Liouville Theorem for Dunkl Polyharmonic Functions

Assume that $f$ is Dunkl polyharmonic in $\mathbb{R}^n$ (i.e. $(\Delta_h)^p f=0$ for some integer $p$, where $\Delta_h$ is the Dunkl Laplacian associated to a root system $R$ and to a multiplicity function $\kappa$, defined on $R$ and invariant with respect to the finite Coxeter group). Necessary and successful condition that $f$ is a polynomial of degree $\le s$ for $s\ge 2p-2$ is proved. As a direct corollary, a Dunkl harmonic function bounded above or below is constant.


Introduction
The classical Liouville theorem for harmonic functions states that a harmonic function on R n must be a constant if it is bounded or nonnegative. Nicolesco [14] extended the Liouville theorem to polyharmonic functions with the Pizetti formula as a starting point (see also [8]). Kuran [9], Armitage [2], and Futamura, Kishi, and Mizuta [6] proved further extensions and showed that if f is a polyharmonic function on R n and the growth of the positive part of f is suitably restricted, then f must be a polynomial. Their starting point is the Almansi decomposition theorem for polyharmonic functions. We also refer to [10,11] for the extension of Liouville theorems for conformally invariant fully nonlinear equations. Recently, Gallardo and Godefroy [7] showed that if f is a bounded Dunkl harmonic function in R n , then it is a constant. However, their approach is not adaptable to Dunkl polyharmonic functions.
The purpose of this article is to establish the Liouville theorem for Dunkl polyharmonic functions. To achieve this, we shall resort to the Almansi decomposition for Dunkl polyharmonic functions [15]. As a direct corollary of our results, a Dunkl harmonic function bounded below or above is actually constant, which extends the corresponding result of Gallardo and Godefroy for the bounded case [7]. In the Dunkl ananlysis, the multiplicity function is usually restricted to be non-negative. We shall discuss in the final section the possible extension of our main result to the case when the multiplicity function is negative.
A root system R is a finite set of nonzero vectors in R m such that σ v R = R and R∩Rv = {±v} for all v ∈ R.
The Coxeter group G (or the finite reflection group) generated by the root system R is the subgroup of the orthogonal group O(n) generated by {σ u : u ∈ R}.
The positive subsystem R + is a subset of R such that R = R + ∪ (−R + ), where R + and −R + are separated by some hyperplane through the origin.
A multiplicity function Fix a positive subsystem R + of R and denote Let D j be the Dunkl operator associated to the group G and to the multiplicity function κ, defined by We call ∆ h = n j=1 D j 2 the Dunkl Laplacian. We always assume that κ υ ≥ 0.
Let dσ be the Lebesgue surface measure in the unit sphere and h k ( The mean value property holds for Dunkl harmonic functions f , i.e., for some constant c > 0. This property for polynomials is implicit in the orthogonality relation in [3]. Then one only needs a limiting argument for arbitrary Dunkl harmonic functions. See also [13,12]. Our main theorem is as follows. (ii) f is a polynomial of degree s if and only if As a direct corollary, a Dunkl harmonic function on R n must be constant if it is bounded below or above. Indeed, let f be Dunkl harmonic in R n and set s = 0. If f ≥ 0, then f + = f and so that the mean value property shows that Therefore, Theorem 1 shows that f is a polynomial of degree less than or equal to s = 0, which means that f must be a constant. If f is bounded above or below, multiplication by −1 if necessary makes it bounded below, adding a constant if necessary makes it positive. Thus the known result for positive functions shows that f is a constant.

Homogeneous expansions
The non-Dunkl case of the following homogeneous expansions is well-known (see [ the series converging absolutely and uniformly on compact subsets of the unit ball. Proof . The formula can be verified as in the classical case of Corollary 5.34 in [1] by using the formulae corresponding to the classical case in [5]. Indeed, notice that u is harmonic on the closed unit ball. Theorem 5.31 in [5] and the preceding statement show that |u(y)|h 2 κ (y)dσ(y), ∀ x ∈ R n , and thus the series m p m converges absolutely and uniformly to u on compact subsets of the unit ball.
where C 1 = cϕ p−1 (0) for s = 2(p − 1) and C 1 = 0 for s > 2(p − 1). Since ϕ m are Dunkl harmonic in R n , by Lemma 1 we can write where g m,j are Dunkl harmonic homogeneous polynomials of degree j and the convergence of the series is uniform on compact subsets of the unit ball. We claim that g m,j = 0 when 2m + j > s. With this claim, we have so that f (x) is a polynomial of degree no more than s.
On considering item m = m 0 , the above integral is of the form r 2γ P (r) with P (r) a polynomial of r with degree at least 2m 0 + 2j 0 + n − 1, which is strictly larger than s + j 0 + n − 1. This contradicts the fact that The last step used (3) and (6).
We have proved that f is a polynomial of degree ≤ s if and only if (3) holds. With this result, we know that the statements (i) and (ii) in Theorem 1 are equivalent.
We now claim that g m,j = 0 if 2m + j ≥ s. Indeed, if not, then there exists g m 0 ,j 0 ≡ 0 such that 2m 0 + j 0 ≥ s. However, |y|=r f (y)g m 0 ,j 0 (y)h 2 k dσ(y) = |y|=r p−1 m=0 |y| 2m ∞ j=0 g m,j (y)g m 0 ,j 0 (y)h 2 k (y)dσ(y), and the above integral is of the form r 2γ Q(r), where Q(r) is a polynomial of r with degree no less than 2m 0 + 2j 0 + n − 1 ≥ s + j 0 + n − 1. Since r 2γ Q(r) is not an o(r s+j 0 +n−1+2γ ), we arrive at a contradiction. From the claim and (5), we see that f is a polynomials of degree less than s.