Partial slice regularity and Fueter's Theorem in several quaternionic variables

We extend some definitions and give new results about the theory of slice analysis in several quaternionic variables. The sets of slice functions which are respectively slice, slice regular and circular w.r.t. given variables are characterized. We introduce new notions of partial spherical values and derivative for functions of several variables that extend those of one variable. We recover some of their properties and find new ones. Finally, we prove a generalization of Fueter's Theorem in this context.


Introduction
Slice regular functions were firstly introduced in [6] by Gentili and Struppa for quaternionvalued functions, defined over Euclidean balls with real centre.They exploited the complex-slice structure of the quaternion algebra H and, following an idea of Cullen [3], they defined slice regular (or Cullen regular) functions as real differentiable functions, which are slice by slice holomorphic.The main purpose of this new hypercomplex theory was to overcome the problem encountered by the theory of quaternionic functions already well established by Fueter [4], in which the class of regular functions does not contain polynomials.On the contrary, the class of slice regular functions contains all the power series with right quaternionic coefficients.The two theories are indeed very skew, since, in general, only constant functions are both Fueter and Cullen regular, even though they present some connections, as Fueter's Theorem suggests.We refer the reader to the monograph [5] for a comprehensive treatment of the theory of slice regular functions of one quaternionic variable and to [10], [13] for Fueter regular functions.
Interest in this new subject grew rapidly and a large number of papers were published.The theory was soon generalized to more general domains of definition, the so called slice domains [1] and extended to octonions [7] and Clifford algebras [2].A new viewpoint took place after the work of Ghiloni and Perotti [8] with the introduction of stem functions, already used by Fueter to generate axially monogenic functions through Fueter's map [4].This approach allows to define slice functions, in which no regularity is needed, over any axially symmetric domain and to extend the theory uniformly in any real alternative * -algebra with unity.
The stem function approach suggested the way to construct a several variable analogue of the theory in the foundational paper [9], to which the present article contributes to develop some ideas introduced therein.In that paper, the importance of partial slice regularity has been pointed out.Indeed, it is possible to interpret the slice regularity of an n variables slice function in terms of the one-variable slice regularity of 2 n − 1 slice functions [9,Theorem 3.23], obtained as all possible iterations of partial spherical values and derivatives of that function.This result establishes a bridge between the one and several variables theories, which has been frequently exploited, for example in [12], where local slice analysis was naturally extended from one to several quaternionic variables.But, the study of partial slice regularity, as well as partial spherical values and derivatives was not developed further and a more detailed study deserved attention, leading to this work.
We describe the structure of the paper.After recalling briefly the theory of slice regular functions of one and several quaternionic variables, we focus on the study of partial slice properties, i.e. sliceness, slice regularity or circularity w.r.t. a specific subset of variables ( §3).More precisely, given a set of variables {x h } h∈H , we characterize (Propositions 3.1, 3.2 and 3.4) the sets S H , SR H and S c,H of slice functions which are, respectively, slice, slice regular and circular w.r.t.all the variables x h .The stem function's approach is fundamental as all those characterizations are given through conditions over stem functions.Furthermore, we show that for every choice of H ∈ P(n), the set S c,H forms a subalgebra of (S, ⊙) (Corollary 3.5), however this does not happens for S H and SR H .
In Chapter 4, we define partial spherical values and derivatives for functions of several variables, which extend the one-variable analogues.We recover some of their main properties such as harmonicity (Proposition 4.9), representation and Leibniz formulas (19), (20) and we find new ones, peculiar of the several variables setting (Proposition 4.4), through the characterizations of Chapter 3. Finally, thanks to the harmonicity of the partial spherical derivatives, we prove a generalization of Fueter's Theorem for slice regular functions of several quaternionic variables (Theorem 4.10), which estends the link between slice regular and axially monogenic functions in higher dimensions.

Preliminaries
We breafly recall the main definitions of the theory of slice regular functions of one and several quaternionic variables.We state here the definitions of [8] and [9], reduced to the quaternionic setting.

Slice regular functions of one quaternionic variable
Let H denote the algebra of quaternions with basis elements {1, i, j, k}.We can embed R ⊂ H as the subalgebra generated by 1, while Im(H) :=< i, j, k >, whence They are unique if we require β > 0. Every such q generates a sphere we denote with S q = S α,β = {α + Iβ : Every slice function can be completely recovered by its value over one slice C J , with a representation formula [8, Proposition 6]: let I, J ∈ S H , then for every x = α + Iβ it holds Given a slice function f , we define its spherical value and its spherical derivative respectively as Note that the spherical value and the spherical derivative are both slice functions, as . Moreover, applying (1) with I = J we get We can define a product over slice functions.Let F, G be two stem functions with , which happens to be a stem function.Now, if f = I(F ) and g = I(G), define f ⊙ g = I(F ⊗ G).With respect to this product, the spherical derivative satisfies a Lebniz rule: Let F be a C Finally, a slice function f = I(F ) is said to be slice regular if ∂f /∂x c = 0 or equivalently if ∂F/∂z = 0. Note that [8, Proposition 8], if Ω D ∩ R = ∅, the definition of slice regular function coincide with the one given by Gentili and Struppa [6], namely that, for every J ∈ S H the restriction of f , f J : Ω D ∩ C J → H is holomorphic w.r.t. the complex structure defined by multiplication by J.

Slice regular functions of several quaternionic variables
Let n be a positive integer and let P(n) denote all possible subsets of {1, ..., n}.Given an ordered set K = {k 1 , ..., k p } ∈ P(n), with k 1 < ... < k p and an associated p-tuple (q k1 , ..., q kp ) ∈ H p , we define q K := q k1 • ... • q kp (with q ∅ := 1) and for any q ∈ H, [q K , q] := q K • q.Given z = (z 1 , ..., z n ) ∈ C n , set z h := (z 1 , ..., z h−1 , z h , z h+1 , ..., z n ), ∀h = 1, ..., n.A set D ⊂ C n is called invariant w.r.t.complex conjugation whenever z ∈ D if and only if z h ∈ D for every h ∈ {1, ..., n}.We define its circularization Ω D ⊂ H n as and we call circular those sets Ω such that Ω = Ω D for some D ⊂ C n , invariant w.r.t.complex conjugation.From now on, we will always assume D an invariant subset of C n w.r.t.complex conjugation and Ω D a circular set of H n .
Let {e 1 , ..., e n } be an orthonormal frame of R n and denote with {e ( Write Stem(D) for the set of all stem functions from D to where x = (x 1 , ..., x n ), with 2) is necessary to make slice functions well defined.We say that f is induced by F .S(Ω D ) will denote the set of all slice functions from Ω D to H and I : Stem(D) → S(Ω D ) will be the map sending a stem function to its induced slice function.By [9, Proposition 2.12], every slice function is induced by a unique stem function, so the map I is injective.
We can define slice functions through a commutative diagram too: for any J 1 , ..., Given F ∈ Stem(D), we can define its induced slice function f = I(F ) as the unique slice function that makes the following diagram commutative for any J 1 , ..., J n ∈ S H : where H∆K = (H ∪ K) \ (H ∩ K) and extended by linearity to all R 2 n .This product induces a product on where a H b K is just the usual H-product.Furthermore, we can define a product between stem functions as the pointwise product induced by The advantage of this definition is that the product of two stem functions is again a stem function [9,Lemma 2.34] and this allows to define a product on slice functions, too.Let f, g ∈ S(Ω D ), with f = I(F ) and g = I(G), then define the slice tensor product f ⊙ g between f and g as Equip R 2 n with the family of commutative complex structures J = J h : R 2 n → R 2 n n h=1 , where each J h is defined over any basis element e K of R 2 n as and extend it by linearity to all R 2 n .J induces a family of commutative complex structure on H ⊗ R 2 n (by abuse of notation, we use the same symbol) according to the formula We can associate two Cauchy-Riemann operators to each complex structure , we can define the partial derivatives for every h = 1, ..., n and it is called holomorphic if it is h-holomorphic for every h = 1, ..., n.Finally, given a holomorphic stem function F , the induced slice function I(F ) will be called slice regular function.The set of all slice regular functions from Ω D to H will be denoted by SR(Ω D ).By [9, Proposition 3.13], f ∈ SR(Ω D ) if and only if ∂f ∂x c h = 0 for every h = 1, ..., n.We recall two other operators on H, known as Cauchy-Riemann-Fueter operators: where α, β, γ and δ denotes the four real components of a quaternion x = α + iβ + jγ + kδ.Functions in the kernel of ∂ CRF are usually called Fueter regular (or monogenic in the context of Clifford algebras).We can extend these operators to H n : for a slice function f : Ω D → H, we define, for any h = 1, ..., n, ∂ x h and ∂ x h as the Cauchy-Riemann-Fueter operators w.r.t.
x h := α h + iβ h + jγ h + kδ h : The importance of these operators is evident as they factorize the Laplacian in each variable, indeed for any h = 1, ..., n where the set of monogenic functions w.r.t x h and let   (5) In particular, for any h ∈ {1, ..., n}, Equivalently, Then we have and where we have used (2).Thus, the right hand side of (7) becomes Comparing ( 8) and ( 9), ( 7) is satisfied if and only if Since ( 7) is assumed to be true for every I, J, J 1 , ..., J n ∈ S H and every z ′ , w, z ′′ , (10) holds if and only if ∀K ⊂ {1, ..., n} \ {h} Indeed, if (11) were not true, there would be a K ⊂ P({1, ..., n} \ {h}) such that but for J 1 = ... = J n = J = I we would have and this implies that F P ∪{h}∪Q ≡ 0. Indeed, if F P ∪{h}∪Q = 0, the previous equation would reduce to J P I = −IJJ P J which does not hold for every choice of I, J, J P .⇐) Vice versa, suppose F takes the form Following the notation above, it holds Thus, consider the function G y h is a one-variable stem function, indeed, and f y h = I(G y h ), by construction, so f ∈ S h (Ω D ).
Remark 1.By the previous proof, we can better understand the set S H (Ω D ): let f = I(F ) ∈ S H (Ω D ), then for any x ∈ Ω D with x = (φ J1 × ... × φ Jn )(z), f (x) takes the form Moreover, for any h ∈ H and any y = (y 1 , ..., y n ), f y h is a one-variable slice function, induced by the stem function G y h , with components where z = (z ′ , z h , z") and y = (φ J1 × ... × φ Jn )(z).Now, we deal with partial slice regularity.Proposition 3.2.For every H ∈ P(n) it holds thanks to (6).For any y ∈ Ω D , f y h is induced by the stem function By definition, f ∈ SR h (Ω D ) means that ∀y ∈ Ω D , the stem function G y h is holomorphic, i.e. recalling (12) it must hold that for every z = (z ′ , z h , z") ∈ D, w ∈ D h (z) and , where in the above sums P ∈ P(h−1) and Q ⊂ {h+1, ..., n}.Now, since that system is true for every choice of imaginary unit J j , proceeding as in the proof of Proposition 3.1 we can deduce that an equivalence between each term of the sum holds.Let any Q ⊂ {h+ 1, ..., n}: if P = ∅, equality can sussist only if ∂ α h F P ∪Q = ∂ β h F P ∪Q = 0 and this trivially proves that the components F P ∪Q satisfies (3), since F P ∪{h}∪Q = 0, by (6).Otherwise, let P = ∅, then the previous system becomes 13) and (3).As in the proof of Proposition 3.1, represent K = P ⊔ Q, with P ∈ P(h − 1) and Q ⊂ {h + 1, ..., n}.Since, by (13), F P ∪{h}∪Q ≡ 0, ∀P ∈ P(h − 1) \ {∅}, ∀Q ⊂ {h + 1, ..., n} the h-holomorphicity of F reduces to the following conditions: where y = (φ J1 × ... × φ Jn )(z), z = (z 1 , ..., z n ), z j = α j + iβ j .Let us prove the first row of the system.Using the first two equation of ( 14) and splitting K = P ⊔ Q, we can write the left side as The second equation is proved in the same way.
Let any y = (y 1 , ..., y n ) ∈ Ω D , with y j := α j + J j β j , z j := α j + iβ j , set z ′ = (z 1 , ..., z h−1 ) and z" = (z h+1 , ..., z n ).f ∈ S c,h (Ω D ) if for every x = a + Ib, f y h (x) does not depend on I. Let w := a + ib, M p := J p if p = h and M h = I, then It is clear that f y h (a + Ib) does not depend on I if and only if F K∪{h} = 0 for every K ∈ P(n).Finally, comparing ( 5) and (15) we see that S c,H (Ω D ) ⊂ S H (Ω D ).
Note that functions of the form (15) were introduced in [9] as H c -reduced slice functions, hence we can say that f ∈ S c,H (Ω D ) if and only if it is H c -reduced.It is easy now to prove the following property.
Proof.We only need to show that f, g ∈ S c,H (Ω D ), implies f ⊙ g ∈ S c,H (Ω D ).Let f = I(F ) and g ∈ I(G), with F = K⊂H c e K F K and G = T ⊂H c e T G T , by (15).Then Note that the previous result does not apply to ∈ S 2 (Ω D ).Slice regularity and circularity are hardly compatible.
, by Proposition 3.2, so by ( 3) Thus, f does not depend neither on α h and β h and so it is locally constant w.r.t.x h .
Example 1.Consider the following polynomial function f : 3 k, which happens to be a slice regular function, [9, Proposition 3.14].We claim that f ∈ SR 2 (Ω D ).Let us explicit the components of the stem function inducing Thus, F has the structure required by (6) for h = 2, so f ∈ S 2 (Ω D ).Moreover, for K = ∅, {1}, {3}, {1, 3} it holds . We could have proven the claim by definition, through Remark 1, which explicitly gives us the stem function that induces the corresponding one variable slice function, for every choice of y.Fix any y = (y 1 , y 2 , y 3 ) ∈ H 3 , then f y 2 is a slice regular function, induced by the holomorphic stem function
Lemma 4.1.For every H ∈ P(n), F • H and F ′ H are well defined stem functions on D and D \R H , respectively.
Proof.Firstly, let us prove that F • H and F ′ H are well defined, i.e. their definition does not depend on the order of H. Indeed, for any i, j = 1, ..., n it holds and analogously for (F • i ) • j .Without loss of generality, assume H = {h}, for some = 1, ..., n.F • h is trivially a stem function because its non zero components are the same of F .Let us explicit we will show that every component of F ′ h satisfies (2).Let us consider only the components G K , with h / ∈ K, otherwise (2) is trivial.For any m = h we have The previous Lemma allows to make the following Definition 4.2.Let f = I(F ) ∈ S(Ω D ).For h ∈ {1, ..., n}, we define its spherical x h -value and x h -derivative rispectively as We stress that the terms spherical value and spherical derivatives have been already used in [9, §2.3] in the context of slice functions of several quaternionic variables, but they refer to different objects.With respect to our definition, spherical values and derivatives are more related to the truncated spherical derivatives D ǫ (f ) [9,Definition 2.24].
The following proposition justifies the names given to f • s,h and f ′ s,h , comparing them to their one-variable analogues ( §2.1).Note that we have to assume f ∈ h (Ω D ), in order for the spherical derivative to agree with it.
In particular, if we assume f ∈ S 1 (Ω D ), then we can extend the definition of f ′ s,h to all Ω D , thanks to [8,Proposition 7,(2)].
Proof.Let f = I(F ), with F = K∈P(n) e K F K .Then for any z ∈ D and x = (φ J1 × ...× φ Jn )(z) we get Now, assume f ∈ S h (Ω D ), then by ( 6) and so On the other hand, let x = (φ J1 × ... × φ Jn )(z), then by (2) we have from which We extend from [11] properties of the spherical derivative of one-variable slice regular functions to several variables.Lemma 4.3.If f ∈ SR h (Ω D ), the following hold: Proof.
1.The first claim follows directly from Proposition 3.4.

By Proposition
In particular, f ′ s,H = 0.

It follows from (1) and (3).
Partial spherical derivatives do not affect regularity in other variables.
As recalled in §2.1, every one variable slice function f can be decomposed as f . We now give a similar decomposition for every variable, through the slice product.
Proposition 4.6.Let f ∈ S(Ω D ), then for any h = 1, ..., n we can decompose Proof.Let f = I(F ), with F = K∈P(n) e K F K .Suppose first x ∈ R h , i.e.Im(x h )(x) = 0, then by (2), with the usual notation, we have Next proposition shows that the partial spherical derivatives satisfies a Leibniz-type formula, analogue to the one-dimensional case.
Proof.Let f = I(F ) and g = I(G).We have to show that (F On the other hand, ⊗ G = K∈P(n) e K (F ⊗ G) K , where Corollary 4.8.Let f ∈ S(Ω D ) and g ∈ S c,H (Ω D ) for some H ∈ P(n), then (f ⊙g) ′ s,H = f ′ s,H ⊙g.Proof.We proceed by induction over |H|.Suppose first |H| = 1, then it follows from Proposition 4.7 and Proposition 4.4 (3).Now, suppose by induction that (f ⊙ g) ′ s,H = f ′ s,H ⊙ g and let h / ∈ H, then in the same way we have The next result highlights a fundamental property of partial spherical derivatives, i.e. harmonicity.The only requirement is regularity in such variable.This extends the result for onevariable slice regular functions [11,Theorem 6.3,(c)].Proposition 4.9.Let f ∈ S 1 (Ω D ).Suppose that f ∈ ker(∂/∂x c h ), for some h = 1, ..., n.Then ∆ h f ′ s,h = 0. Proof.Let us introduce a slightly different notation: let x = (x 1 , ..., x n ) ∈ Ω D , with x l = α l + iβ l + jγ l + kδ l = α l + J l b l , where Thus, it is enough to prove that Our last application is a generalization to several variables of Fueter's Theorem, which is a fundamental result in hypercomplex analysis.In modern language, it states that, given a slice regular function f : Ω D ⊂ H → H, its laplacian generates an axially monogenic function, i.e.
∂ CRF ∆f = 0. Theorem 4.10.Let Ω D ⊂ H n be a circular set and let f ∈ SR h (Ω D ) be a slice function, which is slice regular w.r.t.x h , for some h = 1, ..., n.Then ∆ h f is an axially monogenic function w.r.t.x h , i.e.

Proposition 3 . 4 .
For every H ∈ P(n) it holds S c,H (Ω D ) = I(F ) : F ∈ Stem(D), F = K⊂H c e K F K .