Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions

We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair h1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_1$$\end{document}, h2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_2$$\end{document} of zonal harmonic functions such that f=h1-x¯h2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=h_1-\bar{x} h_2$$\end{document}. We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.


Introduction
In this note we prove a quaternionic Almansi-type Theorem for polynomials with quaternionic coefficients and, more generally, for slice-regular functions of a quaternionic variable (see Sect. 3 for definitions and references about this function theory).
We show that to every slice-regular function f defined on a domain Ω of the skew field H of quaternions, it is possible to associate two uniquely defined harmonic functions h 1 , h 2 such that f (x) = h 1 (x)−xh 2 (x) on Ω. This result allows to apply well-known results of harmonic function theory on R 4 to obtain a mean value property for slice-regular functions and a Poisson-type formula over three-dimensional spheres. An interesting aspect of these last results is their validity over any sphere in Ω, without requiring any symmetry with respect to the real axis.
An Almansi-type Theorem for slice-regular functions on real Clifford algebras R n (also called slice-monogenic functions [3]) has been proved in [14]. The quaternionic algebra can be identified with the Clifford algebra R 2 . However, slice-monogenic functions are naturally defined on the paravector space R n+1 . Therefore the results of [14] for n = 2 are related to harmonic function theory in R 3 instead of the one in R 4 H. The quaternionic results obtained in the present paper present some similarities with those obtained in [14] for n = 4, since slice-regular functions on R 3 , as those on H, are linked to harmonic functions of four real variables (see [13]).
We recall that similar Almansi-type decompositions have been established in other settings, for example in Clifford analysis [10], for Dunkl operators [15] and in the umbral calculus setting [4].
The paper is structured in the following way. In Sect. 2 we prove the Almansi decomposition for quaternionic polynomials, then in Sect. 3 we recall the basic notions of slice function theory on H and we prove the quaternionic Almansi Theorem for slice-regular functions. Finally, in Sect. 4 we deduce the mean value formulas and the Poisson formula for slice-regular functions.

An Almansi Decomposition for Quaternionic Polynomials
We consider quaternionic polynomials of the form with quaternionic coefficients on the right and quaternionic variable x = x 0 + ix 1 + jx 2 + kx 3 . It is well-known (see, e.g., [13,Theorem 3.6.3]) that the four real components of these polynomials, considered as functions of the four real variables x 0 , x 1 , x 2 , x 3 , are biharmonic with respect to the standard Laplacian Δ of R 4 , i.e., Δ 2 P = 0 on R 4 .
We recall the classical result of Emilio Almansi, published in 1899 [1], which dealt more generally with polyharmonic functions on R n . Theorem 1. (Almansi [1]). If Δ p u = 0 on a starlike domain D ⊆ R n with centre 0, then there exist unique harmonic functions u 0 , . . . , u p−1 in D such that Almansi's Theorem implies that the four real components of a quaternionic polynomial can be expressed in terms of pairs of harmonic functions. Observe that every polynomial P is more than biharmonic, since it belongs to the kernel of a third order operator. If ∂ CRF denotes the Cauchy-Riemann-Fueter operator Let B be the open unit ball in R 4 and let S 3 = ∂B the three-dimensional unit sphere. Let Z k (x, a) denote the real four-dimensional (solid) zonal harmonic of degree k and pole a ∈ S 3 (see, e.g., [2,Ch.5]). From the uniqueness properties of zonal harmonics and their invariance with respect to fourdimensional rotations, it holds Z k (x, a) = Z k (xa, 1) for every x ∈ H and a ∈ S 3 . Let Z k (x) be the real-valued zonal harmonic defined on H R 4 by Remark 1. The restriction of Z k (x) to the unit sphere S 3 is equal to the Gegenbauer (or Chebyshev of the second kind ) polynomial C We recall the following result from [13, Corollary 3.6.7]. It gives the zonal harmonics decomposition of quaternionic powers.

Proposition 1. ([13]
). For every k ∈ N, it holds (1) In the next statement we will consider polynomials in the four real variables x 0 , x 1 , x 2 , x 3 of the form with quaternionic coefficients a k . All these harmonic polynomials have an axial symmetry with respect to the real axis. For every orthogonal transformation T of H R 4 that fixes 1, it holds H • T = H. These polynomials will be called zonal harmonic polynomials with pole 1.
From formula (1) we easily deduce the Almansi Theorem for quaternionic polynomials, which was already stated in [12].
There exist two zonal harmonic polynomials h 1 , h 2 with pole 1, of degrees d and d − 1 respectively, such that The restrictions of h 1 and h 2 to the unit sphere S 3 are H-valued spherical harmonics depending only on x k a k . Formula (2) follows immediately from (1) by setting These formulas imply immediately the last statement.

An Almansi Decomposition for Slice-regular Functions
Polynomials P (x) = k x k a k with quaternionic coefficients do not belong to the kernel of the Cauchy-Riemann-Fueter operator. Instead they are examples of slice-regular functions. This function theory was introduced in 2006-2007 by Gentili and Struppa [6] for functions of one quaternionic variable. We refer to [5,[7][8][9] for precise definitions and for more properties and results about this class of functions. Here we recall only the basic facts that are necessary to state and prove the Almansi decomposition for slice-regular functions. Slice function theory is based on the "slice" decomposition of H. For each quaternion J in the sphere of imaginary units C be the subalgebra generated by J. Then A differentiable function f : Ω ⊆ H → H is called (left) slice-regular [6] on the open set Ω if, for each J ∈ S H , the restriction f |Ω∩CJ : Ω ∩ C J → H is holomorphic with respect to the complex structure defined by left multiplication by J.

Slice Functions
Let D ⊆ C, invariant with respect to conjugation. If a function F : Slice functions were introduced in [7]. They are exactly the quaternionic functions compatible with the slice structure of the skew-algebra H, i.e., the ones corresponding to a commutative diagram for every If the domain D intersects the real axis, this definition of slice regularity is equivalent to the one proposed by Gentili and Struppa [6].
The function f = I(F ) is called slice-preserving if F 1 and F 2 are realvalued. This is equivalent to the condition f (x) = f (x) for every x ∈ Ω D . If f is slice-preserving, then f · g coincides with the pointwise product of f and g. If f, g are slice-regular on Ω, then also their slice product f · g is slice-regular on Ω. For example, the polynomial P (x) given in Example 3 is the slice product

Operations on Slice Functions
The slice derivatives ∂f ∂x , ∂f ∂x c of a slice functions f = I(F ) are defined by means of the complex Cauchy-Riemann operators Note that f is slice-regular on Ω if and only if ∂f ∂x c = 0 and if f is slice-regular on Ω then also ∂f ∂x is slice-regular on Ω. We recall other two useful concepts introduced in [7]. The function f • s : Ω → H, called spherical value of f , and the function f s : Ω \ R → H, called spherical derivative of f , are defined as where Im(x) = 1 2 (x − x) = ix 1 + jx 2 + kx 3 . The functions f • s and f s are slice functions, constant on 2-spheres S x = α + S H β for any x = α + Jβ ∈ Ω \ R, such that for every x ∈ Ω \ R. For any slice-regular function f on Ω, f s extends as the slice derivative ∂f ∂x on Ω ∩ R. Note that for each

Almansi Type Decomposition
We recall a result proved in [13, Corollary 3.6.2 and Theorem 3.6.3] about some formulas linking the spherical value and the spherical derivative of slice functions with the Cauchy-Riemann-Fueter operator. We are now able to generalize Theorem 2 to every slice-regular function.
The functions h 1 and h 2 can be computed through differentiation:

Moreover, f is slice-preserving if and only if h 1 and h 2 are real-valued.
Proof. If f : Ω → H is slice-regular, then it holds, from (3),  From point 2(b) of Theorem 3 it follows that also the function h 1 is harmonic on Ω. It can also be observed, using the Leibniz-type formula for spherical derivative (see [7, §5]), that Since also the slice product x · f is slice-regular, this gives again the harmonicity of h 1 , this condition is equivalent to having h 1 and h 2 real-valued.
It remains to prove that h 1 and h 2 are uniquely determined by f . Assume that f (x) = h 1 (x) − xh 2 (x) with h 1 , h 2 axially symmetric with respect to the real axis. A direct computation gives f s (x) = h 2 (x) for every x ∈ Ω \ R and then h 1 (x) = f (x) + xf s (x).
We now prove a converse of the preceding theorem.
Proof. Let Ω = Ω D . In view of the symmetry of h 1 and h 2 , given z = x 0 +iβ ∈ D and x = x 0 + Jβ ∈ Ω, the function , is a C 1 stem function on D that induces the slice function g. Equation (4) for h 1 , h 2 are equivalent to the Cauchy-Riemann equations for G. The last statement follows from the uniqueness of the harmonic decomposition given in Theorem 4.
Example 4. Consider again the polynomial P (x) = x 2 of Example 2. According to Theorem 4, we can obtain the harmonic functions h 1 , h 2 applying the Cauchy-Riemann-Fueter operator. Since Example 5. Let f (x) = Log(x) be the slice-preserving slice-regular function on Ω = H \ {x ∈ R | x ≤ 0} induced by the (principal) complex logarithm Observe that h 1 extends continuously to the real points x 0 ∈ {x ∈ R | x > 0} as log(x 0 ) + 1, while h 2 extends to the same set as 1/x 0 .

Mean Value Property and Poisson Formula
As an application of the quaternionic Almansi Theorem, we give a mean value property for slice-regular functions. Let σ denote the normalized rotationinvariant surface-area measure on the unit sphere S 3 of H R 4 , such that σ(S 3 ) = 1. A slice-regular function f , being biharmonic, satisfies also a mean value formula of the form (see, e.g., [11,Chap. 7]). Comparing this formula with the one obtained before, we get the following.  Let P (x, ζ) = (1 − |x| 2 )/|x − ζ| 4 be the Poisson kernel of the unit ball B in R 4 .

Proposition 3. (Poisson formula). Let f be slice-regular on a circular set Ω.
Let the open ball B(a, r) have closure contained in Ω. Then, for every x ∈ B, it holds Proof. We can proceed as in the proof of Proposition 2, using the Poisson formula for the quaternionic-valued harmonic components h 1 and h 2 of f .
Observe that the Poisson kernel itself is related to the spherical derivative of a slice-regular function, the quaternionic Koebe function f (x) = (1 − x) −2 x (see [13,Cor. 3.6.7]).
When the centre a is a real point, the ball B(a, r) is a circular set. In this case we can obtain two formulas in which the spherical derivative of f does not appear. (2) for every x ∈ B.
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