Almansi decomposition for slice regular functions of several quaternionic variables

In this paper, we propose a generalization of the Almansi decomposition for slice regular functions of several quaternionic variables. Our method yields $2^n$ distinct and unique decompositions for any slice function with domain in $\mathbb{H}^n$. Depending on the choice of the decomposition, every component is given explicitly, uniquely determined and exhibits desirable properties, such as harmonicity and circularity in the selected variables. As consequences of these decompositions, we give another proof of Fueter's Theorem in $\mathbb{H}^n$, establish the biharmonicity of slice regular functions in every variable and derive mean value and Poisson formulas for them.


Introduction
In 1989, Emilio Almansi [1] proved that any polyharmonic function f of degree m defined on a star-like domain centered at the origin of some R n could be written as a combination of m harmonic functions {S i (f )} m−1 i=0 as This theorem plays a central role in the theory of polyharmonic functions, establishing a bridge between the theory of harmonic and polyharmonic functions.We refer to the monograph [2] and the references therein for applications of the theorem in the classical case of several real or complex variables.Generalizations of Almansi decomposition has been studied both concerning other type of iterated differential operators [18], both for more general classes of functions, especially in the hypercomplex settings of slice regular [15,14] and monogenic functions [13].But, as far as we know, no Almansi decomposition has been provided for functions of several hypercomplex variables.
We are strongly motivated by the work of A. Perotti [15], in which an Almansi decomposition holds for slice regular functions of one quaternionic variable.
Theorem 1.1.[15,Theorem 4] Let f be a slice regular function on a circular set, then there exist two unique circular harmonic functions The aim of the present paper is to extend the Almansi decomposition of slice regular functions to the context of several variables.The extension to higher dimensions poses some new challenges, one of which is the exponential growth of possible decompositions.Indeed, for slice functions of n variables, we obtain 2 n decompositions (Theorem 3.1), as the cardinality of all possible subsets of {1, ..., n}, that correspond to choices of sets of variables.Moreover, if we consider a slice regular function, every component is harmonic and radially symmetric in the imaginary part w.r.t. the chosen variables that determine the decomposition.Every component of each decomposition is given explicitly, completely determined by the original function through its partial spherical derivatives, similar to the one variable setting.We also prove the unique character of this decomposition (Proposition 3.2), i.e. that the functions performing the decomposition are unique, if specific symmetry properties are required.
Among these, we select the class of ordered decompositions, corresponding to integer intervals of the form {1, 2, ..., m} (Corollary 3.3).Such special components are sufficient for characterizing the slice regularity of the slice function they decompose (Proposition 3.4), in analogy with the one variable interpretation of slice regularity proposed in [12, §3.4].In this case, components can also be obtained applying iteratively the Cauchy-Riemann-Fueter operator ∂ x h and they can be used to define strongly slice regular functions of several variables, which are a generalization of slice regular functions defined on a not necessarily axially symmetric domain [17].
We describe the structure of the paper.In Section 2 we recall the main features of slice regular functions of several quaternionic variables and some properties from [4] we will need through the paper.Section 3 is divided into three parts: in the first one we state the main results described above, whose proofs are postponed in the third part, while in the second one we show some preliminary results.
The last section is devoted to applications of Almansi decomposition.First, we apply Theorem 3.1 to slice regular polynomials, where the components are given through zonal harmonics, which are some particular rotational invariant harmonic functions.Then, we give a different proof of the several quaternionic variables version of Fueter's Theorem and prove the biharmonicity of any slice regular function w.r.t. each variable (Corollary 4.6), justifying a posteriori the existence of an Almansi-type decomposition.Moreover, we are able to give examples of monogenic functions of several variables, in the spirit of Fueter's Theorem (Proposition 4.3) generated by the components of Almansi decomposition.Finally, we derive various formulas ( §4.1), such as mean value and Poisson formulas.

Preliminaries
The study of quaternionic analysis has led to the development of various classes of functions that generalize complex analysis to higher dimensions.Among them, the notion of slice regularity has received particular attention in the last years.Slice analysis was firstly introduced by Gentili and Struppa in [9], where they defined a new class of regular quaternionic functions as real differentiable functions which are holomorphic in every complex slice of H.They initially referred to that class as C-regular functions, in honor of Cullen [6], who previously conceived that definition.We refer the reader to [8] for a comprehensive treatment of the theory of slice regular functions of one quaternionic variable.
Various attempts have been made to generalize this new hypercomplex theory and more general algebras (see e.g.[10] and [5]) has been considered.Then, a new approach based on the concept of stem functions was introduced in [11] by Ghiloni and Perotti, giving slice analysis a crucial development: they extended the theory to any real alternative * -algebra with unity, embodying the aforementioned generalizations, as well as the quaternionic case.Moreover, from this formulation, it was possible to give a definition of slice function without requiring any regularity assumption.The stem function approach paved the way to achieve an analogous theory in several variables [12].Here we recall the main features of the theory of slice regular functions of several variables.We take them from [12] and [4], focusing on the quaternionic case.
With S H we denote the set of square roots of −1 in H, i.e. S H := {J ∈ H | J 2 = −1} ⊂ Im (H).Note that every element q ∈ H \ R can be uniquely represented as q = α + Jβ, with α ∈ R, β ∈ R + and J ∈ S H .For such q = α + Jβ, define S q = S α,β := {α where z = (α 1 + iβ 1 , ..., α n + iβ n ) ∈ D. Stem(D) will denote the set of stem functions F : D → H ⊗ R 2 n , S(Ω D ) the set of slice functions f : Ω D → H and I : Stem(D) → S(Ω D ) the map sending a stem function to its induced slice function.A slice function is slice preserving whenever the components of its inducing stem function are real valued.S R (Ω D ) will denote the set of slice preserving functions.We can define a product between stem functions.Given two stem functions F and G, with The product of stem functions is again a stem function, [12,Lemma 2.34] and this allows to define a product between slice functions: given Equip H ⊗ R 2 n with the commuting complex structures J = {J h } n h=1 , where J h (q ⊗ e K ) = (−1) |K∩{h}| q⊗e K∆{h} , for every q ∈ H, K ∈ P(n), then define for every h = 1, ..., n the operators A slice function induced by a holomorphic stem function is called slice regular.The set of slice regular functions over Ω D will be denoted by By [12,Proposition 3.13], f ∈ SR(Ω D ) if and only if ∂f /∂x c h = 0, for every h = 1, ..., n.Thus, (SR(Ω D ), ⊙) is a real subalgebra of (S(Ω D ), ⊙), as consequence of Leibniz rule [12,Proposition 3.25].
Fix h ∈ {1, ..., n} and for any y x, y h+1 , ..., y n ) is a one-variable slice (resp.slice regular) function.We call it circular w.r.t.x h if f y h is constant over the sets S α,β for any fixed α, β ∈ R and y ∈ Ω D .With S h (Ω D ), SR h (Ω D ) and S c,h (Ω D ) we will denote respectively the subset of S(Ω D ) of function that are slice, slice regular and circular w.r.t.x h .For every H ∈ P(n), we also denote where z = (z 1 , ..., z n ) and where Ω DH = Ω D \ R H .Note that these definitions are well posed, since they do not depend on the order of where is defined on the all Ω D , for every H ∈ P(n).In the next Proposition we recap the main properties of the spherical value and the spherical derivative we will need through the paper.Their proofs can be found in [4, §4].

For every
2. Leibniz formula In particular, if We can give (2) and (3) through stem functions: let F, G ∈ Stem(D) and h ∈ {1, ..., n}, then for every z = (z 1 , ..., z n ) ∈ D it holds where Im(z h ) = I −1 (Im(x h )) and Let us represent any x = (x 1 , ..., x n ) ∈ H n with real coordinates as x h = α h + iβ h + jγ h + kδ h , then recall two differential operators for any h = 1, ..., n They extend to several variables the well known Cauchy-Riemann-Fueter operators ∂ CRF and ∂ CRF [16, §6].Sometimes notation can be slightly different and the factor 1/2 can be omitted.These operators factorize the Laplacian, indeed ∀h = 1, ..., n where . Functions in the kernel of ∂ x h are usually called monogenic (or Fueter regular) w.r.t.x h .Furthermore we call x h -axially monogenic those functions monogenic w.r.t.x h which are slice, too, i.e.
3 Almansi decomposition , .., m} is an integer interval from 1 to some m ∈ {1, ..., n}, we can write where χ K is the characteristic function of the set K. Note that in this case we can use the ordinary pointwise product as well as the slice product [12, Proposition 2.52].If f = I(F ), every S H K (f ) is induced by the stem function where , inducing the monomial x j , for any j = 1, ..., n.
We can now formulate our main result.
Theorem 3.1.Let Ω D ⊂ H n be a circular set and let f ∈ S(Ω D ) be a slice function.Fix any H ∈ P(n), then 1. we can decompose f as For any Moreover, we can characterize the slice regularity and slice preservingness of f : For any H ∈ P(n) we can define the H-right linear operator , where p := min H c and its restriction .
For every choice of H ∈ P(n), the functions S H K (f ) are the only H-circular functions that realize decomposition (8).Proposition 3.2.Let f ∈ S(Ω D ) and fix H ∈ P(n).Suppose that there exist functions Then h K = S H K (f ).From Theorem 3.1, we emphasize the case in which H = 1, m , that leads to what we call an ordered decomposition of f .The next results in this particular situation were proved independently by Perotti in [17].

Preliminary results
Lemma 3.6.For every m = 1, ..., n, it holds Proof.Since x m = I(Z m ) and x m = I(Z m ), it is enough to prove the properties for the slice funtions or for the stem functions.By Proposition 2.1 (4), since Proposition 3.7.Let F ∈ Stem(D) and fix H ∈ P(n).For every K ⊂ H, the functions G H K (F ) satisfy the following properties: 1.For every m / ∈ H, it holds

For every h ∈ H it holds
Proof. 1. Apply ( 4), ( 5) and Lemma 3.6, then Now, suppose that (11) holds for some H ∈ P(n) and let us prove it for Suppose now m ∈ K, then Now, we show that decomposition (8) holds for stem functions, too.This will immediately prove (8).
Lemma 3.9.For every H ∈ P(n) and every where, if H = {1, ..., n} we mean Proof.Let us proceed by induction over |H|.First, suppose H = {h}, for any h = 1, ..., n, then K = ∅, {h}.If K = ∅, we have where (5) and Lemma 3.6 has been used.If K = {h}, immediately we get Now, suppose that (13) holds for some H ∈ P(n) and let us prove it for H ′ = H ∪ {m}, with any m / ∈ H and any K ⊂ H ′ .Let us split m ∈ K and m / ∈ K.If m ∈ K, let us set K ′ := K \ {m}, then we have e T F K∪T .

Proofs
Proof of Theorem 3.1.
. ⇐) It is a particular case of Proposition 3.4.

5.
f is slice preserving if and only if F K is real ∀K ∈ P(n), which by ( 14) is equivalent for G {1,...,n} K (F ) = S n K (f ) to be real valued for every K ∈ P(n).
Proof of Proposition 3.2.Apply Proposition 2.1 (2) and the hypothesis h T ∈ S c,H (Ω D ) in the following computation . So, the unique non trivial element of the sum refers to T = K, for which we have where we have used Proposition 2.1 (2) and Lemma 3.6 (1).
Proof of Proposition 3.4.
Proof of Corollary 3.5.By Corollary 3.3 and Lemma 3.10 we have

Applications
The first application we give of Theorem 3.1 concerns quaternionic (ordered) polynomials with right coefficients, in which the components of the decomposition are given through zonal harmonics.Note that Theorem 3.1 can be fully applied, since every polynomial with right coefficients is a slice regular function [12,Proposition 3.14].Before, we recall a result from [16,Corollary 6.7].
Lemma 4.1.For every m ≥ 0, consider the slice regular power x m : H → H. Then it holds where ) is the real valued zonal harmonic of R 4 with pole 1 (see [3,Ch. 5]).
The components of the ordered decomposition provide examples of axially monogenic functions.Proof.By Corollary 3.3, S h K (f ) is harmonic w.r.t.x j , for any j = 1, ..., h, ∀K ∈ P(h), so and by Proposition 3.4 The following result highlights a difference between the one and several variables slice regular functions: in the first case the Laplacian is always an axially monogenic functions (this is Fueter's Theorem [7]), in the latter this happens just for the first variable, but in general we can only write it as sum of axially monogenic functions.

) Remark 2 .
The previous characterization resembles the one given in[12, Theorem 3.23], in which iterations of spherical values and spherical derivatives (also referred as truncated spherical derivatives D ǫ (f )) has been used.It is easy to see that you can express truncated spherical derivatives as real combinations of the components S m K (f ) and viceversa, making the two characterizations equivalent.Corollary 3.5.Let f ∈ S h (Ω D ), then the ordered decomposition of f of order h reduces to f = K∈P(h−1)
and holomorphic if it is hholomorphic for every h = 1, ..., n.By [12, Lemma 3.12], F = K∈P(n) e K F K is h-holomorphic if and only if it satisfies the following Cauchy-Riemann system