic function is Dunkl polyharmonic of degree if . If , it is called Dunkl harmonic function.

Let be the identity operator. For any with we define the operator by

(1.11)

If is Dunkl harmonic in , then so is . For any , by assumption (1.5) we can introduce the operator:

(1.12)

For any and , we denote

(1.13)

Our main result is the following theorem.

Theorem 1.2.

Assume that is a root system in and its associated complex Coxter group. Let be a -invariant convex domain in including . If is a Dunkl polyharmonic function in of degree , then there exist uniquely Dunkl harmonic functions such that

(1.14)

Moreover the Dunkl harmonic functions are given by the following formulae:

(1.15)

Conversely, the sum in (1.14), with Dunkl harmonic in , defines a Dunkl polyharmonic function in of degree .

Remark 1.3.

By the Scheme in (1.4), we know that the formulae of above play the role of Taylor coefficient formulae. These formulae are new even in the classical case .

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 is a finite set of nonzero vectors in such that and for all .

The positive subsystem is a subset of such that , where and are separated by a hyperplane through the origin.

For a nonzero vector , the reflection in the hyperplane orthogonal to is defined by

(2.1)

where the symbol and .

The Coxeter group (or the finite reflection group) generated by the root system is the subgroup of the unitary group generated by .

A multiplicity function is a -invariant complex valued function defined on , that is, for all .

Notice that Dunkl operators were studied in literature for .

The Dunkl operator , associated with the Coxeter group and the multiplicity function , is the first-order differential-difference operator. The remarkable property of Dunkl operators is that they are commutative:

(2.2)

The Dunkl Laplacian can be split into three parts

(2.3)

with

(2.4)

When , the Dunkl Laplacian is just the ordinary complex Laplacian .

Consider the natural action of on functions , given by . The Dunkl Laplacian is -invariant, that is,

(2.5)

Example 2.1.

Let be an integer and . Since we need to consider the sum and runs from 1 to , this forces Take the Coxeter group , which is the symmetric group in elements, acting on by permuting the standard basis (see [17, page 289]). We regard the transposition in as a reflection such that

(2.6)

Therefore, is a finite reflection generated by with a root system

(2.7)

As all transpositions are conjugate in , the vector space of multiplicity function is one dimensional. The complex Dunkl operators associated with the multiplicity parameters are given by

(2.8)

where acts on by interchanging and ; more precisely, with , , and for any .

In this case, the condition (1.5) of the main theorem reduces to

(2.9)

Example 2.2.

In the one-dimensional case , the root system is of type , the reflection group , and the multiplicity function is given by a single parameter . The Dunkl operator and the Dunkl Laplacian are given, respectively, by

(2.10)

If is an even function, then the third term in the formula of 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

(3.1)

We write instead of when .

Lemma 3.1.

If , , and , then

(3.2)

Proof.

For any , and ,

(3.3)

By direct calculation

(3.4)

where . Therefore

(3.5)

From the above two identities and the definitions of and , we have and

Lemma 3.2.

If , then for any , , and

(3.6)

Proof.

By definition, we have for a.e.

(3.7)

Similarly

(3.8)

It is also easy to see

(3.9)

Since , it follows that for a.e. . From the regularity property of Dunkl operators, maps into . By the continuity, Lemma 3.2 follows.

Lemma 3.3.

Let . If and as in (1.12), then

(3.10)

Proof.

Note that (3.6) implies

(3.11)

Indeed, from Lemma 3.1, . As direct consequence of (3.6) and (3.11), we find that and are Dunkl harmonic, whenever is Dunkl harmonic. From the definition of , we thus obtain .

Lemma 3.4.

Let , , Then for any

(3.12)

Proof.

For any

(3.13)

Take and apply identities and to yield

(3.14)

By our assumption . Therefore

(3.15)

As a result,

(3.16)

for any and . Indeed

(3.17)

Then

(3.18)

By definition, we have

(3.19)

Since , summing up the above identity leads to identity (3.12) for .

Lemma 3.5.

For any complex Dunkl harmonic function in ,

(3.20)

Proof.

From (1.12) and Lemma 3.1, we know that

(3.21)

Denote . Then is Dunkl harmonic in due to (3.10), and

(3.22)

We need to show

(3.23)

for any Dunkl harmonic function in and .

Let be Dunkl harmonic in and . Then Lemma 3.4 shows

(3.24)

We use induction on to prove (3.23). It is easy to prove when . For the general case, from (3.24) we have

(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 . It is sufficient to show that

(3.26)

where . Notice that Lemma 3.5 states that

(3.27)

We split the proof into two parts.

(i). Since , we need only to show . For any , by (3.27) and (3.10) we have

(3.28)

(ii). For any , we have the decomposition

(3.29)

We will show that the first summand above is in and the item in the braces of the second summand is in . This can be verified directly. First,

(3.30)

Next, since and , we have , as desired.This proves that . By induction, we can easily deduce that .

Next we prove that for any the decomposition

(3.31)

is unique. In fact, for such a decomposition, applying on both sides we obtain

(3.32)

Therefore

(3.33)

so that

(3.34)

Thus the uniqueness follows by induction.

To prove the converse, we see from (3.23) that, for any Replacing by , we have

(3.35)

for any .

4. Gauss Decomposition

As a direct consequence of Theorem 1.2, we can get the extended Fischer decomposition theorem. Let denote the space of homogeneous polynomials of degree in . Notice that maps into , so that . If , then

(4.1)

and so that

(4.2)

where and . Denote

(4.3)

Corollary 4.1.

Let be a homogeneous polynomial of degree in . Then there exist uniquely Dunkl harmonic homogeneous polynomials of degree such that

(4.4)

Moreover the Dunkl harmonic functions are given by the following formulae:

(4.5)

Proof.

Let , then is Dunkl harmonic of degree , so that Theorem 1.2 gives the decomposition of as in (4.4). It remains to check the formulae of . We only consider the formula of , since the others are similar. That is, we need to show

(4.6)

Notice that for any , , so that (4.2) implies

(4.7)

Therefore

(4.8)

and also

(4.9)

The remaining proof follows by induction.