Almansi-Type Decomposition for Slice Regular Functions of Several Quaternionic Variables

Abstract

In this paper we propose an Almansi-type 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.

1 Introduction

In 1899, Emilio Almansi [1] proved that any polyharmonic function f of degree m, defined on a star-like domain centered at the origin of $\mathbb {R}^n$ could be written as a combination of m harmonic functions $\{\mathcal {S}_i(f)\}_{i=0}^{m-1}$ as

$$\begin{aligned} f(x)=\mathcal {S}_0(f)(x)+|x|^2\mathcal {S}_1(f)(x)+\dots +|x|^{2(m-1)}\mathcal {S}_{m-1}(f)(x). \end{aligned}$$

This theorem plays a central role in the theory of polyharmonic functions, establishing a bridge between the theories of harmonic and polyharmonic functions. We refer to the monograph [3] and the references therein for applications of the theorem in the classical case of several real or complex variables.

Generalizations of Almansi decomposition have been studied both concerning other type of iterated differential operators (e.g. in [27] for the Dunkl Laplacian), both for more general classes of functions, especially in the hypercomplex settings of slice regular [23, 24] and monogenic functions [22]. But, as far as we know, no Almansi decomposition has been provided for functions of several hypercomplex variables.

Our starting point is the work of Perotti [23], in which an Almansi-type decomposition holds for slice regular functions of one quaternionic variable:

Theorem 1.1

[23, Theorem 4] Let f be a slice regular function defined on a circular set $\Omega \subset \mathbb {H}$, then there exist two unique, circular and harmonic functions $h_1,h_2$ such that

$$\begin{aligned} f(x)=h_1(x)-\overline{x}h_2(x),\qquad \forall x\in \Omega . \end{aligned}$$

The functions are given by $h_1=(xf)'_s=-\overline{\partial }_{CRF}(xf)$ and $h_2=f'_s=-\overline{\partial }_{CRF}(f)$, where $g'_s(x):=[2{\text {Im}}(x)]^{-1}(g(x)-g(\overline{x}))$ is the spherical derivative of a slice function g, which, up to a factor, agrees on slice regular functions with the Cauchy–Riemann–Fueter operator $\overline{\partial }_{CRF}=\frac{1}{2}(\partial _{\alpha }+i\partial _\beta +j\partial _\gamma +k\partial _\delta )$.

The aim of the present paper is to extend the previous Almansi-type 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 all possible decompositions. Indeed, for slice functions of n variables, we obtain $2^n$ decompositions (Theorem 3.1), as the cardinality of all possible choices of variables between $x_1, \ldots ,x_n$. Every component is radially symmetric in the imaginary part of the chosen variables that determine the decomposition; if, moreover, the decomposing function is slice regular they are harmonic in the same variables, too. Every component of each decomposition is given explicitly and it is completely determined by the original function through its partial spherical derivatives, as in Theorem 1.1. We also prove the unique character of these decompositions (Proposition 3.2), namely the functions performing the decomposition are unique, if specific symmetry properties are required.

Among these, we point out the class of ordered decompositions, corresponding to integers intervals of the form $\{1,2, \ldots ,m\}$ (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 [17, §3.4]. In this case they can also be given applying iteratively the Cauchy–Riemann–Fueter operator $\overline{\partial }_{x_h}$, instead of the partial spherical derivatives. The components of the ordered decompositions have already been exploited 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 [26].

We describe the structure of the paper. In Sect. 2 we recall from [5, 17] the main features of slice regular functions of several quaternionic variables and some properties of partial spherical values and derivatives. Section 3 is divided into three parts: in the first one we state the main results described above, whose proofs are postponed to Sect. 3.3, while in the second one we show some preliminary results.

Last section is devoted to applications of Almansi-type decomposition. First, we apply Theorem 3.1 to slice regular polynomials, where the components are given through zonal harmonics, namely, rotational invariant harmonic functions. Then, by the harmonicity of the components of Almansi-type decompositions, we give a different proof of the several variables version of Fueter’s Theorem ([5, Theorem 4.10]) and prove the biharmonicity of any slice regular function w.r.t. each variable (Corollary 4.7), 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 4.4) generated from the components of Almansi-type decomposition. Finally, we derive various formulas (Sect. 4.1), such as mean value and Poisson formulas for slice regular functions.

2 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 [15], where they defined a new class of regular quaternionic functions as real differentiable functions which are holomorphic in every complex slice of $\mathbb {H}$. They initially referred to that class as C-regular functions, in honor of Cullen [11], who previously conceived that definition. We refer the reader to [2, 8, 9, 14] 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, 16]) has been considered. Then, a new approach based on the concept of stem functions was introduced in [19] by Ghiloni and Perotti, using an idea that goes back to Fueter [13]. This gave slice analysis a crucial development extending 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 [17], which is now of great interest (see e.g. [5,6,7, 12, 18, 20, 21]). Here we recall from [5, 17], the main features of the theory of slice regular functions of several variables, focusing on the quaternionic case.

2.1 Slice Regular Functions of Several Quaternionic Variables

For any $n\in \mathbb {N}$, let $\mathcal {P}(n):=\mathcal {P}(\{1, \ldots ,n\})$, with $\mathcal {P}(U):=\{V\subset U\}$. Given $K=\left\{ k_1, \cdots ,k_p\right\} \in \mathcal {P}(n)$, with $k_1< \cdots <k_p$, and $(q_{k_1}, \cdots ,q_{k_p})\subset \mathbb {H}^p$, denote the usual ordered $\mathbb {H}$-product of its elements by $q_K:=q_{k_1}\cdot \ldots \cdot q_{k_p}$ (if $K=\emptyset$, we set $q_\emptyset :=1$). Let $\mathbb {S}_\mathbb {H}$ be the set of square roots of $-1$ in $\mathbb {H}$, i.e. $\mathbb {S}_\mathbb {H}:=\{J\in \mathbb {H}\mid J^2=-1\}\subset {\text {Im}}\left( \mathbb {H}\right)$. Note that every element $q\in \mathbb {H}{\setminus }\mathbb {R}$ can be uniquely represented as $q=\alpha +J\beta$, with $\alpha \in \mathbb {R}$, $\beta \in \mathbb {R}^+$ and $J\in \mathbb {S}_\mathbb {H}$. For such $q=\alpha +J\beta$, define $\mathbb {S}_q=\mathbb {S}_{\alpha ,\beta }:=\{\alpha +J\beta \mid J\in \mathbb {S}_\mathbb {H}\}\subset \mathbb {H}$. For any $h\in \{1, \ldots ,n\}$, let $\mathbb {R}_h:=\{(x_1, \ldots ,x_n)\in \mathbb {H}^n\mid x_h\in \mathbb {R}\}$ and if $H\in \mathcal {P}(n)$, let $\mathbb {R}_H:=\bigcap _{h\in H}\mathbb {R}_h$.

A continuous function $F=\sum _{K\in \mathcal {P}(n)}e_KF_K:D\subset \mathbb {C}^n\rightarrow \mathbb {H}\otimes \mathbb {R}^{2^n}$ is called stem function if the set D is invariant under complex conjugation, (i.e. $z\in D$ if and only if $\overline{z}^h\in D$, where $\overline{z}^h:=(z_1, \ldots ,z_{h-1},\overline{z}_h,z_{h+1}, \ldots ,z_n)$) and if $F_K$ satisfies

$$\begin{aligned} F_K(\overline{z}^h)=(-1)^{|K\cap \{h\}|}F_K(z),\qquad \forall K\in \mathcal {P}(n),h=1, \ldots ,n. \end{aligned}$$

The circularization $\Omega _D\subset \mathbb {H}^n$ of an invariant set D is defined as $\Omega _D:=\{(\alpha _1+J_1\beta _1, \cdots ,\alpha _n+J_n\beta _n)\mid (\alpha _1+i\beta _1, \cdots ,\alpha _n+i\beta _n)\in D, J_1, \cdots ,J_n\in \mathbb {S}_\mathbb {H}\}$. Any stem function F uniquely induces a slice function $f:=\mathcal {I}(F):\Omega _D\rightarrow \mathbb {H}$, defined for every $x=(\alpha _1+J_1\beta _1, \cdots ,\alpha _n+J_n\beta _n)\in \Omega _D$ as

$$\begin{aligned} f(x):=\displaystyle \sum _{K\in \mathcal {P}(n)}[J_K,F_K(z)], \end{aligned}$$

where $z=(\alpha _1+i\beta _1, \cdots ,\alpha _n+i\beta _n)\in D$ and $[J_K,F_K(z)]:=J_KF_K(z)=J_{k_1} \cdots J_{k_p}F_{\{k_1, \cdots ,k_p\}}(z)$ if $K=\{k_1, \cdots ,k_p\}$, with $k_1< \cdots <k_p$. Stem(D) will denote the set of stem functions $F:D\rightarrow \mathbb {H}\otimes \mathbb {R}^{2^n}$, $\mathcal {S}(\Omega _D)$ the set of slice functions $f:\Omega _D\rightarrow \mathbb {H}$ and $\mathcal {I}:Stem(D)\rightarrow \mathcal {S}(\Omega _D)$ the map sending a stem function to its induced slice function. A slice function is called slice preserving whenever the components of its inducing stem function are real valued, we use the symbol $\mathcal {S}_\mathbb {R}(\Omega _D)$ to denote the set of slice preserving functions.

We can define a product between stem functions: given two stem functions F and G, with $F=\sum _{K\in \mathcal {P}(n)}e_KF_K$, $G=\sum _{H\in \mathcal {P}(n)}e_HG_H$, define

$$\begin{aligned} (F\otimes G)(z):=\displaystyle \sum _{H,K\in \mathcal {P}(n)}(-1)^{|H\cap K|}e_{K\Delta H}F_K(z)G_H(z), \end{aligned}$$

with $K\Delta H:=(K\cup H){\setminus }(K\cap H)$. The product of stem functions is again a stem function, [17, Lemma 2.34] and this allows to define a product between slice functions: given $f=\mathcal {I}(F),g=\mathcal {I}(G)\in \mathcal {S}(\Omega _D)$, define $f\odot g:=\mathcal {I}(F\otimes G)$. Thus, $\mathcal {I}:(Stem(D),\otimes )\rightarrow (\mathcal {S}(\Omega _D),\odot )$ is an algebra isomorphism.

Equip $\mathbb {H}\otimes \mathbb {R}^{2^n}$ with the commuting complex structures $\mathcal {J}=\{\mathcal {J}_h\}_{h=1}^n$, where $\mathcal {J}_h(q\otimes e_K)=(-1)^{|K\cap \{h\}|}q\otimes e_{K\Delta \{h\}}$, for every $q\in \mathbb {H}$, $K\in \mathcal {P}(n)$, then define for every $h=1, \ldots ,n$ the operators

$$\begin{aligned} \partial _h:=\dfrac{1}{2}\left( \dfrac{\partial }{\partial \alpha _h}-\mathcal {J}_h\dfrac{\partial }{\partial \beta _h}\right) ,\qquad \overline{\partial }_h:=\dfrac{1}{2}\left( \dfrac{\partial }{\partial \alpha _h}+\mathcal {J}_h\dfrac{\partial }{\partial \beta _h}\right) . \end{aligned}$$

A stem function $F\in \mathcal {C}^1(D)$ is h-holomorphic when $\overline{\partial }_hF=0$ and holomorphic if it is h-holomorphic for every $h=1, \ldots ,n$. By [17, Lemma 3.12], $F=\sum _{K\in \mathcal {P}(n)}e_KF_K$ is h-holomorphic if and only if it satisfies the following Cauchy–Riemann system

$$\begin{aligned} \dfrac{\partial F_K}{\partial \alpha _h}=\dfrac{\partial F_{K\cup \{h\}}}{\partial \beta _h},\qquad \dfrac{\partial F_K}{\partial \beta _h}=-\dfrac{\partial F_{K\cup \{h\}}}{\partial \alpha _h},\qquad \forall K\in \mathcal {P}(n),h\notin K. \end{aligned}$$

A slice regular function is a slice function induced by a holomorphic stem function. The set of slice regular functions over $\Omega _D$ will be denoted by $\mathcal {S}\mathcal {R}(\Omega _D)$. Since $\partial _h F$ and $\overline{\partial }_hF$ are stem functions if F is so, we can define

$$\begin{aligned} \dfrac{\partial f}{\partial x_h}:=\mathcal {I}(\partial _hF),\qquad \dfrac{\partial f}{\partial x_h^c}:=\mathcal {I}(\overline{\partial }_hF). \end{aligned}$$

By [17, Proposition 3.13], $f\in \mathcal {S}\mathcal {R}(\Omega _D)$ if and only if $\partial f/\partial x_h^c=0$, for every $h=1, \ldots ,n$. Thus, $(\mathcal {S}\mathcal {R}(\Omega _D),\odot )$ is a real subalgebra of $(\mathcal {S}(\Omega _D),\odot )$, as consequence of Leibniz rule [17, Proposition 3.25].

Fix $h\in \{1, \ldots ,n\}$ and for any $y=(y_1, \cdots ,y_n)\in \Omega _D$, let $\Omega _{D,h}(y):=\{x\in \mathbb {H}\mid (y_1, \cdots ,y_{h-1},x, y_{h+1}, \cdots ,y_n)\in \Omega _D\}.$ We say that a function $f\in \mathcal {S}(\Omega _D)$ is slice (resp. slice regular) w.r.t. $x_h$ if for every $y\in \Omega _D$ its restriction $f^y_h:\Omega _{D,h}(y)\subset \mathbb {H}\rightarrow \mathbb {H}$, $f^y_h:x\mapsto f(y_1, \cdots ,y_{h-1},x,y_{h+1}, \cdots ,y_n)$ is a one-variable slice (resp. slice regular) function. We call f circular w.r.t. $x_h$ if $f^y_h$ is constant over the sets $\mathbb {S}_{\alpha ,\beta }$ for any fixed $\alpha ,\beta \in \mathbb {R}$ and $y\in \Omega _D$. With $\mathcal {S}_h(\Omega _D)$, $\mathcal {S}\mathcal {R}_h(\Omega _D)$ and $\mathcal {S}_{c,h}(\Omega _D)$ we will denote respectively the subset of $\mathcal {S}(\Omega _D)$ of function that are slice, slice regular and circular w.r.t. $x_h$. For every $H\in \mathcal {P}(n)$, we also denote $\mathcal {S}_H(\Omega _D):=\bigcap _{h\in H}\mathcal {S}_h(\Omega _D)$, $\mathcal {S}\mathcal {R}_H(\Omega _D):=\bigcap _{h\in H}\mathcal {S}\mathcal {R}_h(\Omega _D)$ and $\mathcal {S}_{c,H}(\Omega _D):=\bigcap _{h\in H}\mathcal {S}_{c,h}(\Omega _D)$. In [5, Proposition 3.1, 3.2, 3.4] those sets are characterized:

Proposition 2.1

For any $H\in \mathcal {P}(n)$, it holds

$$\begin{aligned} \mathcal {S}_H(\Omega _D)=\left\{ \mathcal {I}(F): F\in Stem(D), F=\sum _{K\subset H^c}e_KF_K+\sum _{h\in H}e_{\{h\}}\sum _{Q\subset \{ h+1, \ldots ,n\}{\setminus } H}e_QF_{\{h\}\cup Q}\right\} , \end{aligned}$$

$$\begin{aligned} \mathcal {S}\mathcal {R}_H(\Omega _D)=\mathcal {S}_H(\Omega _D)\bigcap _{h\in H}\ker (\partial /\partial x_h^c)\subset \mathcal {S}_H(\Omega _D), \end{aligned}$$

(1)

$$\begin{aligned} \mathcal {S}_{c,H}(\Omega _D)=\left\{ \mathcal {I}(F): F\in Stem(D), F=\sum _{K\subset H^c}e_KF_K\right\} \subset \mathcal {S}_H(\Omega _D). \end{aligned}$$

Given $f\in \mathcal {S}(\Omega _D)$, induced by $F=\sum _{K\in \mathcal {P}(n)}e_KF_K$, we define its $x_h$-spherical value $f^\circ _{s,h}=\mathcal {I}(F^\circ _h)$ and its $x_h$-spherical derivative $f'_{s,h}=\mathcal {I}(F'_h)$, where $F^\circ _h\in Stem(D)$ and $F'_h\in Stem (D{\setminus }\mathbb {R}_h)$ are defined by

$$\begin{aligned} F^\circ _h(z)=\displaystyle \sum _{K\in \mathcal {P}(n),h\notin K}e_KF_K(z),\qquad F'_h(z)=\beta _h^{-1}\displaystyle \sum _{K\in \mathcal {P}(n),h\notin K}e_KF_{K\cup \{h\}}(z), \end{aligned}$$

where $z=(z_1, \ldots ,z_n)=(\alpha _1+i\beta _1, \cdots ,\alpha _n+i\beta _n)$ and $\beta _h={\text {im}}(z_h)\in \mathbb {R}$. If $H=\{h_1,h_2, \cdots ,h_p\}\in \mathcal {P}(n)$ define also $f^\circ _{s,H}:=(\cdots (f^\circ _{s,h_1})^\circ _{s,h_2}\cdots )^\circ _{s,h_p}\in \mathcal {S}(\Omega _D)$ and $f'_{s,H}:=(\cdots (f'_{s,h_1})'_{s,h_2}\cdots )'_{s,h_p}\in \mathcal {S}(\Omega _{D_H})$, where $\Omega _{D_H}=\Omega _D{\setminus }\mathbb {R}_H$. Note that these definitions are well posed, since they do not depend on the order of $\{h_j\}_{h=1}^p$ [5, Lemma 4.1]. Moreover, $f^\circ _{s,H}=\mathcal {I}(F^\circ _H)$, $f'_{s,H}=\mathcal {I}(F'_H)$, with

$$\begin{aligned} F^\circ _H(z)=\sum _{K\subset H^c}e_KF_K(z),\qquad F'_H(z)=\beta _H^{-1}\sum _{K\subset H^c}e_KF_{K\cup H}(z), \end{aligned}$$

where $\beta _H:=\prod _{h\in H}\beta _h$. Note that, if $f\in \mathcal {S}^1(\Omega _D)$, then $f'_{s,H}$ is defined over all $\Omega _D$, for every $H\in \mathcal {P}(n)$. In the next Proposition we recap the main properties of the spherical values and the spherical derivatives we will need through the paper. Their proofs can be found in [5, §4].

Proposition 2.2

Let $f,g\in \mathcal {S}(\Omega _D)$, $h\in \{1, \ldots ,n\}$ and $H\in \mathcal {P}(n)$, then the following holds true.

• 1.
  $$\begin{aligned} f=f^\circ _{s,h}+{\text {Im}}(x_h)\odot f'_{s,h}, \end{aligned}$$

  (2)

  where ${\text {Im}}(x_h):\mathbb {H}^n\rightarrow \mathbb {H}$ is the slice function $(\alpha _1+J_1\beta _1, \cdots ,\alpha _n+J_n\beta _n)\mapsto J_h\beta _h$;

• 2. Leibniz formula

  $$\begin{aligned} (f\odot g)'_{s,h}=f'_{s,h}\odot g^\circ _{s,h}+f^\circ _{s,h}\odot g'_{s,h}. \end{aligned}$$

  (3)

  In particular, if $g\in \mathcal {S}_{c,H}(\Omega _D)$, then $(f\odot g)'_{s,H}=f'_{s,H}\odot g$.

• 3. $f'_{s,H}\in \mathcal {S}_{c,H}(\Omega _{D_H})\cap \mathcal {S}_p(\Omega _{D_H})$, where $p=\min H^c=\min \{\{1, \ldots ,n\}{\setminus } H\}$;

• 4. if $f\in \mathcal {S}_{c,h}(\Omega _D)$, then $f^\circ _{s,h}=f$ and $f'_{s,h}=0$;

• 5. if $h\in H$, $H\cap \{1, \ldots ,h-1\}\ne \emptyset$ and $f\in \mathcal {S}_h(\Omega _D)$, then $f'_{s,H}=0$;

• 6. if $f\in \ker (\partial /\partial x_t^c)$ for some $t\ne h$, then $f'_{s,h}\in \ker (\partial /\partial x_t^c)$;

• 7. if $f\in \ker (\partial /\partial x_h^c)$, then $\Delta _hf'_{s,h}=0$, where $\Delta _h:=\frac{\partial ^2}{\partial \alpha ^2_h}+\frac{\partial ^2}{\partial \beta ^2_h}+\frac{\partial ^2}{\partial \gamma ^2_h}+\frac{\partial ^2}{\partial \delta ^2_h}$ is the Laplacian of $\mathbb {R}^4$ w.r.t. the variable $x_h=\alpha _h+i\beta _h+j\gamma _h+k\delta _h$.

We can give (2) and (3) through stem functions: let $F,G\in Stem(D)$ and $h\in \{1, \ldots ,n\}$, then it holds

$$\begin{aligned} F=F_h^\circ +{\text {Im}}(Z_h)\otimes F'_h, \end{aligned}$$

(4)

where ${\text {Im}}(Z_h)=\mathcal {I}^{-1}({\text {Im}}(x_h))\in Stem(\mathbb {C}^n)$, ${\text {Im}}(Z_h)(z_1,\dots ,z_n)=e_h\beta _h$, if $z_h=\alpha _h+i\beta _h$ and

$$\begin{aligned} (F\otimes G)'_h=F'_h\otimes G^\circ _h+F^\circ _h\otimes G'_h. \end{aligned}$$

(5)

Let us represent any $x=(x_1, \ldots ,x_n)\in \mathbb {H}^n$ with real coordinates as $x_h=\alpha _h+i\beta _h+j\gamma _h+k\delta _h$, then recall two differential operators for any $h=1, \ldots ,n$

$$\begin{aligned}{} & {} \partial _{x_h}:=\dfrac{1}{2}\left( \dfrac{\partial }{\partial \alpha _h}-i\dfrac{\partial }{\partial \beta _h}-j\dfrac{\partial }{\partial \gamma _h}-k\dfrac{\partial }{\partial \delta _h}\right) ,\\{} & {} \overline{\partial }_{x_h}:=\dfrac{1}{2}\left( \dfrac{\partial }{\partial \alpha _h}+i\dfrac{\partial }{\partial \beta _h}+j\dfrac{\partial }{\partial \gamma _h}+k\dfrac{\partial }{\partial \delta _h}\right) . \end{aligned}$$

They extend to several variables the well known Cauchy–Riemann–Fueter operators $\partial _{CRF}$ and $\overline{\partial }_{CRF}$ [25, §6]. Sometimes notation can be slightly different and the factor 1/2 can be omitted. These operators factorize the Laplacian, indeed $\forall h=1, \ldots ,n$

$$\begin{aligned} 4\partial _{x_h}\overline{\partial }_{x_h}=4\overline{\partial }_{x_h}\partial _{x_h}=\Delta _h. \end{aligned}$$

(6)

Functions in the kernel of $\overline{\partial }_{x_h}$ are usually called monogenic (or Fueter regular) w.r.t. $x_h$. By (6), we get that $x_h$-monogenic functions are, in particular, harmonic w.r.t. $x_h$. Furthermore we call $x_h$-axially monogenic those slice functions which are monogenic w.r.t. $x_h$, too, i.e. $\mathcal {A}\mathcal {M}_h(\Omega _D):=\{f\in \mathcal {S}(\Omega _D)\mid f\in \mathcal {C}^1(\Omega _D), \ \overline{\partial }_{x_h}f=0\}$. Finally, $\overline{\partial }_{x_h}f$ and $f'_{s,h}$ are closely related, whenever $f\in \mathcal {S}\mathcal {R}_h(\Omega _D)$, indeed by [5, Lemma 4.3] it holds

$$\begin{aligned} \overline{\partial }_{x_h}f=-f'_{s,h}. \end{aligned}$$

(7)

3 Almansi-Type Decomposition

3.1 Main Results

Definition 3.1

Let $\Omega _D\subset \mathbb {H}^n$, $f\in \mathcal {S}(\Omega _D)$ and $H\in \mathcal {P}(n)$. For every $K=\{k_1, \cdots ,k_p\}\subset \mathbb {H}$, with $k_1< \cdots <k_p$, define over $\Omega _{D_H}$ the slice functions

$$\begin{aligned} \mathcal {S}^H_K(f):=\left( x_K\odot f\right) '_{s,H}=\left( \prod _{i=1}^px_{k_i}\odot f\right) '_{s,H}, \end{aligned}$$

and set $\mathcal {S}^\emptyset _\emptyset (f):=f$. If $H=\llbracket m\rrbracket :=\{1,2, \ldots ,m\}$ is an integers interval from 1 to some $m\in \{1, \ldots ,n\}$, we can write $\forall K\in \mathcal {P}(m)$

$$\begin{aligned} \mathcal {S}^{\llbracket m\rrbracket }_K(f):=(x_m^{\chi _K(m)}\dots (x_1^{\chi _K(1)}f)'_{s,1}\dots )'_{s,m}, \end{aligned}$$

where $\chi _K$ 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 [17, Proposition 2.52]. If $f=\mathcal {I}(F)$, every $\mathcal {S}^H_K(f)$ is induced by the stem function

$$\begin{aligned} G^H_K(F):=\left( Z_K\otimes F\right) '_H, \end{aligned}$$

where $Z_j\in Stem(\mathbb {C}^n)$ is the stem function $Z_j(\alpha _1+i\beta _1, \cdots ,\alpha _n+i\beta _n):=\alpha _j+e_j\beta _j$, inducing the monomial $x_j\in \mathcal {S}(\mathbb {H}^n)$, for any $j=1, \ldots ,n$.

We can now formulate our main result.

Theorem 3.1

Let $\Omega _D\subset \mathbb {H}^n$ be a circular set and let $f\in \mathcal {S}(\Omega _D)$ be a slice function. Fix any $H\in \mathcal {P}(n)$, then

• 1. we can decompose f as

  $$\begin{aligned} f(x)=\displaystyle \sum _{K\subset H}(-1)^{|H{\setminus } K|}\left( \overline{x}\right) _{H{\setminus } K}\odot \mathcal {S}^H_K(f)(x). \end{aligned}$$

  (8)

• 2. $\mathcal {S}^H_K(f)\in \mathcal {S}_{c,H}(\Omega _{D_H})\cap \mathcal {S}_p(\Omega _{D_H})$, where $p=\min H^c$, $\forall K\subset H$;

• 3. if $f\in \mathcal {S}\mathcal {R}(\Omega _D)$, then $\Delta _h\mathcal {S}^H_K(f)=0$, $\forall h\in H$, $\forall K\subset H$;

• 4. $f\in \mathcal {S}\mathcal {R}(\Omega _D)$ if and only if $\mathcal {S}^H_K(f)\in \mathcal {S}\mathcal {R}_p(\Omega _D)$, $\forall H\in \mathcal {P}(n)$, $K\subset H$, $p=\min H^c$;

• 5. $f\in \mathcal {S}_\mathbb {R}(\Omega _D)$ if and only if $\mathcal {S}^{\llbracket 1,n\rrbracket }_K(f)$ is real valued, $\forall K\in \mathcal {P}(n)$.

Remark 1

For any $H\in \mathcal {P}(n)$ we can define the $\mathbb {H}$-right linear operator

$$\begin{aligned} \mathcal {S}^H:\mathcal {S}(\Omega _D)\ni f\mapsto \left\{ \mathcal {S}^H_K(f)\right\} _{K\subset H}\in \left( \mathcal {S}_{c,H}(\Omega _{D_H})\cap \mathcal {S}_p(\Omega _{D_H})\right) ^{2^{|H|}}, \end{aligned}$$

where $p:=\min H^c$ and its restriction

$$\begin{aligned} \mathcal {S}^H:\mathcal {S}\mathcal {R}(\Omega _D)\rightarrow \left( \mathcal {S}_{c,H}(\Omega _D)\cap \mathcal {S}\mathcal {R}_p(\Omega _D)\bigcap _{h\in H}\ker \Delta _h\right) ^{2^{|H|}}. \end{aligned}$$

We highlight the unique character of the decomposition, indeed for every choice of $H\in \mathcal {P}(n)$, the functions $\mathcal {S}^H_K(f)$ are the only H-circular functions that realize decomposition (8).

Proposition 3.2

Let $f\in \mathcal {S}(\Omega _D)$ and fix $H\in \mathcal {P}(n)$. Suppose that there exist functions $\{h_K\}_{K\subset H}$ such that $h_K\in \mathcal {S}_{c,H}(\Omega _D)$, $\forall K\subset H$ and

$$\begin{aligned} f(x)=\displaystyle \sum _{K\subset H}(-1)^{|H{\setminus } K|}\left( \overline{x}\right) _{H{\setminus } K}\odot h_K(x). \end{aligned}$$

Then $h_K=\mathcal {S}^H_K(f)$.

From Theorem 3.1, we emphasize the case in which $H=\llbracket m\rrbracket =\{1,2, \ldots ,m\}$, that leads to what we call an ordered decomposition of f.

Corollary 3.3

Let $f\in \mathcal {S}(\Omega _D)$ and fix any $m\in \{1, \ldots ,n\}$, then we can orderly decompose f as

$$\begin{aligned} f(x)=\displaystyle \sum _{K\in \mathcal {P}(m)}(-1)^{|K^c|}\left( \overline{x}\right) _{K^c}\mathcal {S}^{\llbracket m\rrbracket }_K(f)(x), \end{aligned}$$

(9)

where $K^c=\{1, \ldots ,m\}{\setminus } K$. Moreover,

• 1. if $m<n$, $\mathcal {S}^{\llbracket m\rrbracket }_K(f)\in \mathcal {S}_{c,\{1, \ldots ,m\}}(\Omega _{D_{\llbracket m\rrbracket }})\cap \mathcal {S}_{m+1}(\Omega _{D_{\llbracket m\rrbracket }})$, for any $K\in \mathcal {P}(m)$, while $\mathcal {S}^{\llbracket 1,n\rrbracket }_K(f)\in \mathcal {S}_{c,\{1, \ldots ,n\}}(\Omega _{D_{\llbracket 1,n\rrbracket }})$;

• 2. if $f\in \mathcal {S}\mathcal {R}(\Omega _D)$, then $\Delta _h\mathcal {S}^{\llbracket m\rrbracket }_K(f)=0$, $\forall h\le m$, $\forall K\in \mathcal {P}(m)$.

We point out that formula (9) holds with the ordinary pointwise product [17, Proposition 2.52]. On the contrary, in (8) the slice product is necessary.

We now give a one variable interpretation of slice regularity in terms of partial slice regularity of the functions $\mathcal {S}^{\llbracket m\rrbracket }_K(f)$.

Proposition 3.4

Let $f\in \mathcal {S}(\Omega _D)$, then $f\in \mathcal {S}\mathcal {R}(\Omega _D)$ if and only if $\mathcal {S}^{\llbracket m\rrbracket }_K(f)\in \mathcal {S}\mathcal {R}_{m+1}(\Omega _D)$, $\forall m=0, \ldots ,n-1$, $K\in \mathcal {P}(m)$. In that case we can write $\mathcal {S}^{\llbracket m\rrbracket }_K(f)$ as

$$\begin{aligned} \mathcal {S}^{\llbracket m\rrbracket }_K(f)=(-1)^m\overline{\partial }_{x_m}(x_m^{\chi _K(m)}\dots \overline{\partial }_{x_1}(x_1^{\chi _K(1)}f)\dots ). \end{aligned}$$

(10)

Remark 2

The previous characterization resembles the one given in [17, Theorem 3.23], in which iterations of spherical values and spherical derivatives (also referred as truncated spherical derivatives $\mathcal {D}_\epsilon (f)$) have been used. It is easy to see that truncated spherical derivatives can be expressed as real combinations of the components $\mathcal {S}^{\llbracket m\rrbracket }_K(f)$ and viceversa, making the two characterizations equivalent.

Example 1

• 1. Let $f\in \mathcal {S}\mathcal {R}(\mathbb {H}^2)$, $f(x_1,x_2):=x_1 x_2$. We can give $2^2$ decompositions of f for $H=\emptyset ,\{1\},\{1,2\},\{2\}$: let $x=(\alpha _1+J_1\beta _1,\alpha _2+J_2\beta _2)$, then

  $$\begin{aligned} \begin{aligned} f(x)&=\mathcal {S}^\emptyset _\emptyset (f)(x)=f(x)\\&=\mathcal {S}^{\llbracket 1\rrbracket }_{\{1\}}(f)(x)-\overline{x}_1\mathcal {S}^{\llbracket 1\rrbracket }_\emptyset (f)(x)=2\alpha _1x_2-\overline{x}_1 x_2\\&=\mathcal {S}^{\llbracket 1,2\rrbracket }_{\{1,2\}}(f)(x)-\overline{x}_1\mathcal {S}^{\llbracket 1,2\rrbracket }_{\{2\}}(f)(x)-\overline{x}_2\mathcal {S}^{\llbracket 1,2\rrbracket }_{\{1\}}(f)(x)+\overline{x}_1\overline{x}_2\mathcal {S}^{\llbracket 1,2\rrbracket }_\emptyset (f)(x)\\&=4\alpha _1\alpha _2-2\alpha _2\overline{x}_1-2\alpha _1\overline{x}_1+\overline{x}_1\overline{x}_2\\&=\mathcal {S}^{\{2\}}_{\{2\}}(f)(x)-\overline{x}_2\odot \mathcal {S}^{\{2\}}_\emptyset (f)(x)=2\alpha _2x_1-\overline{x}_2\odot x_1=2\alpha _2x_1-x_1\overline{x}_2. \end{aligned} \end{aligned}$$

  Note that in the first three decompositions the slice product is not needed. On the contrary, the last one, corresponding to $H=\{2\}$ needs the slice product. Moreover, $\mathcal {S}^\emptyset _\emptyset (f)=f\in \mathcal {S}\mathcal {R}_1(\Omega _D)$ and $\mathcal {S}^{\llbracket 1\rrbracket }_\emptyset (f),\mathcal {S}^{\llbracket 1\rrbracket }_{\{1\}}(f)\in \mathcal {S}\mathcal {R}_2(\Omega _D)$, as $f\in \mathcal {S}\mathcal {R}(\Omega _D)$.

• 2. Let $g\in \mathcal {S}\mathcal {R}(\mathbb {H}^3)$, $g(x_1,x_2,x_3)=e^{x_1}x_2x_3^3$. Now we have $2^3$ decompositions for $H\in \mathcal {P}(3)$. Let $x=(\alpha _1+J_1\beta _1,\alpha _2+J_2\beta _2,\alpha _3+J_3\beta _3)$, so aside from the trivial decomposition corresponding to $H=\emptyset$, we have the ordered decompositions for $H=\{1\},\{1,2\},\{1,2,3\}$

  $$\begin{aligned} f(x)&=\mathcal {S}^{\llbracket 1\rrbracket }_{\{1\}}(f)(x)-\overline{x}_1\mathcal {S}^{\llbracket 1\rrbracket }_\emptyset (f)(x)=e^{\alpha _1}(\cos \beta _1+\alpha _1/\beta _1\sin \beta _1)x_2x_3^3\\&\quad -\overline{x}_1e^{\alpha _1}\sin (\beta _1)/\beta _1x_2x_3^3\\&=\mathcal {S}^{\llbracket 1,2\rrbracket }_{\{1,2\}}(f)(x)-\overline{x}_1\mathcal {S}^{\llbracket 1,2\rrbracket }_{\{2\}}(f)(x)-\overline{x}_2\mathcal {S}^{\llbracket 1,2\rrbracket }_{\{1\}}(f)(x)+\overline{x}_1\overline{x}_2\mathcal {S}^{\llbracket 1,2\rrbracket }_\emptyset (f)(x)\\&=e^{\alpha _1}(\cos \beta _1+\alpha _1/\beta _1\sin \beta _1)2\alpha _2x_3^3-\overline{x}_1e^{\alpha _1}\sin (\beta _1)/\beta _12\alpha _2x_3^3+\\&\quad -\overline{x}_2e^{\alpha _1}(\cos \beta _1+\alpha _1/\beta _1\sin \beta _1)x_3^3 +\overline{x}_1\overline{x}_2e^{\alpha _1}\sin (\beta _1)/\beta _1x_3^3\\&=\mathcal {S}^{\llbracket 1,3\rrbracket }_{\{1,2,3\}}(f)(x)-\overline{x}_1\mathcal {S}^{\llbracket 1,3\rrbracket }_{\{2,3\}}(f)(x)-\overline{x}_2\mathcal {S}^{\llbracket 1,3\rrbracket }_{\{1,3\}}(f)(x)-\overline{x}_3\mathcal {S}^{\llbracket 1,3\rrbracket }_{\{1,2\}}(f)(x)+\\&\quad +\overline{x}_1\overline{x}_2\mathcal {S}^{\llbracket 1,3\rrbracket }_{\{3\}}(f)(x)+\overline{x}_1\overline{x}_3\mathcal {S}^{\llbracket 1,3\rrbracket }_{\{2\}}(f)(x)\\&\quad +\overline{x}_2\overline{x}_3\mathcal {S}^{\llbracket 1,3\rrbracket }_{\{1\}}(f)(x)\\&\quad -\overline{x}_1\overline{x}_2\overline{x}_3\mathcal {S}^{\llbracket 1,3\rrbracket }_\emptyset (f)(x)\\&=e^{\alpha _1}(\cos \beta _1+\alpha _1/\beta _1\sin \beta _1)2\alpha _24\alpha _3(\alpha _3^2-\beta _3^2)\\&\quad -\overline{x}_1e^{\alpha _1}\sin (\beta _1)/\beta _12\alpha _24\alpha _3(\alpha _3^2-\beta _3^2)+\\&\quad -\overline{x}_2e^{\alpha _1}(\cos \beta _1+\alpha _1/\beta _1\sin \beta _1)4\alpha _3(\alpha _3^2-\beta _3^2)+\\&\quad -\overline{x}_3e^{\alpha _1}(\cos \beta _1+\alpha _1/\beta _1\sin \beta _1)2\alpha _2(3\alpha _3^2-\beta _3^2)\\&\quad +\overline{x}_1\overline{x}_2e^{\alpha _1}\sin (\beta _1)/\beta _14\alpha _3(\alpha _3^2-\beta _3^2)+\\&\quad +\overline{x}_1\overline{x}_3e^{\alpha _1}\sin (\beta _1)/\beta _12\alpha _2(3\alpha _3^2-\beta _3^2)\\&\quad +\overline{x}_2\overline{x}_3e^{\alpha _1}(\cos \beta _1+\alpha _1/\beta _1\sin \beta _1)(3\alpha _3^2-\beta _3^2)+\\&\quad -\overline{x}_1\overline{x}_2\overline{x}_3e^{\alpha _1}\sin (\beta _1)/\beta _1(3\alpha _3^2-\beta _3^2) \end{aligned}$$

  and the remaining decompositions for $H=\{2\},\{3\},\{1,3\},\{2,3\}$

  $$\begin{aligned} f(x)&=\mathcal {S}^{\{2\}}_{\{2\}}(f)(x)-\overline{x}_2\odot \mathcal {S}^{\{2\}}_\emptyset (f)(x)=e^{x_1}2\alpha _2x_3^3-\overline{x}_2\odot e^{x_1}x_3^3\\&=\mathcal {S}^{\{3\}}_{\{3\}}(f)(x)-\overline{x}_3\odot \mathcal {S}^{\{3\}}_\emptyset (f)(x)\\&\quad =e^{x_1}x_24\alpha _3(\alpha _3^2-\beta _3^2)-\overline{x}_3\odot e^{x_1}x_2(3\alpha _3^2-\beta _3^2)\\&=\mathcal {S}^{\{1,3\}}_{\{1,3\}}(f)(x)-\overline{x}_1\odot \mathcal {S}^{\{1,3\}}_{\{3\}}(f)(x)\\&\quad -\overline{x}_3\odot \mathcal {S}^{\{1,3\}}_{\{1\}}(f)(x)+\overline{x}_1\overline{x}_3\odot \mathcal {S}^{\{1,3\}}_\emptyset (f)(x)\\&=e^{\alpha _1}(\cos \beta _1+\alpha _1/\beta _1\sin \beta _1)x_24\alpha _3(\alpha _3^2-\beta _3^2)\\&\quad -\overline{x}_1\odot e^{\alpha _1}\sin (\beta _1)/\beta _1x_24\alpha _3(\alpha _3^2-\beta _3^2)+\\&\quad -\overline{x}_3\odot e^{\alpha _1}(\cos \beta _1+\alpha _1/\beta _1\sin \beta _1)x_2(3\alpha _3^2-\beta _3^2)\\&\quad +\overline{x}_1\overline{x}_3\odot e^{\alpha _1}\sin (\beta _1)/\beta _1x_2(3\alpha _3^2-\beta _3^2)\\&=\mathcal {S}^{\{2,3\}}_{\{2,3\}}(f)(x)-\overline{x}_2\odot \mathcal {S}^{\{2,3\}}_{\{3\}}(f)(x)\\&\quad -\overline{x}_3\odot \mathcal {S}^{\{2,3\}}_{\{2\}}(f)(x)+\overline{x}_2\overline{x}_3\odot \mathcal {S}^{\{2,3\}}_\emptyset (f)(x)\\&=e^{x_1}2\alpha _24\alpha _3(\alpha _3^2-\beta _3^2)-\overline{x}_2\odot e^{x_1}4\alpha _3(\alpha _3^2-\beta _3^2)\\&\quad -\overline{x}_3\odot e^{x_1}2\alpha _2(3\alpha _3^2-\beta _3^2)++\overline{x}_2\overline{x}_3\odot e^{x_1}(3\alpha _3^2-\beta _3^2). \end{aligned}$$

3.2 Preliminary Results

Lemma 3.5

For every $m=1, \ldots ,n$, it holds

• 1. $(x_m)'_{s,h}=(Z_m)'_h=\delta _{h,m}$ and $(\overline{x}_m)'_{s,h}=(\overline{Z}_m)'_h=-\delta _{h,m}$, where $\delta _{i,j}$ is the Kronecker symbol and $\overline{Z}_m=\mathcal {I}^{-1}(\overline{x}_m)\in Stem(\mathbb {C}^n)$, $\overline{Z}_m(\alpha _1+i\beta _1,\dots ,\alpha _n+i\beta _n)=\alpha _m-e_m\beta _m$;

• 2. $(x_m)^\circ _{s,h}=x_m$, $(\overline{x}_m)^\circ _{s,h}=\overline{x}_m$, $(Z_m)^\circ _h=Z_m$, $(\overline{Z}_m)^\circ _h=\overline{Z}_m$ if $h\ne m$ and $(x_m)^\circ _{s,m}=(\overline{x}_m)^\circ _{s,m}={\text {Re}}(x_m)$, $(Z_m)^\circ _m=(\overline{Z}_m)^\circ _m={\text {Re}}(Z_m)$, where ${\text {Re}}(Z_m)(\alpha _1+i\beta _1,\dots ,\alpha _n+i\beta _n)=\alpha _m$ and ${\text {Re}}(x_m)=\mathcal {I}({\text {Re}}(Z_m))$.

Proof

Since $x_m=\mathcal {I}(Z_m)$ and $\overline{x}_m=\mathcal {I}(\overline{Z}_m)$, it is enough to prove the properties for the slice funtions or for the stem functions. By Proposition 2.2 (4), since $x_m,\overline{x}_m\in \mathcal {S}_{c,h}(\Omega _D)$, $\forall h\ne m$, immediately it holds $(x_m)^\circ _{s,h}=x_m$, $(\overline{x}_m)^\circ _{s,h}=\overline{x}_m$ and $(x_m)'_{s,h}=(\overline{x}_m)'_{s,h}=0$. Finally, by direct computation, $(Z_m)^\circ _m(z)=(\overline{Z}_m)^\circ _m(z)=\alpha _m={\text {Re}}(Z_m)(z)$, if $z=(\alpha _1+i\beta _1,\dots ,\alpha _n+i\beta _n)$ and $(Z_m)'_m(z)={\text {im}}(z_m)^{-1}{\text {im}}(z_m)=1$, $(\overline{Z}_m)'_m(z)={\text {im}}(z_m)^{-1}(-{\text {im}}(z_m))=-1$. $\square$

Proposition 3.6

Let $F\in Stem(D)$ and fix $H\in \mathcal {P}(n)$. For every $K\subset H$, the functions $G^H_K(F)$ satisfy the following properties:

• 1. For every $m\notin H$, it holds

  $$\begin{aligned} G^H_K(F)=G^{H\cup \{m\}}_{K\cup \{m\}}(F)-\overline{Z}_m\otimes G^{H\cup \{m\}}_K(F). \end{aligned}$$

  (11)

• 2. For every $h\in H$ it holds

  $$\begin{aligned} G^H_K(F)=\left( Z_h^{\chi _K(h)}\otimes G^{H{\setminus }\{h\}}_{K{\setminus }\{h\}}(F)\right) '_h. \end{aligned}$$

• 3. Explicitely, for $z=(z_1, \cdots ,z_n)\in D{\setminus }\mathbb {R}_H$, with $z_j=\alpha _j+i\beta _j$, we have

  $$\begin{aligned} G^H_K(F)(z)=\sum _{T\subset H^c}e_T\left( \displaystyle \sum _{L\subset K}\alpha _{K{\setminus } L}\beta ^{-1}_{H{\setminus } L}F_{(T\cup H){\setminus } L}(z)\right) . \end{aligned}$$

  (12)

• 4. If F is holomorphic, then every $G^H_K(F)$ is m-holomorphic, $\forall m\notin H$.

Proof

1. Apply (4), (5) and Lemma 3.5, then

$$\begin{aligned} \begin{aligned}&G^{H\cup \{m\}}_{K\cup \{m\}}(F)-\overline{Z}_m\otimes G^{H\cup \{m\}}_K(F)=\left( Z_{K\cup \{m\}}\otimes F\right) '_{H\cup \{m\}}-\overline{Z}_m\otimes \left( Z_K\otimes F\right) '_{H\cup \{m\}}\\&\quad =\left( \left( Z_{K\cup \{m\}}\otimes F\right) '_m\right) '_H-\overline{Z}_m\otimes \left( \left( Z_K\otimes F\right) '_m\right) '_H\\&\quad =\left( \left( Z_{K\cup \{m\}}\right) '_m\otimes F^\circ _m+\left( Z_{K\cup \{m\}}\right) ^\circ _m\otimes F'_m\right) '_H-\overline{Z}_m\otimes \left( \left( Z_K\right) '_m\otimes F^\circ _m+(Z_K)^\circ _m\otimes F'_m\right) '_H\\&\quad =\left( Z_K\otimes F^\circ _m+(Z_m)^\circ _m\otimes Z_K\otimes F'_m\right) '_H-\overline{Z}_m\otimes \left( Z_K\otimes F'_m\right) '_H\\&\quad =\left( Z_K\otimes F^\circ _m+{\text {Re}}(Z_m)\otimes Z_K\otimes F'_m-\overline{Z}_m\otimes Z_K\otimes F'_m\right) '_H\\&\quad =\left( Z_K\otimes \left( F^\circ _m+({\text {Re}}(Z_m)-\overline{Z}_m)\otimes F'_m\right) \right) '_H=\left( Z_K\otimes \left( F^\circ _m+{\text {Im}}(Z_m)\otimes F'_m\right) \right) '_H\\&\quad =\left( Z_K\otimes F\right) '_H=G^H_K(F). \end{aligned} \end{aligned}$$

2. It follows immediately by definition of $G^H_K(F)$.

3. We procede by induction over |H|. Suppose first $|H|=1$, i.e. $H=\{h\}$ for some $h\in \{1, \ldots ,n\}$, then we have two components $G^{\{h\}}_\emptyset (F)$ and $G^{\{h\}}_{\{h\}}(F)$. Let us compute them explicitely: for any $z=(\alpha _1+i\beta _1,\dots ,\alpha _n+i\beta _n)\in D{\setminus }\mathbb {R}_h$

$$\begin{aligned} G^{\{h\}}_\emptyset (F)(z)=F'_h(z):=\displaystyle \sum _{T\in \mathcal {P}(n),h\notin T}e_T\beta _h^{-1}F_{T\cup \{h\}}(z), \end{aligned}$$

$$\begin{aligned} \begin{aligned} G^{\{h\}}_{\{h\}}(F)(z)&=\left( Z_h\otimes F\right) '_h(z)=\left( Z_h\right) '_h(z)\otimes F^\circ _h(z)+\left( Z_h\right) ^\circ _h(z)\otimes F'_h(z)\\&=\displaystyle \sum _{T\in \mathcal {P}(n),h\notin T}e_TF_T(z)+\sum _{T\in \mathcal {P}(n),h\notin T}e_T\alpha _h\beta ^{-1}_hF_{T\cup \{h\}}(z). \end{aligned} \end{aligned}$$

Now, suppose that (12) holds for some $H\in \mathcal {P}(n)$ and let us prove it for $H'=H\cup \{m\}$, for any $m\notin H$. Suppose first $m\notin K$, then for $z=(z_1, \ldots ,z_n)\in D{\setminus }\mathbb {R}_H$, with $z_j=\alpha _j+i\beta _j$, $j=1,\dots ,n$ we have

$$\begin{aligned} \begin{aligned} G^{H'}_K(F)(z)&=\left( \left( Z_K\otimes F\right) '_H\right) '_m(z)=\left( \sum _{T\subset H^c}e_T\left( \displaystyle \sum _{L\subset K}\alpha _{K{\setminus } L}\beta ^{-1}_{H{\setminus } L}F_{(T\cup H){\setminus } L}(z)\right) \right) '_m\\&=\beta _m^{-1}\displaystyle \sum _{T\subset H^c{\setminus }\{m\}}e_T\left( \displaystyle \sum _{L\subset K}\alpha _{K{\setminus } L}\beta ^{-1}_{H{\setminus } L}F_{(T\cup H\cup \{m\}){\setminus } L}(z)\right) \\&=\displaystyle \sum _{T\subset (H\cup \{m\})^c}e_T\left( \displaystyle \sum _{L\subset K}\alpha _{K{\setminus } L}\beta ^{-1}_{(H\cup \{m\}){\setminus } L}F_{(T\cup H\cup \{m\}){\setminus } L}(z)\right) . \end{aligned} \end{aligned}$$

Suppose now $m\in K$, then

$$\begin{aligned} \begin{aligned} G^{H'}_K(F)(z)&=\left( Z_m(z)\otimes \left( Z_{K{\setminus }\{m\}}\otimes F(z)\right) '_H\right) '_m\\&=\left( Z_m(z)\otimes \sum _{T\subset H^c}e_T\left( \displaystyle \sum _{L\subset (K{\setminus }\{m\})}\alpha _{(K{\setminus }\{m\}){\setminus } L}\beta ^{-1}_{H{\setminus } L}F_{(T\cup H){\setminus } L}(z)\right) \right) '_m\\&=\left( \sum _{T\subset H^c}e_T\left( \displaystyle \sum _{L\subset (K{\setminus }\{m\})}\alpha _{(K{\setminus }\{m\}){\setminus } L}\beta ^{-1}_{H{\setminus } L}F_{(T\cup H){\setminus } L}(z)\right) \right) ^\circ _m\\&\quad +\alpha _m\left( \sum _{T\subset H^c}e_T\left( \displaystyle \sum _{L\subset (K{\setminus }\{m\})}\alpha _{(K{\setminus }\{m\}){\setminus } L}\beta ^{-1}_{H{\setminus } L}F_{(T\cup H){\setminus } L}(z)\right) \right) '_m\\&=\sum _{T\subset (H^c{\setminus }\{m\})}e_T\left( \displaystyle \sum _{L\subset (K{\setminus }\{m\})}\alpha _{(K{\setminus }\{m\}){\setminus } L}\beta ^{-1}_{H{\setminus } L}F_{(T\cup H){\setminus } L}(z)\right) \\&\quad +\beta _m^{-1}\alpha _m\sum _{T\subset (H^c){\setminus }\{m\}}e_T\left( \displaystyle \sum _{L\subset (K{\setminus }\{m\})}\alpha _{(K{\setminus }\{m\}){\setminus } L}\beta ^{-1}_{H{\setminus } L}F_{(T\cup H\cup \{m\}){\setminus } L}(z)\right) \\&=\sum _{T\subset (H\cup \{m\})^c}e_T\left( \displaystyle \sum _{L\subset (K{\setminus }\{m\})}\alpha _{K{\setminus }(\{m\}\cup L)}\beta ^{-1}_{(H\cup \{m\}{\setminus } (L\cup \{m\})}F_{(T\cup H\cup \{m\}){\setminus } (L\cup \{m\})}(z)\right) \\&\quad +\sum _{T\subset (H\cup \{m\})^c}e_T\left( \displaystyle \sum _{L\subset (K{\setminus }\{m\})}\alpha _{K{\setminus } L}\beta ^{-1}_{(H\cup \{m\}){\setminus } L}F_{(T\cup H\cup \{m\}){\setminus } L}(z)\right) \\&=\displaystyle \sum _{T\subset H'}e_T\left( \displaystyle \sum _{L\subset K}\alpha _{K{\setminus } L}\beta ^{-1}_{H'{\setminus } L}F_{(T\cup H'){\setminus } L}(z)\right) . \end{aligned} \end{aligned}$$

4. F and $Z_K$ are holomorphic, so is $Z_K\otimes F$. Finally, $G^H_K(F)=(Z_K\otimes F)'_H$ is holomorphic w.r.t. $z_m$ for every $m\notin H$, by (6) of Proposition 2.2. $\square$

Next Lemma will be used to prove 5. of Theorem 3.1.

Lemma 3.7

For every $H\in \mathcal {P}(n)$, every $K\subset H$ and every $z=(\alpha _1+i\beta _1,\dots ,\alpha _n+i\beta _n)\in D{\setminus }\mathbb {R}_H$, it holds

$$\begin{aligned} \beta _K^{-1}\displaystyle \sum _{T\subset H^c}e_TF_{K\cup T}(z)=\displaystyle \sum _{T\subset H{\setminus } K}(-1)^{|T|}\alpha _TG^H_{H{\setminus }(K\cup T)}(F)(z), \end{aligned}$$

(13)

where, if $H=\{1, \ldots ,n\}$ we mean

$$\begin{aligned} \beta _K^{-1}F_K(z)=\displaystyle \sum _{T\subset K^c}(-1)^{|T|}\alpha _TG^{\{1, \ldots ,n\}}_{(K\cup T)^c}(F)(z). \end{aligned}$$

(14)

Proof

Let us proceed by induction over |H|. First, suppose $H=\{h\}$, for any $h=1, \ldots ,n$, then $K=\emptyset ,\{h\}$. If $K=\emptyset$, we have

$$\begin{aligned} \begin{aligned}&\displaystyle \sum _{T\subset \{h\}}(-1)^{|T|}\alpha _TG^{\{h\}}_{\{h\}{\setminus } T}(F)(z)\\&\quad =G^{\{h\}}_{\{h\}}(F)(z)-\alpha _hG^{\{h\}}_\emptyset (F)(z)= \left( Z_h\otimes F\right) '_h(z)-\alpha _h F'_h(z)\\&\quad =F^\circ _h(z)+\alpha _hF'_h(z)-\alpha _hF'_h(z)=\displaystyle \sum _{T\in \{h\}^c}e_TF_T(z), \end{aligned} \end{aligned}$$

where (5) and Lemma 3.5 has been used. If $K=\{h\}$, immediately we get

$$\begin{aligned} G^{\{h\}}_\emptyset (F)(z)=F'_h(z)=\beta _h^{-1}\displaystyle \sum _{T\subset \{h\}^c}e_TF_{\{h\}\cup T}(z). \end{aligned}$$

Now, suppose that (13) holds for some $H\in \mathcal {P}(n)$ and let us prove it for $H'=H\cup \{m\}$, with any $m\notin H$ and any $K\subset H'$. Let us split $m\in K$ and $m\notin K$. If $m\in K$, let us set $K':=K{\setminus }\{m\}$, then we have

$$\begin{aligned} \begin{aligned}&\displaystyle \sum _{T\subset H'{\setminus } K}(-1)^{|T|}\alpha _TG^{H'}_{H'{\setminus }(K\cup T)}(F)(z)=\displaystyle \sum _{T\subset H{\setminus } K'}(-1)^{|T|}\alpha _T\left( Z_{H{\setminus }(K'\cup T)}(z)\otimes F(z)\right) '_{H\cup \{m\}}\\&\quad =\left( \displaystyle \sum _{T\subset H{\setminus } K'}(-1)^{|T|}\alpha _T\left( Z_{H{\setminus }(K'\cup T)}(z)\otimes F\right) '_H(z)\right) '_m\\&\quad =\left( \displaystyle \sum _{T\subset H{\setminus } K'}(-1)^{|T|}\alpha _TG^H_{H{\setminus }(K'\cup T)}(F)(z)\right) '_m\\&\quad =\left( \beta _{K'}^{-1}\displaystyle \sum _{T\subset H^c}e_TF_{K'\cup T}(z)\right) '_m=\beta _K^{-1}\displaystyle \sum _{T\subset (H')^c}e_TF_{K\cup T}(z). \end{aligned} \end{aligned}$$

Finally, if $m\notin K$

$$\begin{aligned}{} & {} \displaystyle \sum _{T\subset H'{\setminus } K}(-1)^{|T|}\alpha _TG^{H'}_{H'{\setminus }(K\cup T)}(F)(z)=\displaystyle \sum _{T\subset H{\setminus } K}(-1)^{|T|}\alpha _TG^{H'}_{H'{\setminus }(K\cup T)}(F)(z)+\\{} & {} \qquad -\displaystyle \sum _{T\subset H{\setminus } K}(-1)^{|T|}\alpha _m\alpha _TG^{H'}_{H{\setminus }(K\cup T)}(F)(z)\\{} & {} \quad =\displaystyle \sum _{T\subset H{\setminus } K}(-1)^{|T|}\alpha _T\left( \left( Z_m(z)\otimes (Z_{H{\setminus } (K\cup T)}(z)\otimes F(z))\right) '_m\right) '_H\\{} & {} \qquad -\displaystyle \sum _{T\subset H{\setminus } K}(-1)^{|T|}\alpha _m\alpha _T \left( Z_{H{\setminus }(K\cup T)}(z)\otimes F(z)\right) '_{H\cup m}\\{} & {} \quad =\displaystyle \sum _{T\subset H{\setminus } K}(-1)^{|T|}\alpha _T\left( \left( Z_{H{\setminus }(K\cup T)}(z)\otimes F(z)\right) ^\circ _m+\alpha _m\left( Z_{H{\setminus }(K\cup T)}(z)\otimes F(z)\right) '_m\right) '_H+\\{} & {} \qquad -\displaystyle \sum _{T\subset H{\setminus } K}(-1)^{|T|}\alpha _m\alpha _T \left( Z_{H{\setminus }(K\cup T)}(z)\otimes F(z)\right) '_{H\cup m}\\{} & {} \quad =\left( \displaystyle \sum _{T\subset H{\setminus } K}(-1)^{|T|}\alpha _T G_{H{\setminus }(K\cup T)}^H(F)(z)\right) ^\circ _m=\left( \beta _K^{-1}\displaystyle \sum _{T\subset H^c}e_TF_{K\cup T}(z)\right) ^\circ _m\\{} & {} \quad =\beta _K^{-1}\displaystyle \sum _{T\subset (H')^c}e_TF_{K\cup T}(z). \end{aligned}$$

$\square$

3.3 Proofs of Main Results

Proof of Theorem 3.1

• 1. Let us prove that decomposition (8) holds for stem functions, too, namely that for any $H\in \mathcal {P}(n)$ we have

  $$\begin{aligned} F=\displaystyle \sum _{K\subset H}(-1)^{|H{\setminus } K|}\overline{Z}_{H{\setminus } K}\otimes G^H_K(F). \end{aligned}$$

  (15)

  We proceed by induction over |H|. Suppose first $H=\{h\}$, for some $h=1, \ldots ,n$, then, for any $z=(\alpha _1+i\beta _1,\dots ,\alpha _n+i\beta _n)\in D$ we have

  $$\begin{aligned} \begin{aligned}&G^{\{h\}}_{\{h\}}(F)(z)-\overline{Z}_h(z)\otimes G^{\{h\}}_\emptyset (F)(z)=(Z_h(z)\otimes F(z))'_h-\overline{Z}_h(z)\otimes F'_h(z)\\&\quad =F^\circ _h(z)+\alpha _h F'_h(z)-\alpha _h F'_h(z)+{\text {Im}}(Z_h)(z)\otimes F'_h(z)=F(z), \end{aligned} \end{aligned}$$

  by (3), (4) and Lemma 3.5. Now, suppose (15) holds for some $H\in \mathcal {P}(n)$, let us prove it for $H'=H\cup \{m\}$, with $m\notin H$. We have

  $$\begin{aligned}{} & {} \displaystyle \sum _{K\subset H'}(-1)^{|H'{\setminus } K|}\overline{Z}_{H'{\setminus } K}(z)\otimes G^{H'}_K(F)(z)\\{} & {} =\displaystyle \sum _{K\subset H}(-1)^{|H{\setminus } K|}\overline{Z}_{H{\setminus } K}(z)\otimes G^{H\cup \{m\}}_{K\cup \{m\}}(F)(z)\\{} & {} \qquad -\displaystyle \sum _{K\subset H}(-1)^{|H{\setminus } K|}\overline{Z}_{H{\setminus } K}(z)\otimes \overline{Z}_m(z)\otimes G^{H\cup \{m\}}_K(F)(z)\\{} & {} =\displaystyle \sum _{K\subset H}(-1)^{|H{\setminus } K|}\overline{Z}_{H{\setminus } K}(z)\otimes \left( G^{H\cup \{m\}}_{K\cup \{m\}}(F)(z)-\overline{Z}_m(z)\otimes G^{H\cup \{m\}}_K(F)(z)\right) \\{} & {} =\displaystyle \sum _{K\subset H}(-1)^{|H{\setminus } K|}\overline{Z}_{H{\setminus } K}(z)\otimes G^H_K(F)(z)=F(z), \end{aligned}$$

  by (11) and the inductive hypothesis. Now (8) easily follows, indeed

  $$\begin{aligned} \begin{aligned}&f=\mathcal {I}(F)=\mathcal {I}\left( \displaystyle \sum _{K\subset H}(-1)^{|H{\setminus } K|}\overline{Z}_{H{\setminus } K}\otimes G^H_K(F)\right) \\&\quad =\displaystyle \sum _{K\subset H}(-1)^{|H{\setminus } K|}\mathcal {I}\left( \overline{Z}_{H{\setminus } K}\right) \odot \mathcal {I}\left( G^H_K(F)\right) \\&\quad =\displaystyle \sum _{K\subset H}(-1)^{|H{\setminus } K|}\left( \overline{x}\right) _{H{\setminus } K}\odot \mathcal {S}^H_K(f). \end{aligned} \end{aligned}$$

• 2. For any $K\subset H$, $\mathcal {S}^H_K(f)=(x_K\odot f)'_{s,H}\in \mathcal {S}_{c,H}(\Omega _{D_H})\cap \mathcal {S}_p(\Omega _{D_H})$, by Proposition 2.2 (3).

• 3. Write $\mathcal {S}^H_K(f)=\left( x_h^{\chi _K(h)}\odot \mathcal {S}^{H{\setminus }\{h\}}_{K{\setminus }\{h\}}(f)\right) '_{s,h}$. By hypothesis, $f\in \ker (\partial /\partial x_t^c)$, $\forall t=1, \ldots ,n$, then, by 6. of Proposition 2.2, $\mathcal {S}^{H{\setminus }\{h\}}_{K{\setminus }\{h\}}(f)\in \ker (\partial /\partial x_h^c)$ and thanks to Leibniz formula [17, Proposition 3.25], $x_h^{\chi _K(h)}\odot \mathcal {S}^{H{\setminus }\{h\}}_{K{\setminus }\{h\}}(f)\in \ker (\partial /\partial x_h^c)$. Finally, by Proposition 2.2 7. $\Delta _h\mathcal {S}^H_K(f)=\Delta _h\left( x_h^{\chi _K(h)}\odot \mathcal {S}^{H{\setminus }\{h\}}_{K{\setminus }\{h\}}(f)\right) '_{s,h}=0.$

• 4. $\Rightarrow$) By hypothesis, $x_K\odot f\in \mathcal {S}\mathcal {R}(\Omega _D)$, then $\mathcal {S}^H_K(f)=(x_K\odot f)'_{s,H}\in \ker (\partial /\partial x_t^c)$, for any $t\notin H$. In particular, $\mathcal {S}^H_K(f)=(x_K\odot f)'_{s,H}\in \ker (\partial /\partial x_p^c)\cap \mathcal {S}_p(\Omega _D)=\mathcal {S}\mathcal {R}_p(\Omega _D)$, by (1). $\Leftarrow$) It is a particular case of Proposition 3.4.

• 5. f is slice preserving if and only if $F_K$ is real $\forall K\in \mathcal {P}(n)$, which by (14) is equivalent for $G^{\{1, \ldots ,n\}}_K(F)=\mathcal {S}^{\llbracket 1,n\rrbracket }_K(f)$ to be real valued for every $K\in \mathcal {P}(n)$.

$\square$

Proof of Proposition 3.2

Apply Proposition 2.2 (2) and the hypothesis $h_T\in \mathcal {S}_{c,H}(\Omega _D)$ in the following computation

$$\begin{aligned} \begin{aligned} \mathcal {S}^H_K(f):&=(x_K\odot f)'_{s,H}=\left( x_K\odot \displaystyle \sum _{T\subset H}(-1)^{|H{\setminus } T|}\left( \overline{x}\right) _{H{\setminus } T}\odot h_T\right) '_{s,H}\\&=\displaystyle \sum _{T\subset H}(-1)^{|H{\setminus } T|}\left( x_K\odot \left( \overline{x}\right) _{H{\setminus } T}\odot h_T\right) '_{s,H}\\&=\displaystyle \sum _{T\subset H}(-1)^{|H{\setminus } T|}\left( x_K\odot \left( \overline{x}\right) _{H{\setminus } T}\right) '_{s,H}\odot h_T. \end{aligned} \end{aligned}$$

Now we claim that $\left( x_K\odot \left( \overline{x}\right) _{H{\setminus } T}\right) '_{s,H}=(-1)^{|H{\setminus } K|}\delta _{K,T}$, which would reduce the prevoious equation to $\mathcal {S}^H_K(f)=h_K$. Suppose first that exists $h\in T{\setminus } K\subset H$, then $x_K\odot \left( \overline{x}\right) _{H{\setminus } T}\in \mathcal {S}_{c,h}(\Omega _D)$, thus in particular $\left( x_K\odot \left( \overline{x}\right) _{H{\setminus } T}\right) '_{s,H}=0$. Viceversa, suppose $h\in K{\setminus } T\subset H$, but again $x_K\odot \left( \overline{x}\right) _{H{\setminus } T}\in \mathcal {S}_{c,h}(\Omega _D)$, indeed $x_K\odot \left( \overline{x}\right) _{H{\setminus } T}=x_h\overline{x}_h\odot x_{K{\setminus }\{h\}}\odot \left( \overline{x}\right) _{H{\setminus } (T\cup \{h\})}=(\alpha _h^2+\beta _h^2)x_{K{\setminus }\{h\}}\odot \left( \overline{x}\right) _{H{\setminus } (T\cup \{h\})}\in \mathcal {S}_{c,h}(\Omega _D)$. So, the unique non trivial element of the sum refers to $T=K$, for which we have

$$\begin{aligned} \begin{aligned} \left( x_K\odot \left( \overline{x}\right) _{H{\setminus } K}\right) '_{s,H}&=\left( \left( x_K\odot \left( \overline{x}\right) _{H{\setminus } K}\right) '_{s,K}\right) '_{s,H{\setminus } K}=\left( (x_K)'_{s,K}\odot \left( \overline{x}\right) _{H{\setminus } K}\right) '_{s,H{\setminus } K}\\&=\left( \left( \overline{x}\right) _{H{\setminus } K}\right) '_{s,H{\setminus } K}=(-1)^{|H{\setminus } K|}, \end{aligned} \end{aligned}$$

where we have used Proposition 2.2 (2) and Lemma 3.5 (1). $\square$

Proof of Proposition 3.4

$\implies$) We have already proved in Theorem 3.1 (4).

) For any $m\in \{1, \ldots ,n\}$ consider the ordered Almansi-type decomposition (9) of f. Recall that $\partial \overline{x}_{h}/\partial x_{k}^c=0$, $\forall h,k=1, \ldots ,n$ and $\partial /\partial x_{m+1}^c(\mathcal {S}^{\llbracket m\rrbracket }_K(f))=0$, for every $K\in \mathcal {P}(n)$, so applying [17, (73)] it holds

$$\begin{aligned} \begin{aligned}&\dfrac{\partial f}{\partial x_{m+1}^c}=\dfrac{\partial }{\partial x_{m+1}^c}\left( \displaystyle \sum _{K\in \mathcal {P}(m)}(-1)^{|K^c|}\left( \overline{x}\right) _{K^c}\mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) \\&\quad =\displaystyle \sum _{K\in \mathcal {P}(m)}(-1)^{|K^c|}\left( \overline{x}\right) _{K^c}\otimes \dfrac{\partial \mathcal {S}^{\llbracket m\rrbracket }_K(f)}{\partial x_{m+1}^c}=0. \end{aligned} \end{aligned}$$

This proves the charaterization of $f\in \mathcal {S}\mathcal {R}(\Omega _D)$. Finally, (10) follows from (7). $\square$

4 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 [17, Proposition 3.14]. Before, we recall a result from [25, Corollary 6.7].

Lemma 4.1

For every $m\ge 0$, consider the slice regular power $x^m:\mathbb {H}\rightarrow \mathbb {H}$. Then it holds

$$\begin{aligned} (x^m)'_s=\tilde{\mathcal {Z}}_{m-1}(x), \end{aligned}$$

where

$$\begin{aligned} \tilde{\mathcal {Z}}_k(x):=\left\{ \begin{array}{ll} \dfrac{\mathcal {Z}_k(x,1)}{k+1}&{}\text { if }k\ge 0 \\ 0&{} \text { if }k=-1 \end{array}\right. \end{aligned}$$

and $\mathcal {Z}_k(x,1)$ is the real valued zonal harmonic of $\mathbb {R}^4$ with pole 1 (see [4, Ch. 5]).

Proposition 4.2

Let $P\in \mathbb {H}[X_1, \cdots ,X_n]$ be any quaternionic polynomial with right coefficients

$$\begin{aligned} P(x_1, \cdots ,x_n)=\displaystyle \sum _{k=1}^n\displaystyle \sum _{|\alpha |=k}x^\alpha a_\alpha =\displaystyle \sum _{k=1}^n\displaystyle \sum _{|\alpha |=k}x_1^{\alpha _1} \cdots \ x_n^{\alpha _n} a_\alpha . \end{aligned}$$

Then, for every $H\in \mathcal {P}(n)$ and $K\subset H$

$$\begin{aligned} \mathcal {S}^H_K(P)(x)=\displaystyle \sum _{k=1}^n\displaystyle \sum _{|\alpha |=k}\prod _{j\in H}\tilde{\mathcal {Z}}_{\alpha _j-1+\chi _K(j)}(x_j)\prod _{i\in H^c} x^{\alpha _i}_{i}a_\alpha , \end{aligned}$$

(16)

where $\prod _ix_i^{\alpha _i}$ is an ordered product.

Proof

By linearity of the spherical derivative, we can assume without loss of generality $P(x_1, \ldots ,x_n)=x^\alpha =x_1^{\alpha _1} \ldots \ x_n^{\alpha _n}$. We will proceed by induction over |H|. Suppose $H=\{h\}$, for some $h=1, \ldots ,n$, then, since $x_1^{\alpha _1} \cdots \ x_{h-1}^{\alpha _{h-1}} x_{h+1}^{\alpha _{h+1}} \cdots \ x_n^{\alpha _n}\in \mathcal {S}_{c,h}(\Omega _D)$, we have

$$\begin{aligned} \begin{aligned} \mathcal {S}^{\{h\}}_\emptyset (P)&=(x^\alpha )'_{s,h}=(x_h^{\alpha _h}\odot x_1^{\alpha _1} \cdots \ x_{h-1}^{\alpha _{h-1}} x_{h+1}^{\alpha _{h+1}} \cdots \ x_n^{\alpha _n})'_{s,h}\\&=(x_h^{\alpha _h})'_{s,h}\odot x_1^{\alpha _1} \cdots \ x_{h-1}^{\alpha _{h-1}} x_{h+1}^{\alpha _{h+1}} \cdots \ x_n^{\alpha _n}\\&=\tilde{\mathcal {Z}}_{\alpha _h-1}(x_h) x_1^{\alpha _1} \cdots \ x_{h-1}^{\alpha _{h-1}} x_{h+1}^{\alpha _{h+1}} \cdots \ x_n^{\alpha _n}, \end{aligned} \end{aligned}$$

where we have used Lemma 4.1, Proposition 2.2 (2) and that ${Z}_j$ is real valued. Similarly,

$$\begin{aligned} \mathcal {S}^{\{h\}}_{\{h\}}(P)= & {} (x_h\odot x^\alpha )'_{s,h}\\= & {} (x_h^{\alpha _h+1}\odot x_1^{\alpha _1} \cdots \ x_{h-1}^{\alpha _{h-1}} x_{h+1}^{\alpha _{h+1}} \cdots \ x_n^{\alpha _n})'_{s,h}\\= & {} \tilde{\mathcal {Z}}_{\alpha _h}(x_h) x_1^{\alpha _1} \cdots \ x_{h-1}^{\alpha _{h-1}} x_{h+1}^{\alpha _{h+1}} \cdots \ x_n^{\alpha _n}. \end{aligned}$$

Now, suppose that (16) holds for some $H\in \mathcal {P}(n)$ and let us prove it for $H'=H\cup \{m\}$, for any $m\notin H$. Suppose first $m\notin K$, then, as before

$$\begin{aligned} \begin{aligned} \mathcal {S}^{H'}_K(P)&=(\mathcal {S}^H_K(P))'_{s,m}=\left( \prod _{j\in H}\tilde{\mathcal {Z}}_{\alpha _j-1+\chi _K(j)}(x_j)\prod _{i\in H^c}x_i^{\alpha _i}\right) '_{s,m}\\&=\left( \prod _{j\in H}\tilde{\mathcal {Z}}_{\alpha _j-1+\chi _K(j)}(x_j)x_m^{\alpha _m}\odot \prod _{i\in (H')^c}x_i^{\alpha _i}\right) '_{s,m}\\&=\prod _{j\in H}\tilde{\mathcal {Z}}_{\alpha _j-1+\chi _K(j)}(x_j)\tilde{\mathcal {Z}}_{\alpha _m-1}(x_m)\prod _{i\in (H')^c}x_i^{\alpha _i}\\&=\prod _{j\in H'}\tilde{\mathcal {Z}}_{\alpha _j-1+\chi _K(j)}(x_j)\prod _{i\in (H')^c}x_i^{\alpha _i}. \end{aligned} \end{aligned}$$

If $m\in K$, let $K'=K{\setminus }\{m\}$, then

$$\begin{aligned} \mathcal {S}^{H'}_K(P)= & {} (x_m\odot \mathcal {S}^H_{K'}(P))'_{s,m}=\left( \prod _{j\in H}\tilde{\mathcal {Z}}_{\alpha _j-1+\chi _{K'}(j)}(x_j)x_m^{\alpha _m+1}\odot \prod _{i\in (H')^c}x_i^{\alpha _i}\right) '_{s,m}\\= & {} \prod _{j\in H}\tilde{\mathcal {Z}}_{\alpha _j-1+\chi _{K'}(j)}(x_j)\tilde{\mathcal {Z}}_{\alpha _m}(x_m)\prod _{i\in (H')^c}x_i^{\alpha _i}\\= & {} \prod _{j\in H'}\tilde{\mathcal {Z}}_{\alpha _j-1+\chi _{K}(j)}(x_j)\prod _{i\in (H')^c}x_i^{\alpha _i}. \end{aligned}$$

$\square$

Let us examine the case in which f is slice w.r.t. $x_h$.

Proposition 4.3

Let $f\in \mathcal {S}_h(\Omega _D)$ for some $h\in \{1, \ldots ,n\}$, then

$$\begin{aligned} \mathcal {S}^{\llbracket 1,h\rrbracket }_K(f)=0,\qquad \forall K\in \mathcal {P}(h-1), K\ne \{1, \ldots ,h-1\}. \end{aligned}$$

(17)

In particular, the ordered decomposition of f of order h reduces to

$$\begin{aligned} f=\sum _{K\in \mathcal {P}(h-1)}(-1)^{|K^c|}\overline{x}_{K^c}\mathcal {S}^{\llbracket 1,h\rrbracket }_{K\cup \{h\}}(f)-\overline{x}_h\mathcal {S}^{\llbracket 1,h\rrbracket }_{\{1, \ldots ,h-1\}}(f). \end{aligned}$$

Proof

Assume $K\in \mathcal {P}(h-1)$, with $K\ne \{1, \ldots ,h-1\}$, then there exists $m\in \{1, \ldots ,h-1\}{\setminus } K$. Let $H:=\{m,h\}$, then by (5) of Proposition 2.2 it holds $f'_{s,H}=0$. Since $K\cap H=\emptyset$, $x_K\in \mathcal {S}_{c,H}(\Omega _D)$, then by (2) of Proposition 2.2 we have

$$\begin{aligned} \mathcal {S}^{\llbracket 1,h\rrbracket }_K(f)=\left[ (x_K\odot f)'_{s,H}\right] '_{s,\{1, \ldots ,h\}{\setminus } H}=(x_K\odot f'_{s,H})'_{s,\{1, \ldots ,h\}{\setminus } H}=0. \end{aligned}$$

This proves (17). By this and Corollary 3.3 follows

$$\begin{aligned} \begin{aligned} f&=\sum _{K\in \mathcal {P}(h)}(-1)^{|\{1, \ldots ,h\}{\setminus } K|}\overline{x}_{\{1, \ldots ,h\}{\setminus } K}\mathcal {S}^{\llbracket 1,h\rrbracket }_{K}(f)\\&=\sum _{K\in \mathcal {P}(h-1)}(-1)^{|\{1, \ldots ,h\}{\setminus } K|}\overline{x}_{\{1, \ldots ,h\}{\setminus } K}\mathcal {S}^{\llbracket 1,h\rrbracket }_{K}(f)+\\&\quad +\sum _{K\in \mathcal {P}(h-1)}(-1)^{|\{1, \ldots ,h\}{\setminus } (K\cup \{h\})|}\overline{x}_{\{1, \ldots ,h\}{\setminus } (K\cup \{h\})}\mathcal {S}^{\llbracket 1,h\rrbracket }_{K\cup \{h\}}(f)\\&=-\overline{x}_h\mathcal {S}^{\llbracket 1,h\rrbracket }_{\{1, \ldots ,h-1\}}(f)+\sum _{K\in \mathcal {P}(h-1)}(-1)^{|\{1, \ldots ,h-1\}{\setminus } K|}\overline{x}_{\{1, \ldots ,h-1\}{\setminus } K}\mathcal {S}^{\llbracket 1,h\rrbracket }_{K\cup \{h\}}(f). \end{aligned} \end{aligned}$$

$\square$

The components of the ordered decomposition provide examples of axially monogenic functions.

Proposition 4.4

Let $f\in \mathcal {S}\mathcal {R}(\Omega _D)$, $m=1, \ldots ,n-1$ and recall that $\mathcal {A}\mathcal {M}_m(\Omega _D)$ denotes the set of slice functions monogenic with respect to $x_m$. Then $\forall K\in \mathcal {P}(m)$, $\mathcal {S}^{\llbracket m\rrbracket }_K(f)$ satisfies

• 1. $\partial _{x_m}(\mathcal {S}^{\llbracket m\rrbracket }_K(f))\in \mathcal {A}\mathcal {M}_m(\Omega _D)$;

• 2. $\Delta _{m+1}(\mathcal {S}^{\llbracket m\rrbracket }_K(f))\in \mathcal {A}\mathcal {M}_{m+1}(\Omega _D)$.

Proof

By Corollary 3.3, $\mathcal {S}^{\llbracket 1,h\rrbracket }_K(f)$ is harmonic w.r.t. $x_j$, for any $j=1, \ldots ,h$, $\forall K\in \mathcal {P}(h)$, so

$$\begin{aligned} \overline{\partial }_{x_m}\partial _{x_m}\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) =\dfrac{1}{4}\Delta _m\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) =0 \end{aligned}$$

and by Proposition 3.4

$$\begin{aligned} \overline{\partial }_{x_{m+1}}\Delta _{m+1}\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) =\Delta _{m+1}\overline{\partial }_{x_{m+1}}\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) =-\Delta _{m+1}\left( \mathcal {S}^{\llbracket 1,m+1\rrbracket }_K(f)\right) =0. \end{aligned}$$

$\square$

The following result highlights a difference between the one and several variables slice regular functions: in the first case the Laplacian of a slice regular function is always an axially monogenic functions (this is Fueter’s Theorem [13]), in the latter this happens only for the first variable. But for any variable, we can at least write it as sum of axially monogenic functions.

Lemma 4.5

Let $m=1, \ldots ,n$ and let $f\in \mathcal {S}^1(\Omega _D)\cap \ker (\partial /\partial x_m^c)$, then it holds

$$\begin{aligned} \Delta _mf=-4\displaystyle \sum _{K\in \mathcal {P}(m-1)}(-1)^{|K^c|}\left( \overline{x}\right) _{K^c}\partial _{x_m}\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) . \end{aligned}$$

Proof

By (3) and (6) of Proposition 2.2 it holds $\mathcal {S}^{\llbracket 1,m-1\rrbracket }_K(f)=(x_K\odot f)'_{s,\{1, \ldots ,m-1\}}\in \ker (\partial /\partial x_m^c)\cap \mathcal {S}_m(\Omega _D)=\mathcal {S}\mathcal {R}_m(\Omega _D)$, $\forall K\in \mathcal {P}(m-1)$, then by (6) and (7) it holds

$$\begin{aligned} \Delta _m\mathcal {S}^{\llbracket 1,m-1\rrbracket }_K(f)= & {} 4\partial _{x_m}\overline{\partial }_{x_m}\mathcal {S}^{\llbracket 1,m-1\rrbracket }_K(f)=-4\partial _{x_m}\left[ \left( \mathcal {S}^{\llbracket 1,m-1\rrbracket }_K(f)\right) '_{s,m}\right] \\= & {} -4\partial _{x_m}\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) , \end{aligned}$$

with $\partial _{x_m}\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) \in \mathcal {A}\mathcal {M}_m(\Omega _D)$, by Proposition 4.4, 1. So, applying (9), we have

$$\begin{aligned} \begin{aligned} \Delta _m f&=\Delta _m\left( \displaystyle \sum _{K\in \mathcal {P}(m-1)}(-1)^{|K^c|}(\overline{x})_{K^c}\mathcal {S}^{\llbracket 1,m-1\rrbracket }_K(f)\right) \\&=\displaystyle \sum _{K\in \mathcal {P}(m-1)}(-1)^{|K^c|}(\overline{x})_{K^c}\Delta _m\mathcal {S}^{\llbracket 1,m-1\rrbracket }_K(f)\\&=-4\displaystyle \sum _{K\in \mathcal {P}(m-1)}(-1)^{|K^c|}(\overline{x})_{K^c}\partial _{x_m}\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) . \end{aligned} \end{aligned}$$

$\square$

The issue changes if we assume the function slice regular in that specific variable, as already proven in [5, Theorem 4.9], getting a generalization of Fueter’s Theorem in several variables. We give another proof through the ordered decomposition of Corollary 3.3.

Corollary 4.6

Let $f\in \mathcal {S}\mathcal {R}_m(\Omega _D)$, then $\Delta _mf\in \mathcal {A}\mathcal {M}_m(\Omega _D)$.

Proof

By Proposition 4.4, Lemma 4.5 and (17) we get

$$\begin{aligned} \Delta _mf= & {} -4\displaystyle \sum _{K\in \mathcal {P}(m-1)}(-1)^{|K^c|}(\overline{x})_{K^c}\partial _{x_m}\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) \\= & {} -4\partial _{x_m}\left( \mathcal {S}^{\llbracket m\rrbracket }_{\{1, \ldots ,m-1\}}(f)\right) \in \mathcal {A}\mathcal {M}_m(\Omega _D). \end{aligned}$$

$\square$

Corollary 4.7

Every slice regular function is separately biharmonic in each variable.

Proof

Let $f\in \mathcal {S}\mathcal {R}(\Omega _D)$, then thanks to Corollary 3.3 2, Lemma 4.5 and (6) we have

$$\begin{aligned} \begin{aligned} \Delta _m^2f&=\Delta _m\left( -4\displaystyle \sum _{K\in \mathcal {P}(m-1)}(-1)^{|\{1, \ldots ,m-1\}{\setminus } K|}\overline{x}_{\{1, \ldots ,m-1\}{\setminus } K}\partial _{x_m}\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) \right) \\&=-4\displaystyle \sum _{K\in \mathcal {P}(m-1)}(-1)^{|\{1, \ldots ,m-1\}{\setminus } K|}\overline{x}_{\{1, \ldots ,m-1\}{\setminus } K}\partial _{x_m}\left( \Delta _m\left( \mathcal {S}^{\llbracket m\rrbracket }_K(f)\right) \right) =0. \end{aligned} \end{aligned}$$

$\square$

4.1 Mean Value Properties for Slice Regular Functions

Notation: in the last part of the section, if $a=(a_1, \ldots ,a_n)\in \mathbb {H}^n$ and $\lambda _k\in \mathbb {H}$, for some $k=1, \ldots ,n$, we denote $a+\lambda _k:=(a_1, \ldots ,a_{k-1},a_k+\lambda _k,a_{k+1}, \ldots ,a_n)$. Let $\sigma$ be the surface measure of $\mathbb {S}^3=\partial \mathbb {B}^4\subset \mathbb {H}\cong \mathbb {R}^4$ such that $\sigma \left( \mathbb {S}^3\right) =1$, namely $\sigma (y):=\mathcal {H}^3(y)/\omega _3$, where $\mathcal {H}^3$ denotes the three-dimensional Hausdorff measure of $\mathbb {R}^4$ and $\omega _3:=\mathcal {H}^3\left( \mathbb {S}^3\right) =2\pi ^2$. Again, by $\sigma ^l$ we mean the l-th power of $\sigma$.

Assumption: throughout the section we will always assume that $f\in \mathcal {S}\mathcal {R}(\Omega _D)$ is a slice regular function and for $a=(a_1, \ldots ,a_n)\in \Omega _D$ and $r_1, \ldots ,r_n\in \mathbb {R}^+\cup \{0\}$ it holds $\overline{B_{r_1}(a_1)}\times \ldots \times \overline{B_{r_n}(a_n)}\subset \Omega _D$.

Proposition 4.8

Let $f\in \mathcal {S}\mathcal {R}(\Omega _D)$, then for every $H\in \mathcal {P}(n)$ and $K\subset H$ it holds

$$\begin{aligned} (\mathcal {S}^H_K(f))(a)=\int _{\left( \mathbb {S}^3\right) ^{|H|}}\left( \mathcal {S}^H_K(f)\right) \left( a+\textstyle \sum _{h\in H}r_h\lambda _h\right) d\sigma ^{|H|}(\lambda ). \end{aligned}$$

(18)

Proof

By Theorem 3.1, every $\mathcal {S}^H_K(f)$ is separately harmonic w.r.t. $x_h$, for every $h\in H$, thus, we can apply the classical mean value formula for harmonic functions for such variables

$$\begin{aligned} \left( \mathcal {S}^H_K(f)\right) (a)=\int _{\mathbb {S}^3}\left( \mathcal {S}^H_K(f)\right) (a+r_h\lambda _h)d\sigma (\lambda _h),\qquad \forall h\in H. \end{aligned}$$

Thus, (18) follows applying the previous formula for any $h\in H$. $\square$

Proposition 4.9

(First mean value formula) For any $m=1, \ldots ,n$ it holds

$$\begin{aligned} \begin{aligned} f(a)=\sum _{K\in \mathcal {P}(m)}(-1)^{|K^c|}\overline{a}_{K^c}\int _{\left( \mathbb {S}^3\right) ^m}\mathcal {S}^{\llbracket m\rrbracket }_K(f)(a+\textstyle \sum _{i=1}^mr_m\lambda _m)d\sigma ^m(\lambda ). \end{aligned} \end{aligned}$$

(19)

Proof

Apply (9) and (18) with $H=\{1, \ldots ,m\}$

$$\begin{aligned} f(a)= & {} \displaystyle \sum _{K\in \mathcal {P}(m)}(-1)^{|K^c|}\overline{a}_{K^c}\mathcal {S}^{\llbracket m\rrbracket }_K(f)(a)\\= & {} \sum _{K\in \mathcal {P}(m)}(-1)^{|K^c|}\overline{a}_{K^c}\int _{\left( \mathbb {S}^3\right) ^m}\mathcal {S}^{\llbracket m\rrbracket }_K(f)(a+\textstyle \sum _{i=1}^mr_i\lambda _i)d\sigma ^m(\lambda ). \end{aligned}$$

$\square$

We can give integral formulas through the general decomposition (8), but we must assume that the centers of the spheres are real.

Proposition 4.10

Let $H\in \mathcal {P}(n)$ and assume $a_h\in \mathbb {R}$, for any $h\in H$. Then

$$\begin{aligned} f(a)=\sum _{K\subset H}(-1)^{|H{\setminus } K|}a_{H{\setminus } K}\int _{(\mathbb {S})^{|H|}}\mathcal {S}^H_K(f)(a+\textstyle \sum _{h\in H}r_h\lambda _h)d\sigma ^{|H|}(\lambda ). \end{aligned}$$

Proof

By (8) and (18) we have

$$\begin{aligned} \begin{aligned} f(a)&=\sum _{K\subset H}(-1)^{|H{\setminus } K|}\left[ \overline{x}_{H{\setminus } K}\odot \mathcal {S}^H_K(f)\right] (a)=\sum _{K\subset H}(-1)^{|H{\setminus } K|}a_{H{\setminus } K}\mathcal {S}^H_K(f)(a)=\\&=\sum _{K\subset H}(-1)^{|H{\setminus } K|}a_{H{\setminus } K}\int _{\left( \mathbb {S}^3\right) ^{|H|}}\left( \mathcal {S}^H_K(f)\right) \left( a+\textstyle \sum _{h\in H}r_h\lambda _h\right) d\sigma ^{|H|}(\lambda ), \end{aligned} \end{aligned}$$

where we have used that $a_h\in \mathbb {R}$, $\forall h\in H$. $\square$

We give another integral formula through decomposition (9). For $m\ge 2$, it highly differs from (19), beacause of the components involved and the dimension of the domain of integration. On the contrary, they coincide if $m=1$.

Proposition 4.11

(Second mean value formula) For any $m=1, \ldots ,n$ it holds

$$\begin{aligned} \begin{aligned} f(a)&=\displaystyle \sum _{j=0}^{m-1}r_{\{1, \ldots ,j\}}\int _{\left( \mathbb {S}^3\right) ^{j+1}}\overline{\lambda }_{\{1, \ldots ,j\}}\mathcal {S}^{\llbracket 1,j\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{j+1}r_i\lambda _i)d\sigma ^{j+1}(\lambda )+\\&\quad +r_{\{1, \ldots ,m\}}\int _{\left( \mathbb {S}^3\right) ^m}\overline{\lambda }_{\{1, \ldots ,m\}}\mathcal {S}^{\llbracket m\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^mr_i\lambda _i)d\sigma ^m(\lambda ), \end{aligned} \end{aligned}$$

(20)

where $r_\emptyset =\overline{\lambda }_\emptyset =1$.

Proof

We prove the identity by induction over m, using the corresponding one-variable formula [23, Proposition 2] that we can apply iteratively, since $\mathcal {S}^{\llbracket m\rrbracket }_K(f)\in \mathcal {S}\mathcal {R}_{m+1}(\Omega _D)$. For $m=1$, (20) is precisely [23, Proposition 2], indeed

$$\begin{aligned} \begin{aligned} f(a)&=\int _{\mathbb {S}^3}f(a+r_1\lambda _1)d\sigma (\lambda _1)+r_1\int _{\mathbb {S}^3}\overline{\lambda }_1\left( \mathcal {S}^{\llbracket 1\rrbracket }_\emptyset (f)\right) (a+r_1\lambda _1)d\sigma (\lambda _1)\\&=\int _{\mathbb {S}^3}f(a+r_1\lambda _1)d\sigma (\lambda _1)+r_1\int _{\mathbb {S}^3}\overline{\lambda }_1f'_{s,1}(a+r_1\lambda _1)d\sigma (\lambda _1). \end{aligned} \end{aligned}$$

Now, suppose that the formula holds for some m, then, apply [23, Proposition 2] to $\mathcal {S}^{\llbracket m\rrbracket }_\emptyset (f)^{\tilde{a}}_{m+1}\in \mathcal {S}\mathcal {R}(\Omega _{D,m+1}(\tilde{a}))$, where $\tilde{a}=a+\textstyle \sum _{i=1}^mr_i\lambda _i$:

$$\begin{aligned} f(a)= & {} \displaystyle \sum _{j=0}^{m-1}r_{\{1, \ldots ,j\}}\int _{\left( \mathbb {S}^3\right) ^{j+1}}\overline{\lambda }_{\{1, \ldots ,j\}}\mathcal {S}^{\llbracket 1,j\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{j+1}r_i\lambda _i)d\sigma ^{j+1}(\lambda )\\{} & {} +r_{\{1, \ldots ,m\}}\int _{\left( \mathbb {S}^3\right) ^m}\overline{\lambda }_{\{1, \ldots ,m\}}\mathcal {S}^{\llbracket m\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{m}r_i\lambda _i)d\sigma ^m(\lambda )\\= & {} \displaystyle \sum _{j=0}^{m-1}r_{\{1, \ldots ,j\}}\int _{\left( \mathbb {S}^3\right) ^{j+1}}\overline{\lambda }_{\{1, \ldots ,j\}}\mathcal {S}^{\llbracket 1,j\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{j+1}r_i\lambda _i)d\sigma ^{j+1}(\lambda )\\{} & {} +r_{\{1, \ldots ,m\}}\int _{\left( \mathbb {S}^3\right) ^m}\overline{\lambda }_{\{1, \ldots ,m\}}\left[ \int _{\mathbb {S}^3}\mathcal {S}^{\llbracket m\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{m+1}r_i\lambda _i)d\sigma (\lambda _{m+1})\right] d\sigma ^m(\lambda )\\{} & {} +r_{\{1, \ldots ,m\}}\int _{\left( \mathbb {S}^3\right) ^m}\overline{\lambda }_{\{1, \ldots ,m\}}\left[ r_{m+1}\int _{\mathbb {S}^3}\overline{\lambda }_{m+1}\mathcal {S}^{\llbracket 1,m+1\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{m+1}r_i\lambda _i)d\sigma (\lambda _{m+1})\right] \\{} & {} d\sigma ^m(\lambda ) \\= & {} \displaystyle \sum _{j=0}^{m-1}r_{\{1, \ldots ,j\}}\int _{\left( \mathbb {S}^3\right) ^{j+1}}\overline{\lambda }_{\{1, \ldots ,j\}}\mathcal {S}^{\llbracket 1,j\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{j+1}r_i\lambda _i)d\sigma ^{j+1}(\lambda )\\{} & {} +r_{\{1, \ldots ,m\}}\int _{\left( \mathbb {S}^3\right) ^{m+1}}\overline{\lambda }_{\{1, \ldots ,m\}}\mathcal {S}^{\llbracket m\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{m+1}r_i\lambda _i)d\sigma ^{m+1}(\lambda )\\{} & {} +r_{\{1, \ldots ,m+1\}}\int _{\left( \mathbb {S}^3\right) ^{m+1}}\overline{\lambda }_{\{1, \ldots ,m+1\}}\mathcal {S}^{\llbracket 1,m+1\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{m+1}r_i\lambda _i)d\sigma ^{m+1}(\lambda )\\= & {} \sum _{j=0}^{m}r_{\{1, \ldots ,j\}}\int _{\left( \mathbb {S}^3\right) ^{j+1}}\overline{\lambda }_{\{1, \ldots ,j\}}\mathcal {S}^{\llbracket 1,j\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{j+1}r_i\lambda _i)d\sigma ^{j+1}(\lambda )\\{} & {} +r_{\{1, \ldots ,m+1\}}\int _{\left( \mathbb {S}^3\right) ^{m+1}}\overline{\lambda }_{\{1, \ldots ,m+1\}}\mathcal {S}^{\llbracket 1,m+1\rrbracket }_\emptyset (f)(a+\textstyle \sum _{i=1}^{m+1}r_i\lambda _i)d\sigma ^{m+1}(\lambda ). \end{aligned}$$

$\square$

In the rest of the section we mimic what has been done so far, but with the Poisson kernel: first we find Poisson formulas for the components $\mathcal {S}^{\llbracket m\rrbracket }_K(f)$ and finally two types of formulas for f.

Proposition 4.12

Let $x_1, \ldots ,x_m\in \mathbb {B}_\mathbb {H}$, then it holds

$$\begin{aligned} \begin{aligned} \mathcal {S}^{\llbracket m\rrbracket }_K(f)(a+\textstyle \sum _{i=1}^mr_ix_i)=\displaystyle \int _{(\mathbb {S}^3)^m}\mathcal {S}^{\llbracket m\rrbracket }_K(f)(a+\textstyle \sum _{i=1}^mr_i\xi _i)\prod _{j=1}^mP(x_j,\xi _j)d\sigma ^m(\xi ), \end{aligned} \end{aligned}$$

(21)

where $P(x,\xi ):=\dfrac{1-|x|^2}{|x-\xi |^4}$ is the Poisson kernel of $\mathbb {B}\subset \mathbb {R}^4$.

Proof

By Corollary 3.3, every $\mathcal {S}^{\llbracket m\rrbracket }_K(f)$ is harmonic w.r.t. $x_1, \ldots ,x_m$, so by Poisson integral formula for harmonic functions it holds for any $k=1, \ldots ,m$

$$\begin{aligned} \mathcal {S}^{\llbracket m\rrbracket }_K(f)(a+\textstyle \sum _{i=1}^mr_ix_i)=\displaystyle \int _{\mathbb {S}^3}\mathcal {S}^{\llbracket m\rrbracket }_K(f)(a+\textstyle \sum _{i\ne k}r_ix_i+r_k\xi _k)P(x_k,\xi _k)d\sigma (\xi _k). \end{aligned}$$

Thus, (21) follows by applying the previous formula for $k=1, \ldots ,m$. $\square$

Proposition 4.13

(First Poisson formula) Let $m=1, \ldots ,n$ and $x_1, \ldots ,x_m\in \mathbb {B}_\mathbb {H}$, then it holds

$$\begin{aligned} f(a+\textstyle \sum _{i=1}^mr_ix_i)= & {} \displaystyle \sum _{K\in \mathcal {P}(m)}(-1)^{|K^c|}\left( \overline{a}+r\overline{x}\right) _{K^c}\\{} & {} \int _{(\mathbb {S}^3)^m}\mathcal {S}^{\llbracket m\rrbracket }_K(f)(a+\textstyle \sum _{i=1}^mr_i\xi _i)\prod _{j=1}^mP(x_j,\xi _j)d\sigma ^m(\xi ). \end{aligned}$$

Proof

Apply (9) and (21) to every $\mathcal {S}^{\llbracket m\rrbracket }_K(f)$, to get

$$\begin{aligned} \begin{aligned} f(a+\textstyle \sum _{i=1}^mr_ix_i)&=\displaystyle \sum _{K\in \mathcal {P}(m)}(-1)^{|K^c|}\left( \overline{a}+r\overline{x}\right) _{K^c}\mathcal {S}^{\llbracket m\rrbracket }_K(a+\textstyle \sum _{i=1}^mr_ix_i)\\&=\displaystyle \sum _{K\in \mathcal {P}(m)}(-1)^{|K^c|}\left( \overline{a}+r\overline{x}\right) _{K^c}\int _{(\mathbb {S}^3)^m}\mathcal {S}^{\llbracket m\rrbracket }_K(f)(a+\textstyle \sum _{i=1}^mr_i\xi _i)\\&\quad \prod _{j=1}^mP(x_j,\xi _j)d\sigma ^m(\xi ). \end{aligned} \end{aligned}$$

$\square$

Proposition 4.14

(Second Poisson formula) Let $m=1, \ldots ,n$ and $x_1, \ldots ,x_m\in \mathbb {B}_\mathbb {H}$, then it holds

$$\begin{aligned} \begin{aligned} f(a+\textstyle \sum _{i=1}^mr_ix_i)&=\displaystyle \sum _{j=1}^{m-1}r_{\{1, \ldots ,j\}}\int _{(\mathbb {S}^3)^{j+1}}(\overline{\xi }-\overline{x})_{\{1, \ldots ,j\}}(\mathcal {S}^{\llbracket 1,j\rrbracket }_{\emptyset }(f))\left( a+\textstyle \sum _{i=1}^{j+1}r_i\xi _i\right) \\&\quad \prod _{t=1}^{j+1}P(x_t,\xi _t)d\sigma ^{j+1}(\xi )+\\&+r_{\{1, \ldots ,m\}}\int _{(\mathbb {S}^3)^m}(\overline{\xi }-\overline{x})_{\{1, \ldots ,m\}}\mathcal {S}^{\llbracket m\rrbracket }_\emptyset (f)\left( a+\textstyle \sum _{i=1}^{m}r_i\xi _i\right) \\&\quad \prod _{t=1}^{m}P(x_t,\xi _t)d\sigma ^{m}(\xi ). \end{aligned} \end{aligned}$$

Proof

The proof is analogue of the one of Proposition 4.11, but here apply [23, Proposition 3]. $\square$