A local Cauchy integral formula for slice-regular functions

We prove a Cauchy-type integral formula for slice-regular functions where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. In particular, we get a local Cauchy-type integral formula. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two axially monogenic functions.


I
Cauchy's integral formula is one of the most powerful tools in complex analysis.It plays a key role also in the study of any function theory that aims to extend complex analysis to higher dimensional algebras.In the four dimensional case represented by the quaternionic skew field, there are at least two different generalizations of the concept of holomorphic functions.The first one deals with functions in the kernel of the Cauchy-Riemann-Fueter differential operator := 1 2 0 + 1 + 2 + 3 , where , , are the quaternionic basic imaginary units and = 0 + 1 + 2 + 3 is the real representation of a quaternion .These functions, usually called Fueterregular or monogenic, have been studied extensively for many decades.Primary references are the article of Fueter [7], where the Cauchy's integral theorem was proved, the paper of Sudbery [22] and the monograph [2], where the results were given in their generality in the context of Clifford analysis.The second function theory, introduced in 2006-2007 by Gentili and Struppa [10,11] following an idea of Cullen [5], with the objective to include quaternionic polynomials and series, is the theory of quaternionic slice-regular functions.This function theory is based on the particular complex-slice structure of the quaternionic space H.In Section 2 we briefly recall the definitions and properties of slice functions and slice-regular functions that are used in the subsequent sections.
A Cauchy-type integral formula with slice-regular kernel was proved in [3].In this formula integration is performed over the boundary of a two-dimensional domain having an axial symmetry with respect to the real axis.A volume Cauchytype formula, where integration is made over the boundary of an open axially symmetric domain, was proved in [14] in the more general context of real alternative *-algebras.That result extended to every slice-regular function a similar one obtained by Cullen [5] on the quaternions.
The aim of the present paper is to prove a Cauchy-type integral formula (Theorem 12) for slice-regular functions where the integration is performed on the boundary of a not necessarily axially symmetric open set.As a corollary, we obtain a local Cauchy-type integral formula for slice-regular functions (Corollary 14).The integral kernel is not slice-regular, but it is universal, i.e., not depending on the domain.An unavoidable aspect of the formula is the appearing, along with the boundary values of the slice-regular function, of the values of a complementary function, namely, the slice derivative of a slice primitive of the function.
The Cauchy-type formula is proved using facts from both the above-mentioned quaternionic functions theories.We show (Theorem 8) that every slice-regular function can be expressed as a combination of two axially monogenic functions, to which the Fueter's version of Cauchy's integral formula apply.We recall that an axially monogenic function is a monogenic slice function, i.e., a slice function in the kernel of the operator .We also give a new proof (Theorem 3) of the surjectivity of the Laplacian mapping from the space of slice-regular functions to that of axially monogenic functions (see [4]).This result has a role in proving the uniqueness of the above-mentioned decomposition in terms of axially monogenic functions.

P
The slice function theory of one quaternionic variable [10,11] is based on the slice decomposition of the quaternionic space H.For each element in the sphere of quaternionic imaginary units [11] on the open set Ω if, for each ∈ S, the restriction |Ω∩C : Ω ∩ C → H is holomorphic with respect to the complex structure defined by left multiplication by .We refer the reader to [9] and the references therein for more results in this function theory.
Another approach to slice regularity was introduced in [12,13] (see also [17] for recent developments), making use of the concept of slice functions.We briefly recall their definition and some operations on them.Given a set ⊆ C, invariant with respect to complex conjugation, a function : → H ⊗ C that satisfies ( ) = ( ) for every ∈ (the conjugation in H ⊗ C is induced by complex conjugation in the second factor) is called a stem function on , a concept already present in seminal works of Fueter [7] and Cullen [5].
Let Φ : C → C be the canonical isomorphism that maps The slice function is called slice-regular if is holomorphic w.r.t. the complex structure induced on H ⊗ C by the second factor.If a domain Ω in H is axially symmetric and intersects the real axis, then this definition of slice regularity is equivalent to the one proposed by Gentili and Struppa [11].We will denote by SR (Ω) the right quaternionic module of slice-regular functions on Ω and by S 1 (Ω) the class of slice functions induced by stem functions of the class 1 on Ω.
The slice product of two slice functions = I ( ), = I ( ) on Ω = Ω is defined by means of the pointwise product of the stem functions: ).
The function = I ( ) is called slice-preserving if the H-components 1 and 2 of the stem function are real-valued.This is equivalent to the condition ( ) = ( ) for every ∈ Ω.If is slice-preserving, then • coincides with the pointwise product of and .If , are slice-regular on Ω, then also their slice product • is slice-regular on Ω.
The slice derivatives , of a slice functions = I ( ) are defined by means of the Cauchy-Riemann operators applied to the inducing stem function : It follows that is slice-regular if and only if = 0 and if is slice-regular on Ω then also is slice-regular on Ω.Moreover, the slice derivatives satisfy the Leibniz product rule w.r.t. the slice product.If = , we will say that is a slice primitive of .

A -
Let denote the Cauchy-Riemann-Fueter operator Given an axially symmetric domain Ω of H, let AM (Ω) be the class of axially monogenic functions, i.e., of monogenic slice functions on Ω: There is a result, usually called Fueter's Theorem [7], which in its generalised form can be seen as a bridge between the class of slice-regular functions and the one of monogenic functions.We report this result from [20,Prop. 3.61, Cor.3.6.2and Thm.3.6.3],where some formulas linking the spherical derivative of slice functions with the Cauchy-Riemann-Fueter operator were proved.Let Δ denote the Laplacian operator in R 4 .In the statement of this result we will also need the global differential operator introduced in [15].For slice functions ∈ 1 (Ω), it holds Theorem 1 ([20]).Let Ω be an axially symmetric domain in H. Let : Ω → H be a slice function of class C 1 (Ω).Then In the following we will use also the following result from [19], which shows that the Laplacian of a slice-regular function can be expressed by first order derivatives.

Lemma 2 ([19] Lemma 23). If ∈ SR (Ω), then for every ∈ Ω it holds
It is known that the Laplacian (also called Fueter mapping in this context) maps the space SR (Ω) onto AM (Ω) surjectively (see [4] and [6] and references therein).If Ω is connected, the inverse image in SR (Ω) of a function ∈ AM (Ω) under Δ is unique up to a quaternionic affine function + (see [19, Lemma 23(a)]).Using Lemma 2, we now give an elementary proof of the surjectivity of Δ under suitable topological hypotheses.

Theorem 3. Let Ω = Ω be an axially symmetric open set in H. Assume that every connected component of is simply connected. Then the Laplacian
which implies = −1 2 for each = 0, . . ., 3. Therefore = 3
Our aim is to find a slice-preserving ∈ SR (Ω) such that Im( ) = 2( ′ − ), since then Lemma 2 gives Δ = .As above, the condition = 0 is equivalent to Since any axially symmetric open set is union of a family of slice domains or product domains, we can assume that Ω is a domain of one of these types.
In view of Theorem 1(2c), a right inverse Δ : AM (Ω) → SR (Ω) of Δ can be defined on axially monogenic polynomials as in [19,Prop. 24].It associates for every ∈ Z the slice-regular monomial − 1 4 +2 to the rational function P , defined by The functions P are axially monogenic and then harmonic.They are slicepreserving functions (not slice-regular for ≠ 0) on H.They were computed already by Fueter in [7] (see formula (12) on p. 316) and afterwords used by many authors.For ≥ 0 the functions P are polynomials of degree in 0 , 1 , 2 , 3 .For < 0 they are homogeneous functions on H \ {0}, still of degree .
For ≥ 0, the polynomials P are related to the spherical derivatives Z ( ) := ( +1 ) ′ of quaternionic powers.These functions are harmonic homogeneous polynomials of degree in the four real variables 0 , 1 , 2 , 3 .The polynomials Z are called zonal harmonic polynomials with pole 1, since they have an axial symmetry with respect to the real axis (see [1,Ch.5] and [20,21]).
We are now able to write a decomposition of quaternionic polynomials in terms of a pair of axially monogenic polynomials.for every ∈ H.
Proof.It was proved in [20,Cor. 6.7] that for every ∈ N, it holds Here the pointwise and slice products coincide since is a slice-preserving function.
Taking slice derivatives and using Leibniz property, we get

Corollary 5. Let ∈ H[ ] have degree ≥ 1.
There exist two axially monogenic polynomials 1 , 2 , of degrees and − 1 respectively, such that . The thesis follows immediately from Proposition 4 by setting 1 ( ) Corollary 5 can be generalized to every slice-regular function.Before doing it, we show one more general property of axially monogenic functions.Lemma 6.Let Ω be as in Theorem 3. If both and are axially monogenic on Ω, then is identically zero.
Proof.Let D be the first order linear operator defined for any slice function ℎ of class 2 (Ω) by ( 5) We claim that if ∈ SR (Ω), then D ( ′ ) = ′ .Indeed, it holds where we used Theorem 1(2c).Then Now we extend Corollary 5 to every slice-regular function.Since every monogenic function is harmonic, the result we obtain can be seen as a refinement of the Almansi type decomposition proved in [21,Theorem 4] Theorem 8. Let be slice-regular on an axially symmetric open set Ω. Assume that Ω = Ω , and that every connected component of is simply connected.Then there exist two uniquely determined axially monogenic functions 1 and 2 , such that The functions 1 and 2 can be computed from a slice-regular primitive of .If = on Ω, with ∈ SR (Ω), then Moreover, is slice-preserving if and only if 1 and 2 are slice-preserving.
Proof.Since any axially symmetric open set is union of a family of slice domains or product domains, we can assume that Ω is a domain of one of these types.Assume that there exists ∈ SR (Ω) such that = on Ω.Then, using Theorem 1(2c), with Δ( Log( )) and Δ(Log( )) axially monogenic on H \ { ∈ R | ≤ 0}.The function Δ(Log( )) coincides up to a multiplicative constant with the -primitive ( ) of the Cauchy-Fueter kernel ( ) defined in [22, (5.7)].Here is the conjugated Cauchy-Riemann-Fueter operator.
The statement of Theorem 8 has a converse.In the following proposition we give a differential condition on the pair ( 1 , 2 ) that ensures the slice-regularity of 1 − 2 .Proposition 11.Let 1 and 2 be two axially monogenic functions on an axially symmetric set Ω.If is defined as ( ) = 1 ( ) − 2 ( ) for every ∈ Ω, then is slice-regular on Ω if and only if it holds Proof.Clearly is a slice function on Ω.From Theorem 1(1), is slice-regular if and only if Since again from Theorem 1(1) it holds ( 1 ) ′ = 1 , is slice-regular if and only if (6) holds.

A C -
Let Ω = Ω be an axially symmetric open set.Assume that every connected component of is simply connected.Given ∈ SR (Ω), we know from the proof of Theorem 8 that it is possible to find a slice-regular primitive ∈ SR (Ω).Let ZH (Ω) denote the right H-module of zonal harmonic functions with pole 1 on Ω, i.e., the quaternionic harmonic functions ℎ on Ω, such that ℎ • = ℎ for every orthogonal transformation of H ≃ R 4 that fixes 1.
Using Theorem 1(2a), we can define a linear operator that maps to the spherical derivative ′ = − of any slice-regular primitive of .This map is well-defined since if = ˜ , with , ˜ slice-regular, then − ˜ is locally constant, and then ′ − ˜ ′ = ( ˜ − ) = 0.It holds S ( ) = for any constant ∈ H and S ( ) = ( + 1) −1 Z for every ∈ N. If is slice-preserving, then S is real-valued.Moreover, S is injective, since ) be the Cauchy-Fueter kernel and let be the 3-form with quaternionic coefficients defined as in [22, (2.28)]: We are now in the position to prove a Cauchy-type integral formula for sliceregular functions on Ω where the integration is performed on the boundary of a not necessarily axially symmetric open subset of Ω.As a consequence, we are able to prove a local Cauchy-type integral formula for slice-regular functions.From the Cauchy-Fueter integral formula for monogenic, i.e., Fueter-regular, functions (see [7] for the original proof and also [22], where the formula was proved in its full generality), we get for every ∈ .Now we transform the two integrals using Lemma 2. We obtain where we used the equality 2 = (2 Im( )) −1 (− ′ + ).Using ( 8), ( 9) and ′ = S , we get and the integral formula is proved.
Remark 13.The 3-forms 1 , 2 are real-analytic for ( , ) ∈ H × (H \ R) with ≠ .If the closure of does not intersect the real axis, then the two integrals with kernels 1 and 2 converge also separately and the integral formula (7) can be written as a sum of two integrals The same holds also when the boundary is sufficiently smooth and the intersection ∩ R ≠ ∅ is transversal.Proof.If ⊆ Ω is an open ball centred in ˜ , we can take as the symmetric completion = ∪ ∈ S of , where S = + S for = + ∈ .Then = Ω with ⊂ C having simply connected components ( is a complex disc, or a pair of disjoint conjugate discs, or the union of two intersecting conjugate discs).Since ∈ SR ( ), the thesis follows from Theorem 12.
The proof of the preceding corollary shows that the operator S can be defined on SR (Ω) for every axially symmetric domain Ω = Ω , without further assumptions on .For every pair of slice-regular primitives ∈ SR ( ), ˜ ∈ SR ( ) of , it holds ′ = ˜ ′ on the intersection ∩ .This common value defines S .
In the case of quaternionic polynomials, from the equality S ( ) = ( + 1) −1 Z we obtain a more explicit form of the integral formula: since the function 1 ( ) = 0 + 1 2 and any constant function are Fueter-regular.Observe that if is outside the closure of the integral formula gives a zero value.
Remark 17.The integral formula (7) has an interpretation which shows its analogy with the classical Cauchy formula in complex analysis.If is a complex holomorphic function on a neighbourhood of , and is a holomorphic primitive of on , then the Cauchy's integral formula can be written as where ( , ) = (2 ) −1 ( − ) −1 is the Cauchy kernel and the real differential of at acts on a complex 1-form as multiplication by ′ ( ) = ( ).If is a slice-regular primitive of on Ω, its real differential at ∈ Ω ∩ C is the left C -linear map given by = ( ) • + ′ ( ) • ⊥ , where : H → H denotes the orthogonal projection onto the real vector subspace C , ⊥ = id H − and is the operator of right multiplication by ∈ H (see [8, §3] and [16,Cor. 3.2]).Let : H 2 → H be the left H-linear extension of defined by ( , ) := ( ) ( ) + ′ ( ) ( ).

∫
( 1 ( , ), 2 ( , )) for every ∈ , where the action of the operator is extended linearly to quaternionic 3-forms by making it act on the coefficients of the forms.
(2)If : Ω → H is slice-regular, then it holds: (a) The four real components of ′ are harmonic on Ω.(b) The following generalization of Fueter's Theorem holds: