Hypercomplex analysis and applications

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Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference.

Hypercomplex Analysis and Applications Irene Sabadini Frank Sommen Editors

Editors Irene Sabadini Dipartimento di Matematica Politecnico di Milano Via Bonardi, 9 20133 Milano Italy [email protected]

Frank Sommen Department of Mathematical Analysis Ghent University Galglaan 2 9000 Gent Belgium [email protected]

2010 Mathematical Subject Classification: primary: 30G35; secondary: 32A26, 30C15, 33C80, 58A10, 58C50, 58G03, 76W05

ISBN 978-3-0346-0245-7 DOI 10.1007/978-3-0346-0246-4

e-ISBN 978-3-0346-0246-4

© Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the right of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii C. Bisi and G. Gentili On the Geometry of the Quaternionic Unit Disc . . . . . . . . . . . . . . . . . . . . . . . . . 1 F. Colombo and I. Sabadini Bounded Perturbations of the Resolvent Operators Associated to the F -Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 K. Coulembier, H. De Bie and F. Sommen Harmonic and Monogenic Functions in Superspace . . . . . . . . . . . . . . . . . . . . . . 29 S.-L. Eriksson and H. Orelma A Hyperbolic Interpretation of Cauchy-Type Kernels in Hyperbolic Function Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 M. Ferreira Gyrogroups in Projective Hyperbolic Clifford Analysis . . . . . . . . . . . . . . . . . . 61 P. Franek Invariant Operators of First Order Generalizing the Dirac Operator in 2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 G. Gentili and C. Stoppato The Zero Sets of Slice Regular Functions and the Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 R. Ghiloni and A. Perotti A New Approach to Slice Regularity on Real Algebras . . . . . . . . . . . . . . . . . 109 R.S. Kraußhar On the Incompressible Viscous Stationary MHD Equations and Explicit Solution Formulas for Some Three-dimensional Radially Symmetric Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 R. L´ aviˇcka The Fischer Decomposition for the H-action and Its Applications . . . . . . 139 S. Li and M. Fei Bochner’s Formulae for Dunkl-Harmonics and Dunkl-Monogenics . . . . . . 149 M. Libine An Invitation to Split Quaternionic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .161

vi

Contents

M.E. Luna-Elizarrar´ as, M.A. Mac´ıas-Cede˜ no and M. Shapiro On the Hyperderivatives of Moisil–Théodoresco Hyperholomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 M. Martin Deconstructing Dirac Operators. II: Integral Representation Formulas . 195 H. Orelma and F. Sommen A Differential Form Approach to Dirac Operators on Surfaces . . . . . . . . . 213 P. Somberg Killing Tensor Spinor Forms and Their Application in Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 V. Tuˇcek Construction of Conformally Invariant Differential Operators . . . . . . . . . . 249 D.C. Struppa, A. Vajiac and M.B. Vajiac Remarks on Holomorphicity in Three Settings: Complex, Quaternionic, and Bicomplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 F. Vlacci The Gauss-Lucas Theorem for Regular Quaternionic Polynomials . . . . . . 275

Preface This volume contains some papers written by the participants to the Session “Clifford and Quaternionic Analysis” of the 7th ISAAC Conference (hosted at the Imperial College, London, UK, July 2009). The session proved to be particularly lively, as it hosted more than 35 speakers, and showed how the topic of hypercomplex analysis is fertile. The contents of the contributions cover several different aspects of hypercomplex analysis, ranging from Clifford analysis on superspace to the study of Dirac operators on surfaces, Dunkl monogenic functions, and applications, for example, to operator theory. The contributions represent the most recent achievements in the area and have been refereed. The editors are grateful to the contributors to this volume and to the referees, for their painstaking and careful work. The editors thank the Imperial College in London for hosting the Conference and Professor Michael Ruzhansky, President of the ISAAC and Chairman of the local organising committee.

August 2010,

Irene Sabadini Frank Sommen

On the Geometry of the Quaternionic Unit Disc Cinzia Bisi and Graziano Gentili Abstract. In the space H of quaternions, the natural invariant geometry of the open unit disc ΔH , diffeomorphic to the open half-space H+ via a Cayleytype transformation, has been investigated extensively. This was accomplished by constructing, in a natural geometrical manner, the quaternionic Poincaré distance on ΔH (and H+ ). The open unit disc ΔH also inherits the complex Kobayashi distance when viewed as the open unit ball of C2 ∼ = C + Cj ∼ = H. In this paper we give an original, very simple proof of the fact that there exists no isometry between the quaternionic Poincaré distance of ΔH and the Kobayashi distance inherited by ΔH as a domain of C2 . This is in accordance with the well-known consequence of the classification theorem for the non-compact, rank 1, symmetric spaces. Mathematics Subject Classification (2010). Primary 30G35; Secondary 30C20, 30F45, 53C35. Keywords. Functions of hypercomplex variables, quaternionic M¨ obius transformations, quaternionic Poincaré distance, Kobayashi distance, symmetric spaces.

1. Introduction Let H be the skew field of quaternions. Let ΔH = {q ∈ H : |q| < 1} be the open unit disc and let H+ = {q ∈ H : e(q) > 0} be the half-space, diffeomorphic via a Cayley-type transformation. The groups of Möbius transformations of ΔH and of H+ are the groups of all quaternionic, fractional, linear transformations which leave ΔH and H+ invariant, respectively. These groups are used in [4] to find a direct approach to a geometric definition of the analogue of the Poincaré distance (i.e., the real, hyperbolic distance) and differential metric in the quaternionic setting. These distances and differential metrics allowed the construction and study of the This work was partially supported by Progetto MIUR di Rilevante Interesse Nazionale Proprieta ` geometriche delle variet` a reali e complesse and by GNSAGA – INDAM.

I. Sabadini and F. Sommen (eds.), Hypercomplex Analysis and Applications, Trends in Mathematics, DOI 10.1007/978-3-0346-0246-4_1, © Springer Basel AG 2011

1

2

C. Bisi and G. Gentili

invariant quaternionic geometry of the classical hyperbolic domains ΔH and H+ of H. With respect to the standard basis {1, i, j, k} of H, the identification H ∼ = R + Ri + Rj + Rk ∼ = (R + Ri) + (R + Ri)j ∼ = C + Cj leads to the identification ΔH ∼ = ΔC2 between the open unit disc of H and the open unit ball ΔC2 of C2 . Since the ball ΔC2 is naturally endowed with the Kobayashi distance and (metric), see, e.g., [11, 22], it is natural to ask which is the relationship between the quaternionic Poincaré distance and the Kobayashi distance on ΔH ∼ = ΔC2 . By means of the geometrical approach adopted in this paper, we are able to give an original, very simple proof of a deep result that is classically obtained as a consequence of the classification of non-compact, rank 1, symmetric spaces (see, e.g., [9], [15]). This result states that: Theorem 1.1. There exists no isometry between the quaternionic Poincaré distance and the Kobayashi distance of ΔH ∼ = ΔC2 . Notations and terminology are those used in [4]. The elements of H will be denoted by q = x0 +ix1 +jx2 +kx3 , where the xl are real, and i, j, k, are imaginary units (i.e. their square equals −1) such that ij = −ji = k, jk = −kj = i, and ki = −ik = j. We will denote by S3H the sphere of quaternions of unitary modulus {q ∈ H : |q| = 1} and by S the unit sphere of purely imaginary quaternions, i.e., S = {q = ix1 +jx2 +kx3 : x21 +x22 +x23 = 1}. Notice that if I ∈ S, then I 2 = −1; for this reason the elements of S are called imaginary units. We will also use the fact that for any non-real quaternion q ∈ H\R, there exist, and are unique, x, y ∈ R with y > 0, and I ∈ S such that q = x + yI. The paper is organized as follows. In Section 2 we survey and illustrate the construction and the main features of the quaternionic Poincaré distance and differential metric of ΔH and describe the geometrical character of the group of quaternionic Möbius transformations. In Section 3 we briefly recall the definitions of the Kobayashi distance and differential metric, and the structure theorem for complex M¨ obius transformations in the case of the unit ball of Cn . We then conclude by giving the announced new proof of Theorem 1.1.

2. Basics of quaternionic invariant geometry In [4] the authors found an original geometrical approach to the study of the invariant geometry of the unit open disc of H. The study of the relevant groups of matrices with quaternionic entries is based on the definition of the Dieudonné determinant of a quaternionic 2 × 2 matrix, which is recovered in a very natural way in the same paper [4]: a b is a 2 × 2 matrix with quaternionic entries, then Definition 2.1. If A = c d the (Dieudonné) determinant of A is defined to be the non-negative real number detH (A) = |a|2 |d|2 + |c|2 |b|2 − 2e(cabd). (2.1)

On the Geometry of the Quaternionic Unit Disc

3

This definition (luckily!) agrees with the one given in [3, 8, 12, 21], and allows us to set G = {g(q) = (aq + b)(cq + d)−1 : a, b, c, d ∈ H, g invertible}, GL(2, H) = {A 2 × 2

matrix with quaternionic entries : detH (A) = 0}

and to present the following result for H, already established in [31]: Theorem 2.2. The set G of all quaternionic, fractional, linear transformations is a group with respect to composition. The map a b Φ:A= → LA (q) = (aq + b) · (cq + d)−1 (2.2) c d is a group homomorphism of GL(2, H) onto G whose kernel is the center of GL(2, H), that is the subgroup t 0 : t ∈ R\{0} . 0 t For a detailed, modern proof of this theorem, and for bibliographical references, we refer the reader to [4]. The structure-theorem of the complex, fractional, linear transformations can be extended to the quaternionic environment: Proposition 2.3. The group G is generated by all the similarities, L(q) = aq + b (a, b ∈ H, a = 0) and the inversion R(q) = q −1 . Moreover, all the elements of G turn out to be conformal. The basic geometrical ingredient to construct the quaternionic Poincaré distance (and metric) is the quaternionic cross-ratio: indeed the generalizations of the cross-ratio to higher dimensions in Rn play a crucial role in conformal geometry. In fact L. Ahlfors, while studying the conformal structure of Rn , has given in [2] three different definitions of the cross-ratio of four points of Rn . The one that we adopt here is the one given in [4], that specializes to the quaternionic case the definition given by C. Cao and P.L. Waterman in [5], and that is new with respect to the ones given by Ahlfors. This definition of cross-ratio has the peculiar feature that the quaternionic, fractional, linear transformations act on it transforming its value by (quaternionic) conjugation. In particular it turns out: Proposition 2.4. Let CR(q1 , q2 , q3 , q4 ) := (q1 − q3 )(q1 − q4 )−1 (q2 − q4 )(q2 − q3 )−1 be the cross-ratio of the four quaternions q1 , q2 , q3 , q4 . When the cross ratio of four quaternions is real, then it is invariant under the action of all quaternionic, fractional, linear transformations. The above result has a great deal of interest in view of the fact (already proven in [5] in Clifford algebra setting) that Proposition 2.5. Four pairwise distinct points q1 ,q2 ,q3 ,q4 ∈ H lie on a same (onedimensional) circle or straight line if, and only if, their cross-ratio is real.

4


Adopting the point of view due to A.F. M¨ obius one defines (for i = 3, 2, 1, respectively) the families Fi = Si ∪ Pi , where Si is the family of all i-(real) dimensional spheres and Pi is the family of all i-(real) dimensional affine subspaces of H. With an approach which is different from the one used by Wilker in [31], the authors showed in [4] that Theorem 2.6. The group G of all quaternionic, fractional, linear transformations maps elements of Fi onto elements of Fi , for i = 3, 2, 1. The above result, and the use of the point of view of C.L. Siegel for the homologous problem in the complex case (see [28], [7]), led to the following, genuine geometrical approach to the definition of the quaternionic Poincaré distance on ΔH (often simply called Poincaré distance). Set the non-Euclidean line through two points q1 , q2 ∈ ΔH to be the unique circle, or diameter, containing the two points and intersecting ∂ΔH orthogonally in the two ends q3 , q4 , and give the following Definition 2.7. The (quaternionic) Poincaré distance of ΔH is defined as δΔH (q1 , q2 ) =

1 log(CR(q1 , q2 , q3 , q4 )) 2

(2.3)

where the q3 and q4 are the two ends of the non-Euclidean line through q1 and q2 , and the four points are arranged cyclically on the non-Euclidean line through q1 and q2 . It is very easy to see that Proposition 2.8. On each complex plane LI = R + IR (for any imaginary unit I ∈ S) the quaternionic Poincaré distance coincides with the classical Poincaré distance of ΔI = ΔH ∩ LI . The group of M¨ obius transformations is defined as the subgroup M of G whose elements map ΔH onto itself. It is natural to study how the quaternionic Poincaré distance behaves under the action of the elements of M. To do this, we need to know the structure of the group of Möbius transformations of ΔH , whose study is performed, for example, in [6], and completed in [4], in terms of the (classical) group Sp(1, 1). The group Sp(1, 1) is defined (see, e.g., [13]) as Sp(1, 1) = A ∈ GL(2, H) : t AHA = H (2.4) 1 0 , and it can be written equivalently as (see, e.g., [6]) where H = 0 −1 a b 2 2 : |a| = |d|, |b| = |c|, |a| − |c| = 1, ab = cd, ac = bd . Sp(1, 1) = c d The use of the group Sp(1, 1) allowed us to rephrase and complete a result of [6] as follows:


5

Theorem 2.9. The quaternionic, fractional, linear transformation defined by the formula g(q) = (aq + b)(cq + d)−1 is a M¨ obius transformation of ΔH if and only if a b ∈ Sp(1, 1). Moreover the map c d A=

a c

b d

φ : Sp(1, 1) → M → LA (q) = (aq + b) · (cq + d)−1

(2.5)

is a group homomorphism whose kernel is the center of Sp(1, 1), that is the subgroup 1 0 . ± 0 1 For the purposes of this paper, we need the following characterization of the quaternionic M¨ obius transformations, which closely resembles the classical representation of the complex M¨ obius transformations: Theorem 2.10. Each quaternionic M¨ obius transformation of the form g(q) = (aq + b) · (cq + d)−1 ∈ M can be written uniquely as: g(q) = α(q − q0 )(1 − q0 q)−1 β −1 a d ∈ ∂ΔH , β = ∈ ∂ΔH . where q0 = −a−1 b ∈ ΔH and where α = |a| |d|

(2.6)

A detailed proof of the structure result given in Theorem 2.10 can be found in [4]. This structure result, that was also stated without proof in [14], is different from the one given in a more general setting in [2]. Remark 2.11. Thanks to Lemma 2.3, the M¨ obius transformations are conformal (see also [29], [19], [23], [24], [26], [25], [27], [10]). Since any Möbius transformation maps S3H onto itself, the Möbius transformations map non-Euclidean lines of ΔH onto non-Euclidean lines of ΔH in view of Theorem 2.6. Combining Propositions 2.4, 2.5 and Remark 2.11, one gets the following result, whose statement is implicit in the work of Wilker [31] (see also [16]), and urges a comparison with the complex case. Proposition 2.12. The Poincaré distance of ΔH is invariant under the action of the group of all M¨ obius transformations M and of the map q → q. It is now possible to mimic the definition of the classical, complex Poincaré differential metric to set the length of the vector τ ∈ H for the (quaternionic) Poincaré metric at q ∈ ΔH to be the number |τ | . (2.7)

τ q = 1 − |q|2 The following results, see [4], will be used in the sequel:

6


Theorem 2.13. All the elements of the group M of M¨ obius transformations of ΔH , as well as the map q → q, leave the quaternionic Poincaré differential metric invariant. Proposition 2.14. The quaternionic Poincaré distance δΔH of the unit disc ΔH is the integrated distance of the quaternionic Poincaré differential metric of ΔH .

3. Poincaré and Kobayashi distances on the quaternionic unit disc In this section, we will first of all briefly recall the classical definitions of Kobayashi distance and differential metric for the open unit ball ΔCn of the space Cn . To help the reader to follow the proof of the main result, we will also recall the structure of the group of all complex M¨ obius transformations of ΔCn . Thanks to a result due to L. Lempert, [18], the definition of the Kobayashi distance for a convex set of Cn can be given in the following simple fashion (for the classical general definition see, e.g., [11, 22]). Definition 3.1. Let D be the open, unit disc of C and δD the Poincaré distance of D. The Kobayashi distance between any two points z1 and z2 of the open unit ball ΔCn ⊂ Cn is defined as kΔCn (z1 , z2 ) =

inf {δD (ζ1 , ζ2 ) | ∃f : D → ΔCn holomorphic, with f (ζ1 ) = z1 , f (ζ2 ) = z2 }.

ζ1 ,ζ2 ∈D

The Kobayashi differential metric at a point z ∈ ΔCn is defined, for all w ∈ Cn , by γΔCn (z; w) = inf { ζ∈D

|τ | | ∃f : D → ΔCn holomorphic, with f (ζ) = z, dfζ (τ ) = w}. 1 − |ζ|2

It turns out that, to explicitly compute the distance kΔCn (z1 , z2 ), it is enough to consider the complex line Lz1 ,z2 of Cn that contains the two points z1 , z2 of ΔCn , intersect it with the ball ΔCn and measure the Poincaré distance between z1 and z2 in the complex disc Lz1 ,z2 ∩ ΔCn . This observation will be useful in the sequel. Both the Kobayashi distance and the Kobayashi differential metric are invariant, in the obvious sense, under the action of the group of Möbius transformations of ΔCn , whose structure we are going to describe (see, e.g., [22]). Denote by ·, · the classical Hermitian inner product of Cn . For z0 ∈ ΔCn , let Pz0 be the orthogonal projection of Cn onto the subspace [z0 ] spanned by z0 , i.e., let P0 (z) = 0 if

z0 = 0,

and Pz0 (z) =

z , z0 z0

z0 , z0

if z0 = 0.

(3.1)


7

Set Qz0 = I − Pz0 to be the projection of Cn onto the orthogonal complement of 1 [z0 ]. For sz0 = (1 − |z0 |2 ) 2 define the map ϕz0 : Cn \ {z : z , z0 = 1} → Cn as ϕz0 (z) =

z0 − Pz0 (z) − sz0 Qz0 (z) . 1 − z , z0

(3.2)

Since {z : z , z0 = 1} ∩ ΔCn = ∅, the map ϕz0 defines a holomorphic map from ΔCn to Cn , which turns out to be a holomorphic automorphism of ΔCn . Now, if U(n) = {U ∈ GL(n, C) : U t U = I} denotes the unitary group of Cn , then for U ∈ U(n) and z0 ∈ ΔCn we put MU,z0 (z) = U ϕz0 (z) = U

z0 − Pz0 (z) − sz0 Qz0 (z) . 1 − z , z0

(3.3)

Since the elements of U(n) are C-linear automorphisms of ΔCn , the set M = {MU,z0 : U ∈ U(n), z0 ∈ ΔCn }

(3.4)

obius consists of holomorphic automorphisms of ΔCn , and is called the set of M¨ transformations of ΔCn . The transitivity of the group M and a direct application of the n-dimensional Schwarz Lemma lead to the following well known result (see, e.g., [22]): Theorem 3.2. The group Aut(ΔCn ) of all holomorphic automorphisms of the open unit ball ΔCn of Cn coincides with the group M of all M¨ obius transformations of ΔCn . Moreover, the elements of M are isometries for the Kobayashi distance kΔCn and for the Kobayashi differential metric γΔCn . One of the important properties of the Kobayashi distance is the following: Proposition 3.3. The Kobayashi distance kΔCn of the unit ball ΔCn is the integrated distance of the Kobayashi differential metric γΔCn . ∼ C + Cj, which yields the idenLet us now consider the isomorphism H = tification ΔH ∼ = ΔC2 =: Δ between the open unit disc of H and the open unit ball of C2 . Now that we have given a direct, geometrical definition of the Poincaré distance δΔ of ΔH , the natural question arises to find a direct proof of the fact that there exists no isometry between δΔ and the Kobayashi distance kΔ . To find such a proof we begin with the following remark, whose justification can be found, for example, in [11, 22]: Remark 3.4. Both the Poincaré distance δΔ and the Kobayashi distance kΔ have the property that δΔ (0, q) = kΔ (0, q) = δD (0, |q|) for all q ∈ Δ. Moreover, the Poincaré differential metric of ΔH defined in (2.7) and the Kobayashi differential metric γΔ of ΔC2 both coincide with the Euclidean differential metric at the origin of the open unit disc of H. With this in mind, we will prove the following technical result:

8


Lemma 3.5. If there exists an isometry f : Δ → Δ between the Kobayashi distance kΔ and the Poincaré distance δΔ , then the identity function of Δ is an isometry between kΔ and δΔ , and hence kΔ ≡ δΔ . Proof. If f is the identity function of Δ, then there is nothing to prove. Otherwise, let M ∈ M be a quaternionic, M¨ obius transformation of Δ such that M (f (0)) = 0. By Proposition 2.12, the function M ◦ f is an isometry between kΔ and δΔ that fixes 0. If we identify H with R4 , then Remark 3.4, together with Propositions 2.14 and 3.3, yield that the real differential d(M ◦ f )0 is an orthogonal matrix. Now the geometrical definition of the Poincaré distance δΔ given in Definition 2.7 makes it clear that any orthogonal transformation of Δ is a δΔ -isometry together with its inverse. Therefore the function F = d(M ◦ f )−1 0 ◦ M ◦ f : Δ → Δ is an isometry between kΔ and δΔ , whose differential dF0 is the identity function. Since the geodesic curves of both kΔ and δΔ passing through 0 are the diameters of Δ, then, in view of Remark 3.4, the isometry F itself is the identity map. Given any two points q1 , q2 ∈ Δ there exist a quaternionic M¨ obius transformation M of ΔH and a complex Möbius transformation φ of ΔC2 such that M (q1 ) = 0 = φ(q1 ). Now M and φ leave invariant, respectively, δΔ and kΔ , and we want to investigate the relation between |M (q2 )| and |φ(q2 )|. Consider q1 = α = α + 0j and q2 = βj = 0 + βj with α, β ∈ C. Using Theorem 2.10, choose M ∈ M to be the quaternionic Möbius transformation of ΔH M (q) = (q − α)(1 − αq)−1 and consider the complex Möbius transformation (of ΔC2 ) φ(α,0) ∈ M defined by φ(α,0) (z, w) =

(α, 0) − (z, 0) − (1 − |α|2 )1/2 (0, w) . 1 − zα

We get |M (βj)|2 = |(βj − α)(1 − αβj)−1 |2 =

(|β|2 + |α|2 ) (1 + |α|2 |β|2 )

(3.5)

and |φ(α,0) (0, β)|2 = |(α, −(1 − |α|2 )1/2 β)|2 = |α|2 + (1 − |α|2 )|β|2 .

(3.6)

Since the equality among (3.5) and (3.6) does not hold in general (due to the identity principle for real polynomials), remark 3.4 leads to the following. Lemma 3.6. The identity map of Δ is not an isometry between the Kobayashi distance kΔ and the Poincaré distance δΔ . In particular kΔ and δΔ do not coincide. As a direct corollary of the last two lemmas, and in accordance with a classical consequence of the classification of non compact, rank 1, symmetric spaces (see, e.g., [9], [15]), we can now state the following. Theorem 3.7. There exists no isometry between the quaternionic Poincaré distance and the Kobayashi distance of ΔH ∼ = ΔC2 .


9

It is very easy to prove that, for any I ∈ S, the Poincaré distance and the Kobayashi distance coincide on the subsets of Δ of type ΔI = Δ ∩ LI , where LI = {x + yI : x, y ∈ R}. Thanks to our geometrical approach, it is also direct to verify that all the real, sectional curvatures of the quaternionic, Poincaré differential metric at 0 (and hence by homogeneity at all points of Δ) coincide with a same negative constant. It is known that this is not the case for the Kobayashi differential metric of ΔC2 (see, e.g., [17]), for which only the holomorphic, sectional curvatures at all points coincide with a same negative constant.

References [1] L.V. Ahlfors, M¨ obius Transformations and Clifford Numbers. Differential Geometry and Complex Analysis, H.E. Rauch memorial volume, Springer-Verlag, Berlin, 1985, 65–73. [2] L.V. Ahlfors, Cross Ratios and Schwarzian Derivatives in Rn . In: Complex Analysis, Edited by J.Hersch, A. Huber, Birk¨ auser Verlag, Basel 1988, 1–15. [3] H. Aslaksen, Quaternionic Determinants. Math. Intelligencer, 18(3) (1996), 57–65. [4] C. Bisi, G. Gentili, M¨ obius Transformations and the Poincaré Distance in the Quaternionic Setting. Indiana Math. Journal, 59(6) (2009), 2729–2764. [5] C. Cao, P.L. Waterman, Conjugacy Invariants of M¨ obius Groups. Quasiconformal mappings and analysis (Ann Arbor, MI, 1995), 109–139, Springer, New York, 1998. [6] W. Cao, J.R. Parker, X. Wang, On the Classification of Quaternionic M¨ obius Transformations. Math. Proc. Camb. Phil. Soc., 137 (2004), 349–361. [7] H. Cartan, Cours de Maitrise des Sciences Mathématiques. 1969–1970. [8] N. Cohen, S. De Leo, The Quaternionic Determinant. The Electronic Journal of Linear Algebra, 7 (2000), 100–111. [9] B. Doubrovin, S. Novikov, A. Fomenko, Geometrie Contemporaine. Methodes et Applications. Deuxieme partie. Geometrie et Topologie des varietes. Mir, Moscow, 1985, pp. 371. [10] J. Elstrodt, F. Grunewald, J. Mennicke, Vahlen’s group of Clifford matrices and spin-groups. Math. Z., 196 (3) (1987), 369–390. [11] T. Franzoni, E. Vesentini, Holomorphic Maps and Invariant Distances. Notas de Matematica [Mathematical Notes], 69 North-Holland Publishing Co., Amsterdam, New York, 1980, pp. 226. [12] I. Gelfand, V. Retakh, R. L. Wilson, Quaternionic Quasideterminants and Determinants. Lie groups and symmetric spaces, 111–123, Amer. Math. Soc. Transl. Ser. 2, 210, Amer. Math. Soc., Providence, RI, 2003. [13] V. V. Gorbatsevich, A. L. Onishchik, E. B. Vinberg, Structure of Lie Groups and Lie Algebras. Translated from the Russian by V. Minachin. [Lie groups and Lie algebras. III ], Encyclopaedia Math. Sci., 41, Springer-Verlag, Berlin, 1991.

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[14] R. Heidrich, G. Jank, On the Iteration of Quaternionic M¨ obius Transformations. Complex Variables, 29 (1996), 313–318. [15] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York, San Francisco, London, 1978, pp. 628. [16] R. Kellerhals, Collars in P SL(2, H). Ann. Acad. Sci. Fenn. Math., 26 (2001), 51–72. [17] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry. Vol. II. Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1969, pp. 470. [18] L. Lempert, Holomorphic retracts and intrinsic metrics in convex domain. Analysis Mathematica, 8 (1982), 257–261. ¨ [19] H. Maass, Uber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 121 (1949), 141–183. [20] R.M. Porter, Quaternionic M¨ obius Transformations and Loxodromes. Complex Variables, 36 (1998), 285–300. [21] V. Retakh, R. L. Wilson, Advanced Course on Quasideterminants and Universal Localization. Notes of the course, Centre de Recerca Matematica (CRM), Bellaterra, Barcelona (2007). [22] W. Rudin, Function Theory in the Unit Ball of Cn . Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 241. SpringerVerlag, New York-Berlin, 1980. [23] J. Ryan, The conformal covariance of Huygen’s principle-type integral formulae in Clifford Analysis. Clifford algebras and spinor structures, Math. Appl. 321, Kluwer Acad Publ. Dordrecht, (1995), 301–310. [24] J. Ryan, Some applications of conformal covariance in Clifford Analysis. Clifford algebras in analysis and related topics (Fayetteville, AR, 1993), Stud. Adv. Math., CRC, Boca Raton, FL, (1996), 129–156. [25] J. Ryan, Q. Tao, Conformal transformations and Hardy spaces arising in Clifford Analysis. J. Operator Theory, 35, (1996) (2), 349–372. [26] J. Ryan, Dirac operators, conformal transformations and aspects of classical harmonic analysis. J. Lie Theory 8 (1) (1998), 67–82. [27] J. Xinhua, Q. Tao, J. Ryan, Fourier theory under M¨ obius transformations. Clifford algebras and their applications in math. physics, Vol. 2 (Ixtapa, 1999), Progr. Phys. 19, 57–80, Boston (2000). [28] C.L. Siegel, Topics in Complex Function Theory. Vol. II Automorphic Functions and Abelian Integrals. Wiley-Interscience Tracts in Pure and Applied Mathematics. [29] K. Th. Vahlen, Ueber Bewegungen und complexe Zahlen. Math. Ann., 55 (4),(1902), 585–593. [30] X. Wang, Quaternionic M¨ obius Transformations and Subgroups. Complex Variables, 48 (2003), 599–606. [31] J.B. Wilker, The Quaternion Formalism for M¨ obius Groups in Four or Fewer Dimensions. Linear Algebra Appl., 190 (1993), 99–136.

On the Geometry of the Quaternionic Unit Disc Cinzia Bisi Dipartimento di Matematica Universit` a di Ferrara Via Machiavelli 35 44121 Ferrara Italy e-mail: [email protected] [email protected] Graziano Gentili Dipartimento di Matematica “U. Dini” Universit` a di Firenze Viale Morgagni 67/A 50134 Firenze Italy e-mail: [email protected]

11

Bounded Perturbations of the Resolvent Operators Associated to the F-Spectrum Fabrizio Colombo and Irene Sabadini Abstract. Recently, we have introduced the F-functional calculus and the SC-functional calculus. Our theory can be developed for operators of the form T = T0 + e1 T1 + . . . + en Tn where (T0 , T1 , . . . , Tn ) is an (n + 1)-tuple of linear commuting operators. The SC-functional calculus, which is defined for bounded but also for unbounded operators, associates to a suitable slice monogenic function f with values in the Clifford algebra Rn the operator f (T ). The F-functional calculus has been defined, for bounded operators T , by an integral transform. Such an integral transform comes from the Fueter’s mapping theorem and it associates to a suitable slice monogenic function f n−1 the operator f˘(T ), where f˘(x) = Δ 2 f (x) and Δ is the Laplace operator. Both functional calculi are based on the notion of F-spectrum that plays the role that the classical spectrum plays for the Riesz-Dunford functional calculus. The aim of this paper is to study the bounded perturbations of the SC-resolvent operator and of the F-resolvent operator. Moreover we will show some examples of equations that lead to the F-spectrum. Mathematics Subject Classification (2010). Primary 30G35; Secondary 47A10. Keywords. Functional calculus for n-tuples of commuting operators, F-spectrum, perturbation of the SC -resolvent operator, perturbation of the F-resolvent operator, examples of equations that lead to the F-spectrum.

1. Introduction The recent theory of slice monogenic functions, mainly developed in the papers [2], [3], [8], [9], [10], [12], turned out to be very important because of its applications to the so-called S-functional calculus for n-tuples of non-necessarily commuting operators (bounded or unbounded), see [3] and [11]. The theory admits a quaternionic version of the S-functional calculus for quaternionic linear operators which can be found in [4]. It is crucial to note that slice monogenic functions have a Cauchy formula with slice monogenic kernel that admits two expressions. These I. Sabadini and F. Sommen (eds.), Hypercomplex Analysis and Applications, Trends in Mathematics, DOI 10.1007/978-3-0346-0246-4_2, © Springer Basel AG 2011

13

14

F. Colombo and I. Sabadini

two expressions of the Cauchy kernel are not equivalent when we want to define a functional calculus for non-necessarily commuting operators. In this paper, we will consider the case of commuting operators and the expression of the Cauchy kernel which gives rise to the definition of F -spectrum which is the natural tool to treat the case of commuting operators. Let x = x0 + e1 x1 + . . . + en xn and s = s0 + e1 s1 + . . . + en sn be paravectors in Rn+1 . We consider the Cauchy kernel written in the form SC −1 (s, x) = (s − x¯)(s2 − 2Re[x]s + |x|2 )−1 which is defined for x2 − 2xs0 + |s|2 = 0. Let f : U ⊂ Rn+1 → Rn , where Rn is the real Clifford algebra with n imaginary units, U is a suitable open set that contains the singularities of SC −1 (s, x). Let I be a 1-vector such that I 2 = −1 and let CI be the complex plane that contains 1 and I. Then we have the Cauchy formula 1 f (x) = SC −1 (s, x)dsI f (s), dsI = −Ids, (1.1) 2π ∂(U ∩CI ) where the integral does not depend on the open set U and on the imaginary unit I. In the paper [5] we have introduced the so-called SC-functional calculus (SC stands for slice-commuting), which is defined for bounded but also for unbounded commuting operators, starting from the above Cauchy formula 1 f (T ) = SC −1 (s, T )dsI f (s), dsI = −Ids. (1.2) 2π ∂(U ∩CI ) The definition is well posed because the the integral in (1.2) does not depend on the open set U and on the imaginary unit I. The SC-resolvent operator is defined by SC −1 (s, T ) := (sI − T )(s2 I − s(T + T ) + T T )−1 whose associated spectrum is the F-spectrum of T defined as: σF (T ) = {s ∈ Rn+1 : s2 I − s(T + T ) + T T

is not invertible}.

It is important to point out the meaning of the symbols. We have that, by definition, T = T0 + e1 T1 + . . . + en Tn , T = T0 − e1 T1 − . . . − en Tn , so that T + T = 2T0 , and since the components of T commute, we have T T = T02 + T12 + . . . + Tn2 . In the paper [6] we have proved the Fueter mapping theorem in integral form using the Cauchy formula (1.2). Precisely we have proved that, when n is an odd number, given the slice monogenic function f , we can associate to it the monogenic function f˘(x) by the integral transform n+1 1 γn (s−x)(s2 −s(x+x)+|x|2 )− 2 dsI f (s), dsI = −Ids, (1.3) f˘(x) = 2π ∂(U ∩CI ) where γn ia a given constant. We recall that the Fueter mapping theorem in differential form is given in [19] for n odd and in Qian’s paper [16] in the general case. Later on, Fueter’s theorem has been generalized to the case in which a function f as above is multiplied by a monogenic homogeneous polynomial of degree k, see

Bounded Perturbations of the Resolvent Operators

15

[15], [20] and to the case in which the function f is defined on an open set U not necessarily chosen in the upper complex plane, see [17]. We point out that formula (1.3) allows us to define the F -functional calculus by replacing x = x0 + x1 e1 + . . . + xn en by T = T0 + T1 e1 + . . . + Tn en . More precisely, in [6] we have defined the following version of the monogenic functional calculus by setting 1 ˘ f (T ) = F −1 (s, T )dsI f (s), (1.4) 2π ∂(U ∩CI ) n where

(1.5) Fn−1 (s, T ) := γn (sI − T )(s2 I − s(T + T ) + T T )− 2 . The functional calculus in (1.4) is well defined since the integral does not depend on the open set U and on I ∈ S. The natural notion of spectrum in this case is again the notion of F -spectrum of T as it is suggested by the definition of the F -resolvent operator defined in (1.5). The goal of this paper is to prove that bounded perturbations of the SCresolvent operator and of the F-resolvent operator produce bounded variations of the respective functional calculi. We conclude by recalling that the well known theory of monogenic functions, see [1], [7], is the natural tool to define the monogenic functional calculus which has been well studied and developed by several authors, see the book of B. Jefferies [14] and the literature therein. For the analogies with the Riesz-Dunford functional calculus see for example the classical books [13] and [18]. n+1

2. Preliminary material The setting in which we will work is the real Clifford algebra Rn over n imaginary units e1 , . . . , en satisfying the relations ei ej + ej ei = −2δij . An element in the

Clifford algebra will be denoted by A eA xA where A = i1 . . . ir , i ∈ {1, 2, . . . , n}, i1 < . . . < ir is a multi-index, eA = ei1 ei2 . . . eir and e∅ = 1. In the Clifford algebra Rn , we can identify some specific elements with the vectors in the Euclidean space Rn : an element (x1 , x2 , . . . , xn ) ∈ Rn can be identified with a so-called 1-vector in the Clifford algebra through the map (x1 , x2 , . . . , xn ) → x = x1 e1 + . . . + xn en . An element (x0 , x1 , . . . , xn ) ∈ Rn+1 will be identified with the element x = x0 + n x = x0 + j=1 xj ej called a paravector. The norm of x ∈ Rn+1 is defined as |x|2 = x20 + x21 + . . . + x2n . The real part x0 of x will be also denoted by Re[x]. A function f : U ⊆ Rn+1 → Rn is seen as a function f (x) of x (and similarly for a function f (x) of x ∈ U ⊂ Rn+1 ). Definition 2.1. We will denote by S the sphere of unit 1-vectors in Rn , i.e., S = {x = e1 x1 + . . . + en xn : x21 + . . . + x2n = 1}. Note that S is an (n−1)-dimensional sphere in Rn+1 . The vector space R+IR passing through 1 and I ∈ S will be denoted by CI , while an element belonging

16


to CI will be denoted by u + Iv, for u, v ∈ R. Observe that CI , for every I ∈ S, is a 2-dimensional real subspace of Rn+1 isomorphic to the complex plane. The isomorphism turns out to be an algebra isomorphism. Given a paravector x = x0 + x ∈ Rn+1 let us set x if x = 0, |x| Ix = any element of S otherwise. By definition we have that a paravector x, with x = 0, belongs to CIx . Definition 2.2. Given an element x ∈ Rn+1 , we define [x] = {y ∈ Rn+1 : y = Re[x] + I|x|}, where I ∈ S. Remark 2.3. The set [x] is an (n − 1)-dimensional sphere in Rn+1 . When x ∈ R, then [x] contains x only. In this case, the (n − 1)-dimensional sphere has radius equal to zero. Definition 2.4 (Slice monogenic functions). Let U ⊆ Rn+1 be an open set and let f : U → Rn be a real differentiable function. Let I ∈ S and let fI be the restriction of f to the complex plane CI . We say that f is a (left) slice monogenic function, or s-monogenic function, if for every I ∈ S, we have

1 ∂ ∂ +I fI (u + Iv) = 0. 2 ∂u ∂v We denote by SM(U ) the set of s-monogenic functions on U . The natural class of domains in which we can develop the theory of smonogenic functions are the so-called slice domains and axially symmetric domains. Definition 2.5 (Slice domains). Let U ⊆ Rn+1 be a domain. We say that U is a slice domain (s-domain for short) if U ∩ R is non-empty and if CI ∩ U is a domain in CI for all I ∈ S. Definition 2.6 (Axially symmetric domains). Let U ⊆ Rn+1 . We say that U is axially symmetric if, for every u + Iv ∈ U , the whole (n − 1)-sphere [u + Iv] is contained in U . Let us now introduce the notations necessary to deal with linear operators. By V and by Vn we denote a Banach space over R with norm · and V ⊗ Rn , respectively. We recall that

Vn is a two-sided Banach module over Rn and its elements are of the type A vA ⊗ eA (where A = i1 . . . ir , i ∈ {1, 2, . . . , n}, i1 < . . . < ir is a multi-index). The multiplications element v ∈ Vn

(right and left) of an

with a scalar a ∈ Rn are defined as va

= A vA ⊗ (eA a) and av = A vA ⊗ (aeA ).

For short, in the sequel we will write v e instead of v ⊗ e A A A A . Moreover, A A

we define v2Vn = A vA 2V . Let B(V ) be the space of bounded R-homomorphisms of the Banach space V into itself endowed with the natural norm denoted by · B(V ) . If TA ∈ B(V ), we


17

can

define the operator T = A TA eA and its action on v = B vB eB as T (v) = operators is denoted by Bn (Vn ) A,B TA (vB )eA eB . The set of all such bounded

and the norm is defined by T Bn(Vn ) = A TA B(V ) . Note that, in the sequel, we will omit the subscript Bn (Vn ) in the norm of an operator and note also that n T S ≤ T S. A bounded operator T = T0 + j=1 ej Tj , where Tμ ∈ B(V ) for μ = 0, 1, . . . , n, will be called, with an abuse of notation, an operator in paravector 0,1 form. The set of such operators

n will be denoted by Bn (Vn ). The set of bounded operators of the type T = j=1 ej Tj , where Tμ ∈ B(V ) for μ = 1, . . . , n, will be denoted by Bn1 (Vn ) and T will

be said operator in vector form. We will consider n operators of the form T = T0 + j=1 ej Tj where Tμ ∈ B(V ) for μ = 0, 1, . . . , n for the sake of generality, but when dealing with n-tuples of operators, we will embed them into Bn (Vn ) as operators in vector form, by setting T0 = 0. The subset of those operators in Bn (Vn ) whose components commute among themselves will be denoted by BC n (Vn ). In the same spirit we denote by BC 0,1 n (Vn ) the set of paravector operators with commuting components. We now recall some definitions and results from [5], [6]. Definition 2.7 (The F -spectrum and the F -resolvent sets). Let T ∈ BC 0,1 n (Vn ). We define the F-spectrum of T as: σF (T ) = {s ∈ Rn+1 : s2 I − s(T + T ) + T T is not invertible}. The F -resolvent set of T is defined by ρF (T ) = Rn+1 \ σF (T ). Theorem 2.8 (Structure of the F -spectrum). Let T ∈ BC 0,1 n (Vn ) and let p = p0 + p1 I ∈ [p0 + p1 I] ⊂ Rn+1 \ R, such that p ∈ σF (T ). Then all the elements of the (n − 1)-sphere [p0 + p1 I] belong to σF (T ). Thus the F -spectrum consists of real points and/or (n − 1)-spheres. Theorem 2.9 (Compactness of F -spectrum). Let T ∈ BC 0,1 n (Vn ). Then the Fspectrum σF (T ) is a compact non-empty set. Moreover σF (T ) is contained in {s ∈ Rn+1 : |s| ≤ T }. n+1 be an axially symmetric Definition 2.10. Let T ∈ BC 0,1 n (Vn ) and let U ⊂ R s-domain containing the F-spectrum σF (T ), and such that ∂(U ∩ CI ) is union of a finite number of continuously differential Jordan curves for every I ∈ S. Let W be an open set in Rn+1 . A function f ∈ SM(W ) is said to be locally s-monogenic on σF (T ) if there exists a domain U ⊂ Rn+1 as above such that U ⊂ W . We will denote by SMσF (T ) the set of locally s-monogenic functions on σF (T ).

Definition 2.11 (The SC-resolvent operator). Let T ∈ BC 0,1 n (Vn ) and s ∈ ρF (T ). We define the SC-resolvent operator as SC −1 (s, T ) := (sI − T )(s2 I − s(T + T ) + T T )−1 .

(2.1)

18


Definition 2.12 (The SC-functional calculus). Let T ∈ BC 0,1 n (Vn ) and f ∈ SMσF (T ) . Let U ⊂ Rn+1 be a domain as in Definition 2.10 and set dsI = ds/I for I ∈ S. We define the SC-functional calculus as 1 f (T ) = SC −1 (s, T ) dsI f (s). (2.2) 2π ∂(U ∩CI ) Definition 2.13 (F -resolvent operator). Let n be an odd number and let T ∈ BC 0,1 n (Vn ). For s ∈ ρF (T ) we define the F-resolvent operator as Fn−1 (s, T ) := γn (sI − T )(s2 I − s(T + T ) + T T )− where γn := (−1)(n−1)/2 2(n−1)/2 (n − 1)!

n − 1 2

n+1 2

,

!.

Next we define f˘(T ) when f˘ is a monogenic function which comes from an s-monogenic function f via Fueter’s theorem. The F-functional calculus will be n−1 defined for those monogenic functions that are of the form f˘(x) = Δ 2 f (x), where f is an s-monogenic function. For the functional calculus associated to standard monogenic functions we mention the book [14]. Definition 2.14 (The F -functional calculus). Let n be an odd number and let T ∈ BC 0,1 n (Vn ). Let U be an open set as in Definition 2.10. Suppose that f ∈ SMσF (T ) n−1 and let f˘(x) = Δ 2 f (x). We define the F -functional calculus as 1 ˘ f (T ) = F −1 (s, T ) dsI f (s). (2.3) 2π ∂(U ∩CI ) n Remark 2.15. The definitions of the SC-functional calculus and of the F -functional calculus are well posed since the integrals in (2.2) and in (2.3) are independent of I ∈ S and of the open set U .

3. Examples of equations for the F -spectrum Example (The case of Dirac operator). Let us consider the n-tuple of operators (∂x1 , . . . , ∂xn ), each of them acting on the vector space of functions of class C 2 over an open set U ⊆ Rn+1 . The vector operator associated to them is the Dirac operator T = ∂x1 e1 + . . . + ∂xn en . Let us determine the equation which gives its F -spectrum. We have T = −∂x1 e1 − . . .−∂xn en , and, since ∂xi ∂xj = ∂xj ∂xi , for all i, j = 1, . . . , n we also have T +T = 0 and T T = ∂x21 + . . . + ∂x2n = Δ. The F -spectrum is associated to the equation (s2 I − s(T + T ) + T T )v = 0

for v = 0

which, in this case, becomes (s2 I + Δ)v = 0

for v = 0.

(3.1)


19

The paravector s can be considered as an element belonging to a complex plane s ∈ CI0 , so we can assume that s = s0 + s1 I0 is a solution of (3.1) for some I0 . Then the F -spectrum of T is given by σF (T ) = {s = s0 + s1 I, for all I ∈ S}. s∈CI0 solution of (3.1)

Example (The case of second derivatives). Let us consider the second-order operators (∂x21 ,. . . ,∂x2n ) each of them acting on the vector space of functions of class C 4 over an open set U ⊆ Rn+1 , and let us write T = ∂x21 e1 + . . . + ∂x2n en . Determine the equation which gives its F-spectrum. We have T = −∂x21 e1 − . . . − ∂x2n en , and, since ∂x2i ∂x2j = ∂x2j ∂x2i , for all i, j = 1, . . . , n we also have T + T = 0 and T T = ∂x41 + . . . + ∂x4n . The F -spectrum is associated to the equation (s2 I + ∂x41 + . . . + ∂x4n )v = 0

for v = 0.

(3.2)

We solve the equation (3.2) on the complex plane s ∈ CI0 for some I0 . Then the F -spectrum of T is given by σF (T ) = {s = s0 + s1 I, for all I ∈ S}. s∈CI0 solution of (3.2)

Example (The case of powers of a real matrix). Let A be a matrix n × n with real entries and consider the operators Tj := Aj for j = 1, . . . , n. Determine the equation associated to the F -spectrum. It is well known that Tj Tk = Tk Tj , for j, k = 1, . . . , n. So we consider the operator T = Ae1 + . . . + An en . We have T = −Ae1 − . . . − An en , T + T = 0 and also T T = F -spectrum is associated to the equation (s2 I +

n

A2j )v = 0

for v = 0.

n j=1

A2j . The

(3.3)

j=1

Let us conclude this short list of examples with an explicit computation of the F -spectrum. Example (The case of two triangular commuting matrices). Let us consider a, b, α, β ∈ R and the two matrices: a b α β , T2 = . T1 = 0 a 0 α It is easy to verify that T1 T2 = T2 T1 . So we associate to T1 and T2 the operator ae1 + αe2 be1 + βe2 . T = T1 e1 + T2 e2 = 0 ae1 + αe2

20


We have T =− T + T = 0 and

ae1 + αe2 0

be1 + βe2 ae1 + αe2

,

a2 + α2 2ab + 2αβ . 0 a2 + α2 The F-spectrum is associated to the equation 1 0 a2 + α2 2ab + 2αβ + v=0 s2 0 a2 + α2 0 1

TT =

which becomes 2 s + a2 + α2 0

2ab + 2αβ s2 + a2 + α2

u w

for v = 0

=0

for

u w

= 0.

Consider√a paravector s on the complex plane CI0 : with some calculation we obtain s = ±I0 a2 + α2 . Thus the F -spectrum of T is given by σF (T ) = {±I a2 + α2 for all I ∈ S}.

4. Bounded perturbations of the SC-resolvent Lemma 4.1. The set U(Vn ) of elements in Bn (Vn ) which have inverse in Bn (Vn ) is an open set in the uniform topology of Bn (Vn ). If U(Vn ) contains an element A, then it contains the ball Σ = {B ∈ Bn (Vn ) : A − B < A−1 −1 }. If B ∈ Σ, its inverse is given by the series B −1 = A−1 [(A − B)A−1 ]m .

(4.1)

m≥0

Furthermore, the map A → A−1 from U(Vn ) onto U(Vn ) is a homeomorphism in the uniform operator topology.

Proof. See Lemma 7.1 in [3]. In order to state our results, we need the following definitions:

Definition 4.2. Let T ∈ BC 0,1 n (Vn ). We denote by σL (T ) the so-called left spectrum of T related to the resolvent operator (sI − T )−1 that is defined as σL (T ) = {s ∈ Rn+1 : sI − T

is not invertible in BC 0,1 n (Vn )},

where the notation sI in B R (V ) means that (sI)(v) = sv. Definition 4.3. Let W be a subset of Rn+1 . We denote by B(W, ε), for ε > 0, the ε-neighborhood of W defined as B(W, ε) := {x ∈ Rn+1 :

inf |s − x| < ε}.

s∈W


21

Lemma 4.4. Let T , Z ∈ BC 0,1 n (Vn ), s ∈ σL (T ) ∪ σL (Z) and consider SC (s, T ) = sI − (sI − T ) T (sI − T )−1 ,

(4.2)

SC (s, Z) = sI − (sI − Z) Z (sI − Z)−1 .

(4.3)

Then there exists a strictly positive constant K(s), depending on s and also on the operators T and Z, such that SC (s, T ) − SC (s, Z) ≤ K(s)T − Z.

(4.4)

Proof. Consider the chain of equalities SC (s, T ) − SC (s, Z) = (sI − Z) Z (sI − Z)−1 − (sI − T ) T (sI − T )−1 = (sI − Z) Z (sI − Z)−1 − (sI − T ) Z (sI − Z)−1 + (sI − T ) Z (sI − Z)−1 − (sI − T ) T (sI − T )−1 = (T − Z) Z (sI − Z)−1 + (sI − T )[ Z (sI − Z)−1 − T (sI − T )−1 ] = (T − Z) Z (sI − Z)−1 + (sI − T )[(Z − T ) (sI − Z)−1 + T ( (sI − Z)−1 − (sI − T )−1 )] = (T − Z) Z (sI − Z)−1 + (sI − T ) (Z − T ) (sI − Z)−1 + T (sI − Z)−1 (Z − T ) (sI − T )−1 . By taking the norm and observing that T − Z = T − Z, we have SC (s, T ) − SC (s, Z) ≤ T − Z Z (sI − Z)−1 + sI − T (sI − Z)−1 + T (sI − Z)−1 (sI − T )−1 . If we now set

K(s) := (sI − Z)−1 Z + sI − T 1 + T (sI − T )−1 ,

(4.5)

we have that K(s) > 0 and we get the statement.

Lemma 4.5. Let T , Z ∈ BC 0,1 n (Vn ), s ∈ ρF (T ), s ∈ σL (T ) ∪ σL (Z) and suppose that 1 T − Z < SC −1 (s, T )−1 , K(s) where K(s) is defined in (4.5). Then s ∈ ρF (Z) and [(SC (s, T ) − SC (s, Z))SC −1 (s, T )]m . SC −1 (s, Z) − SC −1 (s, T ) = SC −1 (s, T ) m≥1

(4.6)

22


Proof. Let us recall (4.2) and (4.3) and set A−1 = SC −1 (s, T ).

(4.7)

By formula (4.1) in Lemma 4.1 with B −1 := SC −1 (s, Z), we get [(SC (s, T ) − SC (s, Z))SC −1 (s, T )]m . SC −1 (s, Z) = SC −1 (s, T )

(4.8)

A := SC (s, T ),

B := SC (s, Z),

m≥0

The series in (4.8) converges since (SC (s, T ) − SC (s, Z))SC −1 (s, T ) ≤ K(s)T − ZSC −1 (s, T ) < 1,

so we get the statement.

Theorem 4.6. Let T , Z ∈ BC 0,1 n (Vn ), s ∈ ρF (T ), s ∈ σL (T ) ∪ σL (Z). Let ε > 0 and consider the ε-neighborhood B(σF (T ) ∪ σL (T ), ε) of σF (T ) ∪ σL (T ). Then there exists δ > 0 such that, for T − Z < δ, we have σF (Z) ⊆ B(σF (T ) ∪ σL (T ), ε) and SC −1 (s, Z) − SC −1 (s, T ) < ε, f or s ∈ B(σF (T ) ∪ σL (T ), ε). Proof. Let T , Z ∈ BC 0,1 n (Vn ) and let ε > 0. Thanks to Lemma 4.1 there exists a η > 0 such that if T − Z < η then σL (Z) ⊂ B(σL (T ), ε), where B(σL (T ), ε) is the ε-neighborhood of σL (T ). So we can always choose η such that σL (Z) ⊂ B(σF (T ) ∪ σL (T ), ε). Consider the function K(s) defined in (4.5) and observe that the constant Kε defined by Kε =

sup

K(s)

s∈B(σF (T )∪σL (T ),ε)

is finite since s ∈ B(σF (T )∪σL (T ), ε), for the above observation σL (Z) ⊂ B(σF (T ) ∪ σL (T ), ε) and because lim (sI − Z)−1 = lim (sI − T )−1 = 0.

s→∞

s→∞

Observe that since s ∈ ρF (T ) the map s → SC −1 (s, T ) is continuous and lim SC −1 (s, T ) = 0,

s→∞

for s in the complement set of B(σF (T ) ∪ σL (T ), ε) we have that there exists a positive constant Nε such that SC −1 (s, T ) ≤ Nε . From Lemma 4.5, if δ1 > 0 is such that Z − T < and

1 Kε N ε

:= δ1 , then s ∈ ρF (Z)


23

SC −1 (s, T )2 SC (s, T ) − SC (s, Z) 1 − SC −1 (s, T ) SC (s, T ) − SC (s, Z) Nε2 Kε Z − T ≤ 0. Then there exists δ > 0 such that, for Z − T < δ, we have f ∈ SMσF (Z) and f (Z) − f (T ) < ε. Proof. We recall that operator f (T ) is defined by 1 f (T ) = SC −1 (s, T ) dsI f (s) 2π ∂(U ∩CI ) where U ⊂ Rn+1 is a domain as in Definition 2.10, dsI = ds/I for I ∈ S. Suppose that U is an ε-neighborhood of σF (T ) ∪ σL (T ) and it is contained in the domain in which f is s-monogenic. By Lemma 4.6 there is a δ1 > 0 such that σF (Z) ⊂ U for Z − T < δ1 . Consequently f ∈ SMσF (Z) for Z − T < δ1 . By Lemma 4.6 SC −1 (s, T ) is uniformly near to SC −1 (s, Z) with respect to s ∈ ∂(U ∩ CI ) for I ∈ S if Z − T is small enough, so for some positive δ ≤ δ1 we get 1 f (T ) − f (Z) = [SC −1 (s, T ) − SC −1 (s, Z)] dsI f (s) < ε. 2π ∂(U ∩CI )

5. Bounded perturbations of the F -resolvent Let n be an odd number. For s ∈ ρF (T ) the F -resolvent operator associated to T is n+1 Fn−1 (s, T ) := γn (sI − T )(s2 I − s(T + T ) + T T )− 2 , (5.1) while its inverse is Fn (s, T ) :=

n+1 1 2 (s I − s(T + T ) + T T ) 2 (sI − T )−1 , γn

(5.2)

for s ∈ σL (T ). Analogously for s ∈ ρF (Z) the F -resolvent operator associated to Z is n+1 Fn−1 (s, Z) := γn (sI − Z)(s2 I − s(Z + Z) + ZZ)− 2 , (5.3) and it has the inverse Fn (s, Z) := for s ∈ σL (Z).

n+1 1 2 (s I − s(Z + Z) + ZZ) 2 (sI − Z)−1 , γn

(5.4)

24


Lemma 5.1. Let n be an odd number, T , Z ∈ BC 0,1 n (Vn ) and let s ∈ σL (T ) ∪ σL (Z). Then there exists a positive constant Cn (s) depending on s and also on the operators T and Z such that Fn (s, T ) − Fn (s, Z) ≤ Cn (s)(|s| + ϑ)n−1 T − Z,

(5.5)

where ϑ := max{T , Z}. Proof. For simplicity let us set the positions n+1 := k + 1, for k ∈ N, so that 2 , for n = 1, 3, 5, . . .. The case k = 0 has been studied in the previous k = n−1 2 section. Here we consider k ≥ 1. We set βk := γ2k+1 and we define, for s ∈ ρF (T ), Fk−1 (s, T ) := βk SC −1 (s, T ) (s2 I − s(T + T ) + T T )−k .

(5.6)

The inverse of operator Fk−1 (s, T ) exists for s ∈ σL (T ) and is given by 1 (s2 I − s(T + T ) + T T )k SC (s, T ), Fk (s, T ) = βk

(5.7)

while the inverse of operator Fk−1 (s, Z) exists for s ∈ σL (Z) and is given by 1 (s2 I − s(Z + Z) + ZZ)k SC (s, Z). Fk (s, Z) = βk

(5.8)

Consider (5.7) and (5.8) for k = 1; we have β1 [F1 (s, T ) − F1 (s, Z)] = (s2 I − s(T + T ) + T T ) SC (s, T ) − (s2 I − s(Z + Z) + ZZ) SC (s, Z) = (s2 I − s(T + T ) + T T ) SC (s, T ) − (s2 I − s(T + T ) + T T ) SC (s, Z) + (s2 I − s(T + T ) + T T ) SC (s, Z) − (s2 I − s(Z + Z) + ZZ) SC (s, Z) = (s2 I − s(T + T ) + T T ) [SC (s, T ) − SC (s, Z)] + [−s(T + T ) + T T + s(Z + Z) − ZZ)] SC (s, Z) = (s2 I − s(T + T ) + T T ) [SC (s, T ) − SC (s, Z)] + [s(Z − T + Z − T ) + (T − Z)T + Z(T − Z)] SC (s, Z) and taking the norm we get |β1 |F1 (s, T ) − F1 (s, Z) ≤ (|s|2 + 2|s| T + T T )SC (s, T ) − SC (s, Z) + 2|s| Z − T + T − Z(T + Z) SC (s, Z). ≤ (|s| + ϑ)2 SC (s, T ) − SC (s, Z) + 2(|s| + ϑ)Z − T ) SC (s, Z). Now observe that

(|s| + ϑ)−1 SC (s, Z) ≤ (|s| + ϑ)−1 |s| + (sI − Z) Z (sI − Z)−1 =: M (s) (5.9)


25

where M (s) is a continuous function since s ∈ σL (Z). Using Lemma 4.4 we get 1 [K(s) + 2M (s)](|s| + ϑ)2 Z − T . (5.10) F1 (s, T ) − F1 (s, Z) ≤ |β1 | We now use the induction principle. We assume that the estimate 1 (|s| + ϑ)2k K(s) + 2kM (s) Z − T (5.11) Fk (s, T ) − Fk (s, Z) ≤ |βk | holds for k ≥ 1. Observe that (5.8) implies that the estimate 1 (|s| + ϑ)2k SC (s, Z) Fk (s, Z) ≤ (5.12) |βk | holds. We prove that 1 (|s| + ϑ)2(k+1) K(s) + 2(k + 1)M (s) Z − T . Fk+1 (s, T ) − Fk+1 (s, Z) ≤ |βk+1 | In fact, we have that βk+1 (Fk+1 (s, T ) − Fk+1 (s, Z)) = βk (s2 I − s(T + T ) + T T ) Fk (s, T ) − βk (s2 I − s(Z + Z) + ZZ) Fk (s, Z) = βk (s2 I − s(T + T ) + T T ) [Fk (s, T ) − Fk (s, Z)] − βk [s(Z − T + Z − T ) + (T − Z)T + Z(T − Z)] Fk (s, Z) and taking the norms we have |βk+1 |Fk+1 (s, T ) − Fk+1 s, Z) ≤ |βk |(|s| + ϑ)2 Fk (s, T ) − Fk (s, Z) + 2 βk (|s| + ϑ) Fk (s, Z)Z − T . Using (5.11) and (5.12) we obtain |βk+1 |Fk+1 (s, T ) − Fk+1 s, Z) ≤ (|s| + ϑ)2k+2 K(s) + 2kM (s) Z − T + 2 (|s| + ϑ)2k+1 SC (s, Z)Z − T ≤ (|s| + ϑ)2k+2 K(s) + 2kM (s) + 2 (|s| + ϑ)−1 SC (s, Z) Z − T ≤ (|s| + ϑ)2k+2 K(s) + 2(k + 1)M (s) Z − T . k (s) := Setting C by

1 |βk | [K(s) + 2kM (s)]

Cn (s) :=

the constant Cn (s) in estimate (5.5) is given

1 [K(s) + (n − 1)M (s)]. |γn |

This concludes the proof. BC 0,1 n (Vn ),

(5.13)

Lemma 5.2. Let n be an odd number, T , Z ∈ let s ∈ ρF (T ), s ∈ σL (T ) ∪ σL (Z) and suppose that 1 (|s| + ϑ)−(n−1) Fn−1 (s, T )−1 , T − Z < Cn (s)

26


where Cn (s) is defined in (5.13). Then s ∈ ρF (Z) and Fn−1 (s, Z)−Fn−1 (s, T ) = Fn−1 (s, T ) [(Fn (s, T )−Fn (s, Z))Fn−1 (s, T )]m . (5.14) m≥1

Proof. Let us recall (5.2), (5.4) and set A := Fn (s, T ),

A−1 = Fn−1 (s, T )(s, T ).

B := Fn (s, Z),

By Lemma 4.1, formula (4.1), for B Fn−1 (s, Z) = Fn−1 (s, T )

−1

:=

Fn−1 (s, Z)

(5.15)

we get

[(Fn (s, T ) − Fn (s, Z))Fn−1 (s, T )]m .

(5.16)

m≥0

Using the hypothesis, we have that the series converges since (Fn (s, T ) − Fn (s, Z))Fn−1 (s, T ) ≤ (Fn (s, T ) − Fn (s, Z)) Fn−1 (s, T ) ≤ Cn (s)(|s| + ϑ)n−1 Z − T Fn−1(s, T ) < 1.

Theorem 5.3. Let n be an odd number, T , Z ∈ BC 0,1 n (Vn ), s ∈ ρF (T ), s ∈ σL (T ) ∪ σL (Z). Let ε > 0 and consider the ε-neighborhood B(σF (T ) ∪ σL (T ), ε) of σF (T ) ∪ σL (T ). Then there exists δ > 0 such that, for T − Z < δ, we have σF (Z) ⊆ B(σF (T ) ∪ σL (T ), ε) and

Fn−1 (s, Z) − Fn−1 (s, T ) < ε, f or s ∈ B(σF (T ) ∪ σL (T , ε).

Proof. Let T , Z ∈ BC 0,1 n (Vn ) and let ε > 0. Thanks to Lemma 4.1 there exists a η > 0 such that if T − Z < η, then σL (Z) ⊂ B(σL (T ), ε), where B(σL (T ), ε) is the ε-neighborhood of σL (T ). So we can always choose η such that σL (Z) ⊂ B(σF (T ) ∪ σL (T ), ε). Consider the function Cn (s) defined in (5.13). The constant Cn,ε defined as Cn,ε =

sup

Cn (s)

s∈B(σF (T )∪σL (T ,ε)

is finite because s ∈ B(σF (T ) ∪ σL (T , ε) and lim (sI − Z)−1 = lim (sI − T )−1 = 0.

s→∞

s→∞

Observe that the since s ∈ ρF (T ) map s → Fn−1 (s, T ) is continuous and lim Fn−1 (s, T ) = 0,

s→∞

and so for s in the complement set of B(σF (T ) ∪ σL (T , ε) we have that there exists a positive constant Mε such that Fn−1 (s, T ) ≤ Mε .


27

From Lemma 5.2 if δ1 > 0 is such that Z − T
0 such that, for Z − T < δ, we have f ∈ SMσF (Z) and f˘(Z) − f˘(T ) < ε. Proof. We recall that 1 f˘(T ) = 2π

∂(U ∩CI )

Fn−1 (s, T ) dsI f (s)

and U ⊂ R is a domain as in Definition 2.10, dsI = ds/I for I ∈ S. Let U be an ε-neighborhood of σF (T ) ∪ σL (T ) contained in the domain in which f is s-monogenic. By Lemma 5.3 there is a δ1 > 0 such that σF (Z) ⊂ U for Z − T < δ1 . Consequently, f ∈ SMσF (Z) for Z − T < δ1 . By Lemma 5.3, Fn−1 (s, T ) is uniformly near to Fn−1 (s, Z) with respect to s ∈ ∂(U ∩ CI ) for I ∈ S if Z − T is small enough, so for some positive δ ≤ δ1 we get 1 f˘(T ) − f˘(Z) = [F −1 (s, T ) − Fn−1 (s, Z)] dsI f (s) < ε. 2π ∂(U ∩CI ) n n+1

References [1] F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, Pitman Res. Notes in Math., 76, 1982. [2] F. Colombo, I. Sabadini, A structure formula for slice monogenic functions and some of its consequences, Hypercomplex Analysis, Trends in Mathematics, Birkh¨ auser, 2009, 101–114. [3] F. Colombo, I. Sabadini, The Cauchy formula with s-monogenic kernel and a functional calculus for non-commuting operators, J. Math. Anal. Appl. 373 (2011), 655– 679.

28


[4] F. Colombo, I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal., 19 (2009), 601–627. [5] F. Colombo, I. Sabadini, The F-spectrum and the SC-functional calculus, preprint 2009. [6] F. Colombo, I. Sabadini, F. Sommen, The Fueter’s mapping theorem in integral form and the F-functional calculus, to appear in Math. Methods Appl. Sci. (2010), DOI:10.1002/mma.1315. [7] F. Colombo, I. Sabadini, F. Sommen, D.C. Struppa, Analysis of Dirac Systems and Computational Algebra, Progress in Mathematical Physics, Vol. 39, Birkh¨ auser, Boston, 2004. [8] F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math., 171 (2009), 385–403. [9] F. Colombo, I. Sabadini, D.C. Struppa, Extension properties for slice monogenic functions, Israel J. Math. 177 (2010), 369–389. [10] F. Colombo, I. Sabadini, D.C. Struppa, The Pompeiu formula for slice hyperholomorphic functions, to appear in Michigan Mathematical Journal. [11] F. Colombo, I. Sabadini, D.C. Struppa, A new functional calculus for non-commuting operators, J. Funct. Anal., 254 (2008), 2255–2274. [12] F. Colombo, I. Sabadini, D.C. Struppa, Duality theorems for slice hyperholomorphic functions, J. Reine Angew. Math. 645 (2010), 85–104. [13] N. Dunford, J. Schwartz, Linear operators, part I: general theory , J. Wiley and Sons (1988). [14] B. Jefferies, Spectral properties of non-commuting operators, Lecture Notes in Mathematics, 1843, Springer-Verlag, Berlin, 2004. [15] K. I. Kou, T. Qian, F. Sommen, Generalizations of Fueter’s theorem, Meth. Appl. Anal., 9 (2002), 273–290. [16] T. Qian, Generalization of Fueter’s result to Rn+1 , Rend. Mat. Acc. Lincei, 8 (1997), 111–117. [17] T. Qian, Fourier Analysis on Starlike Lipschitz Surfaces, J. Funct. Anal., 183 (2001), 370-412. [18] W. Rudin, Functional Analysis, Functional analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-D¨ usseldorf-Johannesburg, 1973. [19] M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica, 23 (1957), 220–225. [20] F. Sommen, On a generalization of Fueter’s theorem, Zeit. Anal. Anwen., 19 (2000), 899-902. Fabrizio Colombo and Irene Sabadini Dipartimento di Matematica, Politecnico di Milano Via Bonardi, 9 20133 Milano Italy e-mail: [email protected] [email protected]

Harmonic and Monogenic Functions in Superspace K. Coulembier, H. De Bie and F. Sommen Abstract. The aim of this work is to further extend the analytic theory of monogenic functions in superspace and to construct a set of operators that allow to construct an explicit basis for the spaces of symplectic spherical harmonics. Mathematics Subject Classification (2010). 30G35, 58C50. Keywords. Superanalysis, Clifford analysis, spherical harmonics, Kelvin transformation.

1. Introduction In a previous set of papers (see a.o. [7, 8, 9, 11]) we have developed a theory of harmonic analysis and Clifford analysis in superspace. Superspaces are spaces equipped with both a set of commuting variables and a set of anti-commuting variables (generating a so-called Grassmann algebra). They are usually studied from the point of view of algebraic or differential geometry (see [2, 14]). Our approach, on the other hand, was based on a generalization of harmonic and Clifford analysis by introducing differential operators such as a Dirac and Laplace operator. The null-solutions of the super-Dirac operator ([7]) are called supermonogenic functions. Their properties are similar to those of the classical bosonic monogenic functions. In [8] spherical monogenics were used to obtain a Fischer decomposition when the superdimension is not even and negative. In [4] a Cauchy integral theorem and Morera’s theorem was proven for supermonogenic functions. In this paper we continue the development of monogenic function theory in superspace with Liouville’s theorem, a maximum modulus theorem and the Taylor expansion for monogenic functions. In the second part of this paper we consider a generalization of the Kelvin transformation to the purely fermionic case. We cannot generalize the transform directly, but we obtain a set of operators that exhibit the same behaviour. These I. Sabadini and F. Sommen (eds.), Hypercomplex Analysis and Applications, Trends in Mathematics, DOI 10.1007/978-3-0346-0246-4_3, © Springer Basel AG 2011

29

30

K. Coulembier, H. De Bie and F. Sommen

operators yield a recursive procedure to prove statements concerning fermionic harmonics. They also allow us to construct, in principle, a basis for the space of purely fermionic spherical harmonics. Such a basis is necessary to construct, e.g., a Mehler formula in Grassmann algebras (see [5]). Moreover, the construction then immediately yields a basis for the superspherical harmonics using the results obtained in [11].

2. Preliminaries Superspaces are spaces where one considers not only commuting (bosonic) but also anti-commuting (fermionic) co-ordinates (see a.o. [2, 14]). In our approach to superspace (see [7]), we start with the real algebra P ⊗ C = Alg(xi , x`j ) ⊗ Alg(ei , e`j ) = Alg(xi , ei ; x`j , e`j ), i = 1, . . . , m, j = 1, . . . , 2n generated by • m commuting variables xi and m orthogonal Clifford generators ei • 2n anti-commuting variables xì and 2n symplectic Clifford generators eì subject to the multiplication relations ⎧ ⎪ ⎪ ⎧ ⎪ ⎪ ⎨ ⎨ xi xj = xj xi xì x`j = −x`j xì and ⎪ ⎩ ⎪ xi x`j = x`j xi ⎪ ⎪ ⎩

ej ek + ek ej = −2δjk e`2j e`2k − e`2k e`2j = 0 e`2j−1 e`2k−1 − e`2k−1 e`2j−1 = 0 e`2j−1 e`2k − e`2k e`2j−1 = δjk ej e`k + e`k ej = 0

and where moreover all elements ei , e`j commute with all elements xi , x`j . The algebra generated by all generators ei , e`j is denoted by C. In the case where n = 0 we have that C ∼ = R0,m , the standard orthogonal Clifford algebra with signature (−1, . . . , −1). When m = 0, we have that P ⊗ C = Λ2n ⊗ W2n , with Λ2n the Grassmann algebra generated by the xì and W2n the Weyl algebra generated by the e`j . The most important element of the algebra P ⊗ C is the vector variable x = x + x`with m 2n xi ei , x`= x`j e`j . x= i=1

j=1

n

m 2 The square of x is scalar-valued and equals x2 = j=1 x`2j−1 x`2j − j=1 xj = x`2 + x2 . The bosonic part x2 is invariant under SO(m) while x`2 is invariant under the symplectic group Sp(2n), so x2 is invariant under SO(m) × Sp(2n). On the other hand, the super-Dirac operator is defined as ∂x = ∂x` − ∂x = 2

n

m e`2j ∂x`2j−1 − e`2j−1 ∂x`2j − ej ∂xj .

j=1

n

j=1

2 = 4 j=1 ∂x`2j−1 ∂x`2j − m Its square is the super-Laplace operator Δ = j=1 ∂xj = Δf + Δb . Also, if we let ∂x act on x, we obtain ∂x x = m − 2n = M = (x∂x ) where M is the so-called superdimension. Note that the anti-commuting variables ∂x2

Harmonic and Monogenic Functions in Superspace

31

behave as if their dimension is negative. The numerical parameter M gives a global characterization of our superspace. Furthermore we introduce the super-Euler operator by E = Eb + Ef

=

m

xj ∂xj +

j=1

The operators Δ, x2 and E + M/2 satisfy Δ/2, x2 /2 [Δ/2, E + M/2] 2 x /2, E + M/2

2n

x`j ∂x`j .

j=1

(see [7, 9]) = E + M/2 = 2Δ/2 = −2x2 /2

which are the canonical commutation relations of sl2 , see e.g. [15]. Similar to classical Clifford analysis ∂x and x generate the osp(1|2) Lie superalgebra. This means we have the same computation rules as in Clifford analysis with substitution of the Euclidean dimension by the superdimension. The super-Euler operator allows us to decompose P as ∞ P= Pk , Pk = {p ∈ P | Ep = kp} . k=0

Now we define spherical harmonics in superspace. Definition 2.1. An element F ∈ P is a spherical harmonic of degree k if it satisfies ΔF

=

0

EF

=

kF,

i.e. F ∈ Pk .

Moreover the space of all spherical harmonics of degree k is denoted by Hk . In the purely bosonic case we denote Hk by Hkb , in the purely fermionic case by Hkf . The sl2 -commutation relations lead to the following decomposition (see [8]). Lemma 2.2 (Fischer decomposition 1). If M ∈ −2N, P decomposes as P=

∞

Pk =

∞ ∞

x2j Hk .

(2.1)

j=0 k=0

k=0

If m = 0, then the decomposition is given by ⎛ ⎞ n n−k ⎝ x`2j Hf ⎠ . Λ2n = k

k=0

(2.2)

j=0

In the same way we define the space of spherical monogenics of degree k as Mk = ker ∂x ∩ Pk ⊗ C. It is clear that every spherical monogenic is a spherical harmonic. This allows us to refine the Fischer decomposition, leading to (see [8])

32


Lemma 2.3 (Fischer decomposition 2). If M ∈ −2N, Pk ⊗ C decomposes as Pk ⊗ C =

k

xi Mk−i .

i=0

If m = 0, then the decomposition is given by Pk ⊗ W2n

k

=

xì Mfk−i ,

k≤n

(2.3)

i=0

P2n−k ⊗ W2n

k

=

x`2n−2k+i Mfk−i ,

k ≤ n.

(2.4)

i=0

For the purely fermionic case we thus obtain the full decomposition Λ2n ⊗ W2n

=

n 2n−2k

x`j Mfk .

(2.5)

k=0 j=0

In general a function f ∈ C 1 (Ω)⊗ Λ2n with Ω an open domain in Rm is called left-monogenic (or also monogenic or supermonogenic) in Ω if ∂x f = 0. In [11] the space Hk was decomposed into irreducible pieces under the action of the group SO(m) × Sp(2n). Theorem 2.4 (Decomposition of Hk ). Under the action of SO(m) × Sp(2n) the space Hk decomposes as

Hk =

b Hk−i

⊗

Hif

⊕

i=0

with fk,p,q =

j=0

l=1

min(n,k−1)−1 min(n−j,

min(n,k)

k s=0

k−j 2 )

b fl,k−2l−j,j Hk−2l−j ⊗ Hjf ,

(2.6)

k

(n−q−s)! x2k−2s x`2s . s Γ( m 2 +p+k−s)

Finally we will need the Cauchy-Kowalewskaia extension in superspace. We start from the space Ak = Pk ⊗ C mod x1 Pk−1 ⊗ C, by which we mean the elements of Pk ⊗ C that do not contain x1 . Let f ∈ Ak , then the Cauchy-Kowalewskaia extension is defined as CK(f ) =

k xi

1

i=0

i!

#x )i f (−e1 ∂

#x = ∂x + e1 ∂x .We also define a map R : Mk −→ Ak by with ∂ 1 R(g) = g mod x1 Pk−1 ⊗ C. Now, as was proven in [12], Theorem 2.5. CK is an isomorphism (of right C-modules) between Ak and Mk , with inverse R.


33

3. Monogenic functions theory in superspace In this section we study analytic properties of supermonogenic functions. The classical analogues in bosonic Clifford analysis can be found in [3, 13]. In [4] we already obtained Morera’s theorem and Cauchy integral formulas for monogenic functions in superspace. In this section we will often express superfunctions using the basis provided by the fermionic Fischer decomposition (2.5). Consider a basis for the fermionic spherical monogenics {Mkl } ⊂ Mfk as a right W2n -module. We normalize this basis by Mkl,j = ck,j Mkl , such that ∂x`x`j Mkl,j = x`j−1 Mkl,j−1 . We can then expand a general superfunction f as f (x) =

n 2n−2k j=0

k=0

dim Mfk

fj,k,l (x) x`j Mkl,j .

(3.1)

l=1

Expressing that f is supermonogenic on Ω leads to f

0=

Mk n 2n−2k dim k=0 j=1

f

j−1

fj,k,l (x) x`

Mkl,j

k=0 j=0

l=1

with fj,k,l : Ω ⊂ R

m

−

Mk n 2n−2k dim

(∂x fj,k,l (x)) x`j Mkl,j ,

l=1

→ R0,m . So we find

∂x f2n−2k,k,l = 0 and ∂x fj,k,l = fj+1,k,l for j < 2n − 2k.

(3.2)

This implies that all the bosonic parts are polyharmonic and thus analytic. So all supermonogenic functions on Ω are analytic functions on Ω. In Clifford analysis Liouville’s theorem holds (see [13]), which states that when a monogenic function satisfies |f (x)| ≤ C(1 + |x|)k , then f is a polynomial of degree k, for k ≥ 0. |f (x)| is defined as the norm of f (x) as an element of the 2m -dimensional vector space R0,m for every x ∈ Rm . We first prove a slight generalization of this result. Lemma 3.1. Let f : Rm → R0,m be a function satisfying ∂x f = g in Rm with g ∈ R[x1 , . . . , xm ] ⊗ R0,m . If there exists a constant C and a natural number k such that in Rm |f (x)|

≤ C(1 + |x|)k ,

then f is a polynomial of degree k. Proof. Let l be the degree of the polynomial g. Then by the surjectivity of the Dirac operator, there exists a polynomial h of degree l + 1 for which ∂x h = g. This means that f − h is monogenic and there is a constant D for which |f (x) − h(x)| ≤

C(1 + |x|)k + D(1 + |x|)l+1

≤

(C + D)(1 + |x|)max(k,l+1) .

This means that using Liouville’s theorem, f −h is polynomial of degree max(k, l+ 1). So because h is a polynomial of degree l + 1, f is a polynomial of degree

34


max(k, l + 1). Now because |f (x)| ≤ C(1 + |x|)k , f cannot be a polynomial of degree higher than k. Now we show that Liouville’s theorem also holds for supermonogenic functions. There are several different generalizations of Liouville’s theorem to superspace possible, we will prove two versions. We begin with a definition. Recall that f decomposes as f (x) =

n 2n−2k k=0

j=0

dim Mfk

fj,k,l (x) x`j Mkl,j .

l=1

Then we say that |f (x)|1 ≤ C(1 + |x|)p if and only if |fj,k,l (x)| ≤ C(1 + |x|)p−k−j for all j, k and l. Theorem 3.2 (Liouville). Let f be a superfunction monogenic in Rm . If there exists C ∈ R and p ∈ N such that |f (x)|1 ≤ C(1 + |x|)p in Rm , then f is a superpolynomial of degree p. Proof. We expand f as in (3.1) and because f is monogenic it satisfies the equations in (3.2). From this and the classical Liouville’s theorem we find that f2n−2k,k,l is a polynomial of degree p−2n+k. The rest of the lemma is proved using induction and Lemma 3.1. Indeed, because fj+1,k,l is a polynomial and because ∂x fj,k,l = fj+1,k,l , we have that fj,k,l is a polynomial of degree p − j − k. We could also consider the supermonogenic function f (x) as a bosonic function f (x) : Ω ⊂ Rm → Λ2n ⊗ C. Then we define that |f (x)|2 ≤ C(1 + |x|)p means that with the same expansion as before |fj,k,l (x)| ≤ C(1+|x|)p must hold for every j, k and l. This is equivalent to defining |f (x)|2 as the norm of f (x) in the vector space Λ2n ⊗ C. With exactly the same techniques as in Theorem 3.2 we find Theorem 3.3 (Liouville II). Let f be a superfunction monogenic in Rm . If there exists C ∈ R and p ∈ N such that |f (x)|2 ≤ C(1 + |x|)p in Rm , then f is a superpolynomial of degree p + n. To find a nontrivial maximum modulus theorem, the second interpretation of |f (x)| is necessary. Theorem 3.4 (Maximum Modulus). Let f be a monogenic superfunction in the open and connected set Ω. If there exists a point a ∈ Ω such that |f (x)|2 ≤ |f (a)|2 ,

i.e. |fj,k,l (x)| ≤ |fj,k,l (a)|

∀j, k, l

for all x ∈ Ω, then f is a (monogenic) element of Λ2n ⊗ C.


35

Proof. We start from the maximum modulus theorem for bosonic harmonic functions, which states that if for for a harmonic function g(x), |g(x)| ≤ |g(a)| holds for all x ∈ Ω, then g is a constant function in Ω. In particular we find that the harmonic f2n−2k,k,l and f2n−2k−1,k,l are constants. Because f2n−2k,k,l = ∂x f2n−2k−1,k,l we even find that f2n−2k,k,l is zero. This can be continued using induction, if fj,k,l is a constant, then fj−1,k,l is harmonic and therefore a constant and fj,k,l is zero. So we finally find that f0,k,l is a constant and the other fj,k,l are zero. Now we consider a general monogenic function f on the open ball with radius R, Bm (R), which we expand as in (3.1). Because every fj,k,l is analytic, there exists for each of them an open neighbourhood of the origin Ωj,k,l , where it can be written as a Taylor series ∞ fj,k,l (x) = (Pt fj,k,l )(x) (3.3) t=0

with Pt fj,k,l a polynomial of degree t. The series and all its derived series converge normally in Ωj,k,l . We define the open neighbourhood of the origin Ω = ∩j,k,l Ωj,k,l . (There are only finitely many (j, k, l).) Lemma 3.5. With the notations introduced above the following holds in Ω for every k ≤ n and l, ∂x Pt f2n−2k,k,l = 0

and

∂x Pt+1 fj,k,l = Pt fj+1,k,l for j < 2n − 2k.

Proof. The fact that ∂x Pt f2n−2k,k,l = 0 is exactly Lemma 11.3.3 in [3]. The other cases can be proven using the same technique. For every α with |α| = t, ∂xα ∂x fj,k,l (0) ∂xα ∂x Pt+1 fj,k,l = = ∂xα fj+1,k,l (0) = ∂xα Pt fj+1,k,l ,

which proves the lemma. Now we define Ps f =

j+k≤s

Ps−j−k fj,k,l x`j Mkl,j .

j,k,l

Lemma 3.5 implies ∂x Ps f = 0, so Ps f ∈ Ms and ∞ f= Ps f

(3.4)

s=0

in Ω. This is a vectorial summation of the series in (3.3).

For α = (α2 , · · · , αm ) ∈ Nm−1 , β ∈ {0, 1}2n and |α| + |β| = αi + βi = k we define α! = (α2 !) · · · (αm !) and β2n αm β1 2 xα,β = xα 2 . . . xm x`1 . . . x`2n .

(3.5)

36


Theorem 3.6. For f left monogenic in Bm (R), there exists an open neighbourhood of the origin Ω, for which ∞ 1 f= CK(xα,β )[∂xα,β f ](0) α! s=0 |α|+|β|=s

where ∂xα,β =

∂xα22

2n . . . ∂xαmm ∂xβ`2n

. . . ∂xβ`11 .

Proof. Because of Theorem 2.5 we know that CK(Ps f mod x1 Ps−1 ⊗ C) = Ps f. We also have that for every superpolynomial Ri of degree i 1 α,β x ∂xα,β Ri ≡ Ri mod x0 Pi−1 ⊗ C, α! α,β;|α|+|β|=i

yielding Ps f

=

|α|+|β|=s

=

|α|+|β|=s

1 CK(xα,β )∂xα,β Ps f α! 1 CK(xα,β )[∂xα,β f ](0). α!

The lemma then follows from equation (3.4).

From this theorem we find that the set CK(xα,β ) takes over the role played by the powers of the complex variable z when expanding holomorphic functions in the complex plane into Taylor series. Remark 3.7. In the bosonic case one can prove that the domain of convergence Ω is actually equal to the original Bm (R), see [3]. The proof there can also be done in superspace. When M ∈ −2N, the fundamental solution of the Dirac operator ([10]) can be expanded similarly to the bosonic case (see [6]). The Cauchy integral theorem for monogenic functions (see [4]) then leads to a Taylor expansion for the monogenic function in the entire ball. Because of unicity this has to equal the Taylor expansion in formula (3.4).

4. Basis for the space of symplectic harmonics Theorem 2.4 implies that bases for Hjb and Hlf suffice to obtain a basis for Hk . Bases for Hjb are well known (e.g., [1]), here we will construct a set of operators that allow to generate all elements of Hlf . In this section we will always consider the purely fermionic case, so m = 0 and P = Λ2n . We first introduce a symplectic transformation on the variables. The transformation ˜. : Λ2n → Λ2n is a linear transformation which acts on elements (a, b ∈ Λ2n ) as = ˜b˜ ab a and x `2i−1 = x`2i , x# `2i = −x`2i−1 .


37

2n # In this notation we find x`2 = 12 j=1 x`j x`j = x `2 . One way to construct a basis for the bosonic harmonic polynomials is the Kelvin transformation (see [1]), also known as Maxwell’s representation theorem ([16]). As the Kelvin transformation itself does not generalize to the fermionic case, we introduce operators that have the same effect. Recall that the Kelvin transformation of a (bosonic) function u(x) in Rm is given by x K[u] = |x|2−m u( 2 ). |x| This transformation is clearly an involution (K 2 = 1). It can be proven that α K[∂x |x|2−m ] is a harmonic polynomial of degree j = |α|. We will now deform the results from [1] in a way that will be useful. Inspired by those results we will construct a similar method for the fermionic case. Using the involution property we can write for a certain bosonic spherical harmonic Hjb , ∂xα |x|2−m = K[Hjb ] = |x|2−m−2j Hjb . We can now add one more partial derivative, ∂xk ∂xk ∂xα |x|2−m = |x|2−m−2j−2 (|x|2 ∂xk + (2 − m − 2j)xk )Hjb and see that using the fact that the above is homogeneous of degree 2 − m − j − 1, b for a certain spherical harmonic Hj+1 , b Hj+1

=

K[∂xk ∂xα |x|2−m ]

=

(|x|2 ∂xk + (2 − m − 2j)xk )Hjb

=

(|x|2 ∂xk + (2 − m − 2(Eb − 1))xk )Hjb

=

(|x|2 ∂xk − (2Eb + m − 4))Hjb .

Now we can do something similar in the fermionic case. Using the sl2 -commutation relations we find [Δf , (Ef − n − 2)x`k ] = =

2Δf x`k + Ef [Δf , x`k ] − (n + 2)[Δf , x`k ] 2x`k Δf − (4Ef − 4n)∂x˜`k .

We also used [Δf , x`2j−1 ] = 4∂x`2j−1 ∂x`2j x`2j−1 − x`2j−1 4∂x`2j−1 ∂x`2j = −4∂x`2j which is similar for x`2j , so [Δf , x`k ] = −4∂x˜`k . Again using the sl2 -commutation relations we obtain [Δf , x`2 ∂x˜`k ] = (4Ef − 4n)∂x˜`k , finally yielding [Δf , (Ef − n − 2)x`k + x`2 ∂x˜`k ] = 2x`k Δf .

(4.1)

38


Considering some facts about the operator we created here we can find a basis for Hkf . The first steps are similar to the steps to form a basis for the bosonic harmonic polynomials via the Kelvin transformation ([1]). Definition 4.1. The operator Dk : Λ2n → Λ2n is given by Dk = (Ef − n − 2)x`k + x`2 ∂x˜`k ,

k = 1, . . . , 2n.

Using (4.1) we find that for a spherical harmonic Hl ∈ Hlf , Δf Dk Hl = 0, so f Dk Hl ∈ Hl+1 . Hence we obtained: f Lemma 4.2. The operators from Definition 4.1 map Hlf into Hl+1 : f , Dk Hlf ⊂ Hl+1

∀l ≤ n − 1.

We show that the operators Dj are anticommutative. Lemma 4.3. The operators in Definition 4.1 generate a Grassmann algebra isomorphic to Λ2n , i.e., {Dj , Dk } = 0,

∀j, k ∈ {1, . . . , 2n}.

Proof. The property follows from the calculations x`2 ∂x˜`j x`2 ∂x˜`k + x`2 ∂x˜`k x`2 ∂x˜`j = −x`2 x`j ∂x˜`k + x`k ∂x˜`j and (Ef − n − 2) x`j x`2 ∂x`k + x`2 ∂x`k (Ef − n − 2) x`j = x`2 (Ef − n) x`j ∂x`k + x`2 (Ef − n − 1) ∂x`k x`j = x`2 x`j ∂x`k + x`2 (Ef − n − 1) ∂x`k x`j . So we find that

{Dj , Dk } = x`2 (Ef − n − 1) (∂x`k x`j ) + (∂x`j x`k ) .

The only case where the above is nontrivial is when x`k = ±x`j . This however `j . implies that x`k = ∓x the operators generate a Grassmann algebra, we define the symplectic D2

As n as j=1 D2j−1 D2j . Because D2j−1 D2j 1 = (n + 1)(nx`2j−1 x`2j − x`2 ) we find the important formula D2 1 = 0.

(4.2)

Because of the isomorphism with the Grassmann algebra we can define fermionic polynomials in the operators Dk . This can be used to prove the surjectivity of these operators.


39

Lemma 4.4. For all j ≤ n and for every p ∈ Pj there exists a q ∈ Pj−2 such that p(D)1 = cj (p(x`) + x`2 q(x`)) with cj some nonzero constant, only depending on j. Proof. Because of linearity it only has to be proven for monomials α2n Dα = D1α1 . . . D2n

αj ∈ {0, 1}.

Because the operators Dj generate a Grassmann algebra we find that the order of the operators is unimportant, up to the sign. The case j = 1 is trivial. Using induction we assume the lemma holds for j, and calculate for |α| = j, Dk D α 1

= cj (j − n − 1)x`k [x`α + x`2 q] + x`2 ∂x˜`k Dα 1 = cj (j − n − 1) x`k x`α + x`2 q .

(4.3) (4.4)

So we find that cj+1 = (j − n − 1)cj and the proposed formula still holds for j + 1, this concludes the proof. Corollary 4.5. For a p ∈ Pj , p(D)1 = 0 if and only if p(x`) = x`2 q(x`) for some q ∈ Pj−2 . Proof. The if part follows from formula (4.2) and the only if part from Lemma 4.4. f Now we can prove that the set of operators {Dk } is surjective for Hlf → Hl+1 .

Proposition 4.6. The operators from Definition 4.1 act surjective in the sense that $2n f f k=1 Dk Hl = Hl+1 and span {Dα 1, |α| = l} = Hlf . Proof. For every p(x`) ∈ Hlf we have by Lemma 4.2 that p(D)1 ∈ Hlf . Combining this with Lemma 4.4 and the uniqueness of the Fischer decomposition (2.2) (see [9]) we find that p(D)1 = cl p(x`), so Hlf ⊂ span{Dα 1, |α| = l} ⊂ Hlf . It is now possible to use Proposition 4.6 to construct very explicitly a basis for the space of fermionic harmonics Hkf . As this construction is very tedious and complicated (requiring careful relabelings of the variables), we choose to omit the proof from this paper (although we present a brief sketch below). In some sense, Proposition 4.6 is all one needs in order to prove theorems concerning fermionic harmonics (see, e.g., [5]), as it yields a recursive procedure to reduce statements concerning harmonics of degree k to degree k−1. Moreover, the set of operators Dj are important as they capture the essence of the classical Kelvin transformation in a set of differential operators which allow for more straightforward generalizations. Sketch of construction. Proposition 4.6 combined with Corollary 4.5 implies that f a basis for Λk2n mod x`2 Λk−2 2n is mapped to a basis for Hk under the morphism p(x`) → p(D)1. An important tool in the construction of an explicit basis for Hkf is therefore the following lemma.

40


Lemma 4.7. For a Grassmann algebra Λ2n generated by the x`j , the following relation holds, x`1 x`2 x`3 x`4 · · · x`2j−1 x`2j ≡

(−1)j (x`2j+1 x`2j+2 + · · · + x`2n−1 x`2n )j mod x`2 . j!

It can now be proven that a basis for Λk2n mod x`2 Λk−2 2n is given by the monomials in Λk2n excluding the sets j xì1 xì2 · · · xìk−2j Sn−k+2j (y`1 , y`2 , . . . , y`2n−2k+4j ),

with xìl = x# ìt , 1 ≤ l, t ≤ k − 2j and {y`1 , . . . , y`2n−2k+4j } a relabeling of {x`1 , . . . , x`2n }\ {xì1 , . . . , xìk−2j , xì1 , . . . , xìk−2j } for 1 ≤ j ≤ k/2. j In this notation, the set Sp is defined as follows. Definition 4.8. The set of monomials Spj (y`1 , y`2 , . . . , y`2p ) is the subset of the set βi = j (y`1 y`2 )β1 (y`3 y`4 )β2 · · · (y`2p−1 y`2p )βp , βi ∈ {0, 1}, i

where the powers satisfy βp + βp−1 + · · · + βp−2j+2 = j, or where for some t ∈ {2, 3, . . . , j − 1} and some il , l = 1, 2, . . . , t − 1 with il < il+1 and il < p − 2j + 2l, βi1 βi2 · · · βit−1 (βp + βp−1 + · · · + βp−2j+2t ) = j − t + 1, or where βi1 βi2 · · · βij−1 βij−1 +1 = 1, for some il , l = 1, 2, . . . , j − 1 with il < il+1 and il < p − 2j + 2l. Lemma 4.7 and Definition 4.8 allow us to obtain the following theorem after a laborious procedure which we omit. Theorem 4.9. A basis for the space Hk is given by the set {Dα 1||α| = k} excluding the sets j Di1 Di2 · · · Dik−2j Sn−k+2j (Ds1 , Ds2 , . . . , Ds2n−2k+4j )1

it , 1 ≤ l, t ≤ k − 2j and Ds1 , Ds2 , . . . , Ds the operators from with Dil = D 2n−2k+4j i2 , . . . , D i (D1 , · · · , D2n ) which are not in (Di1 , Di2 , . . . , Dik−2j ) and (Di1 , D ) k−2j in unchanged order, for 1 ≤ j ≤ k/2. Acknowledgment K. Coulembier is supported by a Ph.D. Fellowship of the the Research Foundation – Flanders (FWO). H. De Bie is a Postdoctoral Fellow of the Research Foundation – Flanders (FWO).


41

References [1] S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, second ed., vol. 137 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001. [2] F. Berezin, Introduction to algebra and analysis with anticommuting variables. Moskov. Gos. Univ., Moscow. With a preface by A. A. Kirillov, 1983. [3] F. Brackx, R. Delanghe and F. Sommen, Clifford analysis, vol. 76 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1982. [4] K. Coulembier, H. De Bie and F. Sommen, Integration in superspace using distribution theory. J. Phys. A: Math. Theor. 42, 395206, 2009. [5] K. Coulembier, H. De Bie and F. Sommen, Orthogonality of Hermite polynomials in superspace and Mehler type formulae. arXiv:1002.1118. [6] K. Coulembier, H. De Bie and F. Sommen, Orthosymplectically invariant functions in superspace. Preprint. [7] H. De Bie and F. Sommen, Correct rules for Clifford calculus on superspace. Adv. Appl. Clifford Algebr. 17, 3 (2007), 357–382. [8] H. De Bie and F. Sommen, Fischer decompositions in superspace. In Function spaces in complex and Clifford analysis. National University Publishers Hanoi, 2008, pp. 170–188. [9] H. De Bie and F. Sommen, Spherical harmonics and integration in superspace. J. Phys. A: Math. Theor. 40, 26 (2007), 7193–7212. [10] H. De Bie and F. Sommen, Fundamental solutions for the super Laplace and Dirac operators and all their natural powers. J. Math. Anal. Appl. 338 (2008), 1320–1328. [11] H. De Bie, D. Eelbode and F. Sommen, Spherical harmonics and integration in superspace II. J. Phys. A: Math. Theor. 42 , 245204 (18pp), 2009. [12] H. De Bie, Harmonic and Clifford analysis in superspace. Ph.D. thesis, Ghent university, 2008. [13] R. Delanghe, F. Sommen and V. Souˇcek, Clifford algebra and spinor-valued functions, vol. 53 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1992. [14] B. DeWitt, Supermanifolds. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1984. [15] W. Fulton and J. Harris, Representation theory, vol. 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. [16] C. M¨ uller, Spherical harmonics, Lecture Notes in Mathematics 17 Springer-Verlag, Berlin-New York 1966 iv+45 pp. K. Coulembier, H. De Bie and F. Sommen Krijgslaan 281 9000 Gent Belgium e-mail: [email protected] [email protected] [email protected]

A Hyperbolic Interpretation of Cauchy-Type Kernels in Hyperbolic Function Theory Sirkka-Liisa Eriksson and Heikki Orelma Abstract. In this paper we study a mean-value property for solutions of the Laplace-Beltrami equation ∂h x2n Δh − (n − 1)xn =0 ∂xn with respect to the volume and the surface integral on the Poincaré upper-half = {(x0 , . . . , xn ) ∈ Rn+1 : xn > 0} with the Riemannian metric space Rn+1 + dx2 +dx2 +···+dx2

1 n . We also compute the Cauchy type kernels in terms of g = 0 x2 n the hyperbolic metric.

Mathematics Subject Classification (2010). Primary 30A05; Secondary 30F45. Keywords. Laplace-Beltrami operator, mean value theorem, hypermonogenic function, hyperbolic harmonic function.

1. Introduction The question What is a good generalization for the complex function theory in higher dimensions? has been studied in the preceding century. One of the possible ways is to try to find a generalization for the Cauchy-Riemann system. In the 1950s Elias Stein, in his talk at the world mathematicians congress in Stockholm, announced that there are two good generalizations for the Cauchy-Riemann system, namely the so-called Riesz and Moisil-Theodoresco systems. Let us consider the Riesz system in the form n ∂fk ∂fj ∂fj = 0, = , (1.1) ∂x ∂x ∂x j j k j=0 for k < j. In 1958 Marcel Riesz (see [19]) expressed that the

nsystem∂ (1.1) may be written in the equivalent form D f = 0, where D = j=0 ej ∂xj and This research was partially supported by the Magnus Ehrnrooth foundation.


43

44

S.-L. Eriksson and H. Orelma

f = nj=0 ej fj , and algebraic operations are computed in the Clifford sense, that is, using the relations (1.2) ei ej + ej ei = −2δij , for any i, j = 1, . . . , n and e0 = 1. More precisely the set {e1 , . . . , en } with the relations (1.2) generate the Clifford algebra C0,n . The operator D is called the Dirac or the Cauchy-Riemann operator. It is well known that the function theory related to solutions of the system D f = 0, where f takes values in the Clifford algebra C0,n , is nowadays called Clifford analysis, see [2, 3, 11]. We may also write the system (1.1) more geometrically. If d is the exterior derivative, d∗ its formal adjoint on Rn+1 with the standard metric and ω = dx0 f0 − dx1 f1 − · · · − dxn fn is a one form, then it is easy to see that the system dω = 0, d∗ ω = 0

(1.3)

is equivalent with (1.1), that is, ω is harmonic if and only if its components satisfy the Riesz system. Around 1990 Heinz Leutwiler started to study the system (1.3) in the hyperbolic upper-half space Rn+1 = {(x0 , x1 , . . . , xn ) : xn > 0} with the metric + g=

dx20 + · · · + dx2n . x2n

He deduced that in this case the system (1.3) is equivalent with the so-called modified Riesz system: n ∂fj n−1 − fn = 0, ∂xj xn j=0 ∂fk ∂fj = , for k < j. ∂xj ∂xk

(1.4)

Solutions of the preceding system are called H-solutions. The H-solutions have one really remarkable advantage, namely that the power functions x → xm , where m ∈ Z and x = x0 + x1 e1 + · · · + xn en , are also H-solutions. Hence the classical trigonometric functions are also naturally included in the theory. It is easy to generalize the modified Dirac operator also for Clifford algebra valued functions. A total Clifford algebra-valued generalization of H-solutions was introduced by Leutwiler and the first author in [7]. We briefly recall the definition. Let us generate the Clifford algebra C0,n−1 by the set {e1 , . . . , en−1 } with the relations (1.2). Then we may express C0,n = C0,n−1 ⊕ C0,n−1 en , that is, for any a ∈ Cn there exist b, c ∈ C0,n−1 satisfying a = b + cen . We denote P a = b and Qa = c. Hence every Clifford algebra valued function f admits the so-called P Q-decomposition f = P f + Qf en .

A Hyperbolic Interpretation of Cauchy Kernels

45

In [7] the authors introduced the so-called modified Dirac operator in the form M f = D f +

n−1 Q f, xn

where f takes values in the Clifford algebra C0,n and it is defined on an open subset of the upper-half space. Solutions of the

n equation M f = 0 are called hypermonogenic. One may see that if f = f0 + j=1 fj ej is hypermonogenic then it is also an H-solution. A detailed survey for the preceding remarks is available in [18]. The function theory related to the modified Dirac operator is called hyperbolic function theory. It offers a way to study boundary value problems for the modified Riesz system. For example, every hypermonogenic function admits the Cauchy-type integral representation. Let us now sketch the idea how to find the Cauchy formula. If f is a hypermonogenic function, we derive the Cauchy formula for its P and Q-part separately. Consider the P -part of a hypermonogenic function. The key idea how to find the kernel is the following. It is easy to see that if h is a hyperbolically harmonic real valued function, that is, a solution of the equation Δh −

n − 1 ∂h = 0, xn ∂xn

then the function D h is hypermonogenic, where D =

(1.5) ∂ ∂x0

−

n

∂ j=1 ej ∂xj .

The equation (1.5) may be transformed to the hyperbolic unit ball. Around 1980 Lars Ahlfors studied this equation in [1] and obtained the fundamental solution 1 (1 − s)n−1 ϕ(r) = ds. sn r In [15] Leutwiler transplanted the preceding fundamental solution to the upperhalf space, that is, |a − x| h1 (x, a) = ϕ |x − % a| is a fundamental solution of the equation (1.5) in Rn+1 + . Similar derivation for the preceding functions is also available in [17]. In [8] the first author and Leutwiler obtained that the function 1 p(x, a) = − 2n−1 n D h1 (x, a) 2 an is a hypermonogenic kernel and the corresponding Cauchy formula is dS(x) 2n ann P (p(x, a)ν(x)f (x)) n−1 , P f (a) = ωn+1 ∂K xn where ∂K is a closed hypersurface and ν is its unit normal field.

46


The Cauchy formula for the Q-part of a hypermonogenic function was obtained by the first author in [4], from the function h2 (x, a) =

1 |x −

a|n−1 |x

−% a|n−1

1 and defining the kernel q(x, a) = 2(n−1) D h2 (x, a) (which is (1 − n)-hypermonogenic, see Definition 4.1). The Cauchy formula for the Q-part of a hypermonogenic function is (see [4]) 2n an−1 n Qf (a) = Q(q(x, a)ν(x)f (x))dS(x). ωn+1 ∂K

Using the P Q-decomposition, one obtains the Cauchy formula for each hypermonogenic function with hypermonogenic kernel (see details in [9]). As one can see, the hyperbolic function theory has been studied on the hyperbolic space Rn+1 but hardly with any connections to corresponding (hyperbolic) + geometry. In this paper our aim is to study hyperbolic function theory with hyperbolic geometry. More precisely, we study first some basic geometry and basic geometric objects, e.g., balls in Rn+1 + . Then we express the preceding Cauchy kernels using hyperbolic geometry. This approach opens the way to prove the mean value theorems for P - and Q-part of a hypermonogenic function. In this paper we prove the mean value theorem for the P -part and leave the proof for the Q-part to the forthcoming paper [10].

2. Clifford Numbers The Clifford algebra C0,n is the 2n -dimensional real associative algebra generated by the symbols {e1 , . . . , en } with the multiplication rule ei ej + ej ei = −2δij . A canonical basis of C0,n is given by eA = ea1 · · · eak , where A = {a1 , . . . , ak } ⊂ M = {1, . . . , n} and 1 ≤ a1 < · · · < ak ≤ n. In particular, e∅ = 1 and e{j} = ej . The pseudoscalar is the element eM = e1 · · · en . The space of k-vectors is defined by Ck0,n = span{eA : |A| = k} and then any a ∈ C0,n admits the following multivector decomposition a = [a]0 + [a]1 + · · · + [a]n with [a]k ∈ by

Ck0,n .

There exists the canonical embedding i : Rn+1 → C0,n defined i : (x0 , . . . , xn ) → x0 +

n

ej xj .

j=1

Elements of the set Rn+1 ∼ = i(Rn+1 ) = C00,n ⊕ C10,n are called paravectors. We abbreviate briefly e0 = 1.


47

Let us also define a few involutions. Let {e1 , . . . , en } be an orthonormal basis for Rn and a, b ∈ C0,n . The main involution is a map : C0,n → C0,n defined by the relations e i = −ei and (ab) = a b . The conjugation is an anti-involution − : C0,n → C0,n defined by the relations ei = −ei and ab = ba.

3. On the Poincaré Upper-Half Space We consider the Poincaré upper-half space model (Rn+1 + , g) where the hyperbolic metric is defined by dx2 + dx21 + · · · + dx2n g= 0 . x2n Hyperbolic models have been studied by Ahlfors in [1]. But in these lecture notes Clifford algebras and functions with values in Clifford algebras have not been studied. In general, any oriented smooth Riemannian manifold with the metric g=

n

gij dxi dxj

i,j=0

admits the volume element (see [12]): dVg (x) = det(gij )dx0 ∧ dx1 ∧ · · · ∧ dxn . 2(n+1)

In the canonical coordinates on the upper-half space model, det(gij ) = 1/xn and the volume element on (Rn+1 + , g) is dxh := dVg (x) =

dx , xn+1 n

where dx = dx0 ∧ dx1 ∧ · · · ∧ dxn is the Euclidean volume element. We define the hyperbolic surface element on a smooth manifold-with-boundary U in Rn+1 with the codimension 0 by + dσh =

νdS , xnn

where ν is the unit normal field on U and dS the classical scalar surface element. The metric g allows us to define distances on Rn+1 + . The geodesics are described more detailed in the following theorem. Theorem 3.1. On the Poincaré half-space model Rn+1 geodesics are circles or lines + which meet the boundary orthogonally. Proof. See [21] p. 71 or [13] p. 38.

48


Let us consider distances on the hyperbolic upper-half space. The hyperbolic upper-half space is an immersed submanifold of Rn+1 . Its tangent space at any point x ∈ Rn+1 can nonetheless be viewed as a subspace of Tx Rn+1 . In addition, by + n+1 for each x ∈ Rn+1 → Rn+1 dimensional reasons, Tx Rn+1 = Tx Rn+1 + + . Let ι : R+ n+1 n+1 be the canonical immersion. Then we may identify ι(R+ ) and R+ as sets. That identification allow us to use two different geometric structures on Rn+1 + parallel: the hyperbolic and the Euclidean structures. Hence we may loosely speak of Euclidean distances and balls on Rn+1 + , that is, we may compute the Euclidean n+1 distance |x − y| for x, y ∈ R+ . Lemma 3.2. The hyperbolic distance dh (x, a) between the points x = x0 + x2 e1 + is · · · + xn en and a = a0 + a1 e1 + · · · + an en in Rn+1 + dh (x, a) = arcosh λ(x, a), where 2

λ(x, a) = λ (a, x) =

2

|x − a| |x − a| + 2an xn = + 1. 2xn an 2xn an

Proof. For details see, e.g., [14].

We briefly review the connection between the hyperbolic and the Euclidean distance. Directly computing we obtain the following formulae. Lemma 3.3. If x = x0 + e1 x2 + · · · + xn en and a = a0 + e1 a1 + · · · + an en are points in Rn+1 + , then |x − a|2 = 2xn an (λ(x, a) − 1) , |x − % a|2 = 2xn an (λ(x, a) + 1) ,

|x − a|2 λ(x, a) − 1 2 dh (x, a) = tanh , = |x − a ˆ|2 λ(x, a) + 1 2 where % a = a0 + e1 a1 + · · · + an−1 en−1 − an en . The connection between the hyperbolic and the Euclidean ball in Rn+1 is + stated as follows. Proposition 3.4. Let ι : Rn+1 → Rn+1 be the canonical immersion. Then + ι(Bh (a, Rh )) = Be τ (a, Rh ), Re (a, Rh ) , where τ (a, Rh ) = a0 + a1 e1 + · · · + an−1 en−1 + an en cosh Rh is the Euclidean center and Re (a, Rh ) = an sinh Rh is the corresponding Euclidean radius.


49

Proof. Let x ∈ Bh (a, Rh ), that is dh (x, a) < Rh . This means that λ(x, a) < cosh Rh . Using the above lemma we may write the inequality as |x − a|2 − 1 < cosh Rh . 2xn aa This is equivalent to |x − a|2 < 2xn an (cosh Rh − 1). Since |x − a|2 = |P (x − a)|2 + (xn − an )2 , we obtain |P (x − a)|2 + x2n − 2xn an cosh Rh + a2n < 0. Using x2n − 2xn an cosh Rh = (xn − an cosh Rh )2 − a2n cosh2 Rh , we deduce |P (x − a)|2 + (xn − an cosh Rh )2 < an (cosh2 Rh − 1) = an sinh2 Rh . Denoting τ (a, Rh ) = P a + an en cosh Rh and Re (a, Rh ) = an sinh2 Rh , we obtain |x − τ (a, Rh )| < Re (a, Rh )

completing the proof. Applying the preceding proposition we may abbreviate as follows, Bh (a, Rh ) = Be τ (a, Rh ), Re (a, Rh ) .

Proposition 3.4 says that if x is a boundary point of the hyperbolic ball Bh (a, Rh ), that is dh (a, x) = Rh then the Euclidean distance between the points x and τ (a, Rh ) is |x − τ (a, Rh )| = an sinh Rh . Hence we obtain the following corollary, which plays an important role in the forthcoming sections. Corollary 3.5. If x ∈ Rn+1 and + τ (a, x) = a0 + a1 e1 + · · · + an−1 en−1 + an en cosh dh (a, x), then |x − τ (a, x)| = an sinh dh (x, a).

4. On Hyperbolic Function Theory We briefly recall the definition of k-hypermonogenic functions (see [5]). Let Ω be an open subset of Rn+1 + . We consider functions f : Ω → C0,n , whose components are continuously differentiable. Before the definition we should define the following technical tool. We assume that the Clifford algebra C0,n−1 is generated by {e1 , . . . , en−1 }. Then for each a ∈ C0,n there exist Clifford numbers b and c in C0,n−1 satisfying a = b + cen . Let P a = b and Qa = c. The operator P is obviously a projection but Q is not (since Q2 = 0). The decomposition a = P a + (Qa)en

50


is called the P Q-decomposition. Using this decomposition we may define the involution %: C0,n → C0,n by % a = P a − (Qa)en . The above involution is called the hat-involution. It is straightforward to see that % =% a%b and if {e1 , . . . , en } is the set of generators of C0,n , if a, b ∈ C0,n , then ab 1 δjn then e%j = (−1) ej . Obviously the P - (resp. the Q-) part at the P Q-decomposition is a generalization of the real- (resp. the imaginary-) part of a complex number. The left Dirac operator in C0,n is defined by D f =

n i=0

ei

∂f ∂xi

and the right Dirac operator by Dr f =

n ∂f ei . ∂xi i=0

The operators D and Dr are defined by D f =

n i=0

ei

∂f , ∂xi

Dr f =

n ∂f ei . ∂x i i=0

r Let Ω be an open subset of Rn+1 + . The modified Dirac operators Mk , M k , Mk and r M k are introduced in [5] by

Q f , xn Qf , Mkr f (x) = Dr f (x) + k xn Mk f (x) = D f (x) + k

and Q f , xn Qf r Mk f (x) = Dr f (x) − k , xn

M k f (x) = D f (x) − k

where f is a continuously differentiable function on Ω and

Q f = (Qf ) , P f = (P f ) . The operator Mn−1 is also abbreviated by M . 1 The

symbol δij is the well-known Kronecker delta symbol.


51

Definition 4.1. Let Ω ⊂ Rn+1 be open. A continuously differentiable function + f : Ω → C0,n is left k-hypermonogenic if Mk f (x) = 0 for any x ∈ Ω. The right k-hypermonogenic functions are defined similarly. The (n − 1)-left hypermonogenic functions are briefly called hypermonogenic functions. Paravector-valued hypermonogenic functions are H-solutions introduced by Heinz Leutwiler in [15] and [16]. Clifford algebra-valued hypermonogenic functions are introduced by the first author and Leutwiler in [7]. A nice overview for the latest studies is available in the survey article [9]. We state some main properties of k-hypermonogenic functions. The Laplace-Beltrami operator on (Rn+1 + , g) is (see [1]) ∂f . ∂xn An important fact is that the P -part of a hypermonogenic function is a nullsolution of the Laplace-Beltrami operator and the Q-part is an eigenfunction as follows. Δlb f := x2n Δf − (n − 1)xn

Lemma 4.2 ([7]). Let f : Ω → Cn be twice continuously differentiable. Then n − 1 ∂P f P M M f = P f − , xn ∂xn n − 1 ∂Qf Qf + (n − 1) 2 . Q M M f = Qf − xn ∂xn xn We recall the Cauchy formulae for their P - and Q-parts separately. We need the notion of the manifold-with-boundary, see [20]. For the existence of the outer unit normal, see [12]. Proposition 4.3 ([8]). If f is a hypermonogenic function on Ω and K ⊂ Ω is an oriented (n+1)-dimensional manifold-with-boundary, then for each a ∈ K we have 2n ann dS(x) P f (a) = P (p(x, a)ν(x)f (x)) n−1 ωn+1 ∂K xn where dS is the scalar surface element, ν is the outer unit normal vector field, and 1 1 (1 − s)n−1 ds p(x, a) = − 2n−1 n D |a−x| 2 an sn |x− a| =

(x − a)−1 − (x − % a)−1 xn−1 n . n−1 2an |x − a| |x − % a|n−1

Proposition 4.4 ([4]). If f is a hypermonogenic function on Ω and K ⊂ Ω is an oriented (n+1)-dimensional manifold-with-boundary, then for each a ∈ K we have 2n an−1 n Q(q(x, a)ν(x)f (x))dS(x) Qf (a) = ωn+1 ∂K

52


where dS is the scalar surface element, ν is the outer unit normal vector field, and 1 1 q(x, a) = − D 2(n − 1) |x − a|n−1 |x − % a|n−1 −1 −1 a) 1 (x − a) + (x − % = . 2 |x − a|n−1 |x − % a|n−1

5. Hyperbolic Interpretations of the P - and Q-kernels In this section we will study how we can express the P - and Q-kernels using the hyperbolic metric. First we review some necessary tools. Lemma 5.1. Let Ω be an open subset of Rn+1 . If h : Ω → R and g : (a, b) → R such that h(Ω) ⊂ (a, b) and are differentiable, then D (g ◦ h) (x) = g (h (x)) D h (x) , and similarly

D (g ◦ h) (x) = g (h (x)) D h (x) ,

for any x ∈ Ω. Lemma 5.2. Let x and a be points in Rn+1 + . Then ∂λ (x, a) xi − ai − an (λ (x, a) − 1) δin = . ∂xi xn an Proof. We calculate as usual ∂λ (x, a) xi − ai |x − a|2 = − δin . ∂xi xn an 2x2n an Using Lemma 3.3 we infer ∂λ (x, a) xi − ai 2xn an (λ (x, a) − 1) δin = − ∂xi xn an 2x2n an xi − ai − an (λ (x, a) − 1) δin = , xn an completing the proof.

Lemma 5.3. If a ∈ Rn+1 + and τ (a, Rh ) = a0 + a1 e1 + · · · + an−1 en−1 + an cosh Rh en , then P (x − a) − (xn − an λ (x, a)) en x − τ (a, Rh ) D λ (x, a) = = xn an xn an and |x − a| P (x − a) − (xn − an λ (x, a)) en = D 1 3 |x − % a| (λ(x, a) − 1) 2 (λ(x, a) + 1) 2 xn an =

x − τ (a, Rh ) 1

3

(λ(x, a) − 1) 2 (λ(x, a) + 1) 2 xn an

.


53

Proof. Applying the above lemma we compute

n xi − ai − an (λ (x, a) − 1) δin D λ (x, a) = ei xn an i=0 =

P (x − a) − (xn − an λ (x, a)) en . xn an

Since |x − a| = |x − % a| we infer

'& D

λ(x, a) − 1 λ(x, a) + 1

&

λ(x, a) − 1 , λ(x, a) + 1

( =

=

1 λ(x,a)−1 λ(x,a)+1

2

D λ(x, a)

(λ(x, a) + 1)

P (x − a) − (xn − an λ (x, a)) en 1

3

(λ(x, a) − 1) 2 (λ(x, a) + 1) 2 xn an

.

The proof is complete.

Theorem 5.4. If dh (x, a) is the hyperbolic distance between the points x and a in Rn+1 + , then p(x, a) = =

x − τ (a, x) sinhn+1 dh (x, a)

2n xn an+1 n

x − τ (a, x) 1 , 2n xn |x − τ (a, x)|n+1

where τ (a, x) = a0 + a1 e1 + · · · + an−1 en−1 + an cosh dh (x, a)en . Proof. Using the definition of p(x, a) (see Proposition 4.3) and 1− we obtain p(x, a) = −

1

|a − x|

2

2

|a − y%| '

1

=

4an yn

2,

|a − y%|

1 − s2 sn

n−1

(

D ds |a−x| 22n−1 ann |x− a| n−1 |a−x|2

1 − 2 |a − x| 1 |x− a| D = 2n−1 n |a−x|n 2 an |x − % a| |x− a|n

1 (4xn an )n−1 |a − x|−n |a − x| . = 2n−1 n D −n 2(n−1) 2 an |x − % |x − % a| |x − % a| a|

54


Applying Lemma 3.2 and Lemma 3.3, we infer 2n ann p (x, a) n

n−1 (λ(x, a) + 1) 2 P (x − a) − (xn − an λ (x, a)) en 2xn an = n 1 3 2an xn (λ(x, a) + 1) (λ(x, a) − 1) 2 (λ(x, a) − 1) 2 (λ(x, a) + 1) 2 xn an =

P (x − a) − (xn − an λ (x, a)) en (λ(x, a)2 − 1)

n+1 2

.

xn an

Since λ(x, a)2 − 1 = cosh2 dh (x, a) − 1 = sinh2 dh (x, a), we conclude P (x − a) − (xn − an λ (x, a)) en . 2n xn an+1 sinhn+1 dh (x, a) n Thus the first equality holds. Using Corollary 3.5 we obtain p (x, a) =

p(x, a) =

1 x − τ (a, x) , 2n xn |x − τ (a, x)|n+1

and the result is established.

Remark 5.5. The above theorem gives us an interpretation of the p-kernel. In the x−a Euclidean Clifford analysis the Cauchy kernel is |x−a| n+1 (up to a constant). In the hyperbolic case, the p-kernel is just the Euclidean kernel with the shifted center τ (a, x). Also there is the coefficient 1/xn , which is something we can expect. Our next aim is to compute the q-kernel using hyperbolic tools. We need the following standard lemma. Lemma 5.6. Let Ω ⊂ Rn+1 be an open subset. Let g : (a1 , b1 ) × (a2 , b2 ) → R and fi : (ai , bi ) → R be differentiable for i = 1, 2. Assume that hi (Ω) ⊂ (ai , bi ) for i = 1, 2. Then D (g ◦ (h1 , h2 )) = ∂1 g ◦ (h1 , h2 )D h1 + ∂2 g ◦ (h1 , h2 )D h2 . As an application of the preceding lemma we obtain the following theorem. Theorem 5.7. If dh (x, a) is the hyperbolic distance between the points x and a in Rn+1 + , then q(x, a) = =

(x − τ (a, x)) cosh dh (x, a) − an sinh2 dh (x, a)en (2an xn )n sinhn+1 dh (x, a) x − τ (a, x) 1 1 1 Qτ (a, x) − en , (2xn )n |x − τ (a, x)|n+1 (2xn )n |x − τ (a, x)|n−1

where τ (a, x) = a0 + a1 e1 + · · · + an−1 en−1 + an cosh dh (x, a)en .


55

1 Proof. Recall q(x, a) = − 2(n−1) D H(x, a) where

1

H(x, a) =

n−1 . ((2xn an )(λ(x, a) − 1)) ((2xn an )(λ(x, a) + 1)) 2 n−1 Define the functions g(s1 , s2 ) = s11s2 2 and hi (x) = (2xn an )(λ(x, a) + (−1)i ) for i = 1, 2. Thus H = g ◦ (h1 , h2 ). Then 1 n−1 ∂i g(s1 , s2 ) = − 2 si (s1 s2 ) n−1 2 n−1 2

and using Lemma 5.3 we obtain D hi (x) = 2an xn D λ(x, a) + 2an (λ(x, a) + (−1)i )D xn = 2(x − τ (a, x)) − 2an (λ(x, a) + (−1)i )en . Hence D (g(h1 (x), h2 (x))) = ∂1 g ◦ (h1 (x), h2 (x))D h1 (x) + ∂2 g ◦ (h1 (x), h2 (x))D h2 (x) =−

D h1 (x) D h2 (x) n−1 n−1 − 2 h1 (x)(h1 (x)h2 (x)) n−1 2 h2 (x)(h1 (x)h2 (x)) n−1 2 2

=−

n − 1 h2 (x)D h1 (x) + h1 (x)D h2 (x) . n+1 2 (h1 (x)h2 (x)) 2

Consider the numerator of the above quotient, we compute 2

hi (x)D h3−i (x)

i=1

= 4xn an

2

(λ(x, a) + (−1)i ) (x − τ (a, x)) − an (λ(x, a) + (−1)3−i )en

i=1

= 8xn an λ(x, a)(x − τ (a, x)) − 4xn a2n

2

(λ(x, a) + (−1)i )(λ(x, a) − (−1)i )en .

i=1

The second term is 4xn a2n

2

(λ(x, a) + (−1)i )(λ(x, a) − (−1)i )en

i=1

= 4xn a2n =

2

(λ2 (x, a) − 1)en

i=1 2 8xn an (λ2 (x, a)

− 1)en .

Thus h2 (x)D h1 (x)+h1 (x)D h2 (x) = 8xn an λ(x, a)(x − τ (a, x))−8xn a2n (λ2 (x, a)+1)en .

56


Hence we obtain q(x, a) = =

1 h2 (x)D h1 (x) + h1 (x)D h2 (x) n+1 4 (h1 (x)h2 (x)) 2 1 8xn an λ(x, a)(x − τ (a, x)) − 8xn a2n (λ2 (x, a) − 1)en . n+1 4 (2xn an )n+1 ((λ(x, a) − 1)(λ(x, a) + 1)) 2

Since (λ(x, a) − 1)(λ(x, a) + 1) = λ(x, a)2 − 1 = sinh2 dh (x, a), we have q(x, a) =

(x − τ (a, x)) cosh dh (x, a) − an sinh2 dh (x, a)en , (2xn an )n sinhn+1 dh (x, a)

and the first equality holds. Then q(x, a) =

en (x − τ (a, x)) cosh dh (x, a) − . n−1 (2xn an )n sinhn+1 dh (x, a) (2xn )n an−1 sinh dh (x, a) n

Since cosh dh (x, a) = q(x, a) =

Qτ (a,x) , an

we deduce

x − τ (a, x) en Qτ (a, x) − . n+1 n−1 n sinh dh (x, a) (2xn ) an sinhn−1 dh (x, a)

(2xn )n an+1 n

The second equality follows from Corollary 3.5.

Remark 5.8. The above theorem gives us an interpretation of the q-kernel. Recall that the Newton kernel in the theory of harmonic functions is (up to a constant) 1 |x−a|n−1 . We see that the q-kernel is the linear combination of the Cauchy and the Newton kernels with the shifted center τ (a, x). Moreover, since we consider the kernel of the Cauchy formula for the Q-part of a hypermonogenic function, it can be expected that the coefficient en has a special role.

6. The Mean-Value Theorem for the P -Part of a Hypermonogenic Function Lastly we will study a mean-value property of the P -part of a hypermonogenic function. The hyperbolic machinery developed in the preceding sections has a strong influence in the proof. Theorem 6.1. Let Ω be an open subset of Rn+1 + . If f is hypermonogenic in Ω, then n an dS(x) P f (x) n P f (a) = ωn+1 Ren ∂Bh (a,Rh ) xn for any hyperbolic ball Bh (a, Rh ) with Bh (a, Rh ) ⊂ Ω. Proof. Using Proposition 4.3 we obtain 2n ann dS(x) P f (a) = P (p(x, a)ν(x)f (x)) n−1 . ωn+1 ∂Bh (a,Rh ) xn


57

Since p(x, a) =

x − τ (a) sinhn+1 dh (x, a)

2n xn an+1 n

and

x − τ (a) , Re

ν(x) = we have

|x − τ (a)|2 . 2n xn an+1 Re sinhn+1 dh (x, a) n Moreover since |x − τ (a)|2 = Re2 and Re = an sinh Rh , we obtain p(x, a)ν(x) =

1 . 2n xn Ren

p(x, a)ν(x) = Then P f (a) =

ann ωn+1 Ren

P f (x) ∂Bh (a,Rh )

dS(x) . xnn

The proof is complete. Also we would like to recall the following structure theorem.

Theorem 6.2 ([6]). Let U ⊂ Rn+1 be open. The following properties are equivalent: + (a) h is hyperbolically harmonic on U . (b) h is smooth and 1 h(a) = h(x)dσh (x) ωn sinhn Rh ∂Bh (a,Rh ) for all Bh (a, Rh ) ⊂ U . (c) h is smooth and h(a) =

1 V (Bh (a, Rh ))

h(x)dxh (x) Bh (a,Rh )

for all Bh (a, Rh ) ⊂ U where V (Bh (a, Rh )) = ωn bolic volume of the ball Bh (a, Rh ).

) Rh 0

sinhn tdt is the hyper-

The corresponding theorem is available also for the Q-part of a hypermonogenic function. We shall study these topics in the forthcoming paper [10].

References [1] L.V. Ahlfors, M¨ obius transformations in several dimensions Ordway Professorship Lectures in Mathematics. University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981. [2] F. Brackx, R. Delanghe, F. Sommen, Clifford analysis, Research Notes in Mathematics, 76. Pitman , Boston, MA, 1982.

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[3] R. Delanghe, F. Sommen, V. Souˇcek, Clifford algebra and spinor-valued functions, Mathematics and its Applications, 53. Kluwer Academic Publishers Group, Dordrecht, 1992. [4] S.-L. Eriksson, Integral formulas for hypermonogenic functions, Bull. Bel. Math. Soc. 11 (2004), 705-717. [5] S.-L. Eriksson-Bique, k-hypermonogenic functions; In Progress in Analysis, Vol I, World Scientific (2003), 337-348. [6] S.-L. Eriksson, H. Leutwiler, Hyperbolic harmonic functions and their function theory, Potential Theory and Stochastics in Albac, (2009), 85-100. [7] S.-L. Eriksson-Bique, H. Leutwiler, Hypermonogenic functions, Clifford Algebras and their Applications in Mathematical Physics, Vol. 2, Birkh¨ auser, Boston, 2000, 287302. [8] S.-L Eriksson, H. Leutwiler, Hypermonogenic functions and their Cauchy-type theorems, In Trend in Mathematics: Advances in Analysis and Geometry , Birkh¨ auser, Basel/Switzerland, 2004, 97-112. [9] S.-L. Eriksson, H. Leutwiler, Introduction to hyperbolic function theory, Clifford Algebras and Inverse Problems (Tampere 2008) Tampere Univ. of Tech. Institute of Math. Research Report No. 90 (2009), pp. 1–28 [10] S.-L. Eriksson, H. Orelma, A mean-value theorem for some eigenfunctions of the Laplace-Beltrami operator on the upper-half space, to appear in Ann. Acad. Sci. Fenn. Math. [11] G¨ urlebeck, K., Habetha, K., and Spr¨ ossig, W., Holomorphic Functions in the Plane and n-dimensional Space, Birkh¨ auser, Basel, 2008. [12] J.M. Lee, Introduction to smooth manifolds, Graduate Texts in Mathematics, 218. Springer-Verlag, New York, 2003. [13] J.M. Lee, Riemannian manifolds; An introduction to curvature, Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997. [14] H. Leutwiler, Appendix: Lecture notes of the course “Hyperbolic harmonic functions and their function theory”, Clifford algebras and potential theory, 85–109, Univ. Joensuu Dept. Math. Rep. Ser., 7, Univ. Joensuu, Joensuu, 2004. [15] H. Leutwiler, Modified Clifford analysis, Complex Variables 17 (1992), 153–171. [16] H. Leutwiler, Modified quaternionic analysis in R3 , Complex Variables 20 (1992), 19–51.. [17] Y. Qiao, S. Bernstein, S.-L. Eriksson, J. Ryan, Function theory for Laplace and Dirac-Hodge operators in hyperbolic space, J. Anal. Math. 98 (2006), 43–64. [18] H. Orelma, New perspectives in hyperbolic function theory, Doctoral thesis, Tampere Univ. of Tech. Publication No. 892 (2010) [19] M. Riesz, Clifford numbers and spinors. Edited by Bolinder and Pertti Lounesto and with a preface by Bolinder. Fundamental Theories of Physics, 54. Kluwer Academic Publishers Group, Dordrecht, 1993.


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[20] M. Spivak, Calculus on manifolds, W. A. Benjamin, Inc., New York-Amsterdam 1965 [21] J. Wolf, Spaces of constant curvature, Second edition, Department of Mathematics, University of California, Berkeley, Calif., 1972. Sirkka-Liisa Eriksson and Heikki Orelma Department of Mathematics Tampere University of Technology P.O. Box 553 33101 Tampere Finland e-mail: [email protected] [email protected]

Gyrogroups in Projective Hyperbolic Clifford Analysis Milton Ferreira Abstract. Using the projective hyperbolic model in Clifford analysis we show that different velocities in special relativity theory (non-standard velocities, coordinate velocities and proper velocities) corresponds to taking different realizations of the hyperbolic geometry in the projective model. A full description of the changes is given, together with a proof of the isomorphism of the related gyrogroup structures (M¨ obius, Einstein, and proper velocity gyrogroups). Mathematics Subject Classification (2010). Primary 20N05; Secondary 22E43. Keywords. Gyrogroups, Einstein addition, M¨ obius addition, proper velocity addition, Lorentz group.

1. Introduction Gyrogroups appeared in 1988 [14], after the discovery that Thomas precession of special theory of relativity stores an algebraic structure that turns the set of all relativistically admissible velocities with the Einstein velocity addition law into a grouplike object called gyrogroup. They have been intensively studied by A. Ungar (see [14–22] and the vast list of references in [20] and [22]) due to their interdisciplinary character, spreading from abstract algebra and non-Euclidean geometry to mathematical physics. Ungar’s theory gives a new mathematical formalism, called gyrolanguage, to treat analytic hyperbolic geometry and it provides also an unification between Euclidean and hyperbolic geometries. As a consequence his theory extends Einstein’s special relativity by means of analytic hyperbolic geometry. While Euclidean geometry has a single standard model, hyperbolic geometry is studied in the literature by several models. Three standard models can be considered: the Poincaré ball, the Klein (or Beltrami) ball, and the proper velocity space model or Ungar’s model [20]. These models give different relativistic velocity spaces useful in the study of special relativity theory: I. Sabadini and F. Sommen (eds.), Hypercomplex Analysis and Applications, Trends in Mathematics, DOI 10.1007/978-3-0346-0246-4_5, © Springer Basel AG 2011

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– The Poincaré model is a conformal model also known as the rapidity space [13], because the Poincaré distance from the origin of the ball to any point on the ball coincides with the rapidity of a boost. The Poincaré ball model of hyperbolic geometry is algebraically governed by Möbius addition. This is defined on the relativistic ball of the three dimensional Euclidean space Bc3 = {u ∈ R3 : ||u|| < c} where c is the vacuum speed of light, and is given by u ⊕M v =

(1 +

1 c2

u, v + c12 ||v||2 )u + (1 − c12 ||u||2 )v . 1 + c22 u, v + c14 ||u||2 ||v||2

(1)

Möbius addition satisfies the “gamma identity” * 2 1 (2) γu⊕M v = γu γv 1 + 2 u, v + 4 ||u||2 ||v||2 c c for all u, v ∈ Bc3 , where γu is the Lorentz factor of special relativity theory given by 1 γu = . (3) 2 1 − ||v|| 2 c – The Klein (Beltrami or velocity) model is algebraically governed by Einstein’s addition on Bc3 , which is given by

1 1 1 γu u ⊕E v = u+ v+ 2

u, v u . (4) γu c 1 + γu 1 + u,v 2 c

Einstein’s addition of relativistically admissible velocities was introduced by Einstein in his 1905 famous paper (see [20] for the reference). Einstein’s addition satisfies the “gamma identity”:

u, v γu⊕E v = γu γv 1 + 2 , ∀u, v ∈ Bc3 . (5) c Equation (5) gives rise to a kind of a inner product that was extended by Ungar giving rise to gyrofactors (see [20]). These allow us to construct the gyrogroup extension of a gyrocommutative gyrogroup to a pseudo space-time model. In both cases, in the Newtonian limit, c → ∞, the ball expands to the whole space R3 and both Möbius and Einstein’s additions reduce to the ordinary vector addition in R3 , which allows us to make the link between analytic Euclidean and hyperbolic geometries. – The proper velocity (or Ungar) model is regulated by the proper velocity addition, that is, the relativistic addition of proper velocities rather than the addition of relativistic coordinate velocities as in Einstein addition. The proper velocity addition ⊕U is given by

βu u, v 1 − βv u ⊕U v = u + v + u, (6) + 1 + βu c 2 βv

Gyrogroups in Projective Hyperbolic Clifford Analysis where u, v ∈ R3 and βv , called the β factor, is given by 1 . βv = 2 1 + ||v|| 2 c

63

(7)

The proper velocity addition satisfies the “beta identity”: βu⊕U v =

β u βv 1 + βu βv u,v c2

,

∀u, v ∈ R3 .

(8)

Coordinate time, or observer’s time, is the time t of a moving object measured by an observer at rest. Accordingly, special relativity theory is formulated in terms of coordinate time. In contrast to coordinate time, the proper time, or traveler’s time, is the time τ of a moving object measured by a co-moving obersver. Proper time is useful, for instance, in the understanding of twin paradox. It is well known that the coordinate time t and the proper time τ of a uniformly moving object with relative coordinate velocity v ∈ Bc3 are related by 1 t = γv τ = τ 2 1 − ||v|| c2 and the relative coordinate velocity v ∈ Bc3 and proper velocity w ∈ R3 of an object measured by its coordinate time and proper time, respectively, are related by the equations: 1 1 w = γv v = ∈ R3 and v = βω = ∈ Bc3 . ||v||2 ||w||2 1 − c2 1 + c2 M¨ obius transformations in several dimensions can be defined by using Clifford numbers [1, 2, 5]. Using a projective identification of the points in the Euclidean space Rn with the rays in the null cone in R1,n+1 the group of M¨ obius transformations in Rn coincides with the group Spin(1, n + 1) (see, e.g., [5, 3, 4, 7]). Also related with this approach is the study of Clifford analysis on hyperbolic spaces, due to the fact that the subgroup Spin(1, n) of Möbius transformations leaving the unit sphere invariant is the isometry group of these non-Euclidean geometries (see, e.g., [5, 7]). In this paper we will use the projective hyperbolic model in Clifford analysis to give a geometric comprehension of the three different velocity space representations of special relativity theory. The use of a Clifford algebra representation for relativistic additions proved to be useful in the discovery of new results. In [10] we studied M¨ obius gyrogroups with Clifford algebra techniques which allow us to obtain a characterization of the associativity between elements in the Möbius gyrogroups and to construct quotient Möbius gyrogroups (analogous to quotient groups in group theory) by convenient factorizations of the relativistic ball. These give rise to M¨ obius fiber bundles (see [10]). Applications of gyrogroups in physics can be found in [22, 11] and applications of Möbius gyrogroups in signal processing can be seen in [8].

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2. Gyrogroups In the general sense a gyrogroup is an extension of the notion of group by introducing a gyroautomorphism to compensate the lack of associativity. If the gyroautomorphisms are all reduced to the identity map the gyrogroup becomes a group. Definition 1. [20] A groupoid (G, ⊕) is a gyrogroup if its binary operation satisfies the following axioms: (G1). There is at least one element 0 satisfying 0 ⊕ a = a, for all a ∈ G. (G2). For each a ∈ G there is an element a ∈ G such that a ⊕ a = 0. (G3). For any a, b, c ∈ G there exists a unique element gyr[a, b]c ∈ G such that the binary operation satisfies the left gyroassociative law a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ gyr[a, b]c.

(9)

(G4). The map gyr[a, b] : G → G given by c → gyr[a, b]c is an automorphism of the groupoid (G, ⊕), that is gyr[a, b] ∈ Aut(G, ⊕); (G5). The gyroautomorphism gyr[a, b] possesses the left loop property gyr[a, b] = gyr[a ⊕ b, b].

(10)

Definition 2. [20] A gyrogroup (G, ⊕) is gyrocommutative if its binary operation satisfies a ⊕ b = gyr[a, b](b ⊕ a), ∀ a, b ∈ G. A very simple example of a gyrogroup is the complex unit disc D = {z ∈ C : |z| < 1} endowed with the Möbius addition given by a+z a⊕z = , a, z ∈ D. (11) 1 + az M¨ obius addition on D is neither commutative nor associative but it is gyroassociative and gyrocommutative under gyrations defined by 1 + ab d, a, b, d ∈ D. (12) 1 + ab The gyrosemidirect product is a generalization of the external semidirect product of groups and it gives rise to the construction of groups. gyr[a, b]d =

Proposition 1. [20] Let (G, ⊕) be a gyrogroup, and let Aut0 (G, ⊕) be a gyroautomorphism group of G (any subgroup of Aut(G) which contains all the gyroautomorphisms gyr[a, b] of G, with a, b ∈ G). Then the gyrosemidirect product G×Aut0 (G) is a group, with group operation given by the gyrosemidirect product (a, X)(b, Y ) = (a ⊕ Xb, gyr[a, Xb]XY ).

(13)

In the case of special theory of relativity the gyrosemidirect product structure of the Lorentz group allows time and space in relativistic spacetime to be decoupled similarly as the semidirect product structure of the Galilean group allows time and space in Galilean spacetime to be decoupled.

Gyrogroups in Projective Hyperbolic Clifford Analysis

65

3. The projective hyperbolic space model In this section we review the projective hyperbolic space model. For more details we refer to [3, 4, 7]. Let us consider the real orthogonal space R1,n of signature (1, n) with quadratic form Q(T, X) = T 2 − |X|2 = T 2 −

n

Xi2 ,

∀ (T, X) = (T, X1 , . . . , Xn ) ∈ R1,n . (14)

i=1

Here T is the time coordinate and X = (X1 , . . . , Xn ) is the vector of space coordinates. The null cone N C = {(T, X) ∈ R1,n : Q(T, X) = 0} separates the time-like region (T LR = {(T, X) ∈ R1,n : Q(T, X) > 0}) from the space-like region (SLR = {(T, X) ∈ R1,n : Q(T, X) < 0}). The time-like region itself is the union of the future cone F C and the past cone P C given by F C = {(T, X) ∈ R1,n : Q(T, X) > 0 ∧ T > 0} P C = {(T, X) ∈ R1,n : Q(T, X) > 0 ∧ T < 0}. = To work on the relativistic cone we need to consider the quadratic form Q(T, X) 2 |X| T 2 − 2 . However, to make our exposition simpler we consider c = 1. Whenc ever necessary we will pass to the relativistic cone with the change of variables X = X/c. The group of linear transformations of R1,n preserving the inner product < (T, X), (S, Y ) = T S −

n

Xi Yi

i=1

is the group O(1, n). Some subgroups of O(1, n) are of special importance: SO(1, n) that corresponds to the subgroup of unimodular transformation from O(1, n); O0 (1, n) which is the subgroup of transformations preserving both sheets of the null cone (i.e., preserving the time component); and SO0 (1, n) = SO(1, n) ∩ O0 (1, n) known as the proper Lorentz group is the connected component of SO(1, n) and it is a locally compact group. The upper sheet of the hyperboloid Q(T, X) = 1 will be denoted by H + and it holds H + = SO(1, n)/SO(n), where SO(n) is the maximal compact subgroup of SO0 (1, n) that fixes the vector (1, 0, . . . , 0). Thus, H + is a symmetric Riemannian space of noncompact type also known as the hyperbolic unit ball as will be explained later. Introducing rays on the future cone we obtain a projective model for the hyperbolic unit ball H + . The manifold of rays is given by Ray(F C) = {{λ(T, X) : λ > 0}, (T, X) ∈ F C}. In [23] we can find the Cartan and Iwasawa decompositions of SO0 (1, n). The action of the group SO0 (1, n) on H + determines its action on the manifold of rays Ray(F C). Therefore, the action of SO0 (1, n) on any manifold, lying inside the

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future cone and intersecting every ray from Ray(F C) at one point can also be defined, as we will see in this paper. Working in the manifold of rays as a principal fiber bundle we can obtain other models for the hyperbolic unit ball. These are obtained by intersecting the manifold Ray(F C) with an arbitrary surface inside the future cone, such that each ray intersects the surface in a unique point. This gives rise to different models for the hyperbolic unit ball H + . Three canonical manifolds can be considered in this model: , + 1. the hyperboloid H + = (T, X) ∈ F C : T = 1 + |X|2 ; 2. the disc D = {(T, X)+∈ F C : T = 1}; 3. the paraboloid P = (T, X) ∈ F C : T =

1+|X|2 2

, ∧T )2 + cosh(α) < ω, x > − sinh(α) 1+||x|| 2 . 2 2 1+||a|| 1+||x|| 2||a|| 1−||x||2 a ∓ 1−||a|| 2 1−||a||2 2 ||a|| , x − 2 = 2 .2 . 2 1+||a|| 2||a|| 1+||x||2 a a ||x||2 − , x + , x ∓ 2 2 ||a|| 1−||a|| ||a|| 1−||a|| 2 =

(1 − ||a||2 )(||a||2 + ||x||2 ∓ 2 a, x) (1 ∓ 2 a, x + ||a||2 ||x||2 )(||a||2 + ||x||2 ∓ 2 a, x)

=

1 − ||a||2 . 1 ∓ 2 a, x + ||a||2 ||x||2

Finally, multiplying λ by X and making the substitutions of cosh(α), sinh(α), and ω in X (we are interested in the space coordinate) we obtain λX

=

(1 − ||a||2 )x − (±(1 + ||x||2 ) − 2 a, ω)a 1 − ||a||2 1 ∓ 2 a, x + ||a||2 ||x||2 1 − ||a||2

=

(1 − ||a||2 )x ∓ (1 + ||x||2 ∓ 2 a, x)a . 1 ∓ 2 a, x + ||a||2 ||x||2

We obtained two different transformations depending on the choice of the sign “ + ” or “ − ”. However, they are inverse one of the other, reflecting the fact that in the M¨ obius gyrogroup the inverse of a is −a. Choosing the “ − ” sign we obtain (18). Remark 2. Möbius addition (18) defined on B1n corresponds to a left gyrotranslation of xP ∈ P by a ∈ P. Considering c = 1, u = a and v = xP in (1) both additions coincide. Conversely, considering a = u/c and xP = v/c in (18) we obtain the Möbius addition on the relativistic ball Bcn , i.e., we obtain formula (1). The unit ball B1n endowed with Möbius addition gives rise to the Möbius gyrogroup (B1n , ⊕M ) (see [10] for a detailed proof). In Section 10 we will present


71

M¨ obius gyrations as spin representations of the group Spin(n) using Clifford algebra techniques.

5. The Einstein gyrogroup (B1n , ⊕E ) The Einstein addition is obtained by projection of the boost’s action (17) on the manifold of rays to the disc D. Proposition 4. Let T = 1, X = xD with ||xD || < 1, cosh(α) = √

1 , 1−||a||2

± √ ||a||

1−||a||2

sinh(α) =

, and ω = a/||a||. Then the restriction of the action (17) to the disc D

gives the Einstein addition ⊕E : ' ( 1 − ||a||2 1 2 1 − ||a|| xD +

a, xD a − a . (19) a ⊕E xD = 1 − a, xD 1 + 1 − ||a||2 Proof. Considering in (17) the substitutions mentioned in Proposition 4, together with xD = x, we obtain, by straightforward computations, the time and space components: 1 ∓ a, x T = 1 − ||a||2 and (1 + 1 − ||a||2 ) 1 − ||a||2 x + 1 − ||a||2 a, x a ∓ (1 + 1 − ||a||2 )a . X= (1 + 1 − ||a||2 )( 1 − ||a||2 ) Finally, multiplying X by λ = 1/T (we are interested in the space coordinate) we obtain: ' ( 2 1 − ||a|| 1 λX = 1 − ||a||2 x +

a, x a ∓ a . 1 ∓ a, x 1 + 1 − ||a||2 Here obtained again two different transformations depending on the choice of the sign “ + ” or “ − ”, which are inverse one of the other. This explains the fact that also in the Einstein gyrogroup the inverse of a is −a. Choosing the “ − ” sign we obtain (19). Remark 3. Einstein addition (19) defined on B1n corresponds to a left gyrotranslation of xD ∈ D by a ∈ D Considering c = 1, u = a and v = xD in (4) both additions coincide. Conversely, considering a = u/c and xD = v/c in (19) we obtain the Einstein addition on the relativistic ball Bcn , i.e., we obtain formula (4). The unit ball B1n endowed with Einstein addition gives rise to the Einstein gyrogroup (B1n , ⊕E ). In Section 10 we will compute Einstein gyrations as spin representations of the group Spin(n).

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6. The proper velocity gyrogroup (Rn , ⊕U ) The proper velocity addition was defined by Ungar and is obtained by projection of the boost’s action (17) on the manifold of rays to the hyperboloid H + . 1 + ||xH ||2 , X = xH , with xH ∈ Rn , cosh(α) = Proposition 5. Let T = 1 + ||a||2 , sinh(α) = ±||a||, and ω = a/||a||. Then the restriction of the action (17) to the hyperboloid H + gives the proper velocity addition ⊕U : 1 + ||a||2 − 1

a, xH + 1 + ||xH ||2 a. (20) a ⊕U xH = xH + 2 ||a|| Proof. Considering in (17) the substitutions mentioned in Proposition 5, together with xH = x, we obtain, by straightforward computations, the time and space components: T = 1 + ||a||2 1 + ||x||2 ∓ a, x and

1 + ||a||2 − 1

a, x a ∓ 1 + ||x||2 a. X =x+ 2 ||a||

Since the restriction to the hyperboloid corresponds to the multiplication by λ = 1, the final result is X. The choice of the “ + ” sign gives (20). Also in the proper velocity gyrogroup the inverse of a is −a. Remark 4. The proper velocity addition (20) defined on H + corresponds to a left gyrotranslation of xH ∈ H + by a ∈ H + . Considering c = 1, u = a and v = xH in (6) both additions coincide. Conversely, considering a = u/c and xH = v/c in (6) we obtain the proper velocity addition on the relativistic case, i.e., we obtain formula (4). The space Rn endowed with the proper velocity addition (20) gives rise to the proper velocity gyrogroup (Rn , ⊕U ). In Section 10 we will compute the gyrations for the proper velocity model as spin representations of the group Spin(n).

7. Relation between different velocities In this section we calculate the relation between non-standard velocities aM ∈ B1n (elements of the M¨ obius gyrogroup), and the relativistic velocities, namely, coordinate velocities aE (elements of the Einstein gyrogroup) and proper velocities aU (elements of the proper velocity gyrogroup). This can be easily done from Propositions 3, 4, and 5 since the three different velocities aM , aE , and aU correspond only to different parametrizations of the hyperbolic functions.


73

Proposition 6. The non-standard velocity aM , the Einstein velocity (or coordinate velocity) aE and the proper velocity aU are related by: 2 1 − 1 − ||aE ||2 a ; aE = M aE ; (21) aM = 1 + ||aM ||2 ||aE ||2 −1 + 1 + ||aU ||2 2 aM = aU = aU ; aM ; (22) ||aU ||2 1 − ||aM ||2 aE

=

1 aU ; 1 + ||aU ||2

aU

=

1 aE . 1 − ||aE ||2

(23)

Proof. We only show (21). For the other formulas the reasonings are analogous. Equating both expressions for cosh(α) and sinh(α) from Propositions 3 and 4, we obtain a system of equations from which the solutions give the relations between ||aM || and ||aE || : ⎧ ⎧ 2||aM || 1 + ||aM ||2 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ||aE || = 1 + ||aM ||2 ⎨ 1 − ||aM ||2 = 1 − ||a ||2 E ⇔ ⎪ ⎪ ||a 2||a || || M E 1 − 1 − ||aE ||2 ⎪ ⎪ ⎪ ⎪ = ⎩ . ⎩ ||a || = M 1 − ||aM ||2 1 − ||aE ||2 ||aE || Therefore, aM and aE are related by: 2 aE = aM 1 + ||aM ||2

and

aM =

1−

1 − ||aE ||2 aE . ||aE ||2

(24)

Remark 5. Comparing the scalar factors in formulas (21), (22), and (23) with the factors in Table 2 we conclude that the changing between different velocities aM , aE and aU corresponds exactly with the transition of coordinates between the different models (hyperboloid, disc and paraboloid), in the projective hyperbolic model. Moreover, replacing aM = acM , aE = acE , and aU = acU in (21), (22), and (23), with aM , aE ∈ Bcn , aU ∈ Rn , we obtain the relativistic factors between different velocities in the relativistic case: 2c2 c(c − c2 − || aE ||2 ) aE = a ; aM = M aE ; (25) 2 2 c + || aM || || aE ||2 c(−c + c2 + || aU ||2 ) 2c2 aM = a ; = aM ; (26) a U U || aU ||2 c2 − || aM ||2 c c aE = aU = aU ; aE . (27) c2 + || aU ||2 c2 − || aE ||2 These relativistic factors can be written in a more elegant way using the γ and β functions, (3) and (7), as we will see in Section 9, Table 3.

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8. Gyrovector spaces The concept of gyrovector space was introduced by A. Ungar to develop analytic hyperbolic geometry as an analogue of vector spaces that provides the setting for analytic Euclidean geometry [20]. Some gyrocommutative gyrogroups admit scalar multiplication which obeys the following axioms: (V1 )

1⊗a=a

Identity element

(V2 )

(r + s) ⊗ a = r ⊗ (s ⊗ a)

Scalar distributive law

(V3 )

(rs) ⊗ a = r ⊗ (s ⊗ a)

Scalar associative law

(V4 )

|r|⊗a ||r⊗a||

Scaling property

(V5 )

gyr[u, v](r ⊗ a) = r ⊗ gyr[u, v]a

Gyroautomorphism property

(V6 )

gyr[r ⊗ v, s ⊗ v] = I

Identity automorphism

=

a ||a||

for all r, s ∈ R and a ∈ G. In general, gyroaddition does not distribute with scalar multiplication (r ⊗ (u ⊕ v) = (r ⊗ u) ⊕ (r ⊗ v)). A gyrovector space is a gyrocommutative gyrogroup of a real inner product vector space from which it inherits an inner product and a norm and such that it admits a scalar multiplication. A gyrovector space will be denoted by the triple (G, ⊕, ⊗). When all the gyrations reduce to the identity, the gyrovector space reduces to a vector space. Following Ungar [17, 20], the Möbius and Einstein gyrogroups possess the same scalar multiplication which is defined by r r 1 + ||v|| − 1 − ||v|| c c v r r r ⊗E v = r ⊗M v = c 1 + ||v|| + 1 − ||v|| ||v|| c

c

v ||v|| . = c tanh r tanh−1 c ||v||

(28)

Also the proper velocity gyrogroup admits a scalar multiplication given by /'* (r '* (r 0 v c ||v||2 ||v|| ||v||2 ||v|| r ⊗U v = 1+ 2 + − 1+ 2 − 2 c c c c ||v||

v ||v|| . (29) = c sinh r sinh−1 c ||v|| Ungar showed that Möbius gyrovector spaces form the setting for the Poincaré ball model of hyperbolic geometry while, similarly, Einstein gyrovector spaces form the setting for the Beltrami-Klein ball model of hyperbolic geometry.

9. Gyrovector space isomorphims In this section we explain how to obtain an isomorphism between different gyrovector spaces.


75

Theorem 1. Let (G, ⊕) be a gyrogroup, X an arbitrary space, and φ : X → G a bijection between G and X. Then X endowed with the induced operation a ⊕X b := φ−1 (φ(a) ⊕ φ(b)),

a, b ∈ X

(30)

becomes a gyrogroup. Proof. First we prove the existence of a left identity. Let 0 be the identity element in G and e = φ−1 (0). Then for all a ∈ X we have e ⊕X a = φ−1 (φ(e) ⊕ φ(a)) = φ−1 (0 ⊕ φ(a)) = φ−1 (φ(a)) = a. Thus, e = φ(0) is the identity element in (X, ⊕X ). Now, for each a ∈ X there exists a left inverse given by φ−1 (1 φ(b)). Indeed, we have φ−1 (1 φ(b)) ⊕X b =

φ−1 (φφ−1 (1 φ(b)) ⊕ φ(b))

=

φ−1 (1 φ(b) ⊕ φ(b))

=

φ−1 (0) = e.

Gyrations on X are given by gyr[a, b]d = φ−1 (gyr[φ(a), φ(b)]φ(d)),

a, b, d ∈ X.

(31)

Now we prove the left gyroassociative law: a ⊕X (b ⊕X d) = (a ⊕X b) ⊕X gyr[a, b]d,

a, b, d ∈ X.

We have a ⊕X (b ⊕X d)

= φ(φ−1 (a) ⊕ φ−1 (b ⊕X d)) = φ−1 (φ(a) ⊕ (φ(b) ⊕ φ(d)) = φ−1 ((φ(a) ⊕ φ(b)) ⊕ gyr[φ(a), φ(b)]φ(d)) = φ−1 (φ(a ⊕X b) ⊕ φφ−1 (gyr[φ(a), φ(b)]φ(d))) = φ−1 (φ(a ⊕X b) ⊕ φ(gyr[a, b]d)) = (a ⊕X b) ⊕X gyr[a, b]d.

By (31) and since φ is a bijection between G and X it follows that gyr[a, b]d is an automorphism of (X, ⊕X ). Finally we prove the left loop property gyr[a ⊕X b, b]d = φ−1 (gyr[φ(a ⊕X b), φ(b)]φ(d)) = φ−1 (gyr[φ(a) ⊕ φ(b), φ(b)]φ(d)) = φ−1 (gyr[φ(a), φ(b)]φ(d)) = gyr[a, b]d.

This theorem allows us to construct some new examples of gyrogroups as well as to relate different gyrogroups.

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n Examples 1. 1. Let φ : B n → S+ denote the mapping from B1n (the unit ball n n embedded in Rn+1 . Then we can inin R ) and the upper hemisphere S+ n duce a gyrogroup structure on S+ starting from the Möbius or the Einstein gyrogroups on B1n .

2. The relative coordinate velocity v ∈ Bc3 and proper velocity w ∈ R3 of an object measured by its coordinate time and proper time, respectively, are related by the equations: 1 1 w = γv v = ∈ R3 and v = βω = ∈ Bc3 . 2 ||v|| ||w||2 1 − c2 1 + c2 Thus, defining the bijective mapping φ : R3 → Bc3 , φ(w) = βw w, with inverse φ−1 : Bc3 → R3 , φ−1 (v) = γv v, Einstein and proper velocities are related by w1 ⊕U w2 = φ−1 (φ(w1 ) ⊕E φ(w2 )),

w1 , w2 ∈ R3 .

3. Let (G, ⊕, ⊗) be a gyrovector space. Since scalar multiplication on G defines a bijective mapping, we can consider φ : G → G, ψ(a) = s ⊕ a, a ∈ G, s ∈ R and construct the s-gyrogroup given by a ⊕s b = s−1 ⊗ ((s ⊗ a) ⊕ (s ⊗ b)). In particular, considering our examples a whole class of new gyrogroups can be constructed, called s-Möbius, s-Einstein, and s-proper velocity gyrogroups. The respective gyroadditions are given by a ⊕s,M b = s−1 ⊗M ((s ⊗M a) ⊕M (s ⊗M b));

(32)

a ⊕s,E b = s

−1

⊗E ((s ⊗E a) ⊕E (s ⊗E b));

(33)

a ⊕s,U b = s

−1

⊗U ((s ⊗U a) ⊕U (s ⊗U b)).

(34)

In the case X is a gyrovector space it is possible to establish an isomorphism between gyrovector spaces preserving the gyroaddition, the scalar multiplication and the inner product. Definition 3. [20] Let (G, ⊕G , ⊗G ) and (H, ⊕H , ⊗H ) be two gyrovector spaces. A bijective mapping φ : G → H is an isomorphism from G to H if for all a, b ∈ G and r ∈ R, 1) φ(a ⊕G b) = φ(a) ⊕H φ(b);

(35)

2)

φ(r ⊗G b) = r ⊗H φ(b);

(36)

3)

φ(a), φ(b)

a, b = . ||a|| ||b|| ||φ(a)|| ||φ(b)||

(37)

Two isomorphic gyrovector spaces are two equivalent models of the abstract gyrovector space. Condition (37) ensures that isomorphic gyrovector spaces have equal angle measures. An isomorphism between gyrovector spaces preserves gyrations, i.e., φ(gyrG [a, b]c) = gyrH [φ(a), φ(b)]φ(c). (38)


77

The isomorphisms between the gyrovectors spaces of M¨ obius, Einstein and proper velocities are presented in Table 3 (see [20]). The relativistic factors presented in Table 3 coincide with the relativistic factors (21), (22), and (23). Gyrogroup isomorphism

Formula

φUE : (Bsn , ⊕E , ⊗E ) → (Rn , ⊕U , ⊗U )

φUE : v → γv v

(Bsn , ⊕E , ⊗E )

φEU : v → βv v

φEU : (R , ⊕U , ⊗U ) → n

φME :

(Bsn , ⊕E , ⊗E )

→

(Bsn , ⊕M , ⊗M )

φME : v →

1 2

⊗E v

φEM : (Bsn , ⊕M , ⊗M ) → (Bsn , ⊕E , ⊗E ) φEM : v → 2 ⊗M v φUM : (Bsn , ⊕M , ⊗M ) → (Rn , ⊕U , ⊗U ) φMU : (Rn , ⊕U , ⊗U ) → (Bsn , ⊕M , ⊗M )

φUM : v → 2γv2 v φMU : v →

βv 1+βv v

Table 3. Gyrovector space isomorphisms. From Table 3 we can see that the isomorphism between Einstein addition and M¨ obius addition in the ball Bcn is very simple in the language of gyrovector spaces:

1 1 ⊗M u ⊕M ⊗M v u ⊕E v = 2 ⊗M (39) 2 2 and u ⊕M v = with

1 ⊗E ((2 ⊗E u) ⊕E (2 ⊗E v)) , 2

c(c − c2 − ||v||2 ) 1 1 γv ⊗M v = ⊗E v = v= v 2 2 1 + γv ||v||2

(40)

(41)

and 2 ⊗M v = 2 ⊗E v =

2c2 2γv2 v = v. 2γv2 − 1 c2 + ||v||2

(42)

This explains that the 2-Einstein gyrogroup is the Möbius gyrogroup and reciprocally, the 1/2-Möbius gyrogroup is the Einstein gyrogroup. Moreover, the scalar multiplication ⊗M (or ⊗E ) is a generalization of the relativistic factors (41) and (42). Einstein, Möbius, and proper velocity gyrovector spaces are three equivalent models of the abstract gyrovector space as we have seen in Sections 4, 5, and 6. They provide the setting for three models of the hyperbolic geometry of Bolyai and Lobachevsky. The gyrotrigonometry in each model was already considered by A. Ungar ([20, 22]).

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10. Möbius, Einstein and proper gyrations as spin representation of the group Spin(n) Gyrations play the central role in the gyrogroup theory. Therefore, a simple expression for gyrations is desirable in order to facilitate computations. From the abstract theory of gyrogroups, gyrations on a gyrogroup (G, ⊕) are given in terms of the gyroaddition by gyr[u, v]w = (u ⊕ v) ⊕ (u ⊕ (v ⊕ w)).

(43)

In the case of M¨ obius gyrogroups a simpler expression for M¨ obius gyrations was obtained in [9] and [10] as a spin representation of the group Spin(n), using Clifford algebra theory. In Clifford analysis it is well known that Möbius transformations on the unit ball of B1n (see [1, 2, 5]) can be expressed as fractional linear transformations of the form ϕa (x) = (x + a)(1 + ax)−1 , a, x ∈ B1n (44) or in the case of the relativistic ball Bcn by

−1 x ax = (x + a) 1 + 2 Ψa (x) = c ϕ a , a, x ∈ Bcn . (45) c c c The connection between Möbius transformations and Möbius addition on Bcn is given by a ⊕M b = Ψa (b). (46) In [10], M¨ obius gyrations were computed as spin representation of the form sxs, s ∈ Spin(n), x ∈ Bcn . They are expressed by gyrM [a, b]x =

c2 + ab c2 + ab x , ||c2 + ab|| ||c2 + ab||

a, b, x ∈ Bcn .

(47)

By (30), Table 3, and (46) we can write Einstein and proper velocity additions as fractional linear transformations: a ⊕E b = =

φEM (φME (a) ⊕M φME (b)) ⎛ (−1 ⎞ ' φME (a)φME (b) ⎠ 2 ⊗M ⎝(φME (a) + φME (b)) 1 + c2

(48)

and a ⊕U b

= =

φUM (φMU (a) ⊕M φMU (b)) ⎛ '

φMU (a)φMU (b) φUM ⎝(φMU (a) + φMU (b)) 1 + c2

(−1 ⎞ ⎠.

(49)

From (31), Table 3, and (47), Einstein and proper velocity gyrations can be written as a spin representation of the group Spin(n) : gyrE [a, b]x = φEM (gyrM [φME (a), φME (b)]φME (x)),

a, b, x ∈ Bcn

(50)


79

and gyrU [a, b]x = φUM (gyrM [φMU (a), φMU (b)]φMU (x)),

a, b, x ∈ Rn .

(51)

The gyrosemidirect product between the M¨ obius, Einstein or proper velocity gyrogroup and the spin group Spin(n) gives three different realizations of the proper Lorentz group Spin+ (1, n). Finally we would like to remark that the “gamma” and “beta” identities (2), (5), and (8) can be derived from our considerations and the well-known equality about Möbius transformations: ||Ψa (x)||2 (c2 − ||a||2 )(c2 − ||x||2 ) 1− = . (52) 2 c ||c2 + ax||2 Acknowledgments The research of the author was (partially) supported by CIDMA – “Centro de Investiga¸ca õ e Desenvolvimento em Matem´ atica e Aplica¸c˜ oes” of the University of Aveiro. The author would like to thank Frank Sommen, Uwe Kähler, and Paula Cerejeiras for useful discussions on the hyperbolic projective model.

References [1] L. Ahlfors, M¨ obius transformations in several dimensions, University of Minnesota School of Mathematics, Minneapolis, Minn, 1981. [2] L. Ahlfors, M¨ obius transformations in Rn expressed through 2×2 matrices of Clifford numbers, Complex Variables, 5 (1986), 215-221. [3] P. Cerejeiras, U. K¨ ahler, F. Sommen, Clifford analysis on projective hyperbolic space, J. Nat. Geom. 22 (2002), 19-34. [4] P. Cerejeiras, U. K¨ ahler, F. Sommen, Clifford analysis on projective hyperbolic space II, Math. Methods Appl. Sci. 25 (2002), 1465-1477. [5] J. Cnops, Hurwitz pairs and applications of M¨ obius transformations, Habilitation dissertation, Universiteit Gent, Faculteit van de Wetenschappen, 1994. [6] R. Delanghe, F. Sommen, and V. Sou˘cek, Clifford algebras and spinor-valued functions, Kluwer Acad. Publishers, Dordrecht, 1992. [7] D. Eelbode, Clifford analysis on the hyperbolic unit ball, PhD. Thesis, Ghent, Belgium, 2004. [8] M. Ferreira, Spherical continuous wavelets transforms arising from sections of the Lorentz group, Appl. Comput. Harmon. Anal., 26 (2009), 212-229. [9] M. Ferreira, Factorizations of M¨ obius gyrogroups, Adv. in Appl. Clifford Algebr., 19 (2009), 303-323. [10] M. Ferreira, and G. Ren, M¨ obius gyrogroups: A Clifford algebra approach, J. Algebra (2010), doi:10.1016/j.jalgebra.2010.05.014, in press. [11] Y. Friedman, Physical Applications of Homogeneous Balls, Birkh¨ auser, 2005. [12] I. R. Porteous, Clifford Algebras and the Classical Groups, Cambridge Univ. Press, Cambridge, 1995.

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[13] J. A. Rhodes, and M. D. Semon, Relativistic velocity space, Wigner rotation, and Thomas precession, Am. J. Physics 72(7), (2004), 943-960. [14] A. A. Ungar, Thomas precession and the parametrization of the Lorentz transformation group, Found. Phys. Lett. 1 (1988), 57-89. [15] A. A. Ungar, The abstract Lorentz transformation group, Amer. J. Phys. 60, (1992), 815-828 . [16] A. A. Ungar, Extension of the unit disk gyrogroup into the unit ball of any real inner product space, J. Math. Anal. Appl., 202(3) (1996), 1040-1057. [17] A. A. Ungar, Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics, Found Phys. 27(6) (1997), 881-951. [18] A. A. Ungar, M¨ obius transformations of the ball, Ahlfors’ rotation and gyrovector spaces. In Themistocles M. Rassias (ed.): Nonlinear analysis in geometry and topology, Hadronic Press, Palm Harbor, FL, 2000, 241-287. [19] A. A. Ungar, Beyond the Einstein addition law and its gyroscopic Thomas precession, volume 117 of Fundamental Theories of Physics, Kluwer Academic Publishers Group, Dordrecht, 2001. [20] A. A. Ungar, Analytic Hyperbolic Geometry - Mathematical Foundations and Applications, World Scientific, 2005. [21] A. A. Ungar, From M¨ obius to gyrogroups, Amer. Math. Montly, 115(2) (2008), 138144. [22] A. A. Ungar, Analytic Hyperbolic Geometry and Albert Einsteins Special Theory of Relativity, World Scientific, 2008. [23] N. Vilenkin, and A. Klimyk, Representation of Lie groups and special functions Vol.2, Kluwer Acad. Publishers, Netherlands, 1993. Milton Ferreira School of Technology and Management Polytechnic Institute of Leiria P-2411-901 Leiria Portugal and CIDMA, University of Aveiro, Portugal e-mail: [email protected] [email protected]

Invariant Operators of First Order Generalizing the Dirac Operator in 2 Variables Peter Franek Abstract. In this paper, we analyze differential operators of first order acting between vector bundles associated to G/P where G = Spin(n + 2, 2) and P is a parabolic subgroup. The operators in question are invariant with respect to the group G and we identify them with operators on the flat space that are invariant with respect to its subgroup SL(2) × Spin(n). For n even, the list of all such invariant operators of first order is obtained using certain algebraic conditions on the highest weights of the representations in question. In some cases, an explicit realization of the operator is given in coordinates. Mathematics Subject Classification (2010). Primary 22E46; Secondary 32W99. Keywords. Differential operator, complex, higher spin, Dirac, generalized Verma module.

1. Introduction 1.1. Invariant differential operators Let G be a group acting on two sets F1 and F2 . We will say that a mapping D : F1 → F2 is G-invariant if [D, G] = 0, i.e., for all g ∈ G, x ∈ F1 g · (D(x)) = D(g · x). Further, Fi will be either the vector space of smooth functions between vector spaces, or the space of smooth sections of a vector bundle. If V, W are representations of G, then there is a natural action of G on C ∞ (V, W) defined by (g · f )(v) := g · (f (g −1 · v)), where · is the action of G on C ∞ (V, W), W and V, respectively. Similarly, if P is a subgroup of G and V is a representation of P , the natural action of G on sections of the vector bundle Γ(G ×P V) is (g · s)(g P ) := g · (s(g −1 g P )), whereas the action of G on G ×P V is just g · [g , v]P := [gg , v]P . This work was completed with the support of the grant MSM 0021620839 and GACR 201/08/397.


81

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P. Franek

This corresponds to a change of observer in physics. If the group G is the group of symmetry of a particular theory, then the operators arising are usually invariant with respect to this group. The most classical examples are the following: Gradient ∇ : C ∞ (Rn , R) → C ∞ (Rn , Rn ) is O(n)-invariant, if O(n) acts on n R by the defining representation and trivially on R. In a similar way, the divergence and rotation acting on R3 -valued functions are SO(3)-invariant operators. Laplace Δ : C ∞ (Rn , R) → C ∞ (Rn , R) is O(n)-invariant. All this examples may be constructed from the de Rham differential d : Ωk (M ) → Ωk+1 (M ) acting on k-forms on a manifold that is invariant with respect to the group of all diffeomorphisms of M . Let S be the complex spinor representation of Spin(n), · : Rn ⊗ S → S the Clifford multiplication (this is a homomorphism of Spin-modules, considering Rn as the vector representation of Spin(n)). Then we define the Dirac operator ∂x : C ∞ (Rn , S) → C ∞ (Rn , S),

f →

ei · ∂i f.

i

One can easily show that ∂x is Spin(n)-invariant (it is, however, not Pin(n)invariant). The Dirac

operator may be defined on any Riemannian manifold with a Spinstructure by ei · ∇i , where ∇ is the Levi-Civita connection of the metric that defines the spin structure and {ei } an orthonormal frame. The simplest example is the Spin-structure Spin(n + 1) → Spin(n + 1)/Spin(n) " S n over the sphere S n . It acts between sections Γ(Spin(n + 1) ×Spin(n) S) and is Spin(n + 1)-invariant. 1.2. Dirac operator in k variables Recently, many variations and generalizations of the classical Dirac operator appeared. While mathematical physicists study its spectra on different Riemannian spin-manifolds ([12]) and others construct its analogs in non-Riemannian geometries (see, e.g., [16]), we may define the Dirac operator in several Clifford variables by the following way (compare, e.g., [6]): Definition 1.1. The Dirac operator in k variables is the differential operator Dk : C ∞ ((Rk )∗ ⊗ Rn , S) → C ∞ ((Rk )∗ ⊗ Rn , Ck ⊗ S), where (Rk )∗ ⊗ Rn has coordinates x11 , . . . , xn1 , . . . , x1k , . . . , xnk , defined by Dk : f → ( ei · ∂i1 f, . . . , ei · ∂ik f ), i

i

∂ij

where = ∂/∂xij and we identified the second space with k spinor-valued components. Proposition 1.2. Dk is invariant with respect to SL(k) × Spin(n), considering the natural action of SL(k) × Spin(n) on the spaces (Rk )∗ ⊗ Rn , S " C ⊗ S and Ck ⊗ S.

Operators of First Order Generalizing the Dirac Operator

83

Proof. Each linear operator between function spaces C ∞ (A, V) and C ∞ (A, W) of order at most 1 invariant with respect to a group G is defined by π ◦ ∇, where π is some G-homomorphism A∗ ⊗ V → W ([20]). In our case, G = SL(k) × Spin(n),

A∗ ⊗ V = (Rk ⊗ (Rn )∗ ) ⊗ (C ⊗ S).

Let f1 , . . . , fk be the canonical basis of Rk (and Ck = Rk ⊗R C as well) and f 1 , . . . , f k the dual basis. The SL(k)-part decomposes trivially as Rk ⊗ C " Ck . Therefore, π((fj ⊗v j )⊗(1⊗s)) = fj ⊗˜ π (v j ⊗s), where π ˜ is a Spin(n)-homomorphism n ∗ j n ∗ (R ) ⊗ S → S (v ∈ (R ) , s ∈ S). The Spin(n)-part decomposes (Rn )∗ ⊗ S " S ⊕ T , where T is the twistor module ([3]). The Spin-projection on the S-part is v ∗ ⊗ s → v · s, where s ∈ S, v ∗ ∈ (Rn )∗ , v ∈ Rn corresponds to v ∗ under the isomorphism Rn " (Rn )∗ induced by the standard metric g on Rn and · is the Clifford multiplication. This is because the duality Rn " (Rn )∗ induced by the metric defining Spin(n) is clearly a Spin(n)homomorphism and the Clifford multiplication is a Spin(n)-homomorphism as n i well. i } be the

Leti {e

orthonormal

frame ion R and { } the dual coframe. Then i i π ˜ (( i vj ) ⊗ s) = i vj ei · s = i ei · vj s.

So, ((π ◦ ∇)f )( j f j ⊗ vj ) = j fj ⊗ i ei · ∂ij f which gives the result. For k = 1, D1 is the usual Dirac operator. For n = 2, spinors are just complex numbers and Dk is the first operator (∂¯1 , . . . , ∂¯k ) in the Dolbeault complex ([13]). The resolution of Dk is a generalization of the Dolbeault complex and is not known in general, although some informations are known for particular k and n (see, e.g., [9, 10, 19, 4, 6, 7, 18]). 1.3. Verma modules and invariant operators in parabolic geometry Let g be a Lie algebra, p a parabolic subalgebra, i.e., a subalgebra containing a Borel subalgebra of g. Let U(g) be the universal enveloping algebra of g and V a representation of p. We define the generalized Verma module Mp (V) := U(g) ⊗U (p) V. The action of g is simply left multiplication in U(g). If V is a representation of a group P with Lie algebra p as well, Mp (V) can be given the structure of a (g, P )-module. Let G be a Lie group. A subgroup P is called parabolic, if it containes a Borel subgroup of G, i.e., if its Lie algebra p is a parabolic subalgebra of g. Proposition 1.3. Let V and W be representations of P . There exists a nonzero G-invariant linear differential operator D : Γ(G ×P V) → Γ(G ×P W) if and only if there exists a nonzero (g, P )-homomorphism of generalized Verma modules Mp (W∗ ) → Mp (V∗ ) associated to dual representations W∗ , V∗ . The proof can be found in, e.g., [5]. Further, we will denote Γ(G ×P V) simply as Γ(V). Let G = Spin(n + 2, 2), P the subgroup fixing a 2-plane in the nullset of the metric that defines Spin(n+2, 2), g, p their complexified Lie algebras. As matrix Lie

84

P. Franek

algebras, g " so(n + 4, C) may be realized as the space of matrices anti-symmetric with respect to the anti-diagonal, and we define the Cartan subalgebra h to be the set of all diagonal matrices diag(a1 , . . . , a[n/2]+2 , (0), −a[n/2]+2 , . . . , −a1 ) (the zero is in the middle if n is odd) (see [14] for details). The subalgebra p is then ⎛ ⎞ g 0 g1 g2 p ∼ ⎝ 0 g0 g1 ⎠ , (1.1) 0 0 g0 its reductive part g0 " gl(2, C) × so(n, C). The commutative subalgebra h is also a Cartan subalgebra in p. Elements of h∗ will be called weights and h∗ the weight space. Let us define the basis {i }i of h∗ by i (diag(a1 , . . . , a[n/2]+2 , . . .)) := ai and choose a set of positive roots Φ+ of g as in [14]. Denote by Hγ the coroot corresponding to γ. Each irreducible representation of p contains a one-dimensional weight space of weight λ such that λ(Hγ ) is a nonnegative integer for each positive root γ such that the (−γ)-root space is contained in p (in other words, γ is a root of p). Let us denote the set of such weights by Pp++ . The p+ = g1 ⊕ g2 -part acts trivially on each irreducible p-module, so there is a correspondence between irreducible finite-dimensional pmodules and irreducible finite-dimensional g0 " gl(2, C)×so(n, C)-modules. There exists an irreducible finite-dimensional p-module with highest weight λ if and only if λ ∈ Pp++ and this representation is unique up to isomorphism ([14, 15]). A weight is in Pp++ if and only if its expression in the {i }-basis is λ = [a1 , a2 |b1 , b2 , . . . , b[n/2] ] such that a1 ≥ a2 , a1 − a2 ∈ Z, bi − bj ∈ Z for i = j, and b1 ≥ . . . ≥ b[n/2]−1 ≥ |b[n/2] | for n even, resp. a1 ≥ a2 and b1 ≥ . . . ≥ b[n/2] ≥ 0 for n odd. For example, the module C2 ⊗ C (product of the defining vector representation of gl(2) and trivial representation of so(n)) has highest weight [1, 0|0, . . . , 0], C ⊗ S (product of trivial representation of gl(2) and spinor representation of so(n)) has highest weight [0, 0|1/2, . . . , 1/2], and (C2 )∗ ⊗ S has highest weight [0, −1|1/2, . . . , 1/2].

Let us define the weight ρ := 1/2 φ∈Φ+ φ to be the half sum of all positive roots and the Weyl group to be the finite subgroup of SO(h∗ ) generated by the root reflections {sγ ; γ ∈ Φ+ } (on h∗ , there is a natural g-invariant metric induced by the Killing form, and sγ is the orthogonal transformation fixing the hyperplane γ ⊥ , sγ λ = λ − λ(Hγ )γ, see [15] for details). In our case, ρ = [. . . , 5/2, 3/2, 1/2] for n odd, ρ = [. . . , 2, 1, 0] for n even, and elements of the Weyl group correspond to sign-premutation [a1 , . . . , a[n/2]+2 ] → [±aπ(1) , . . . , ±aπ([n/2]+2) ], where π is a permutation. For n even, the Weyl group consists of all sign-permutations with an even number of sign-changes and for n odd, the Weyl group consists of all sign-permutations. Let us denote by Mp (λ) the generalized Verma module with highest weight λ − ρ. It follows from the Harris-Chandra theorem ([15]) that a homomorphism Mp (μ) → Mp (λ) may exist only if μ and λ are on the same orbit of the Weyl group.


85

Let b be the Borel subalgebra of g, consisting of h and all positive root spaces. It is a minimal parabolic subalgebra and the associated generalized Verma module Mb (V) is called “true” Verma module. Any highest weight module with highest weight λ is a factor of Mb (Cλ ), where Cλ is the one-dimensional representation of b with weight λ. The highest weight module Mp (λ) is a factor of the Verma modules Mb (Cλ−ρ ) =: Mb (λ). Any g-homomorphism of true Verma modules factors to a homomorphism of generalized Verma modules called standard homomorphism of generalized Verma modules. Let λ := [3/2, 1/2| . . . , 5/2, 3/2, −1/2] μ := [3/2, −1/2| . . . , 5/2, 3/2, 1/2] (for n even, for n odd similar). The p modules Vλ and Vμ with highest weight λ − ρ resp. μ − ρ can be considered as P -representations and, as P ss " SL(2) × Spin(n)-modules, are V∗ " C ⊗ S+ , W∗ " C2 ⊗ S− (S± are the irreducibe spinor representations of Spin(n) for n even). It was proved in [9] that there exists a G = Spin(n + 2, 2)-invariant operator D : Γ(V∗ ) → Γ(W∗ ). The parabolic subalgebra p induces a gradation g = ⊕2i=−2 gi such that ⊕i≥0 = p. Then g: = ⊕i 0. We calculate sα (μ) = sα (wλ − γ ) = wλ − (wλ)(Hα )α − (γ − γ (Hα )α). On the other hand, sα (μ) = μ − μ(Hα )α = wλ − γ − μ(Hα )α. Comparing the right-hand sides, we obtain γ (Hα ) = wλ(Hα ) − μ(Hα ).


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Because wλ(Hα ) is a negative integer and μ(Hα ) a positive integer, it follows γ (Hα ) ≤ −2. Recall that α was a simple root and γ (E) = 1. Now let us analyze the roots of the even orthogonal Lie algebra: α is one of the roots [1, −1|0 . . . , 0], [0, 0|1, −1, . . . , 0], [0, 0|0, 1, −1, . . . , 0], . . ., [0, . . . , 1, −1] and [0, . . . , 1, 1] (note that [0, 1| − 1, 0 . . . , 0] is not a root from Φp ). On the other hand, the condition γ (E) = 1 restricts the possibilities for γ to [1, 0|0, . . . , ±1, 0, . . .] or [0, 1| . . . , ±1, . . .]. The action of the reflection sα on it (sign)-interchanges one 1 with the neighbour 0 and γ (Hα )α = γ − sα γ = ±α in all cases, so γ (Hα ) = ±1, contradicting γ (Hα ) ≤ −2. Therefore, the standard homomorphism Mp (μ) → Mp (λ) is nonzero and it follows from Proposition 1.3 that there exists a nonzero linear differential operator Γ(V∗λ ) → Γ(V∗μ ). It was proved in [9] that for λ(E) − μ(E) ∈ {1, 2}, λ(E) − μ(E) is the order of the operator. Let us assume (1), i.e., there exists a nonzero standard differential operator of first order. By [9], λ(E)−μ(E) = 1 and by 1.3, we obtain the existence of a nonzero homomorphism of generalized Verma modules Mp (Vμ ) → Mp (Vλ ). So, the weights λ and μ are connected with a chain of weights λ = λ0 , λ1 = sβ1 λ0 , . . ., λm+1 = sβm+1 λm = μ, λi (Hβi+1 ) ∈ N. The root β1 cannot be in Φp , because otherwise Mb (μ) ⊂ Mb (sβ1 λ) and the homomorphism of generalized Verma modules would be zero as in the previous part. The condition λ(E) − μ(E) = 1 implies β1 (E) = 1. If m > 0, all the further βi must be in Φp , so μ = wsβ1 λ for w ∈ Wp implying w−1 μ = sβ1 λ. This is a contradiction, because μ ∈ Pp++ + ρ, so μ − w−1 μ is a sum of positive roots, whereas we assumed that w−1 μ − μ = sβ1 λ − μ is a sum of positive roots as well. So, m = 0 and there is only one reflection μ = sβ1 λ. Remark 1.5. The first part of the previous theorem cannot be generalized to the odd case, as shows the following counter-example: λ = [3/2, 1/2|5/2, 1/2],

μ = [3/2, −1/2|5/2, 1/2],

λ1 = [3/2, 1/2|5/2, −1/2].

The weights λ and μ are connected by a root reflection, λ(E) − μ(E) = 1 and μ = sγ μ1 , where sγ interchanges the second and fourth coordinate. Moreover, μ1 = sα λ, sα ∈ Wp changes the sign of the last coordinate. Remark 1.6. There are indications that under the assumptions in the theorem above, all invariant first order operators are already standard. However, we are going to prove this.

2. Invariant operators acting between higher spin modules 2.1. Classification of first order operator on G/P in terms of weights Let G = Spin(n + 2, 2), P the parabolic subgroup described in section (1.3) and denote, as before, by Vν the irreducible finite dimensional P -module with highest weight ν − ρ. Assume that n is even.

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Theorem 2.1. There exists a standard linear invariant differential operator Γ(V∗λ−ρ ) → Γ(V∗μ−ρ ) of first order if and only if λ, μ are one of the following weights: (1) λ = [l, m|a1 , . . . , m−1, . . . , as ], μ = [l, m−1|a1 , . . . , m, . . . , as ], λ, μ ∈ Pp++ +ρ (i.e., l − m ∈ Z, l > m, l, m, a1 , . . . , as all integers or all half-integers and a1 > . . . > m > m − 1 > . . . > as−1 > |as | resp. a1 > . . . > as−1 > max{|m|, |m − 1|} if m is on the end). (2) λ = [l, m|a1 , . . . , l − 1, . . . , as ], μ = [l − 1, m|a1 , . . . , l, . . . , as ], λ, μ ∈ Pp++ + ρ. (3) λ = [l, m|a1 , . . . , −(m − 1), . . . , as ], μ = [l, m − 1|a1 , . . . , −m, . . . , as ], λ, μ ∈ Pp++ + ρ. (4) λ = [l, m|a1 , . . . , −(l − 1), . . . , as ], μ = [l − 1, m|a1 , . . . , −l, . . . , as ]. λ, μ ∈ Pp++ + ρ. Proof. It follows from Theorem 1.4 that there exists a linear standard invariant differential operator Γ(V∗λ ) → Γ(V∗μ ) of first order if and only if the weights λ = [k, l|a1 , . . . , as ],

μ = [k , l |a1 , . . . , as ]

are connected by a root reflection sγ , λ(E) − μ(E) = (k + l) − (k + l ) = 1 and μ = λ − γ. Each root reflection corresponds to a transposition, or sign-transposition. Because λ(E) − μ(E) = (μ + γ − μ)(E) = γ(E) = 1, the weight γ may be either of type [1, 0| . . . , ±1, 0, . . .] or [0, 1| . . . , ±1, 0, . . .]. So, sγ transposes or sign-transposes the first or second coordinate with some further coordinate. Obviously, (1), (2) corresponds to such transposition and (3), (4) to sign-transposition. 2.2. Explicit realizations in simple cases As we already mentioned in the introduction, to any of these operators we may assign a differential operator on the flat space, identifying g−2 -invariant sections of G ×P V with V-valued functions on (R2 )∗ ⊗ Rn . The simplest cases λ = [3/2, 1/2| . . . 3/2 , ±1/2] and μ = [3/2, −1/2| . . . , 3/2, ∓1/2] for n even has already been discussed and the corresponding differential operator is the Dirac operator in two variables (the weights are of type (1) or (3) with m = 1/2). Let us consider a slightly generalized case λ = [(2l +1)/2, 1/2| . . . , 3/2, ±1/2], μ = [(2l + 1)/2, −1/2| . . . , 3/2, ∓1/2] for n even. The highest weights of the representation Vλ is λ−ρ = [(−n+2l−1)/2, (−n+1)/2|1/2, . . ., ±1/2]. So, Vλ is a product of a gl(2, C) representation with highest weight (−n + 2l − 1)/2 − (−n + 1)/2 = ± l − 1. As a gss where Vl−1 is the 0 " sl(2, C) × so(n, C)-module, Vλ " Vl−1 ⊗ S ± l-dimensional sl(2, C) module with highest weight l − 1 and S are the irrecudible spinor representations of so(n, C). Similarly, one may show that, as gss 0 -module, Vμ " Vl ⊗ S∓ . We know from Theorem 2.1 that there exists a nonzero invariant differential operator of first order D1 : Γ(V∗λ ) → Γ(V∗μ ), V∗λ

V∗μ

(2.1)

where and are representations of P and " SL(2) × Spin(n). Each irreducible SL(2) module is self-dual (there is only one irreducible representation in each finite dimension) and the Spin(n)-module S+ and S− are dual to each P0ss


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other. As SL(2)-module, Vl " Syml C2 (irreducible SL(2)-modules can be realized as symmetric powers of the defining vector representation). We will try to describe the analog of this operator on flat space in coordinates. Let us identify sections of Γ(V) with V-valued functions on g− and restrict to sections that correspond to functions constant in the g−2 -direction. Because g−1 " (R2 )∗ ⊗ Rn as gss 0 " sl(2, C) × Spin(n)-module, these functions may be considered as function in variables x11 , . . . , xn1 , x12 , . . . , xn2 . Each operator of first order involves only derivatives with respect to the g−1 directions ([9, 20]), so the operator sends g−2 -constant functions into g−2 -constant functions. This operator is still gss 0 " SL(2) × Spin(n)-invariant and we may try to write down its form in coordinates. l−2 Let us choose the basis of Syml−1 C2 by {el−1 · e2 , . . . , el−1 1 , e1 2 } and a l 2 similar basis of Sym C . Then we can identify the space V∗λ with a vector of l spinor components and V∗μ with a vector of l + 1 spinor components. Proposition 2.2. After these identifications the operator corresponding to (2.1) is an operator acting on (R2 )∗ ⊗ Rn and has the following expression: ⎞ ⎛ ⎛ ⎞ ∂x 1 f 0 f0 ⎜ ∂x1 f1 + ∂x2 f0 ⎟ ⎟ ⎜ f1 ⎟ ⎜ ⎟ ⎜ ⎟ ... D1 : ⎝ → ⎜ ⎟ ⎜ ... ⎠ ⎝∂x1 fl−1 + ∂x2 fl−2 ⎠ fl−1 ∂x2 fl−1

i where ∂xi = j ej · ∂j is the Dirac operator in the i-th variable. Proof. We know from [20] that the SL(2) × Spin(n)-invariant first order operator has the form π ◦ ∇, where π : ((R2 )∗ ⊗ Rn )∗ ⊗ (Syml−1 C2 ) ⊗ S) → Syml C2 ⊗ S. Such a projection is of the form π1 ⊗ π2 where π1 is the SL(2)-projection R2 ⊗ Syml−1 C2 → Syml C2 and π2 is the Spin(n)-projection (Rn )∗ ⊗ S → S. Clearly, π1 is just the symmetrization and π2 was discussed in the proof of Proposition 1.2, π2 (i ⊗ v) = ei · v. So, we may write l−1 l j el−1−j · e ⊗ s )) = el−j · ej2 ⊗ (v1 · sj + v2 · sj−1 ) π((e1 ⊗ v1∗ + e2 ⊗ v2∗ ) ⊗ ( j 1 2 1 j=0

j=0

where we define sl = s−1 = 0 and vi ∈ R corresponds to vi∗ ∈ (Rn )∗ under the isomorphism induced by the standard

metric defining Spin(n). So, the (j + 1)-th component of the right-hand side is i v1i ei · sj +

v2i ei · sj−1 (j = 0, . . . , l) and i the (j + 1)-th component of π ◦ ∇ is i ei · ∂1 fj + i ei · ∂2i fj−1 which gives the result. n

Further, consider another simple case of a first-order operator, corresponding to the weights λ = [1/2, −(2l + 1)/2| . . . , 3/2, ±1/2] and μ = [−1/2, −(2l + 1)/2| . . . , 3/2, ∓1/2]. After substracting ρ from these weights, one can easily see that, as sl(2, C) × so(n, C)-modules, V∗λ " Vl ⊗ S± and V∗μ " Vl−1 ⊗ S∓ . Let us

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realize the space Vl again as the symmetric tensor power Syml C2 and Vl−1 as the kernel of the symmetrizing operator S : C2 ⊗ Syml C2 → Syml+1 C2 . Choose a basis of Syml C2 by {el1 , el−1 · e2 , . . . , el2 } and a basis of Ker S by 1 · e2 − el−1 · e22 , . . . , e2 ⊗ e1 · el−1 − e1 · el2 } (2.2) {e2 ⊗ el1 − el1 · e2 , e2 ⊗ el−1 1 1 2 (· is the symmetric tensor product). We again know from Theorem 2.1 that there exists a nonzero invariant differential operator of first order D3 : Γ(V∗λ ) → Γ(V∗μ )

(2.3)

(the weights are of type (2) or (4) with l = 1/2). Again, we may restrict to sections constant in g−2 and identify them with functions on g−1 . Proposition 2.3. After identifying the SL(2) × Spin(n)-module V∗λ with the space of vectors of l + 1 spinor components and V∗μ with the space of vectors of l spinor components via the basis above, the corresponding operator has the following form: ⎞ ⎛ ⎛ ⎞ ∂x2 f0 − (1/l)∂x1 f1 f0 ⎜∂x2 f1 − (2/(l − 1))∂x1 f2 ⎟ ⎟ ⎜ f1 ⎟ ⎜ ⎟ ⎜ ⎟ (2.4) D3 : ⎝ ⎠ , → ⎜ ⎜∂x2 f2 − (3/(l − 2))∂x1 f3 ⎟ ... ⎠ ⎝ ... fl ∂x2 fl−1 − l∂x1 fl where ∂xi is the Dirac operator in the i-th variable. Proof. As before, D3 = π ◦ ∇ where π is a SL(2) × Spin(n)-homomorphism π : (R2 ⊗ (Rn )∗ ) ⊗ (Syml C2 ⊗ S) → Ker S ⊗ S. The projection π is a product of projections π1 ⊗ π2 , where π1 : R2 ⊗ Syml C2 → Ker S. One can easily see that π1 (v ⊗T ) = v ·T −v ⊗T for T ∈ Syml C2 . Symmetric product of symmetric tensors is given by (v1 · v2 · . . . · vk ) · (vk+1 · . . . · wk+l ) =

j 1/(k + l)! π vπ(1) ⊗ . . .⊗ vπ(k+l) . We calculate now coordinates of π1 (ei ⊗ el−j 1 · e2 ) l−j j l−j j l−j j+1 in the basis (2.2) of Ker S. First, π1 (e2 ⊗ e1 · e2 ) = e2 ⊗ e1 · e2 − e1 · e2 which is the (j + 1)-th basis vector of (2.2). For π(e1 ⊗ el−j · ej2 ), we have to find 1 a constant k such that e1 ⊗ el−j · ej2 − e1l−j+1 · ej2 = k(e2 ⊗ e1l−j+1 · ej−1 − e1l−j+1 · ej2 ), 1 2 in other words, a k such that the difference e1 ⊗ el−j · ej2 − k e2 ⊗ e1l−j+1 · ej−1 is 1 2 symmetric. Using the definition of the symmetric product and easy combinatorics, we obtain that k = j/(l + 1 − j). So, identifying Syml C2 with (l + 1)-component vectors and Ker S with l-component vectors via the basis (2.2), we obtain for any vi = vi1 e1 + vi2 e2 , ⎛ ⎞ ⎛ 2 ⎞ v0 − (1/l) v11 v0 ⎜ v1 ⎟ ⎜v12 − 2/(l − 1) v21 ⎟ ⎟ ⎜ ⎟. π1 ⎜ (2.5) ⎝. . .⎠ = ⎝ ⎠ ... 2 1 vl−1 − l vl vl


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The projection π2 is again the Clifford multiplication and writing down the expression π ◦ ∇ gives the result. Remark 2.4. Let λ = [(2l + 1)/2, −1/2| . . . 3/2, ±1/2],

μ = [1/2, −(2l + 1)/2| . . . , 3/2, ±1/2].

Using representation theory and homomophisms of generalized Verma modules, it can be shown that there exists a G-invariant differential operator D2 : Γ(V∗λ ) → Γ(V∗μ ). Moreover, D

D

D

Γ(V1 ) →1 Γ(V2 ) →2 Γ(V3 ) →3 Γ(V4 ) is a complex (D1 is defined by (2.1), D3 by (2.3) and Vi the spaces described above). However, the middle operator is no more of first order. It can be shown ([9]) that the order of D2 is not higher as the difference λ(E)− μ(E) = (2l + 1)/2 − 1/2 − (1/2 − (2l + 1)/2) = 2l. Remark 2.5. If the weights are of type λ = [(2l + 1)/2, 1/2|half − integers, ±1/2],

μ = [(2l + 1)/2, −1/2| . . . , ∓1/2],

the representation spaces can be realized as Syml−1 C2 ⊗ W and Syml C2 ⊗ W where W and W are some higher Spin representations. In the simplest case when l = 1 and the spinor representations W resp. W have highest weights [3/2, 1/2, . . . , ±1/2] resp. [3/2, 1/2, . . . , ∓1/2], the operator corresponding to Theorem 2.1 is the Rarita-Schwinger operator in two variables R1 f f → R2 f , studied in, e.g., [8]. For higher l, the SL(2)-decomposition does not change, so, if V∗λ " Syml−1 C2 ⊗ W, V∗μ " Syml C2 ⊗ W and the operator is ⎛ ⎞ ⎞ ⎛ R1 f0 f0 ⎜R1 f1 + R2 f0 ⎟ ⎜ ⎟ ⎜ f1 ⎟ ⎟ → ⎜R1 f2 + R2 f1 ⎟ ⎜ ⎜ ⎟ ⎝ ... ⎠ ⎝ ⎠ ... fl−1 R2 fl−1 where Ri is the Rarita-Schwinger operator in the i-th variable. Remark 2.6. The operators written down in coordinates are invariant with respect to the group SL(2) × Spin(n) which is a subgroup of G = Spin(n + 2, 2). Choosing the action of the center of G0 in a proper way, these are also GL(2) × Spin(n)invariant. If we identify sections Γ(V) with V-valued functions on the whole g− by (1.2) (not restricting to g−2 -constant functions), we deal with function of 2n + 1 variables (g−2 is one-dimensional in our case). The formulas for the Dirac operator in two variables, or (2.4), (2.5) do not change, because the derivatives are never in the g−2 -directions for a first-order operator. The action of G = Spin(n + 2, 2) is hard to describe on these function (it does not fix the g−2 -constant functions). However, under this action, the operators are G-invariant.

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References [1] I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, Differential Operators on the Base Affine Space and a Study of g-Modules, Lie Groups and Their Representations, I. M. Gelfand, Ed., Adam Hilger, London, 1975 [2] I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, Structure of Representations that are generated by vectors of highest weight, Functional. Anal. Appl. 5 (1971) [3] J. Bureˇs, V. Souˇcek, Regular spinor valued mappings, Seminarii di Geometria, Bologna 1984, ed. S.Coen, Bologna 1986, 7-22 [4] J. Bureˇs, A. Damiano, I. Sabadini, Explicit resolutions for the complex of several Fueter operators, J. Geom. Phys. vol. 57 (2007), 765-775 [5] A. Cap, J. Slovak, Parabolic Geometries I: Background and General Theory, AMS, Mathematical Surveys and Monographs, 2009, 628 pp. [6] F. Colombo, I. Sabadini, F. Sommen, D.C. Struppa, Analysis of Dirac Systems and Computational Algebra, Birkh¨ auser, 2004. [7] F. Colombo, V. Souˇcek, D. Struppa, Invariant resolutions for several Fueter operators, J. Geom. Phys. vol. 56, (2006), 1175-1191 [8] A. Damiano, Algebraic analysis of the Rarita-Schwinger operator in dimension three, Archivum Mathematicum, vol. 42 (2006), issue 5, pp. 197-211 [9] P. Franek, Generalized Dolbeault complex in parabolic geometry, Journal of Lie Theory 18 (2008) No. 4, 757–774 [10] P. Franek, Dirac Operator in Two Variables from the Viewpoint of Parabolic Geometry, Advances in Applied Clifford Algebras, Birkhuser Basel, vol. 17, 2007 [11] P. Franek, Generalized Verma module homomorphisms in singular character, Archivum Mathematicum, Proceedings of the 26th Winter school Geomtry and Physics 2006 [12] T. Friedrich, Dirac-Operatoren in der Riemannschen Geometrie, AMS, 2000 [13] J.E. Gilbert, M.A. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge studies in advanced mathematics, vol. 26, Cambridge, 1991 [14] R. Goodmann, N. R. Wallach, Representations and Invariants of the Classical Groups, Cambridge University Press, Cambridge 1998. [15] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer Verlag, 1980 [16] S. Kr´ ysl, Classification of 1st order symplectic spinor operators over contact projective geometries, Differential Geom. Appl. 26 (2008), no. 5, 553–565. [17] J. Lepowsky, A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra, 49 (1977), 496-511 [18] I. Sabadini, F. Sommen, D.C. Struppa, P. Van Lancker, Complexes of Dirac operators in Clifford algebras, Math. Zeit., 239 no. 2 (2002), 293–320 [19] L. Krump, The generalised Dolbeault complex for four Dirac operators in the stable rank, Proceedings of the ICNAAM, Kos 2008, AIP Conference Proceedings 2008, 670-673


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[20] J. Slov´ ak, V. Souˇcek, Invariant operators of the first order on manifolds with a given parabolic structure (English, French summary), Global analysis and harmonic analysis (Marseille-Luminy, 1999), 251–276, Smin. Congr., 4, Soc. Math. France, Paris, 2000 [21] N. Verma, Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968) Peter Franek Mathematical institute Sokolovsk´ a 83 186 75 Prague Czech Republik e-mail: [email protected]

The Zero Sets of Slice Regular Functions and the Open Mapping Theorem Graziano Gentili and Caterina Stoppato Abstract. A new class of regular quaternionic functions, defined by power series in a natural fashion, has been introduced in [11, 12]. Several results of the theory recall the classical complex analysis, whereas other results reflect the peculiarity of the quaternionic structure. The recent work [1, 2] identified a larger class of domains, on which the study of regular functions is most natural and not limited to the study of quaternionic power series. In the present paper we extend some basic results concerning the algebraic and topological properties of the zero set to regular functions defined on these domains. We then use these results to prove the Maximum and Minimum Modulus Principles and a version of the Open Mapping Theorem in this new setting. Mathematics Subject Classification (2010). 30G35, 30C15, 30C80. Keywords. Functions of one quaternionic variable, zero set, maximum modulus principle, minimum modulus principle, open mapping theorem.

1. Introduction Let H be the real algebra of quaternions. Its elements are of the form q = x0 +ix1 + jx2 + kx3 where the xn are real, and i, j, k, are imaginary units (i.e., their square equals −1) such that ij = −ji = k, jk = −kj = i, and ki = −ik = j. The richness of the theory of holomorphic functions of one complex variable inspired the study of several interesting theories of quaternionic functions during the last century. The most famous was introduced by Fueter, [5, 6], and excellently surveyed in [16]. For recent work on Fueter regularity, see, e.g., [3, 14] and references therein. A different notion of regularity for quaternionic functions, inspired by Cullen [4], has been proposed in [11, 12]. Several classical results in complex analysis have quaternionic analogs, proven in [12] and in the subsequent papers [8, 9, 10, 13, 15] Both authors are partially supported by the GNSAGA of the INdAM and by the PRIN “Propriet` a geometriche delle variet` a reali e complesse”. The second author is also partially supported by the PRIN “Geometria Differenziale e Analisi Globale” of the MIUR.


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for regular functions on open balls B(0, R) = {q ∈ H : |q| < R}. The recent [2] identified a larger class of domains that are the quaternionic analogs of the (complex) domains of holomorphy. The study of regular functions is most natural on these domains, and it does not reduce to the study of quaternionic power series. In the present paper we extend to regular functions defined on these domains the results proven in [8, 9], which concern the (algebraic and) topological structure of the zero set and the openness of regular functions. We also extend the Maximum and Minimum Modulus Principles proven in [12] and [9] respectively. Let us present the basics of the theory, beginning with some notation. Denote by S the two-dimensional sphere of quaternion imaginary units: S = {q ∈ H : q 2 = −1}. For all imaginary units I ∈ S, let LI = R + IR be the complex line through 0, 1 and I. The definition of regularity given in [2] follows. Definition 1.1. Let Ω be a domain in H and let f : Ω → H be a function. For all I ∈ S, we set ΩI = Ω ∩ LI and denote the restriction f|ΩI by fI . The restriction fI is called holomorphic if fI ∈ C 1 (ΩI ) and if

∂ 1 ∂ ¯ +I fI (x + Iy) ∂I f (x + Iy) = (1) 2 ∂x ∂y vanishes identically. The function f is called slice regular (or simply regular ) if fI is holomorphic for all I ∈ S.

For instance, a quaternionic power series n∈N q n an with an ∈ H defines a regular function in its domain of convergence, which proves to be an open ball B(0, R) = {q ∈ H : |q| < R}. Conversely, the following is proven in [2] (generalizing [12]). Theorem 1.2. Let f : Ω → H be a regular function. If p ∈ Ω∩R and if B = B(p, R) is the largest open ball centered at p and included in Ω, then there exist quaternions

an ∈ H such that f (q) = n∈N (q − p)n an for all q ∈ B. In particular, f ∈ C ∞ (B). Expanding a regular function at a non-real point p ∈ H \ R is a much more delicate matter. A detailed study of quaternionic analyticity has been conducted in [7]. For the purpose of the present paper it suffices to know that choosing a “bad” domain of definition Ω may lead to regular functions that are not even continuous: Example 1.3. Let Ω be any neighborhood of S in H which does not intersect the real axis (e.g., Ω = T (S, r) = {q ∈ H : d(q, S) < r} with r < 1/2). Choose I ∈ S and define f : Ω → H by setting fI = f−I ≡ 1, fJ ≡ 0 for all J ∈ S \ {±I}. Since all constant functions are holomorphic, f is regular according to Definition 1.1. Clearly f is not continuous at I (nor at any point of ΩI ). Such pathologies can be avoided by requiring the domain of definition to have certain topological and geometric properties. The first such condition is the following. Definition 1.4. Let Ω be a domain in H, intersecting the real axis. If ΩI = Ω ∩ LI is a domain in LI " C for all I ∈ S, then we say that Ω is a slice domain.

Zero Sets of Slice Regular Functions

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The definition of a slice domain was given in [1, 2], while the following result was proven in [15]. Theorem 1.5 (Identity Principle). Let Ω be a slice domain and let f, g : Ω → H be regular. Suppose that f and g coincide on a subset C of ΩI , for some I ∈ S. If C has an accumulation point in ΩI , then f ≡ g in Ω. Another natural condition for the domain of definition of a regular function is the following symmetry property. Definition 1.6. A subset C of H is axially symmetric if, for all x + yI ∈ C with x, y ∈ R, I ∈ S, the whole set x + yS = {x + yJ : J ∈ S} is contained in C. For the sake of simplicity, we will call such a C a symmetric set. It is worth to point out that symmetric slice domains play the role played by the domains of holomorphy in classical complex analysis, as proven in [2]: Theorem 1.7 (Extension Theorem). Let Ω ⊆ H be a slice domain, and let f : Ω → → H of f H be a regular function. There exists a unique regular extension f : Ω which contains Ω. to the smallest symmetric slice domain Ω The distribution of the values of a regular function on each sphere x + yS contained in its domain of definition is quite special: the following result is proven in [1], generalizing [9], and it has been extended in [2]. Theorem 1.8. Let f be a regular function on a symmetric slice domain Ω ⊆ H. For each x + yS ⊂ Ω there exist constants b(x, y), c(x, y) ∈ H such that for all I ∈ S, f (x + yI) = b(x, y) + Ic(x, y).

(2)

In other words, the function S → H mapping I → f (x + yI) is affine. This immediately implies the following result, proven in [1, 2] extending [12]. Corollary 1.9. Let f be a regular function on a symmetric slice domain Ω ⊆ H and let x + yS ⊂ Ω. If there exist distinct I, J ∈ S such that f (x + yI) = 0 = f (x + yJ) then f ≡ 0 in x + yS. Furthermore, by direct computation, 1 K [f (x + yK) + f (x − yK)] , c(x, y) = [f (x − yK) − f (x + yK)] 2 2 for all K ∈ S, so that b, c are C ∞ functions. Hence, as proven [1], b(x, y) =

Corollary 1.10. If Ω is a symmetric slice domain and f is regular in Ω, then f ∈ C ∞ (Ω). Theorem 1.8 is also the basis for the following extension result, a restatement of Lemma 4.4 in [2]; a similar result, based on the Cauchy representation formula, had been previously proven in [1].

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Lemma 1.11 (Extension Lemma). Let Ω be a symmetric slice domain and choose I ∈ S. If fI : ΩI → H is holomorphic, then setting 1 I (3) f (x + yJ) = [fI (x + yI) + fI (x − yI)] + J [fI (x − yI) − fI (x + yI)] 2 2 extends fI to a regular function f : Ω → H. f is the unique such extension and it is denoted by ext(fI ). The Extension Lemma 1.11 is used in [2] to endow regular functions with a multiplicative operation. Its definition is recalled in Section 2, together with other algebraic tools introduced in the same paper. The original part of this paper is structured as follows. After studying the algebraic properties of the zero set in Section 3, we explore its topology and prove the following result in Section 4: Theorem 1.12 (Structure of the Zero Set). Let Ω ⊆ H be a symmetric slice domain and let f : Ω → H be a regular function. If f does not vanish identically, then the zero set, Zf , of f consists of isolated points or isolated 2-spheres of the form x+yS (with x, y ∈ R and y = 0). Even though the statement above replicates the one established for quaternionic power series in their domain of convergence, its proof requires a different approach that relies upon extension results proven in [2]. This leads, in particular, to a stronger version of the Identity Principle: Theorem 1.13 (Strong Identity Principle). Let f, g be regular functions on a symmetric slice domain Ω. If there exists a 2-sphere (or a singleton) S = x + yS ⊂ Ω such that the zeros of f − g contained in Ω \ S accumulate to a point of S, then f ≡ g on the whole Ω. Section 5 is devoted to the Maximum and Minimum Modulus Principles. Proving them we have to face the peculiarities of the quaternionic context. The approach is different from the one used in [12]. Theorem 1.14 (Maximum Modulus Principle). Let Ω be a slice domain and let f : Ω → H be regular. If |f | has a relative maximum at p ∈ Ω, then f is constant. Theorem 1.15 (Minimum Modulus Principle). Let Ω be a symmetric slice domain and let f : Ω → H be a regular function. If |f | has a local minimum point p ∈ Ω, then either f (p) = 0 or f is constant. As one may expect, these principles are the main tools in the investigation of the topological properties of regular functions. In Section 6 we define the degenerate set of f as the union Df of the 2-spheres x + yS such that f|x+yS is constant. It turns out that Df has no interior points and that f is open on the rest of the domain: Theorem 1.16 (Open Mapping Theorem). Let f be a regular function on a symmetric slice domain Ω and let Df be its degenerate set. Then f : Ω \ Df → H is open.


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Removing the degenerate set is necessary, as shown by a counterexample. Under this point of view, the theory of quaternionic regular functions differs from that of holomorphic complex functions. This depends on the fact that the zero set of a holomorphic function is discrete, while a regular quaternionic function may vanish on a whole 2-sphere as explained above.

2. Preliminary results The set of regular functions on a symmetric slice domain Ω becomes an algebra over R when endowed with the usual addition + and an appropriate multiplicative operation denoted by ∗ and called regular multiplication (indeed, the pointwise multiplication does not preserve regularity). In the special case where the domain is a ball centered at 0, we can make use of the power series expansion and define the ∗-multiplication by the formula ' ( ' ( n n n q an ∗ q bn = qn ak bn−k n∈N

n∈N

n∈N

k=0

(see [8] for details). The general definition, given in [2], is based on the following result. Lemma 2.1 (Splitting Lemma). Let Ω ⊆ H be a symmetric slice domain and let f : Ω → H be a regular function. For any I, J ∈ S, with I ⊥ J there exist holomorphic functions F, G : ΩI → LI such that for all z ∈ ΩI , fI (z) = F (z) + G(z)J.

(4)

In order to define the regular product of two regular functions f, g on a symmetric slice domain Ω, let I, J ∈ S, with I ⊥ J, and choose holomorphic functions F, G, H, K : ΩI → LI such that for all z ∈ ΩI , fI (z) = F (z) + G(z)J,

gI (z) = H(z) + K(z)J.

(5)

Let fI ∗ gI : ΩI → LI be the holomorphic function defined by fI ∗ gI (z) = [F (z)H(z) − G(z)K(¯ z )] + [F (z)K(z) + G(z)H(¯ z )]J.

(6)

Using the Extension Lemma 1.11, the following definition is given in [2]: Definition 2.2. Let Ω ⊆ H be a symmetric slice domain and let f, g : Ω → H be regular. The function f ∗ g(q) = ext(fI ∗ gI )(q) defined as the extension of (6) is called the regular product of f and g. Remark 2.3. The ∗-multiplication is associative, distributive but, in general, not commutative.

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G. Gentili and C. Stoppato The regular conjugate of a power series, is defined in [8] by the formula ' (c n q an = qn a ¯n . n∈N

n∈N

In [2] this concept is extended as follows. Definition 2.4. Let f be a regular function on a symmetric slice domain Ω and suppose f splits on ΩI as in formula (5), fI (z) = F (z) + G(z)J. We consider the holomorphic function fIc (z) = F (¯ z ) − G(z)J

(7)

and define, according to the Extension Lemma 1.11, the regular conjugate of f by the formula f c (q) = ext(fIc )(q).

(8)

Furthermore, the following definition is given under the same assumptions. Definition 2.5. The symmetrization of f is defined as f s = f ∗ f c = f c ∗ f.

(9)

In the case of power series, ' (s n q n an = qn ak a ¯n−k , n∈N

n∈N

k=0

n

where k=0 ak a ¯n−k ∈ R for all n ∈ N. In the general case, when f splits on ΩI as in formula (5), then fI ∗ fIc = (F (z) + G(z)J) ∗ (F (¯ z ) − G(z)J) = F (z)F (¯ z ) + G(z)G(¯ z ).

(10)

Hence f s (q) = ext(fI ∗ fIc )(q).

(11)

Finally, in Section 5, we will make use of the following operation, defined in [2], to derive the Minimum Modulus Principle from the Maximum Modulus Principle. Definition 2.6. Let f be a regular function on a symmetric slice domain Ω. The regular reciprocal of f is the function f −∗ : Ω \ Zf s → H defined by the equation f −∗ (q) =

1 f c (q). f s (q)

(12)

As observed in [2], by direct computation f −∗ is the inverse of f with respect to ∗-multiplication, i.e., f ∗ f −∗ = f −∗ ∗ f ≡ 1 on Ω \ Zf s .


101

3. Algebraic properties of the zero set We now study the correspondences among the zeros of f and g and those of the product f ∗ g, the conjugate f c and the symmetrization f s . We begin by recalling an alternative expression of the regular product f ∗ g, proven in [2]. Proposition 3.1. Let Ω ⊆ H be a symmetric slice domain and let f, g : Ω → H be regular functions. For all q ∈ Ω, if f (q) = 0, then f ∗ g(q) = 0, else f ∗ g(q) = f (q) g(f (q)−1 qf (q)).

(13)

Corollary 3.2. Let Ω ⊆ H be a symmetric slice domain and let f, g : Ω → H be regular functions. Then f ∗ g(q) = 0 if and only if f (q) = 0 or f (q) = 0 and g(f (q)−1 qf (q)) = 0. In particular, for each zero of f ∗ g in S = x + yS there exists a zero of f or a zero of g in S. However, [8] presented examples of products f ∗ g whose zeros were not in one-to-one correspondence with the union of the zero sets of f and g. We now study the relation between the zeros of f and those of f c and f s . We need two preliminary steps. Lemma 3.3. Let Ω ⊆ H be a symmetric slice domain and let f : Ω → H be a regular function such that f (ΩI ) ⊆ LI for all I ∈ S. If f (x0 + y0 I0 ) = 0 for some I0 ∈ S, then f (x0 + y0 I) = 0 for all I ∈ S. Proof. The fact that f (ΩI ) ⊆ LI for all I ∈ S implies that f (x) is real for all x ∈ Ω ∩ R. We now have a holomorphic function fI0 : ΩI0 → LI0 mapping Ω ∩ R to R. By the (complex) Schwarz Reflection Principle, f (x + yI0 ) = f (x − yI0 ) for all x + yI0 ∈ ΩI0 . Since f (x0 + y0 I0 ) = 0, we conclude that f (x0 − y0 I0 ) = 0, and Corollary 1.9 allows us to deduce the thesis. Lemma 3.4. Let Ω ⊆ H be a symmetric slice domain, let f : Ω → H be a regular function and let f s be its symmetrization. Then f s (ΩI ) ⊆ LI for all I ∈ S. Proof. As observed in [2], it follows by direct computation from equation (10).

Proposition 3.5. Let Ω ⊆ H be a symmetric slice domain, let f : Ω → H be regular and choose S = x0 +y0 S ⊂ Ω. The zeros of f in S are in one-to-one correspondence with those of f c . Furthermore, f s vanishes identically on S if and only if f s has a zero in S, if and only if f has a zero in S (if and only if f c has a zero in S). Proof. If q0 = x0 +y0 I0 is a zero of f , then f s = f ∗f c vanishes at q0 by Proposition 3.1. According to Lemmas 3.3 and 3.4, f s (x0 + y0 I) = 0 for all I ∈ S. By Corollary 3.2, the fact that f s (¯ q0 ) = f s (x0 − y0 I0 ) = 0 implies that either c −1 f (¯ q0 ) = 0 or f (f (¯ q0 ) q¯0 f (¯ q0 )) = 0. In the first case we conclude that f vanishes identically on S, which implies that f c vanishes on S because of formula (7). In the second case, f c vanishes at the point f (¯ q0 )−1 q¯0 f (¯ q0 ) = x0 −y0 [f (¯ q0 )−1 I0 f (¯ q0 )] ∈ S. s We have proven that if f has a zero in S, then f has a zero in S, which leads to the vanishing of f s on the whole S, which implies the existence of a zero of f c in S. Since (f c )c = f , exchanging the roles of f and f c proves the thesis.

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4. Topological properties of the zero set We now study the distribution of the zeros of regular functions on symmetric slice domains. In order to obtain a full characterization of the zero set of a regular function, we first deal with a special case that will be crucial in the proof of the main result. Lemma 4.1. Let Ω ⊆ H be a symmetric slice domain and let f : Ω → H be a regular function such that f (ΩI ) ⊆ LI for all I ∈ S. If f ≡ 0, the zero set of f is either empty or it is the union of isolated points (belonging to R) and isolated 2-spheres of the type x + yS. Proof. We know from Lemma 3.3 that the zero set of such an f consists of real points and 2-spheres of the type x + yS. Now choose I in S and notice that the intersection of LI with the zero set of f consists of all the real zeros of f and of exactly two zeros for each sphere x + yS on which f vanishes (namely, x + yI and x − yI). If f ≡ 0, then, by Theorem 1.5, the zeros of f in LI must be isolated. Hence the zero set of f consists of isolated real points and isolated 2-spheres. We now state and prove the result on the topological structure of the zero set of regular functions. Theorem 4.2 (Structure of the Zero Set). Let Ω ⊆ H be a symmetric slice domain and let f : Ω → H be a regular function. If f does not vanish identically, then the zero set of f consists of isolated points or isolated 2-spheres of the form x + yS. Proof. Consider the symmetrization f s of f : by Lemma 3.4, f s fulfills the hypotheses of Lemma 4.1. Hence the zero set of f s consists of isolated real points or isolated 2-spheres. According to Proposition 3.5, the real zeros of f and f s are exactly the same. Furthermore, each 2-sphere in the zero set of f s corresponds either to a 2-sphere of zeros, or to a single zero of f . This concludes the proof. As an immediate consequence of the previous result, we can strengthen the Identity Principle 1.5. Theorem 4.3 (Strong Identity Principle). Let f, g be regular functions on a symmetric slice domain Ω. If there exists a 2-sphere (or a singleton) S = x + yS ⊂ Ω such that the zeros of f − g contained in Ω \ S accumulate to a point of S, then f ≡ g on the whole Ω.

5. The Maximum and Minimum Modulus Principles The Maximum Modulus Principle is a consequence of the analogous result for holomorphic functions. Our proof uses the Identity Principle 1.5, thus we must work on a slice domain. Theorem 5.1 (Maximum Modulus Principle). Let Ω be a slice domain and let f : Ω → H be regular. If |f | has a relative maximum at p ∈ Ω, then f is constant.


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Proof. If f (p) = 0, then the thesis is obvious. Else we may suppose f (p) ∈ R, f (p) > 0, possibly multiplying f by f (p). Let I, J ∈ S be such that p ∈ LI and I ⊥ J; let F, G : ΩI → LI be holomorphic functions such that fI = F + GJ. Then |F (p)|2 = |fI (p)|2 ≥ |fI (z)|2 = |F (z)|2 + |G(z)|2 ≥ |F (z)|2 for all z in a neighborhood UI of p in ΩI . Hence |F | has a relative maximum at p and the Maximum Modulus Principle for holomorphic functions of one complex variable allows us to conclude that F is constant. Namely, F ≡ f (p). Now, |G(z)|2 = |fI (z)|2 −|F (z)|2 = |fI (z)|2 −|fI (p)|2 ≤ |fI (p)|2 −|fI (p)|2 = 0 for all z ∈ UI . Hence fI = F ≡ f (p) in UI . Since Ω is a slice domain, the Identity Principle 1.5 allows us to conclude that f ≡ f (p) in Ω. The Minimum Modulus Principle proven [9] for power series extends to all regular functions on symmetric slice domains with a very similar proof, which we repeat for the sake of completeness. We first find an alternative expression of the regular reciprocal f −∗ . Proposition 5.2. Let f be a regular function on a symmetric slice domain Ω. Then, for all q ∈ Ω \ Zf s , 1 , (14) f −∗ (q) = f (Tf (q)) where Tf : Ω \ Zf s → Ω \ Zf s is defined by Tf (q) = f c (q)−1 qf c (q). Furthermore, Tf and Tf c are mutual inverses so that Tf is a diffeomorphism. Proof. As explained in Proposition 3.5, if f s (q) = 0, then f c (q) = 0. Hence Tf is well defined on Ω\ Zf s . According to Proposition 3.1, f c (q)∗ g(q) = f c (q)g(Tf (q)). We compute: f −∗ (q) = f s (q)−1 f c (q) = [f c ∗ f (q)] = [f c (q)f (Tf (q))]

−1

−1

f c (q)

f c (q) = f (Tf (q))−1 f c (q)−1 f c (q) = f (Tf (q))−1 .

Moreover, Tf : Ω \ Zf s → H maps any 2-sphere (or real singleton) x + yS to itself. In particular Tf (Ω \ Zf s ) ⊆ Ω \ Zf s (indeed, Zf s is symmetric as explained in Proposition 3.5). The conjugacy operation is an involution, i.e., (f c )c = f ; thus Tf c (q) = f (q)−1 qf (q). For all q ∈ Ω \ Zf s , set p = Tf (q) and notice that Tf c ◦ Tf (q) = Tf c (p) = f (p)−1 pf (p) −1 = f (p)−1 f c (q)−1 qf c (q) f (p) = [f c (q)f (p)] q [f c (q)f (p)] where

f c (q)f (p) = f c (q)f (f c (q)−1 qf c (q)) = f c ∗ f (q) = f s (q).

Hence

Tf c ◦ Tf (q) = f s (q)−1 qf s (q) = q, where the last equality holds because f s (q) and q commute, since they always lie in the same complex line by Lemma 3.4. The regular reciprocal f −∗ allows the proof of the following theorem.

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Theorem 5.3 (Minimum Modulus Principle). Let Ω be a symmetric slice domain and let f : Ω → H be a regular function. If |f | has a local minimum point p ∈ Ω, then either f (p) = 0 or f is constant. Proof. Consider a regular f : Ω → H whose modulus has a minimum point p ∈ Ω with f (p) = 0. Such an f does not vanish on the sphere S = x + yS through p. Indeed, if f vanished at a point p ∈ S, then |f|S | would have a relative minimum at two distinct points: p and p . On the contrary, |f|S | has one global minimum, one global maximum and no other extremal point: by Theorem 1.8, I → f (x + yI) is affine. Hence f does not have zeroes in S, nor does f s . Hence the domain Ω = Ω \ Zf s of the regular reciprocal f −∗ includes S. Thanks to Proposition 5.2, 1 |f −∗ (q)| = |f (Tf (q))| for all q ∈ Ω . By Proposition 5.2, if |f | has a minimum at p ∈ x + yS ⊆ Ω then |f ◦ Tf | has a minimum at p = Tf c (p) ∈ Ω . As a consequence, |f −∗ | has a maximum at p . By the Maximum Modulus Principle 5.1, f −∗ is constant on Ω . This implies that f is constant in Ω and, thanks to the Identity Principle 1.5, in the whole domain Ω. As in the case of power series, the Minimum Modulus Principle is the basis for the proof of the Open Mapping Theorem.

6. The Open Mapping Theorem We are now ready to extend the Open Mapping Theorem, proven in 6.4 for power series, to all regular functions on symmetric slice domains. We begin with the following result. Theorem 6.1. Let Ω be a symmetric slice domain and let f : Ω → H be a nonconstant regular function. If U is a symmetric open subset of Ω, then f (U ) is open. In particular, the image f (Ω) is open. Proof. Let p0 ∈ f (U ). Choose q0 = x0 + y0 I ∈ U such that f (q0 ) = p0 , so that f (q) − p0 has a zero on S = x0 + y0 S ⊆ U . For r > 0, consider the symmetric neighborhood of S defined by T (S, r) = {q ∈ H : d(q, S) < r}. There exists r > 0 such that T (S, r) ⊆ U and f (q) − p0 = 0 for all q ∈ T (S, r) \ S. Let ε > 0 be such that |f (q) − p0 | ≥ 3ε for all q such that d(q, S) = r. For all p ∈ B(p0 , ε) and all q such that d(q, S) = r we get |f (q) − p| ≥ |f (q) − p0 | − |p − p0 | ≥ 3ε − ε = 2ε > ε ≥ |p0 − p| = |f (q0 ) − p|. Thus |f (q) − p| must have a local minimum point in T (S, r). By the Minimum Modulus Principle 5.3, there exists q ∈ T (S, r) such that f (q) − p = 0. Definition 6.2. Let Ω be a symmetric slice domain and let f : Ω → H be a regular function. We define the degenerate set of f as the union Df of the 2-spheres x + yS (with y = 0) such that f|x+yS is constant.


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Theorem 6.1 allows the study of the topological properties of the degenerate set. Theorem 6.3. Let Ω be a symmetric slice domain and let f : Ω → H be a nonconstant regular function. The degenerate set Df is closed in Ω \ R and it has empty interior. Proof. As we saw in Theorem 1.8 and in the following discussion, there exist b, c ∈ C ∞ such that f (x + yI) = b(x, y) + Ic(x, y). Clearly, the union Γ of the 2-spheres (or real singletons) x + yS such that c(x, y) = 0 is closed in Ω. If the interior of Γ were not empty, then it would be a symmetric open set having non-open image: indeed, f (x + yI) = b(x, y) for all x + yI ∈ Γ and the image through b of a non-empty subset of R2 cannot be open in H. By Theorem 6.1, f would have to be constant, a contradiction with the hypothesis. Finally, we observe that Df = Γ \ R, so that Df must be closed in Ω \ R and have empty interior. We are now ready for the main result of this section. Theorem 6.4 (Open Mapping Theorem). Let f be a regular function on a symmetric slice domain Ω and let Df be its degenerate set. Then f : Ω \ Df → H is open. Proof. Let U be an open subset of Ω \ Df and let p0 ∈ f (U ). We will show that the image f (U ) contains a ball B(p0 , ε) with ε > 0. Choose q0 ∈ U such that f (q0 ) = p0 . Since U does not intersect any degenerate sphere, by Theorem 4.2 the point q0 must be an isolated zero of the function f (q) − p0 . We may thus choose r > 0 such that B(q0 , r) ⊆ U and f (q)− p0 = 0 for all q ∈ B(q0 , r) \ {q0 }. Let ε > 0 be such that |f (q) − p0 | ≥ 3ε for all q such that |q − q0 | = r. For all p ∈ B(p0 , ε) we get |f (q) − p| ≥ |f (q) − p0 | − |p − p0 | ≥ 3ε − ε = 2ε for |q − q0 | = r, while |f (q0 ) − p| = |p0 − p| ≤ ε. Thus |f (q0 ) − p| < min|q−q0 |=r |f (q) − p| and |f (q) − p| must have a local minimum point in B(q0 , r). Since f (q) − p is not constant, it must vanish at the same point by Theorem 5.3. Hence there exists q ∈ B(q0 , r) ⊆ U such that f (q) = p. As in [9], the non-degeneracy hypothesis cannot be removed. Example 6.5. The 2-sphere S of imaginary units is degenerate for f (q) = q −2 + 1, since f (I) = 0 for all I ∈ S. We can easily prove that f : H \ {0} → H is not open by choosing an I ∈ S and observing that the image of the open ball B = B(I, 1/2) centered at I is not open. Indeed, 0 ∈ f (B) and f (B) ∩ LJ ⊆ R when J ∈ S is orthogonal to I.

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This phenomenon does not arise in the complex case. Checking the proofs shows that the fact of the matter is that a regular quaternionic function may vanish on a whole 2-sphere while the zero set of a non-constant holomorphic function is discrete.

References [1] F. Colombo, G. Gentili, and I. Sabadini. A Cauchy kernel for slice regular functions. Ann. Global Anal. Geom., 37(4):361–378, 2010. [2] F. Colombo, G. Gentili, I. Sabadini, and D. Struppa. Extension results for slice regular functions of a quaternionic variable. Adv. Math., 222:1793–1808, 2009. [3] F. Colombo, I. Sabadini, F. Sommen, and D. C. Struppa. Analysis of Dirac systems and computational algebra, volume 39 of Progress in Mathematical Physics. Birkh¨ auser Boston Inc., Boston, MA, 2004. [4] C. G. Cullen. An integral theorem for analytic intrinsic functions on quaternions. Duke Math. J., 32:139–148, 1965. [5] R. Fueter. Die Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Variablen. Comment. Math. Helv., 7(1):307–330, 1934. ¨ [6] R. Fueter. Uber die analytische Darstellung der regul¨ aren Funktionen einer Quaternionenvariablen. Comment. Math. Helv., 8(1):371–378, 1935. [7] G. Gentili and C. Stoppato. Power series and analyticity over the quaternions. Math. Ann., to appear. [8] G. Gentili and C. Stoppato. Zeros of regular functions and polynomials of a quaternionic variable. Michigan Math. J., 56(3):655–667, 2008. [9] G. Gentili and C. Stoppato. The open mapping theorem for regular quaternionic functions. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), VIII(4):805–815, 2009. [10] G. Gentili, C. Stoppato, D. C. Struppa, and F. Vlacci. Recent developments for regular functions of a hypercomplex variable. In I. Sabadini, M. Shapiro, and F. Sommen, editors, Hypercomplex analysis, Trends in Mathematics, pages 165–186. Birkh¨ auser Verlag, Basel, 2009. [11] G. Gentili and D. C. Struppa. A new approach to Cullen-regular functions of a quaternionic variable. C. R. Math. Acad. Sci. Paris, 342(10):741–744, 2006. [12] G. Gentili and D. C. Struppa. A new theory of regular functions of a quaternionic variable. Adv. Math., 216(1):279–301, 2007. [13] G. Gentili and F. Vlacci. Rigidity for regular functions over Hamilton and Cayley numbers and a boundary Schwarz Lemma. Indag. Math. (N.S.), 19(4):535–545, 2008. [14] V. V. Kravchenko and M. V. Shapiro. Integral representations for spatial models of mathematical physics, volume 351 of Pitman Research Notes in Mathematics Series. Longman, Harlow, 1996. [15] C. Stoppato. Poles of regular quaternionic functions. Complex Var. Elliptic Equ., 54(11):1001–1018, 2009.


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[16] A. Sudbery. Quaternionic analysis. Math. Proc. Cambridge Philos. Soc., 85(2):199– 224, 1979. Graziano Gentili and Caterina Stoppato Universit` a degli Studi di Firenze Dipartimento di Matematica “U. Dini” Viale Morgagni, 67/A 50134 Firenze Italy e-mail: [email protected] [email protected]

A New Approach to Slice Regularity on Real Algebras Riccardo Ghiloni and Alessandro Perotti Abstract. We expose the main results of a theory of slice regular functions on a real alternative algebra A, based on a well-known Fueter construction. Our general theory includes the theory of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits us to extend the range of these function theories and to obtain new results. In particular, we show that a fundamental theorem of algebra with multiplicities holds for an ample class of polynomials with coefficients in A. We give several examples to illustrate some interesting aspects of the theory. Mathematics Subject Classification (2010). Primary 30C15; Secondary 30G35, 32A30. Keywords. Functions of a hypercomplex variable, quaternions, octonions, Clifford algebras, fundamental theorem of algebra.

1. Introduction In this survey paper, we propose a new approach to the concepts of “slice regularity” for functions of one quaternionic, octonionic or Clifford variable which have been recently introduced by Gentili and Struppa in [18, 19] and by Colombo, Sabadini and Struppa in [8]. Actually, the starting point for our approach is not new: it dates back to a paper of Rudolf Fueter [12], in which he proposed a simple method, which is now known as Fueter’s Theorem, to generate quaternionic regular functions (cf. [35] and [25] for the theory of Fueter regular functions) by means of complex holomorphic functions. Given a holomorphic “stem function” F (z) = u(α, β) + i v(α, β)

(z = α + iβ complex, u, v real-valued)

Partially supported by MIUR (Project “Proprietà geometriche delle varietà reali e complesse”) and GNSAGA of INdAM.


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in the upper complex half-plane, real-valued on R, the formula f (q) := u (q0 , | Im(q)|) +

Im(q) v (q0 , | Im(q)|) | Im(q)|

(with q = q0 + q1 i + q2 j + q3 k ∈ H, Im(q) = q1 i + q2 j + q3 k) defines a function on H, whose Laplacian is Fueter regular. Fueter’s construction was later extended to higher dimensions by Sce [32], Qian [30] and Sommen [34] in the setting of octonionic and Clifford analysis. We extend Fueter’s construction in order to develop a theory of slice regular functions on a real alternative algebra A with a fixed antiinvolution. These functions will be obtained by taking A-valued components u, v of the stem function F . The domains on which slice regular functions can be defined are open subsets of what we call the quadratic cone of the algebra. This cone is the whole algebra only in the case in which A is a real division algebra (i.e. the complex numbers, the quaternions H or the octonions O). We refer the reader to the article [22] for complete proofs of the main results stated in the present paper. If A is the algebra of quaternions, we get the theory of slice regular (or Cullen regular ) functions of a quaternionic variable introduced by Gentili and Struppa [18, 19]. If A is the algebra of octonions, we obtain the corresponding theory of regular functions already considered in [16, 17, 23]. If A is the Clifford algebra Cl0,n = Rn , the quadratic cone is a real algebraic (proper for n ≥ 3) subset of Rn , containing the subspace of paravectors. By restricting the Clifford variables to the paravectors, we get the theory of slice monogenic functions introduced by Colombo, Sabadini and Struppa in [8]. In Section 2, we define the normal cone and the quadratic cone of an algebra A and prove that the quadratic cone is a union of complex planes of A. This property is the starting point for the extension of Fueter’s construction. We compute the cones for some relevant algebras. In particular, we show that the quadratic cone can be a semi-algebraic set (e.g. in Clifford algebras with non-definite signature) and depends on the antiinvolution chosen in A. We also find the dimensions of the cones. In Section 3, we introduce complex intrinsic functions with values in the complexified algebra A ⊗R C and use them as stem functions to generate A-valued (left) slice functions. This approach does not require the holomorphy of the stem function. Moreover, slice functions can be defined also on domains which do not intersect the real axis of A. In Section 4, we restrict our attention to slice functions with holomorphic stem function, what we call (left) slice regular functions on A. These functions form a real vector space that is not closed w.r.t. the pointwise product in A. The pointwise product for stem functions induces a natural product on slice functions (cf. Section 5), that generalizes the usual product of polynomials and power series. In Section 6, we recall some properties of the zero set of slice functions. We restrict our attention to admissible slice regular functions, which preserve many relevant properties of classical holomorphic functions. We generalize a structure

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theorem for the zero set proved by Pogorui and Shapiro [29] for quaternionic polynomials and by Gentili and Stoppato [15] for quaternionic power series. A Remainder Theorem (Theorem 11) gives us the possibility to define a notion of multiplicity for the zeros of an admissible slice regular function. Polynomials with right coefficients in A are slice regular functions on the quadratic cone. We obtain a version of the fundamental theorem of algebra for slice regular admissible polynomials. This theorem was proved for quaternionic polynomials by Eilenberg and Niven [10, 28] and for octonionic polynomials by Jou [26]. See also [11, pp. 308ff] for a topological proof of the theorem valid for a class of real algebras including C, H and O, and [36], [31] and [21] for other proofs. Gordon and Motzkin [24] proved, for polynomials on a (associative) division ring, that the number of conjugacy classes containing zeros of p cannot be greater than the degree m of p. This estimate was improved on H by Pogorui and Shapiro [29]: if p has s spherical zeros and l non-spherical zeros, then 2s + l ≤ m. Gentili and Struppa [20] showed that, using the right definition of multiplicity, the number of zeros of p equals the degree of the polynomial. In [23], this strong form was generalized to the octonions. Recently, Colombo, Sabadini and Struppa [8, 9] and Yang and Qian [37] proved some results on the structure of the set of zeros of a polynomial with paravector coefficients in a Clifford algebra. We obtain a strong form of the fundamental theorem of algebra (Theorem 14), which contains and generalizes the above results. We show that the sum of the multiplicities of the zeros of a slice regular admissible polynomial is equal to its degree. The last section contains several examples that illustrate the relevance of the quadratic cone and of the condition of admissibility for the algebraic and topological properties of the zero set of a polynomial. We see that the quadratic cone is sufficiently large to contain the “right” number of (isolated or spherical) zeros, and sufficiently small to exclude “wild” sets of zeros of the polynomial. Outside the quadratic cone, an admissible polynomial can have infinite non-spherical zeros. If the polynomial is not admissible, its zero set loses its regular structure.

2. The quadratic cone of a real alternative algebra Let A be a finite-dimensional alternative real algebra with a unity, of dimension d > 1. We will identify the field of real numbers with the subalgebra of A generated by the unity. In a real algebra, we can consider the imaginary space consisting of all non-real elements whose square is real. Definition 1. Let Im(A) := {x ∈ A | x2 ∈ R, x ∈ / R \ {0}}. The elements of Im(A) are called purely imaginary elements of A. In general, the imaginary space Im(A) is not a vector subspace of A. In what follows, we will assume that on A an antiinvolution is fixed. It is a linear map x → xc of A into A satisfying the following properties: (xc )c = x

∀x ∈ A,

(xy)c = y c xc

∀x, y ∈ A,

xc = x for every real x.

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Definition 2. For every element x of A, the trace of x is t(x) := x + xc ∈ A and the (squared) norm of x is n(x) := xxc ∈ A. Definition 3. We call normal cone of the algebra A the subset NA := {0} ∪ {x ∈ A | n(x) = n(xc ) ∈ R \ {0}}, and quadratic cone of A the set QA := R ∪ {x ∈ A | t(x) ∈ R, n(x) ∈ R, 4n(x) > t(x)2 }. We also set SA := {J ∈ QA | J 2 = −1} ⊆ Im(A). Elements of SA will be called square roots of −1 in the algebra A. For every J ∈ SA , we will denote by CJ := 1, J " C the subalgebra of A generated by J. Proposition 1. Let A be an alternative real algebra with a fixed antiinvolution x → xc . The following statements hold. (1) Every x ∈ QA satisfies the real quadratic equation x2 − x t(x) + n(x) = 0. (2) x ∈ QA is equivalent to xc ∈ QA . Moreover, QA ⊆ NA . −1 (3) Every nonzero x ∈ NA is invertible: x−1 = n(x) xc . c (4) J = −J for every J ∈ SA , i.e., t(J) = 0, n(J) = 1. (5) QA = A if and only if A is isomorphic to one of the division algebras C, H or O with the usual conjugation mapping. (6) For every x ∈ QA , there exist uniquely determined elements x0 ∈ R, y ∈ Im(A) 3 ∩ QA , with t(y) = 0, such that x = x0 + y. (7) QA = J∈SA CJ and CI ∩ CJ = R for every I, J ∈ SA , I = ±J. Using the notation of the above proposition, for every x ∈ QA , we set c x−xc Re(x) := x0 = x+x 2 , Im(x) := y = 2 . As shown by the examples below, the quadratic cone QA needs not be a subalgebra or a subspace of A. Examples 1. (1) Let A be the algebra H of the quaternions or the algebra O of the octonions. Let xc = x ¯ be the usual conjugation mapping. Then Im(A) is a subspace, A = R ⊕ Im(A) and QH = NH = H, QO = NO = O. In these cases, SH is a two-dimensional sphere and SO is a six-dimensional sphere. (2) Let A be the real Clifford algebra Clp,q = Rp,q , with the conjugation xc = ([x]0 + [x]1 + [x]2 + [x]3 + [x]4 + · · · )c = [x]0 − [x]1 − [x]2 + [x]3 + [x]4 − · · · , where [x]k denotes the k-vector component of x in Rp,q (cf. for example [7, §4.1] or [25, §3.2]).

Let n := p + q. An element x of Rp,q can be represented in the form x = K xK eK , with K = (i1 , . . . , ik ) an increasing multiindex of length k, 0 ≤ k ≤ n, eK = ei1 · · · eik , e∅ = 1, xK ∈ R, x∅ = x0 , e1 , . . . , en basis elements. Let Rn := R0,n . In this case, the quadratic cone Qn := QRn is the real algebraic (proper for n ≥ 3) subset of Rn defined in terms of the euclidean scalar product x · y by the equations xK = 0, x · (xeK ) = 0 for every eK = 1 such that e2K = 1.


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The subspace of paravectors Rn+1 := {x ∈ Rn | [x]k = 0 for every k > 1} is contained in Qn . For x ∈ Rn+1 , t(x) = 2x0 ∈ R and n(x) = |x|2 ≥ 0 (the euclidean norm). The (n − 1)-dimensional sphere S = {x = x1 e1 + · · · + xn en ∈ Rn+1 | x21 + · · · + x2n = 1} of unit 1-vectors is (properly) contained in SRn . The normal cone Nn := NRn contains also the Clifford group Γn and its subgroups Pin(n) and Spin(n). If n ≥ 3, Im(A) is not a subspace of A. (3) The case of R3 . A direct computation shows that N3 = {x ∈ R3 | x0 x123 + x2 x13 − x1 x23 − x3 x12 = 0} and the quadratic cone is the 6-dimensional real algebraic set Q3 = {x ∈ R3 | x123 = 0, x2 x13 − x1 x23 − x3 x12 = 0}.

Finally, SR3 = {x ∈ Q3 | x0 = 0, i x2i + j,k x2jk = 1} ⊂ Im(R3 ) ∩ Q3 is the intersection of a 5-sphere with the hypersurface x2 x13 − x1 x23 − x3 x12 = 0. (4) The case of R4 . The normal cone N4 is the 11-dimensional real algebraic set with equations x1 x1234 + x124 x13 − x12 x134 − x123 x14 = 0, x2 x1234 + x124 x23 − x12 x234 − x123 x24 = 0, x3 x1234 + x134 x23 − x13 x234 − x123 x34 = 0, x4 x1234 − x14 x234 + x134 x24 − x124 x34 = 0, x134 x2 − x1 x234 − x124 x3 + x123 x4 = 0, x0 x1234 − x14 x23 + x13 x24 − x12 x34 = 0, x0 x234 + x3 x24 − x2 x34 − x23 x4 = 0, x0 x134 + x3 x14 − x1 x34 − x13 x4 = 0, x0 x124 + x2 x14 − x1 x24 − x12 x4 = 0, x0 x123 + x2 x13 − x1 x23 − x12 x3 = 0, while the quadratic cone Q4 is the 8-dimensional real algebraic set defined by x14 x23 − x13 x24 + x12 x34 = x3 x24 − x2 x34 − x23 x4 = x3 x14 − x1 x34 − x13 x4 = 0, x2 x14 − x1 x24 − x12 x4 = x2 x13 − x1 x23 − x12 x3 = x123 = x124 = x134 = x234 = x1234 = 0. (5) Let A = R1,2 . Then N1,2 is the 7-dimensional real semi-algebraic set {x ∈ R1,2 |x0 x123 + x2 x13 − x1 x23 − x3 x12 = 0, x20 + x22 + x23 + x223 = x21 + x212 + x213 + x2123 } ∪ {0} and the quadratic cone Q1,2 is the 6-dimensional real semi-algebraic set Q1,2 = R ∪ {x ∈ R1,2 | x123 = 0, x2 x13 − x1 x23 − x3 x12 = 0, x22 + x23 + x223 > x21 + x212 + x213 }. (6) The quadratic cone and the normal cone depend on the antiinvolution chosen in the algebra. For example, in the Clifford algebras Rn , it is possible to take the reversion x → x∗ in place of the conjugation. The reversion is defined by x∗ = ([x]0 + [x]1 + [x]2 + [x]3 + [x]4 + · · · )∗ = [x]0 + [x]1 − [x]2 − [x]3 + [x]4 − · · · .

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N2∗

= {x | x1 = x2 = 0} ∪ {x | x0 = x12 = 0}, Q∗2 = {x ∈ R2 | x1 = x2 = 0} " C.

Here SA = {±e12 }. In R3 the normal cone (w.r.t. reversion) is the 5-dimensional real semialgebraic set N3∗ = R ∪ {x | x2 x12 + x3 x13 = 0, x1 x13 + x2 x23 = 0, x1 x12 − x23 x3 = 0, x0 x1 + x123 x23 = 0, x0 x2 − x123 x13 = 0, x0 x3 + x12 x123 = 0, x20 + x212 + x213 + x223 = x21 + x22 + x23 + x2123 } and the quadratic cone is the 4-dimensional plane Q∗3 = {x | x1 = x2 = x3 = x123 = 0} = 1, e12 , e13 , e23 " H.

3. Slice functions Let AC = A ⊗R C be the complexification of A. We will use the representation AC = {w = x + iy | x, y ∈ A}

(i2 = −1).

AC is an alternative complex algebra with a unity w.r.t. the product: (x + iy)(x + iy ) = xx − yy + i(xy + yx ). The algebra A can be identified with the real subalgebra A := {w = x+iy | y = 0} of AC . In AC two commuting operators are defined: the complex-linear antiinvolution w → wc = (x + iy)c = xc + iy c and the complex conjugation defined by w = x + iy = x − iy. Definition 4. Let D ⊆ C be an open subset. If a function F : D → AC is complex intrinsic, i.e., it satisfies the condition F (z) = F (z) for every z ∈ D such that z ∈ D, then F is called an A-stem function on D. In the preceding definition, there is no restriction to assume that D is symmetric w.r.t. the real axis, i.e., D = conj(D) := {z ∈ C | z¯ ∈ D}. Remarks 1. (1) A function F is an A-stem function if and only if the A-valued components F1 , F2 of F = F1 + iF2 form an even-odd pair w.r.t. the imaginary part of z. (2) By means of a basis B = {uk }k=1,...,d of A, F can be identified with a complex intrinsic curve in Cd . Given an open subset D of C, let ΩD be the relatively open subset of QA obtained by the action on D of the square roots of −1: ΩD := {x = α + βJ ∈ CJ | α, β ∈ R, α + iβ ∈ D, J ∈ SA }. Sets of this type will be called circular sets in A.


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Definition 5. Any stem function F : D → AC induces a left slice function f = I(F ) : ΩD → A. If x = α + βJ ∈ DJ := ΩD ∩ CJ , we set f (x) := F1 (z) + JF2 (z) (z = α + iβ). There is an analogous definition for right slice functions when J is placed on the right of F2 (z). In what follows, the term slice functions will always mean left slice functions. We will denote the real vector space of (left) slice functions on ΩD by S(ΩD ). Examples 2. Assume A = H or A = O, with the usual conjugation mapping. (1) For any element a ∈ A, F (z) := z n a = Re(z n )a + i (Im(z n )a) induces the monomial f (x) = xn a ∈ S(A).

n j (2) By linearity, we get all the standard polynomials p(x) = j=0 x aj with right quaternionic

j or octonionic coefficients. More generally, every convergent power series j x aj is a slice function on an open ball of A centered in the origin with (possibly infinite) radius. The above examples generalize to standard polynomials in x = I(z) and xc = I(¯ z ) with coefficients in A. The domain of slice polynomial functions or series must be restricted to subsets of the quadratic cone. + For an element J ∈ SA , let C+ J denote the upper half plane CJ = {x = α + βJ ∈ A | β ≥ 0}.

Proposition 2. Let J, K ∈ SA with J −K invertible. Every slice function f ∈ S(ΩD ) + is uniquely determined by its values on the two distinct half planes C+ J and CK . Moreover, the following representation formula holds: f (x) = (I − K) (J − K)−1 f (α + βJ) − (I − J) (J − K)−1 f (α + βK) (3.1) for every I ∈ SA and for every x = α + βI ∈ DI = ΩD ∩ CI . Representation formulas for quaternionic Cullen regular functions appeared in [3, 4], and for slice monogenic functions of a Clifford variable in [5, 6]. Definition 6. Let f ∈ S(ΩD ) be a slice function. The spherical value of f in x ∈ ΩD is vs f (x) := 12 (f (x) + f (xc )). The spherical derivative of f in x ∈ ΩD \ R is ∂s f (x) := 12 Im(x)−1 (f (x) − f (xc )). In this way, we get two A-valued slice functions associated with f , constant on every “sphere” Sx := {y ∈ QA | y = α + βI, I ∈ SA } (x = α + βJ, J ∈ SA ), such that ∂s f (x) = 0 if and only if f is constant on Sx . In this case, f has value vs f (x) on Sx . If ΩD ∩ R = ∅, under mild regularity conditions on F , we get that ∂s f can be continuously extended as a slice function on ΩD . By definition, the following identity holds for every x ∈ ΩD : f (x) = vs f (x) + Im(x) ∂s f (x). Proposition 3. Let f = I(F ) ∈ S(ΩD ) be a slice function. Then the following statements hold: (1) If F ∈ C 0 (D), then f ∈ C 0 (ΩD ).

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(2) If F ∈ C 2s+1 (D) for a positive integer s, then f is of class C s (ΩD ). As a consequence, if F ∈ C ∞ (D), then f is of class C ∞ (ΩD ). (3) If F ∈ C ω (D), then f is of class C ω (ΩD ). We will denote by S 1 (ΩD ) := {f = I(F ) ∈ S(ΩD ) | F ∈ C 1 (D)} the real vector space of slice functions with stem function of class C 1 . Definition 7. Let f = I(F ) ∈ S 1 (ΩD ). We set

∂f ∂F ∂f ∂F := I , . := I ∂x ∂z ∂xc ∂ z¯ These functions are continuous slice functions on ΩD .

4. Slice regular functions Left multiplication by i defines a complex structure on AC . With respect to this structure, a C 1 function F = F1 + iF2 : D → AC is holomorphic if and only if its components F1 , F2 satisfy the Cauchy–Riemann equations. Definition 8. A (left) slice function f ∈ S 1 (ΩD ) is (left) slice regular if its stem function F is holomorphic. We will denote the real vector space of slice regular functions on ΩD by SR(ΩD ) := {f ∈ S 1 (ΩD ) | f = I(F ), F : D → AC holomorphic}. Polynomials with right coefficients in A can be considered as slice regular functions on the quadratic cone. Assume that on A is defined a positive scalar product x · y whose associated norm satisfies an inequality |xy| ≤ C|x||y| (C > 0) and such that |x|2 = n(x)

for every x ∈ QA . In this case we can consider also convergent power series k xk ak as slice regular functions on the intersection of the quadratic cone with a ball centered in the origin. See for example [25, §4.2] for the quaternionic and Clifford algebra cases, where we can take as product x · y the n euclidean product in R4 or R2 , respectively. Proposition 4. Let f = I(F ) ∈ S 1 (ΩD ). Then f is slice regular on ΩD if and only if the restriction fJ := f |CJ ∩ΩD : DJ = CJ ∩ ΩD → A is holomorphic for every J ∈ SA with respect to the complex structures defined by left multiplication by J. Proposition 4 implies that if A is the algebra of quaternions or octonions, and D intersects the real axis, then f is slice regular on ΩD if and only if it is Cullen regular in the sense introduced by Gentili and Struppa in [18, 19, 17, 16]. If A is the real Clifford algebra Rn , slice regularity generalizes the concept of slice monogenic functions introduced by Colombo, Sabadini and Struppa in [8]. If f = I(F ) ∈ SR(ΩD ), F ∈ C 1 (D) and D intersects the real axis, then the restriction of f to the subspace of paravectors is a slice monogenic function. Conversely, every slice monogenic function is the restriction of a unique slice regular function.


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5. Product of slice functions In general, the pointwise product of two slice functions is not a slice function. However, pointwise product in the algebra AC of A-stem functions induces a natural product on slice functions. Definition 9. Let f = I(F ), g = I(G) ∈ S(ΩD ). The product of f and g is the slice function f · g := I(F G) ∈ S(ΩD ). The preceding definition is well posed, since the pointwise product of F and G is complex intrinsic. The product is distributive and also associative if A is an associative algebra. If the components F1 , F2 of the first stem function F are realvalued, then (f · g)(x) = f (x)g(x) for every x ∈ ΩD . In this case, we will use also the notation f g in place of f · g. Definition 10. A slice function f = I(F ) is called real if the A-valued components F1 , F2 of its stem function are real-valued. Equivalently, f is real if the spherical value vs f and the spherical derivative ∂s f are real-valued. Real slice functions are characterized by the following property: for every J ∈ SA , the image f (CJ ∩ ΩD ) is contained in CJ .

Let f (x) = j xj aj and g(x) = k xk bk be polynomials or convergent power series with coefficients aj , bk ∈ A. The usual product of polynomials, where x is considered to be a commuting variable (cf. for example [27] and [14, 13]), can be extended to power series (cf. [15, 20] for the quaternionic case) by setting:

(f ∗ g)(x) := n xn j+k=n aj bk . Proposition 5. Let f and g be polynomials or convergent power series. Then the product f · g coincides with the star product f ∗ g, i.e., I(F G) = I(F ) ∗ I(G). We now associate to every slice function the normal function, which is useful when dealing with zero sets. Our definition is equivalent to the symmetrization of quaternionic power series given in [15]. c

Definition 11. Let f = I(F ) ∈ S(ΩD ). Then also F c (z) := F (z)c = F1 (z) + iF2 (z)c is an A-stem function. We set f c := I(F c ), CN (F ) := F F c = n(F1 ) − n(F2 ) + i t(F1 F2 c ) and N (f ) := f · f c = I(CN (F )) ∈ S(ΩD ). The slice function N (f ) will be called the normal function of f . If f is slice regular, then also f c and N (f ) are slice regular. If A is the algebra of quaternions or octonions, then CN (F ) is complex-valued and then the normal function N (f ) is real. For a general algebra A, this is not true for every slice function. This is the motivation for the following definition. Definition 12. A slice function f = I(F ) ∈ S(ΩD ) is called admissible if vs f (x) ∈ NA for every x ∈ ΩD and the real vector subspace vs f (x), ∂s f (x) ⊆ NA for every x ∈ ΩD \ R. Equivalently, F1 (z), F2 (z) ⊆ NA for every z ∈ D.

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If A = H or O, then every slice function is admissible. Moreover, if f is real, then f c = f , N (f ) = f 2 and f is admissible. If f is admissible, then CN (F ) is complex-valued and then N (f ) is real. Example 3. Consider the Clifford algebra R3 with the usual conjugation. Its

normal cone contains the subspace R4 of paravectors. Every polynomial p(x) = n xn an with paravectors coefficients is admissible. The polynomial p(x) = xe23 + e1 is an example of a non-admissible slice regular function. Theorem 6. Let A be associative or A = O. Then N (f · g) = N (f ) N (g) for every admissible f, g ∈ S(ΩD ). Corollary 7. Let A be associative or A = O. Assume that n(x) = n(xc ) = 0 for every x ∈ A \ {0} such that n(x) is real. If f and g are admissible slice functions, then also the product f · g is admissible. Example 4. Let A = R3 . Consider the admissible polynomials f (x) = xe2 + e1 , g(x) = xe3 + e2 . Then (f · g)(x) = x2 e23 + x(e13 − 1) + e12 is admissible, N (f ) = N (g) = x2 + 1 and N (f · g) = (x2 + 1)2 .

6. Zeros of slice functions The zero set V (f ) = {x ∈ ΩD | f (x) = 0} of an admissible slice function f has a particular structure. For every fixed x ∈ QA , the “sphere” Sx is entirely contained in V (f ) or it contains at most one zero of f . Moreover, if f is not real, there can be isolated, non-real zeros. These different types of zeros of a slice function correspond to the existence of zero-divisors in the complexified algebra AC . Theorem 8 (Structure of V (f )). Let f ∈ S(ΩD ) an admissible slice function. Let x = α + βJ ∈ ΩD and z = α + iβ ∈ D. Then one of the following mutually exclusive statements holds: (1) Sx ∩ V (f ) = ∅. (2) Sx ⊆ V (f ). In this case x is called a real (if x ∈ R) or spherical (if x ∈ / R) zero of f . (3) Sx ∩ V (f ) consists of a single, non-real point. In this case x is called an SA -isolated non-real zero of f . Moreover, a real slice function has no SA -isolated 3 non-real zeros, and for every admissible slice function f , we have V (N (f )) = x∈V (f ) Sx . Theorem 9. Let ΩD be connected. If f is slice3regular and admissible on ΩD , and N (f ) does not vanish identically, then CJ ∩ x∈V (f ) Sx is closed and discrete in DJ = CJ ∩ ΩD for every J ∈ SA . If ΩD ∩ R = ∅, then N (f ) ≡ 0 if and only if f ≡ 0. In the quaternionic case, the structure theorem for the zero set of slice regular functions was proved by Pogorui and Shapiro [29] for polynomials and by Gentili and Stoppato [15] for power series.


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Remark 2. If ΩD does not intersect the real axis, a not identically zero slice regular function f can have normal function N (f ) ≡ 0. For example, let J ∈ SH be fixed. The admissible slice regular function defined on H \ R by f (x) = 1 − IJ for x ∈ C+ I = {x = α + βI ∈ A | β ≥ 0} has zero normal function. Now we state a remainder theorem, which generalizes a result proved by Beck [1] for quaternionic polynomials and by Serôdio [33] for octonionic polynomials. Definition 13. For any y ∈ QA , the characteristic polynomial of y is the slice regular function on QA Δy (x) := N (x − y) = (x − y) · (x − y c ) = x2 − x t(y) + n(y). Proposition 10. The characteristic polynomial Δy of y ∈ QA is real. Two characteristic polynomials Δy and Δy coincide if and only if Sy = Sy . Moreover, V (Δy ) = Sy . Theorem 11. Let f ∈ SR(ΩD ) be an admissible slice regular function. Let y ∈ V (f ) = {x ∈ QA | f (x) = 0}. Then the following statements hold. (1) If y is a real zero, then there exists an admissible g ∈ SR(ΩD ) such that f (x) = (x − y) g(x). (2) If y ∈ ΩD \ R, then there exists an admissible h ∈ SR(ΩD ) and a, b ∈ A such that a, b ⊆ NA and f (x) = Δy (x) h(x) + xa + b. Moreover, • y is a spherical zero of f if and only if a = b = 0; • y is an SA -isolated non-real zero of f if and only if a = 0 (in this case y = −ba−1 ). If f is real, then g, h are real and a = b = 0. For every non-real y ∈ V (f ), the element a ∈ NA which appears in the statement of the preceding theorem is the spherical derivative of f at x ∈ Sy . Corollary 12. Let f ∈ SR(ΩD ) be admissible. If Sy contains at least one zero of f , of whatever type, then Δy divides N (f ). Definition 14. Let f ∈ SR(ΩD ) be admissible, with N (f ) ≡ 0. Given a nonnegative integer s and an element y of V (f ), we say that y is a zero of f of multiplicity s if Δsy | N (f ) and Δs+1 N (f ). We will denote the integer s, called y multiplicity of y, by mf (y). In the case of y real, the preceding condition is equivalent to (x − y)s | f and (x − y)s+1 f . If y is a spherical zero, then mf (y) ≥ 2. In the case of quaternionic polynomials, the definition is equivalent to the one given in [2] and in [20]. Proposition 13. Let A be associative. Let f, g ∈ S(ΩD ). Then V (f ) ⊆ V (f · g). As shown in [33] for octonionic polynomials and in [23] for octonionic power series, if A is not associative the statement of the proposition is no more true. However, we can still say something about the location of the zeros of f · g (cf. [23] for the precise relation linking the zeros of f and g to those of f · g). For the associative case, see also [27, §16].

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Now we focus

our attention on the zero set of slice regular admissible polym nomials. If p(x) = j=0 xj aj is an admissible polynomial of degree m with coefficients aj ∈ A, then the normal polynomial

c N (p)(x) = (p ∗ pc )(x) = n xn j+k=n aj ak has degree 2m and real coefficients. A sufficient condition for the admissibility of p is that the real vector subspace a0 , . . . , am is contained in NA . Theorem

m j 14 (Fundamental Theorem of Algebra with multiplicities). Let p(x) = j=0 x aj be an admissible polynomial of degree m > 0 with coefficients in A. Then V (p) = {y ∈ QA | p(y) = 0} is non-empty. More precisely, there are distinct “spheres” Sx1 , . . . , Sxt such that V (p) ⊆

t

Sxk = V (N (p)),

V (p) ∩ Sxj = ∅

for every j,

k=1

and, for any choice of zeros y1 ∈ Sx1 , . . . , yt ∈ Sxt of p, it holds

t k=1

mp (yk ) = m.

Remark 3. If r denotes the number of real zeros of the polynomial p, i the number of SA -isolated non-real zeros of p and s the number of “spheres” Sy (y ∈ / R) containing spherical zeros of p, we have that r + i + 2s ≤ deg(p).

7. Examples The following examples show the relevance of the quadratic cone and of admissibility for the algebraic and

topological properties of the zero set of a polynomial. m (1) Every polynomial j=0 xj aj , with paravector coefficients aj in the Clifford algebra Rn (with conjugation as antiinvolution), has m roots (counted with their multiplicities) in the quadratic cone Qn . If the coefficients are real, then it has at least one root in the paravector space Rn+1 , since every “sphere” Sy intersects Rn+1 (cf. [37, Theorem 3.1]). (2) In R3 , the admissible polynomial p(x) = x2 + xe3 + e2 (cf. [37, Ex. 3]) has two isolated zeros y1 = 12 (1 − e2 − e3 + e23 ), y2 = 12 (−1 + e2 − e3 + e23 ) in Q3 \ R4 . They can be computed by solving the complex equation CN (P ) = z 4 + z 2 + 1 = 0 to find the two “spheres” Sy1 , Sy2 and then using the Remainder Theorem (Theorem 11) with Δy1 = x2 − x + 1 and Δy2 = x2 + x + 1. / Q3 . (3) In R3 , the polynomial p(x) = xe23 + e1 vanishes only at y = e123 ∈ Note that p is not admissible: e1 , e23 ∈ N3 , but e1 + e23 ∈ / N3 . (4) An admissible polynomial of degree m, even in the case of non-spherical zeros, can have more than m roots in the whole algebra. For example, p(x) = x2 −1 has four roots in R3 , two in the quadratic cone (x = ±1) and two outside it (x = ±e123 ).


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(5) Outside the quadratic cone, an admissible polynomial can have infinite non-spherical zeros. For example, in R1,2 the polynomial p(x) = x2 − 1 has zeros ±1 in Q1,2 , while in R1,2 \ Q1,2 it vanishes on the 4-dimensional set {x | x0 = x123 = 0, x2 x13 − x1 x23 − x3 x12 = 0, x21 + x212 + x213 − x22 − x23 − x223 = 1}. (6) In R√3 , the admissible polynomial p(x) = x3 −1 has zero set V (p) = {1}∪Sy (y = − 12 + 23 J, J ∈ SR3 ) in Q3 , while in R3 \ Q3 the polynomial p vanishes on the two 2-spheres S± = {x = 14 +x1 (e1 ±e23 )+x2 (e2 ±e13 )+x3 (e3 ±e12 )± 34 e123 | x21 +x22 +x23 =

3 16 }.

(7) In the algebras H, O, R3 , the solutions of the equation x2 = −1 are exactly the elements of SR3 , i.e., they all belong to the quadratic cone. This is not the case for other algebras. In R1,2 , e2123 = −1, but e123 ∈ / Q1,2 . In R4 , the equation x2 = −1 has many solutions: the square roots of −1 in the “sphere” SR4 , but also infinite other points outside the quadratic cone: x=

4 i=1

xi ei + x1234 e1234 , with

4

x2i − x21234 = 1.

i=1

(8) The admissibility of a slice function depends on the algebra. For example, the polynomial p(x) = x2 + xe1 + 2 is admissible on the algebra R3 (w.r.t. conjugation), where its zero set is the union of V (p) = {e1 , −2e1 } ⊆ Q3 and the subset {− 21 e1 ± 32 e23 } of R3 \ Q3 . The same polynomial is not admissible on the algebra R1,2 , and it has no zeros in Q1,2 . (9) The admissibility depends also on the antiinvolution chosen in the algebra. For example, in the algebra Rn with the reversion as fixed antiinvolution, a polynomial with paravector coefficient can be not admissible. For n = 2, p(x) = xe1 + 1 is not admissible, since 1, e1 ⊆ N2∗ . This property reflects in the fact that the unique zero of p in R2 does not belong to the quadratic cone Q∗2 = {x ∈ R2 | x1 = x2 = 0}. The same holds in every Rn . Instead, the polynomials with coefficients in the real vector subspace of 1-vectors are admissible in Rn (w.r.t. reversion). (10) The reality of N (p) is not sufficient to get the admissibility of p. For example, in R3 , the polynomial p(x) = x2 e123 + x(e1 + e23 ) + 1 has real normal function N (p) = (x2 + 1)2 , but the spherical derivative ∂s p = t(x)e123 + e1 + e23 has N (∂s p) not real. In particular, ∂s p(J) = e1 + e23 ∈ / N3 for every J ∈ SR3 . The non-admissibility of p is reflected by the existence of a S 1 of distinct zeros on SR3 , where p does not vanish identically.

References [1] B. Beck. Sur les équations polynomiales dans les quaternions. Enseign. Math. (2), 25 (3-4) (1980), 1979, 193–201. [2] U. Bray and G. Whaples. Polynomials with coefficients from a division ring. Canad. J. Math., 35 (3) 1983, 509–515.

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[3] F. Colombo, G. Gentili, and I. Sabadini. A Cauchy kernel for slice regular functions. Ann. Glob. Anal. Geom., 37 2010, 361–378. [4] F. Colombo, G. Gentili, I. Sabadini, and D. C. Struppa. Extension results for slice regular functions of a quaternionic variable. Adv. Math., 222 (5) 2009, 1793–1808. [5] F. Colombo and I. Sabadini. The Cauchy formula with s-monogenic kernel and a functional calculus for noncommuting operators. J. Math. Anal. Appl., 373 2011, 655–679. [6] F. Colombo and I. Sabadini. A structure formula for slice monogenic functions and some of its consequences. In Hypercomplex analysis, Trends Math., pages 101–114. Birkhäuser, Basel, 2009. [7] F. Colombo, I. Sabadini, F. Sommen, and D. C. Struppa. Analysis of Dirac systems and computational algebra, volume 39 of Progress in Mathematical Physics. Birkhäuser Boston Inc., Boston, MA, 2004. [8] F. Colombo, I. Sabadini, and D. C. Struppa. Slice monogenic functions. Israel J. Math., 171 2009, 385–403. [9] F. Colombo, I. Sabadini, and D. C. Struppa. An extension theorem for slice monogenic functions and some of its consequences. Israel J. Math., 177 2010, 369–389. [10] S. Eilenberg and I. Niven. The “fundamental theorem of algebra” for quaternions. Bull. Amer. Math. Soc., 50 1944, 246–248. [11] S. Eilenberg and N. Steenrod. Foundations of algebraic topology. Princeton University Press, Princeton, New Jersey, 1952. [12] R. Fueter. Die Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Variablen. Comment. Math. Helv., 7 (1) 1934, 307–330. [13] I. Gelfand, S. Gelfand, V. Retakh, and R. L. Wilson. Factorizations of polynomials over noncommutative algebras and sufficient sets of edges in directed graphs. Lett. Math. Phys., 74 (2) 2005, 153–167. [14] I. Gelfand, V. Retakh, and R. L. Wilson. Quadratic linear algebras associated with factorizations of noncommutative polynomials and noncommutative differential polynomials. Selecta Math. (N.S.), 7 (4) 2001, 493–523. [15] G. Gentili and C. Stoppato. Zeros of regular functions and polynomials of a quaternionic variable. Michigan Math. J., 56 (3) 2008, 655–667. [16] G. Gentili, C. Stoppato, D. C. Struppa, and F. Vlacci. Recent developments for regular functions of a hypercomplex variable. In Hypercomplex analysis, Trends Math., pages 165–186. Birkhäuser, Basel, 2009. [17] G. Gentili and D. C. Struppa. Regular functions on the space of Cayley numbers. Rocky Mountain J. Math. 40 (1) 2010, 225–241. [18] G. Gentili and D. C. Struppa. A new approach to Cullen-regular functions of a quaternionic variable. C. R. Math. Acad. Sci. Paris, 342 (10) 2006, 741–744. [19] G. Gentili and D. C. Struppa. A new theory of regular functions of a quaternionic variable. Adv. Math., 216 (1) 2007, 279–301. [20] G. Gentili and D. C. Struppa. On the multiplicity of zeroes of polynomials with quaternionic coefficients. Milan J. Math., 76 2008, 15–25. [21] G. Gentili, D. C. Struppa, and F. Vlacci. The fundamental theorem of algebra for Hamilton and Cayley numbers. Math. Z., 259 (4) 2008, 895–902.


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[22] R. Ghiloni and A. Perotti. Slice regular functions on real alternative algebras. Adv. Math. DOI: 10.1016/j.aim.2010.08.015, 2010. [23] R. Ghiloni and A. Perotti. Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect. To appear in Ann. Mat. Pura Appl.. DOI: 10.1007/s10231-010-0162-1. [24] B. Gordon and T. S. Motzkin. On the zeros of polynomials over division rings. Trans. Amer. Math. Soc., 116 1965, 218–226. [25] K. Gürlebeck, K. Habetha, and W. Sprößig. Holomorphic functions in the plane and n-dimensional space. Birkhäuser Verlag, Basel, 2008. [26] Y.-L. Jou. The “fundamental theorem of algebra” for Cayley numbers. Acad. Sinica Science Record, 3 1950, 29–33. [27] T. Y. Lam. A first course in noncommutative rings, volume 131 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. [28] I. Niven. Equations in quaternions. Amer. Math. Monthly, 48 1941, 654–661. [29] A. Pogorui and M. Shapiro. On the structure of the set of zeros of quaternionic polynomials. Complex Var. Theory Appl., 49 (6) 2004, 379–389. [30] T. Qian. Generalization of Fueter’s result to Rn+1 . Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 8 (2) 1997, 111–117. [31] H. Rodríguez-Ordóñez. A note on the fundamental theorem of algebra for the octonions. Expo. Math., 25 (4) 2007, 355–361. [32] M. Sce. Osservazioni sulle serie di potenze nei moduli quadratici. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8), 23 1957, 220–225. [33] R. Serôdio. On octonionic polynomials. Adv. Appl. Clifford Algebr., 17 (2) 2007, 245–258. [34] F. Sommen. On a generalization of Fueter’s theorem. Z. Anal. Anwendungen, 19 (4) 2000, 899–902. [35] A. Sudbery. Quaternionic analysis. Math. Proc. Cambridge Philos. Soc., 85 (2) 1979, 199–224. [36] N. Topuridze. On the roots of polynomials over division algebras. Georgian Math. J., 10 (4) 2003, 745–762. [37] Y. Yang and T. Qian. On sets of zeroes of Clifford algebra-valued polynomials. Acta Math. Sin. Ser. B Engl. Ed., 30 (3) 2010, 1004–1012. Riccardo Ghiloni and Alessandro Perotti Department of Mathematics University of Trento Via Sommarive 14 I 38123 Povo-Trento Italy e-mail: [email protected] [email protected]

On the Incompressible Viscous Stationary MHD Equations and Explicit Solution Formulas for Some Three-dimensional Radially Symmetric Domains Rolf Sören Kraußhar Abstract. The quaternionic calculus is a powerful tool to treat many complicated systems of linear and non-linear PDEs in higher dimensions. In this paper we apply these new techniques to treat the stationary incompressible viscous magnetohydrodynamic equations. For the highly viscous case, in which the convective terms are negligibly small we present explicit analytic representation formulas for some three-dimensional radially symmetric domains. Then we look at the fully non-linear case for which we propose a fixed point algorithm. In this more complicated context, the solutions of the simpler linear problems treated in the first part of the paper need to be used to solving the corresponding equations in each step of the proposed iteration. Mathematics Subject Classification (2010). Primary 30G35; Secondary 76W05. Keywords. Incompressible viscous magnetohydrodynamic equations, hypercomplex integral operator calculus, Bergman projection, radially symmetric domains, Dirac operators.

1. Introduction The magnetohydrodynamic equations (MHD equations) represent a combination of the Navier-Stokes system with the Maxwell system. They describe fluid dynamical processes under the influence of a present electromagnetic field. The inviscid MHD equations play an important role in the description of the dynamics of astrophysical plasmas, see for instance [1, 16] among many other references. Also the viscous MHD equations have attracted a growing interest by mathematicians and physicists, cf. for example [4, 15, 23, 24, 33]. I. Sabadini and F. Sommen (eds.), Hypercomplex Analysis and Applications, Trends in Mathematics, DOI 10.1007/978-3-0346-0246-4_9, © Springer Basel AG 2011

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In this paper we look at the three-dimensional incompressible viscous MHD equations ∂u 1 1 − Δu + (u grad) u + grad p = rotB × B in G ∂t Re μ0 ∂B 1 − ΔB + (u grad) B − (B grad)u = 0 in G ∂t Rm div u = 0 in G div B u = 0, B

= 0 in G = h on ∂G.

(1.1) (1.2) (1.3) (1.4) (1.5)

Here, G is some strongly Lipschitz domain lying in R3 . Furthermore, u represents the velocity of the flow, p the pressure and B the magnetic field. μ0 is the magnetic permeability of the vacuum and Re and Rm the fluid mechanical resp. magnetic Reynolds number. The expression (u grad) u needs to be understood in the sense that one first forms the scalar product between the formal expressions

algebraically ∂ u and ∇, i.e., 3i=1 ui ∂x . This new operator then is applied from the left to u. i In a similar way the other expressions in equation (1.2) need to be interpreted. The first equation basically looks like the Navier-Stokes equation. However, the external driving force is an unknown magnetic entity that also needs to be computed. Together with the second equation the dynamics of the magnetic field, the velocity and the pressure is described. The third equation is the continuity equation. It represents the incompressibility of the flow. The forth equation states the non-existence of magnetic monopoles. Finally, the remaining equations in (1.5) represent the given measured data for the velocity and for the magnetic field on the boundary Γ = ∂G of the domain G. In the above cited works some numerical algorithms are proposed to compute weak solutions to this complicated PDE system in some special cases. However, the proposed numerical algorithms are often directly coupled with the special geometry of the domain. Other very efficient algorithms of complexity rate O(n) that use standard classes of wavelets, as proposed for instance in [32], are particularly designed for domains that are constructed as tensor products of simpler elementary domains, such as rectangular domains. Furthermore, in the case of dealing with large temporal distances the classical time stepping methods, where small time steps are required, do not always lead to the desired result. Also some first results on the global existence for weak solutions have been presented. Simultaneously, in the recent time the quaternionic operator calculus proposed by K. G¨ urlebeck, W. Sprößig, M. Shapiro, V.V. Kravchenko, P. Cerejeiras, U. K¨ ahler and by their collaborators and students, see for instance [2, 3, 21, 22, 27, 28], turned out to provide a new analytic toolkit to treat the Navier-Stokes system and the Maxwell system. The quaternionic calculus lead to further explicit global criteria concerning the regularity, the existence and the uniqueness of the solutions. Some strong points of the quaternionic operator calculus are the relatively independence of the geometry and that one can derive from it discretized

On the Incompressible Viscous Stationary MHD Equations

127

versions that yield strong convergence instead of only weak convergence. Based on the new theoretical results also new numerical algorithms could be developed, see for instance [14, 18]. In the numerical algorithm proposed in these works for these difficult non linear problems one only needs to solve a series of linear Stokesor Poisson-type problems in each iterative step. To do this one can use standard solvers that converge with a complexity rate O(n). Another added value of the quaternionic approach is that also fully analytic representation formulas for the solutions to the Navier-Stokes equations, and the Maxwell and Helmholtz systems could be established for a number of special classes of domains. See for instance [9, 10, 11, 12]. Since the quaternionic calculus turned out to be useful in the treatment of both the Navier-Stokes system as also the Maxwell system, it is natural to expect similar insightful results for the MHD system. The latter one is just a coupling of both systems. In [20] we described how to model the stationary inviscid MHD equations in the quaternionic setting. In this paper we now focus on the stationary incompressible viscous case. First we study the case where the viscous terms are strongly dominating. In this context the convective terms are negligibly small. Based on the closed formulas for the Bergman kernel that we obtained in our previous papers [5, 7, 8, 9, 10] for rectangular, triangular, radially symmetric domains and hyperbolic polyhedron type domains, one can set up explicit representation formulas for the solutions of this system in these classes of domains. Here in this paper we summarize some of the representation formulas for three-dimensional spherical and radially symmetric domains. Then we turn to the case where the convective terms are not negligibly small. Here we propose to use a similar adapted fixed point algorithm as used in [9] for the Navier-Stokes system with heat transfer in order to compute the solutions to the non-linear stationary incompressible viscous MHD equations. These formulas hold universally, independently from the special geometry of the domain. In each step of the iteration one has to solve a linear problem of exactly that type treated in the previous section. Using adapted parabolic versions of the Dirac operator (working in an appropriate Witt basis) and related integral transforms as used for example in [3], the time-dependent system over any arbitrary bounded and unbounded Lipschitz domain can be treated in the same way as well. This, however, will be done in our follow-up paper [26]. Summarizing these results should be regarded as a starting point to study efficiently with hypercomplex methods the more complicated inviscid case, as proposed in [20].

2. Preliminaries 2.1. The quaternionic operator calculus Let e1 , e2 , e3 denote the standard basis of the Euclidean vector space R3 . To endow the space R3 with an additional multiplicative structure, we embed it into the

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algebra of Hamiltonian quaternions. The latter is denoted by H. A quaternion is an element of the form x = x0 +x := x0 +x1 e1 +x2 e2 +x3 e3 where x0 , . . . , x3 are real numbers. x0 is called the real part of the quaternion and will be denoted by (x). x is the vector part of x. In the quaternionic setting the standard unit vectors play the role of imaginary units, i.e., e2i = −1 for i = 1, 2, 3. Their mutual multiplication coincides with the usual vector product, i.e., e1 e2 = e3 , e2 e3 = e1 , e3 e1 = e2 and ei ej = −ej ei for i = j. The generalized conjugation anti-automorphism in H is Euclidean norm extends to a norm defined by ab = b a, ei = −ei , i = 1, 2, 3. The

3 2 on the whole quaternionic algebra, viz a := i=0 ai . The additional multiplicative structure of the quaternions allows us to describe all C 1 -functions f : R3 → R3 that satisfy both div f = 0 and rot f = 0 equivalently in a compact form as null-solutions to one differential opera single 3 ∂ tor, namely the three-dimensional Dirac operator D := i=1 ∂x ei . The Euclidean i Dirac operator in turn coincides with the usual gradient operator when it is applied to a scalar-valued function. If U ⊆ R3 is an open subset, then a real differentiable function f : U → H is called left quaternionic holomorphic or left monogenic in U , if Df = 0. In the quaternionic calculus, the square of the Euclidean Dirac operator gives the Euclidean Laplacian up to a minus sign; we have D2 = −Δ. Hence, every real component of a left monogenic function is harmonic. Conversely, following for instance [13], if f is a solution to the Laplacian in U , then in any open ball B(˜ x, r) ⊆ U there exist two left monogenic functions f0 and f1 , such that f = f0 + xf1 in B(˜ x, r). This property allows us to treat harmonic functions with the function theory of the Dirac operator offering generalizations of many powerful theorems used in complex analysis. In fact, quaternionic analysis can be regarded as a refinement of harmonic analysis in four dimensions. For particular details on quaternionic function and operator theory, we refer the reader for instance to [13, 21, 22]. For our needs we recall the following ones. Adapting from [22] and [29] we have the following Theorem 2.1 (Borel–Pompeiu). Let G ⊂ R3 be a bounded Lipschitz domain with a strongly Lipschitz boundary Γ = ∂G. Then for all u ∈ C 1 (G, H) ∩ C(G, H), q0 (x − y)n(y)u(y)dΓy − q0 (x − y)(Du)(y)dV (y) = 4πu(x), x ∈ G. Γ

G

x 1 Here, q0 (x) := − x 3 stands (up to the factor 4π ) for the fundamental solution to the Euclidean Dirac operator. The expression n(y) denotes the exterior unit normal vector at y ∈ Γ.

) 1 If u ∈ Ker D, then (FΓ u)(x) := 4π q (x − y)n(y)u(y)dΓy = u(x). This Γ 0 is the well-known generalized Cauchy integral formula for quaternionic monogenic functions.


129

1 Remark. Following [19], in the unbounded case the expression 4π q0 (x − y) is 1 1 ˜ is x − y) where x substituted by the modified Cauchy kernel 4π q0 (x − y) − 4π q0 (˜ an arbitrary point from the complement of the closure of G.

The inverse operator of D is induced by the Teodorescu transform, which is defined for all u ∈ C(G, H) by 1 (TG u)(x) := − q0 (x − y)u(y)dV (y) 4π G in the case where G is bounded. Again, following [19], in the unbounded case the Teodorescu transformation can be defined by 1 (TG u)(x) := − q0 (x − y) − q0 (˜ x − y) u(y)dV (y), 4π G ˜ is an arbitrary point taken from the complement of the closure of where again x G. For all u ∈ C 2 (G, H) ∩ C(G, H), one has (DTG u)(x) = u(x) for all x ∈ G. Conversely, all u ∈ C 1 (G, H) ∩ C(G, H) satisfy (FΓ u)(x) + (TG Du)(x) = u(x) where we again assume that x ∈ G. For all that follows, it is important to mention that this formula also remains valid for weakly differentiable functions, for instance for functions u ∈ W 2,1 (G). Here W 2,1 (G) is the space of differentiable functions in the sense of Sobolev, whose first partial derivatives belong to L2 (G). See for instance [2, 22]. The following direct decomposition of the space L2 (G) into the subspace of functions that are square-integrable and left monogenic in the inside of G and its complement will be crucially applied in this paper: Theorem 2.2. [22]. Let G ⊆ R3 be a bounded domain. ◦ 2,1

Then L2 (G) = B(G, H) ⊕ D W (G) where B(G, H) := L2 (G)∩ Ker D is ◦ 2,1

the Bergman space of left monogenic functions, and where W (G) is the subset of W 2,1 (G) with vanishing boundary data. P : L2 (G) → B(G, H) denotes the orthogonal Bergman projection while ◦ 2,1

Q : L2 (G) → D W (G) stands for the projection into the complementary space in all that follows. One has Q = I − P, I standing for the identity operator. The Bergman space of left monogenic functions is a Hilbert space with a uniquely defined reproducing kernel function. The latter is called the Bergman kernel and is denoted by B(x, y). The orthogonal Bergman projection P : L2 (G) → B(G, H) is given by the convolution with the Bergman kernel (Pu)(x) = B(x, y)u(y)dV (y), u ∈ L2 (G). G

In particular, one has (Pu)(x) = u(x) for all u ∈ B(G, H).

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3. The incompressible stationary MHD equations revisited in the quaternionic calculus In all that follows we assume that G ⊂ R3 is a bounded Lipschitz domain. As a first step in our consideration we look at the stationary viscous incompressible MHD equations. In the classical vector calculus they have the form 1 1 − Δu + (u grad) u + grad p = rotB × B in G (3.1) Re μ0 1 ΔB − (u grad) B + (B grad)u = 0 in G (3.2) − Rm div u = 0 in G (3.3) div B u = 0, B

= =

0 in G h on ∂G

(3.4) (3.5)

with given boundary data g = 0 and h. To apply the quaternionic integral operator calculus we first reformulate this system into the quaternionic setting. As a first step we reformulate this system in the quaternionic calculus. We recall that we have for a quaternionic function f (x0 + x) : R4 → R4 the relation Df = grad f0 + rot f − div f where f0 = (f ) is the scalar part of f and

3 ∂ is where f = Vec(f ) ∈ R3 is the vector part of f . Furthermore, D := i=0 ei ∂x i the quaternionic Cauchy-Riemann operator. Its vector part will be denoted by D, which is the three-dimensional Euclidean Dirac operator introduced before. In the case where f is a vector valued function, i.e., a function defined in an open subset of R3 with values in R3 , we have Df = rot f − div f . If p is a scalar valued function defined in an open subset of R3 , then we have Dp = grad p. When applying these rules to the magnetic vector field B ∈ R3 , we thus have DB = rot B − div B. In view of equation (1.4), which expresses that there are no magnetic monopoles, this equation simply reduces to DB = rot B. Furthermore, we have (DB) × B = Vec((DB) · B) in terms of the quaternionic product ·. The divergence of an R3 -valued vector field f can be expressed as div f = (Df ). The

3 ∂ 2 three-dimensional Laplacian Δ = i=1 ∂x 2 can be expressed in terms of the Dirac i

operator by Δ = −D2 , applying the multiplication rule e2i = −1 for all i = 1, 2, 3. The system (3.1)–(3.6) can thus be reformulated in the quaternions following way: 1 2 1 D u + (u D) u + D p = Vec((DB) · B) in G Re μ0 1 D2 B + (u D) B − (B D)u = 0 in G Rm (Du) = 0 in G (DB) = u = 0, B

=

0 in G h on ∂G.

in the (3.6) (3.7) (3.8) (3.9) (3.10)


131

To avoid confusions with the ordinary scalar product, we mean by · here the quaternionic product. The aim is now to apply the previously introduced hypercomplex integral operators in order to get representation formulas for the magnetic field B, the velocity u and the pressure p.

4. The highly viscous case ◦ 2,1

In all that follows we suppose that p ∈ L2 (G), u ∈ W (G) and B ∈ W 2,1 (G) and that h ∈ W 2,3/2 (∂G). In this section we first show how the quaternionic operator calculus can be successfully applied to set up fully explicit analytic solution representations for the highly viscous case in which the convective terms (u grad) u, (B grad) u and (u grad) B are negligibly small. In this case the stationary viscous incompressible MHD equations simplify to the system 1 2 D u+Dp Re 1 D2 B Rm (Du) (DB) u = 0, B

=

1 Vec((DB) · B) in G μ0

(4.1)

= 0 in G

(4.2)

= 0 in G

(4.3)

= 0 in G = h on ∂G

(4.4) (4.5)

represented in the quaternionic formalism. Remember that · stands for the quaternionic product. The magnetic field thus is described by the simple Dirichlet problem 1 D2 B = 0 in G (4.6) Rm B = h on ∂G. (4.7) According to [21, 22], the solutions for B can be represented in terms of the quaternionic operators in the form B = FΓ h + TG PG H,

(4.8)

where H is a W 2,3/2 (G, H) extension of h to the interior of G. After having applied this formula the right-hand side of (3.6) is known. The velocity u and the pressure p can now be computed from the system 1 2 D u+Dp Re (Du) u

1 Vec((DB) · B) in G μ0 = 0 in G

(4.10)

= 0 on ∂G,

(4.11)

=

(4.9)

132

R.S. Kraußhar

where we can replace B = FΓ h + TG PG H. Applying the Teodorescu transform to (4.9) leads to the equation 1 1 (TG D)(Du) + TG Dp = TG (Vec((DB) · B)). (4.12) Re μ0 Applying the Borel-Pompeiu formula to (4.12) leads to 1 1 (Du − FΓ Du) + p − FΓ p = TG (Vec((DB) · B)). (4.13) Re μ0 Next we apply the orthoprojector Q to this equation. This yields 1 1 (QDu − QFΓ Du) + Qp − QFΓ p = QTG (Vec((DB) · B)). (4.14) Re μ0 Since the Cauchy integral produces a monogenic function, we get QFΓ Du = 0 and QFΓ p = 0. So, equation (4.14) simplifies to 1 1 QDu + Qp = QTG (Vec((DB) · B)). (4.15) Re μ0 Now we again apply the Teodorescu transform to equation (4.15). This yields 1 1 TG QDu + TG Qp = TG QTG (Vec((DB) · B)). (4.16) Re μ0 In view of Du ∈ im(Q) we get TG QDu = TG Du. Applying again the BorelPompeiu formula leads to 1 1 1 u− FΓ u + TG Qp = TG QTG (Vec((DB) · B)). (4.17) Re Re μ0 The Cauchy integral projects any L2 function to a function that belongs to the space L2 (G)∩Ker D. The expression QFΓ u naturally vanishes, because of the imposed boundary condition u|Γ = 0. Summarizing we obtain the following representation formula for the velocity field u: Re u= TG QTG Vec((DB) · B) − Re TG Qp, (4.18) μ0 where the values for magnetic field vector can be recovered from (4.8). The pressure p can be obtained from the continuity equation (Du) = 0. Applying the continuity equation to (4.18) leads to 1 (Qp) = (QTG Vec((DB) · B)), (4.19) μ0 in view of the identity DTG f = f . Summarizing we have got the following representation formulas for the solutions to the system (3.6)–(3.10): B

= FΓ h + TG PG H 1 (Qp) = (QTG Vec((DB) · B)) μ0 Re TG QTG Vec((DB) · B) − Re TG Qp. u = μ0

(4.20) (4.21) (4.22)


133

We can solve Equation (4.21) for the pressure by applying the same algebraic trace formulas as in [21] pp. 102/103 used for the simpler Stokes problem. Adapting formula (4.60) on p. 102 one obtains p = (TG b) − (FΓ trΓ TG Vec FΓ )−1 trΓ TG Vec TG b + C,

(4.23)

where we set b :=

1 Vec((DB) · B), μ0

(4.24)

where C is some additive constant and where tr denotes the usual trace operator representing the restriction to the boundary. B is computed in the first equation (4.20). Plugging its solution into equation (4.23) gives p up to a constant (after applying the substitution (4.24)). Finally, inserting the obtained values for B and p into the equation (4.22) gives u. The problem is reduced to apply the integral operators TG , FΓ and Q. The integral operators TG and FΓ have a universal integral kernel for all bounded domains, 1 namely the Cauchy kernel 4π q0 (x − y). In the case of unbounded domains the Cauchy kernel has to be modified in the sense of substituting it by the expression 1 x − y)), as mentioned in the preliminary section. The third 4π (q0 (x − y) − q0 (˜ operator Q can be expressed by the Bergman projection viz Q = I − P. However, the Bergman kernel differs from domain to domain. In this direction D. Constales and the author were able to set up explicit formulas for large classes of elementary domains that are bounded by spheres and hyperplanes. The crucial idea is to use a new class of hypercomplex Eisenstein series on arithmetic subgroups of the orthogonal group. See [25] for their fundamental theory. The domains for which we got explicit formulas for the monogenic Bergman kernel include for instance rectangular domains [5], wedge shaped domains and orthogonal ball sectors [7], finite and infinite cylinders [6], triangular domains, [9], hyperbolic polyhydron domains [8] and radially symmetric domains [10]. The kernel can be expressed as multipole series with poles that are located in a set that is invariant under a discrete arithmetical group. These formulas also expose a reasonable numerical convergence behavior of convergence order O(n log n), as manifested in [30] in connection with 3D mapping problems. Hence, they are useful to evaluate the formulas (4.20), (4.21) and (4.22). For convenience we recall some of the most elementary examples with radial symmetry: • For the annulus of radii μ and 1 with 0 < μ < 1 the Bergman kernel reads, cf. [10]: Bμ,1;0 (z, w) =

μk 1 Dw , Dz 4π 1 + μ2k zw k∈Z

(4.25)

134

R.S. Kraußhar

• For the annulus of the unit half-ball with radii μ and 1 the kernel reads, cf. [10]: / μk 1 μk Bμ,1:0,1/2 (z, w) = + Dz 4π w − μ2k z 1 + μ2k zw k∈Z μk D . − w 1 + μ2k zw • After having adapted the formula from [6] to the vector formalism, the Bergman kernel of the cylinder Cd defined by x21 + x22 = 1 and 0 < x3 < d reads as follows: +∞ +∞ 1 BCd (z, w) = −Dz (ei(πm/d)(x3 −w3 ) − ei(πm/d)(−x3 −w3 ) ) 2d n=0 m=−∞ I1/2+n ((πm/d)w)I1/2+n ((πm/d)z) w1/2+n z1/2+n K1/2+n (πm/d) Pn (z, w) × I1/2+n (πm/d) ∞ 1 1 Dw . − 4π m=−∞ z + 2md + w ×

Here Pn is the homogeneous part of total degree n of the Poisson kernel in R3 defined by 1/2 + n zn wn Cn1/2 (cos θ), Pn (z, w) = 1/2 1/2

where Cn

denotes a Gegenbauer polynomial, see [17].

5. Outlook for the non-linear case We round off by outlining how to treat the non-linear stationary case. In the case where the convective terms are not negligibly small but still relatively small in comparison with the viscous terms we propose to apply the following fixed point algorithm, similarly as already successfully performed for the Navier-Stokes system combined with the heat equation in [21, 22]. We refer the interested reader also to our recent paper [9] in which we treated the the Navier-Stokes system with heat transfer in some of the special domains that we described in the previous section. When we apply to the system (3.6)–(3.10) the same calculations and arguments as performed in the previous section to the highly viscous case, we obtain the following representation formulas for the velocity u, B and p. For the velocity we obtain the equation Re u= TG QTG Vec((DB) · B) − (uD)u − ReTG Qp. μ0

(5.1)


135

Again, by combining this equation with the continuity equation we get the following formula for determining the pressure: 1 (Qp) = QTG Vec((DB) · B) − (uD)u . (5.2) μ0 Finally, when applying the operators TG QTG to equation (12) we obtain for the magnetic field the equation B = RmTGQTG (BD)u − (uD)B + FΓ h + TG PG H. (5.3) These equations also remain true when the convective terms are high in comparison with the viscous terms. In the case where the viscous terms dominate the convective terms we propose to apply the following fixed point algorithm to Re un = TG QTG Vec((DBn−1 ) · Bn−1 ) − (un−1 D)un−1 − ReTG Qpn μ0 1 (Qpn ) = QTG Vec((DBn−1 ) · Bn−1 ) − (un−1 D)un−1 μ0 Bn = RmTG QTG (Bn D)un − (un D)Bn + FΓ h + TG PG H. To compute Bn we propose to apply the following inner iteration: Bn (i) = RmTG QTG (Bn (i−1) D)un − (un D)Bn (i−1) + FΓ h + TG PG H. Notice that in every step of the outer iteration procedure we have to solve a simpler uncoupled linear problem of the type discussed in Section 4. The precise convergence proof of this fixed point algorithm and the uniqueness arguments involving concrete conditions on the operator norms are discussed in the follow-up separate paper [26]. As already mentioned in the introduction, we can handle the time dependent case in the same flavour when applying the appropriate parabolic versions of the Dirac operator and the corresponding integral operators, working in an appropriate Witt basis, as proposed in [3]. Also this will be treated in our follow-up separate paper [26]. Acknowledgements The author is very thankful to the anonymous referee for the very helpful suggestions and comments.

References [1] A. Belien, M. Botchev, J. Goedbloed, B. van der Holst and R. Keppens, FINESSE: Axisymmetric MHD Equilibria with Flow, J. Comp. Phys., 182 (2002), 91–117. [2] P. Cerejeiras and U. K¨ ahler, Elliptic boundary value problems of fluid dynamics over unbounded domains, Math. Methods Appl. Sci., 23 (2000), 81–101. [3] P. Cerejeiras, U. K¨ ahler and F. Sommen, Parabolic Dirac operators and the NavierStokes equations over time-varying domains, Math. Methods Appl. Sci., 28 (2005), 1715–1724.

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[4] Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919–930. [5] D. Constales and R.S. Kraußhar, The Bergman Kernels for rectangular domains and Multiperiodic Functions in Clifford Analysis, Math. Methods Appl. Sci., 25 (2002), 1509 – 1526. [6] D. Constales and R.S. Kraußhar, Hilbert Spaces of Solutions to Polynomial Dirac equations, Fourier Transforms and Reproducing Kernel Functions for Cylindrical Domains, Z. Anal. Anw., 24 (2005), 611–636. [7] D. Constales and R.S. Kraußhar, The Bergman Kernels for the half-ball and for fractional wedge-shaped domains in Clifford Analysis, Forum Math., 17, No. 5 (2005), 809–821. [8] D. Constales and R.S. Kraußhar, Bergman Spaces of higher dimensional hyperbolic polyhedron type domains I, Math. Methods Appl. Sci., 29, (2006), 85–98. [9] D. Constales and R.S. Kraußhar, On the Navier-Stokes equation with Free Convection in three-dimensional triangular channels, Math. Methods Appl. Sci., 31 (2008), 735 – 751. [10] D. Constales, D. Grob and R.S. Kraußhar, Explicit formulas for the Green’s function and the Bergman kernel for monogenic functions in annular shaped domains in Rn+1 , to appear in Results Math. [11] D. Constales, D. Grob and R.S. Kraußhar, Reproducing kernel functions of solutions to polynomial Dirac equations in the annulus of the unit ball in Rn and applications to boundary value problems, J. Math. Anal. Appl., 358 (2009), 281–293. [12] D. Constales, D. Grob and R.S. Kraußhar, On generalized Helmholtz type equations in concentric annular domains in R3 , Math. Methods Appl. Sci., 33 No. 4 (2010), 431-438. [13] R. Delanghe, F. Sommen and V. Souˇcek, Clifford Algebra and Spinor Valued Functions, Kluwer, Dortrecht-Boston-London, 1992. [14] N. Faustino, K. G¨ urlebeck, A. Hommel and U. K¨ ahler, Difference potentials for the Navier-Stokes equations in unbounded domains, J. Difference Equ. Appl., 12 (2006), 577–595. [15] S. Gala, Extension criterion on regularity for weak solutions to the 3D MHD equations, to appear in Math. Methods Appl. Sci. [16] H. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics, Cambridge University Press, 2004. [17] I. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1980. [18] K. G¨ urlebeck and A. Hommel, On discrete Stokes and Navier-Stokes equations in the plane, in: Clifford algebras. Applications to mathematics, physics, and engineering (eds. R. Ablamowicz), Progress in Mathematical Physics, 34, Birkh¨ auser, Boston, 2004, 35–58. [19] K. G¨ urlebeck, U. K¨ ahler, J. Ryan and W. Spr¨ oßig, Clifford analysis over unbounded domains, Adv. Appl. Math., 19 (1997), 216–239.


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[20] K. G¨ urlebeck, R.S. Kraußhar and S. Poedts, A quaternionic approach to treat the ideally stationary magnetohydrodynamic equations, American Institute of Physics Conference Proceedings, 1168 (2009), 789–792. [21] K. G¨ urlebeck and W. Spr¨ oßig, Quaternionic analysis and elliptic boundary value problems, Basel: Birkh¨ auser, 1990. [22] K. G¨ urlebeck and W. Spr¨ oßig, Quaternionic and Clifford Calculus for Physicists and Engineers, John Wiley & Sons, Chichester-New York, 1997. [23] M. Gunzburger, A. Meir and J. Peterson, On the existence, uniqueness and finite element approximation of the equations of stationary, incompressible magnetohydrodynamics, Math. Comput., 56 (1991), 523–563. [24] C. He, Y. Wang, Remark on the regularity for weak solutions to the magnetohydrodynamic equations, Math. Methods Appl. Sci., 31 (2008), 1667–1684. [25] R. S. Kraußhar, Generalized Analytic Automorphic Forms in Hypercomplex Spaces, Frontiers in Mathematics, Birkh¨ auser, Basel, 2004. [26] R. S. Kraußhar and K. G¨ urlebeck, Applications of parabolic Dirac operators to the instationary viscous MHD equations in bounded and unbounded time-varying domains of R3 , preprint 2010. [27] V. Kravchenko, Applied quaternionic analysis, Research and Exposition in Mathematics 28, Heldermann Verlag, Lemgo, 2003. [28] V. Kravchenko and M. Shapiro, Integral representations for spatial models of mathematical physics, Addison Wesley Longman, Harlow, 1996. [29] C. Li, A. McIntosh and S. Semmes, Convolution singular integrals on Lipschitz surfaces, J. Amer. Math. Soc. 5 (1992), 455–481. [30] J. R¨ usges, Bergmankerne und Abbildungen auf die Einheitskugel, Diplomarbeit, Lehrstuhl II f¨ ur Mathematik, RWTH Aachen, Aachen, 2006. [31] M. Shapiro and N. Vasilevski, On the Bergman kernel function in the Clifford analysis, in: Clifford algebras and their applications in mathematical physics, Fund. Theor. Phys. 55, Kluwer, Dortrecht 1993, 183 – 192. [32] R. Stevenson, Divergence-free wavelet bases on the hypercube: free-slip boundary conditions, and applications for solving the instationary Stokes equations, to appear in Math. Comp. [33] J. Wu, Viscous and inviscid magneto-hydrodynamics equations, J. d’Analyse Math., 73 (1997), 251–265. Rolf S¨ oren Kraußhar Fachbereich Mathematik Technical University of Darmstadt Schlossgartenstraße 7 D-64289 Darmstadt Germany e-mail: [email protected]

The Fischer Decomposition for the H-action and Its Applications Roman Láviˇcka Abstract. Recently the Fischer decomposition for the H-action of the Pin group on Clifford algebra-valued polynomials has been obtained. We apply this tool to get various decompositions of special monogenic and inframonogenic polynomials in terms of two-sided monogenic ones. Mathematics Subject Classification (2010). Primary 30G35; Secondary 58A10. Keywords. Fischer decomposition, special monogenic polynomials, inframonogenic polynomials, two-sided monogenic polynomials.

1. Introduction The classical Fischer decomposition of Clifford algebra-valued (or spinor-valued) polynomials is one of the most important facts concerning solutions of the Dirac equation in the Euclidean space Rm . It leads immediately to the right form of the Taylor (and Laurent) series for monogenic functions and has further important applications in Clifford analysis (see, e.g., [2, 9]). As is well known, the classical Fischer decomposition can be understood as an irreducible decomposition with respect to the L-action (see [3]). On the other hand, on the space of Clifford algebra-valued polynomials we can consider yet another action, the so-called Haction. A natural question arises what the corresponding irreducible decomposition looks like in this case. Actually, the Fischer decomposition for the H-action has been obtained recently in [5] using results from [11]. Furthermore, it turns out that the Fischer decomposition for the H-action can be viewed even as a refinement of that for the L-action (see [5]). The main aim of the paper is to describe an application of the Fischer decomposition for the H-action to inframonogenic functions introduced recently by I acknowledge the financial support from the grant GA 201/08/0397. This work is also a part of the research plan MSM 0021620839, which is financed by the Ministry of Education of the Czech Republic.


139

140

R. Láviˇcka

H. Malonek, D. Pe˜ na Pe˜ na and F. Sommen in papers [14, 15]. In Section 2, we give a short review of the Fischer decomposition for the H-action. In Section 3, we summarize the Fischer decomposition for special monogenic functions, that is, for those solutions of the Dirac equation taking values in a given H-invariant subspace V of the Clifford algebra R0,m . In particular, we describe quite explicitly the Fischer decomposition for solutions of generalized Moisil-Théodoresco systems. These systems have been recently carefully studied and the corresponding Fischer decomposition and the analogues of the Howe dual pairs have been described in a series of papers [7, 8, 4, 6]. In Section 4, we study spaces of homogeneous inframonogenic polynomials and we obtain their Fischer decomposition (see Theorem 4). As is explained in detail in [1], these results can be easily translated into the language of differential forms.

2. The Fischer Decomposition for the H-action In this section, we describe the Fischer decomposition for the H-action obtained recently in [5] using results from [11]. Before doing so we recall the well-known decomposition of spinor-valued polynomials. Let us denote by S a basic spinor representation for the Pin group P in(m) of the Euclidean space Rm and by P(S) the space of S-valued polynomials in the vector variable x = (x1 , . . . , xm ) of Rm . On the space P(S), we can consider the so-called L-action of P in(m) given by [L(r)(P )](x) = r P (r−1 x r), r ∈ P in(m), P ∈ P(S) and x ∈ Rm .

(1)

Let the vectors e1 , . . . , em form the standard basis of R . Then it is easily seen that the multiplication by the vector variable x = e1 x1 + · · · + em xm and the Dirac operator ∂ = e1 ∂x1 + · · · + em ∂xm (both applied from the left) are examples of invariant linear operators on the space P(S) with the L-action. Actually, the invariant operators x and ∂ are basic in the sense that they generate the Lie superalgebra osp(1|2) which gives the hidden symmetry of the space P(S), see [6] for details. Using this observation, the Fischer decomposition for spinor-valued polynomials is given in [3] as a special case of the general theory of the Howe duality (see [12]). Indeed, denote by Mk (S) the space of k-homogeneous polynomials P ∈ P(S) which are left monogenic, that is, those satisfying the Dirac equation ∂P = 0. Then in this case the Fischer decomposition reads as follows: ∞ ∞ P(S) = xp Mk (S). (2) m

k=0 p=0

In addition, we know that, under the L-action, all submodules Mk (S) are irreducible and mutually inequivalent. Recall that the decomposition (2) follows

The Fischer Decomposition

141

easily from the classical Fischer decomposition of the space P of scalar-valued polynomials in Rm which is given by ∞ ∞ |x|2p Kerk Δ. (3) P= k=0 p=0

Here |x| = + ··· + = −x2 , Δ = ∂x21 + · · · + ∂x2m = −∂ 2 is the Laplace operator and Kerk Δ is the space of k-homogeneous polynomials P ∈ P such that ΔP = 0. Indeed, we get easily the decomposition (2) from (3) using the fact that 2

x21

x2m

(Kerk Δ) ⊗ S = Mk (S) ⊕ xMk−1 (S).

(4)

Now we are going to deal with the case of Clifford algebra-valued polynomials. Let R0,m be the real Clifford algebra over Rm satisfying the relations ei ej + ej ei = −2δij . The Clifford algebra R0,m can be viewed naturally as the graded associative algebra R0,m =

m

Rs0,m

s=0

where Rs0,m denotes the space of s-vectors in R0,m . As usual we identify R10,m with Rm . For an 1-vector u and an s-vector v, the Clifford product uv splits into the sum of an (s − 1)-vector u • v and an (s + 1)-vector u ∧ v. Indeed, we have that 1 1 uv = u • v + u ∧ v with u • v = (uv − (−1)s vu) and u ∧ v = (uv + (−1)s vu). 2 2 By linearity, we extend the so-called inner product u • v and the outer product u ∧ v for an 1-vector u and an arbitrary Clifford number v ∈ R0,m . In what follows, we deal with the space P ∗ of R0,m -valued polynomials in m R . Each polynomial P ∈ P ∗ is of the form aα xα , x ∈ Rm P (x) = α

where the sum is taken over a finite subset of multi-indices α = (α1 , . . . , αm ) of α1 α αm ∗ Nm 0 , all coefficients aα belong to R0,m and x = x1 · · · xm . Denote by Pk the ∗ s space of k-homogeneous polynomials of P and by Pk the space of s-vector-valued polynomials of Pk∗ . In general, for V ⊂ P ∗ put Vk = V ∩ Pk∗ and Vks = V ∩ Pks . As we have mentioned above, on the space P(S) with the L-action, ∂ and x are basic invariant operators. On the space P ∗ of Clifford algebra-valued polynomials we can consider yet another action, namely, the so-called H-action given by [H(r)P ](x) = r P (r−1 x r) r−1 , r ∈ P in(m), P ∈ P ∗ and x ∈ Rm .

(5)

Then a natural question arises which invariant operators are basic under the Haction. It is not difficult to see that, under the H-action, the operators ∂ and x are still invariant. On the other hand, we can split left multiplication by the 1-vector x into the outer multiplication x ∧ and the inner multiplication x •, that is, x= x∧+ x•.

142

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Analogously, the Dirac operator ∂ can be split also into two parts ∂ = ∂ + + ∂ − where m m ej ∧ (∂xj P ) and ∂ − P = ej • (∂xj P ). ∂+P = j=1

j=1

Actually, the operators ∂ + , ∂ − , x ∧ and x • are basic invariant operators for the H-action. Indeed, these operators generate the Lie superalgebra sl(2|1) which gives the hidden symmetry of the space P ∗ , see [6] for details. As suggested by the general theory of the Howe duality, the corresponding Fischer decomposition can be obtained even in this case (see [5]). Recall that the space of spinor-valued polynomials decomposes into the direct sum of multiples of spaces of homogeneous solutions of the Dirac equation by non-negative integer powers of x, see (2). It turns out that, for the H-action, basic building blocks are spaces of ‘homogeneous’ solutions of the Hodge-de Rham system of equations ∂ + P = 0, ∂ − P = 0

(6)

and their multiples by non-trivial words in the letters x ∧ and x • . Note that (∂ + )2 = 0, (∂ − )2 = 0, x ∧ x ∧ = 0 and x • x • = 0. In particular, we have that the set Ω of all non-trivial words in the letters x ∧ and x • looks like Ω = {1, x ∧, x •, x ∧ x •, x • x ∧, x ∧ x • x ∧, x • x ∧ x •, . . .}.

(7)

Let H be the space of R0,m -valued polynomials P in R satisfying the Hodge-de Rham system (6). Recall that Hks is then the subset of s-vector-valued polynomials P of H which are homogeneous of degree k. The Fischer decomposition for the H-action reads as follows (see [5]). m

Theorem 1. The space P ∗ of R0,m -valued polynomials in Rm decomposes as P∗ =

∞ m

wHks .

(8)

s=0 k=0 w∈Ω

Remark 1. (i) In addition, we have that Hks = {0} just for s ∈ {0, m} and k ≥ 1, H00 = R and H0m = Re1 e2 · · · em . Moreover, under the H-action, all non-trivial modules Hks are irreducible and mutually inequivalent. (ii) It is easy to see that wHks = {0} if either s = 0 and the word w begins with the letter x • or s = m and the word w begins with the letter x ∧ . Otherwise, wHks " Hks . Now we sketch a proof of Theorem 1, see [5] for details. By the classical Fischer decomposition (3), it is easy to see that m ∞ ∞ P∗ = |x|2p Kersk Δ. (9) s=0 k=0 p=0

where Δ = {P ∈ : ΔP = 0}. Realizing that Δ = −(∂ + ∂ − + ∂ − ∂ + ) and 2 |x| = −(x ∧ x • + x • x ∧), the decomposition (8) then easily follows from the next result obtained by Y. Homma in [11]. Kersk

Pks


143

Theorem 2. Let 0 ≤ s ≤ m and k ∈ N0 . Then, under the H-action, Kersk Δ decomposes into inequivalent irreducible pieces as s−1 s+1 Kersk Δ = Hks ⊕ x ∧ Hk−1 ⊕ x • Hk−1 ⊕ Wks

where s Wks = ((k − 2 + m − s) x ∧ x • −(k − 2 + s) x • x ∧)Hk−2

for 1 ≤ s ≤ m − 1 and k ≥ 2, and Wks = {0} otherwise. Here Hks = {0} unless 0 ≤ s ≤ m and k ∈ N0 . In Section 4, we obtain analogous decompositions for inframonogenic polynomials, see Theorem 4 below.

3. Special Monogenic Polynomials In this section, we study special polynomial solutions of the Dirac equation. In what follows, special solutions are just those taking values in a given subspace V of the Clifford algebra R0,m . Since we consider the H-action on the space P ∗ it is natural to assume that the subspace V is invariant under the both side action of the Pin group P in(m), that is, rV r−1 ⊂ V for each r ∈ Pin(m). Obviously, in this case, for some S ⊂ {0, . . . , m}, we have that V = RS0,m with RS0,m = Rs0,m s∈S

Rs0,m

because the spaces of s-vectors are all irreducible and mutually inequivalent with respect to the given action. Denote by PkS the space of RS0,m -valued polynomials P in Rm which are homogeneous of degree k. In general, for V ⊂ P ∗ put VkS = V ∩ PkS as usual. ˜ the spaces of left and Furthermore, let us denote by, respectively, M and M right monogenic polynomials, that is, ˜ = {P ∈ P ∗ : P ∂ = 0}. M = {P ∈ P ∗ : ∂P = 0} and M Recall that H = {P ∈ P ∗ : ∂ + P = 0, ∂ − P = 0} is the space of polynomial solutions of the Hodge-de Rham system. It is easy to see and well known that the space H is formed just by all two-sided monogenic polynomials, that is, ˜ H = M ∩ M. Indeed, for P ∈ Pks , we have that ˜ P ∂ = (−1)s ∂P where ∂˜ = ∂ + − ∂ − is the so-called modified Dirac operator. Moreover, it is easy to see that m ˜ = 0} and Hk = ˜ = {P ∈ P ∗ : ∂P M Hks . s=0

See [1] for details.

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Given a set S ⊂ {0, . . . , m}, we are mainly interested in the space MSk of left monogenic polynomials in Rm which are homogeneous of degree k. Let us give a few known examples of such spaces.

RS0,m -valued

˜ s = Hs . In other words, for Example 1. For S = {s}, we have that Msk = M k k s-vector-valued functions, all three notions of monogeneity coincide. As we shall see again below the spaces Hks of homogeneous solutions of the Hodge-de Rham system (6) are basic building blocks for the H-action. Example 2. For S = {0, . . . , m}, we have that MSk = Mk . In [5], as an easy application of Theorem 1, the following multiplicity free irreducible decomposition of the space Mk has been obtained. Theorem 3. Under the H-action, the space Mk decomposes into inequivalent irreducible pieces as ( 'm−1 ( ' m s s Mk = Hk ⊕ ((k − 1 + m − s)x • −(k − 1 + s)x ∧)Hk−1 . s=0

s=1

Remark 2. We now rewrite the result of Theorem 3 as in [13]. Let us define the Euler operator E and fermionic Euler operators ∂ + and ∂ − % by m m m E= xj ∂xj , ∂ + = − ej ∧ ej • and ∂ − % = − ej • ej ∧ . (10) j=1

j=1

j=1

For P ∈ Pks , it is easy to see that EP = kP,

∂ + P = sP and ∂ − %P = (m − s)P.

See [1] for details. Putting A = E + ∂ + and B = E + ∂ − %, Theorem 3 tells us that Mk = Hk ⊕ XHk−1 with X = x ∧ A − x • B. (11) 0 m Here XHk−1 = {XP : P ∈ Hk−1 }. Notice that XHk−1 = {0} and XHk−1 = {0}. Moreover, we can easily obtain an analogous decomposition for right monogenic polynomials. Indeed, we have that ˜ k = Hk ⊕ XH ˜ k−1 with X ˜ = x ∧ A + x • B. M (12) Example 3. Assume that r, p and q are non-negative integers such that p < q and r + 2q ≤ m. Putting S = {r + 2p, r + 2p + 2, . . . , r + 2q}, we call ∂f = 0 for RS0,m -valued functions f the generalized Moisil-Théodoresco system of type (r, p, q). In [4], the space MSk of k-homogeneous solutions of this system is decomposed into a direct sum of pieces isomorphic to spaces Hks of homogeneous solutions of the Hodge-de Rham system. In Corollary 1 below, we describe more explicitly the pieces of the decomposition of the space MSk even in the general case. Since MSk is always an invariant subspace of the space Mk the next result follows easily from Theorem 3.


145

Corollary 1. Let S ⊂ {0, . . . , m} and S = {s : s ± 1 ∈ S}. Under the H-action, the space MSk decomposes into inequivalent irreducible pieces as ( ' ( ' S s s Hk ⊕ ((k − 1 + m − s)x • −(k − 1 + s)x ∧)Hk−1 . Mk = s∈S

s∈S

S where X is as in (11). In particular, we have that MSk = HkS ⊕ XHk−1

4. Inframonogenic Polynomials In [14, 15], inframonogenic functions have been recently introduced and studied. In particular, in [14], an analogue of the Fischer decomposition for inframonogenic polynomials is given. Recall that an R0,m -valued polynomial P in Rm is said to be inframonogenic if ∂P ∂ = 0. ˜ by Defining the modified Laplace operator Δ + ˜ = −(∂ ∂ − − ∂ − ∂ + ), Δ ˜ = 0, see it is well known that a polynomial P is inframonogenic if and only if ΔP s [1]. Indeed, for P ∈ Pk , we have that ˜ ∂P ∂ = (−1)s ΔP ˜ is scalar in the sense that it preserves the order of the multiand the operator Δ plicative function on which it acts. Moreover, let us remark that, for P ∈ Pks , we have that xP x = (−1)s (x • x ∧ − x ∧ x•)P. See [1] for details. In [14], the next decomposition has been obtained (cf. (9)): m ∞ ∞ ˜ p P∗ = xp (Kersk Δ)x (13) s=0 k=0 p=0

˜ = 0}. In Theorem 4 below, we present decomposi˜ = {P ∈ P s : ΔP Δ where k tions of homogeneous inframonogenic polynomials in terms of two-sided monogenic ones which are quite analogous to the decompositions of homogeneous harmonic polynomials given in Theorem 2. Moreover, decompositions of homogeneous polynomials which are harmonic and inframonogenic at the same time are given. Kersk

Theorem 4. Let 0 ≤ s ≤ m and k ∈ N0 . ˜ decomposes into inequivalent irreducible pieces (i) Under the H-action, Kersk Δ as ˜ ks ˜ = Hks ⊕ x ∧ Hs−1 ⊕ x • Hs+1 ⊕ W Kersk Δ k−1 k−1 where, for 1 ≤ s ≤ m − 1 and k ≥ 2, we have that s ˜ ks = ((c1 + 1)c2 x ∧ x • +(c2 + 1)c1 x • x ∧)Hk−2 W ˜ s = {0} otherwise. with c1 = k − 2 + s and c2 = k − 2 + m − s, and W k

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Here Hks = {0} unless 0 ≤ s ≤ m and k ∈ N0 . (ii) Moreover, we have that ˜ = Hs ⊕ x ∧ Hs−1 ⊕ x • Hs+1 . Kersk Δ ∩ Kersk Δ k k−1 k−1 Proof. (a) For a proof of the statement (ii), see [13, proof of Theorem 1]. (b) For 1 ≤ s ≤ m − 1 and k ≥ 2, we show now that s s ˜ ∩ (x ∧ x • Hk−2 ˜ ks = Kersk Δ ⊕ x • x ∧ Hk−2 ). W

To do this, let a polynomial P belong to the space s s ⊕ x • x ∧ Hk−2 . x ∧ x • Hk−2 s such that Then there are uniquely determined polynomials P1 , P2 ∈ Hk−2

P = x ∧ x • P1 + x • x ∧ P2 . ˜ = 0 if and only if the polynomial P lies in Obviously, it remains to show that ΔP s ˜ Wk . To show this we use the following well-known relations (see, e.g., [1]): Lemma 1. If we put {T, S} = T S + ST for linear operators T and S on the space P ∗ , then we have that

m {x ∧, x ∧} = 0, {x •, x •} = 0, {x ∧, x •} = − j=1 x2j = −|x|2 , {∂ + , ∂ + } = 0,

{∂ − , ∂ − } = 0,

{∂ + , ∂ − } = −Δ,

{x •, ∂ + } = −A, {x ∧, ∂ − } = −B, {x •, ∂ − } = 0 = {x ∧, ∂ + }. ˜ = −2c1 (c2 + 1)P1 + 2(c1 + 1)c2 P2 Obviously, Lemma 1 gives us that ΔP because ˜ x • x ∧ P2 = 2(c1 + 1)c2 P2 ˜ x ∧ x • P1 = −2c1 (c2 + 1)P1 and Δ Δ ˜ = 0 if with c1 = k − 2 + s and c2 = k − 2 + m − s. Whence we conclude that ΔP nad only if c1 (c2 + 1) P2 = P1 , c2 (c1 + 1) which finishes the proof. (c) We prove the statement (i). Assume that k ≥ 2 and 1 ≤ s ≤ m − 1. Otherwise, we can argue in an analogous way. Denoting s−1 s+1 ˜ s, Nks = Hks ⊕ x ∧ Hk−1 ⊕ x • Hk−1 ⊕W k

˜ Moreover, by (13), we have it is easy to see, by (ii) and (b), that Nks ⊂ Kersk Δ. that [k/2] s ˜ p. xp (Kersk−2p Δ)x Pk = p=0


147

˜ it is sufficient to show that Hence to prove the opposite inclusion Nks ⊃ Kersk Δ

[k/2]

Pks

=

s xp Nk−2p xp .

(14)

p=0

To do this notice that, putting w1 = x ∧ x • and w2 = x • x ∧, we get s−1 s−1 s+1 s+1 )xp = w1p x ∧ Hk−2p−1 and xp (x • Hk−2p−1 )xp = w2p x • Hk−2p−1 . xp (x ∧ Hk−2p−1 s ˜ s = w1 Hs ⊕ w2 Hs we have that, for p ≥ 1, x⊕W Moreover, since x Hk−2 k k−2 k−2 s s s s ˜ k−2p+2 xp Hk−2p xp ⊕ xp−1 W xp−1 = w1p Hk−2p ⊕ w2p Hk−2p .

Finally, Theorem 1 gives us that

[k/2]

Pks = Hks ⊕

p=1

[(k−1)/2] s s (w1p Hk−2p ⊕ w2p Hk−2p )⊕

s−1 s+1 (w1p x ∧ Hk−2p−1 ⊕ w2p x • Hk−2p−1 ).

p=0

Using these observations, we get easily the decomposition (14), which completes the proof of (i). Acknowledgment I am grateful to R. Delanghe and V. Souˇcek for useful conversations.

References [1] F. Brackx, R. Delanghe and F. Sommen, Differential forms and/or multi-vector functions, CUBO 7 (2005), 139-170. [2] F. Brackx, R. Delanghe and F. Sommen, Clifford analysis, Pitman, London, 1982. [3] F. Brackx, H. De Schepper, D. Eelbode and V. Souˇcek, The Howe dual pair in hermitian Clifford analysis, Rev. Mat. Iberoamericana 26 (2010)(2), 449–479. [4] R. Delanghe, R. L´ aviˇcka and V. Souˇcek, On polynomial solutions of generalized Moisil-Théodoresco systems and Hodge systems, arXiv:0908.0842 [math.CV], 2009 (to appear in Adv. appl. Clifford alg.). [5] R. Delanghe, R. L´ aviˇcka and V. Souˇcek, The Fischer decomposition for Hodge-de Rham systems in Euclidean spaces, preprint. [6] R. Delanghe, R. L´ aviˇcka and V. Souˇcek, The Howe duality for Hodge systems, In: Proceedings of 18th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering (ed. K. Grlebeck and C. Knke), Bauhaus-Universitt Weimar, Weimar, 2009. [7] R. Delanghe, On homogeneous polynomial solutions of the Riesz system and their harmonic potentials, Complex Var. Elliptic Equ. 52 (2007), no. 10-11, 1047–1061. [8] R. Delanghe, On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space, preprint. [9] R. Delanghe, F. Sommen, V. Souˇcek, Clifford Algebra and Spinor-valued Functions, Mathematics and Its Applications 53, Kluwer Academic Publishers, 1992, 485 pp.

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[10] J. E. Gilbert and M. A. M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, Cambridge, 1991. [11] Y. Homma, Spinor-valued and Clifford algebra-valued harmonic polynomials, J. Geom. Phys. 37 (2001), 201-215. [12] R. Howe, Remarks on classical invariant theory, Trans. Am. Math. Soc. 313 (1989) (2), 539-570. [13] R. L´ aviˇcka, On the Structure of Monogenic Multi-Vector Valued Polynomials, In: ICNAAM 2009, Rethymno, Crete, Greece, 18-22 September 2009 (ed. T. E. simos, G. Psihoyios and Ch. Tsitouras), AIP Conf. Proc. 1168 793 (2009), pp. 793-796. [14] H. R. Malonek, D. Pe˜ na Pe˜ na and F. Sommen, Fischer decomposition by inframonogenic functions, arXiv:0911.0070 [math.CV], 2009 (to appear in CUBO). [15] H. R. Malonek, D. Pe˜ na Pe˜ na and F. Sommen, A Cauchy-Kowalevski theorem for inframonogenic functions, arXiv:0911.0716 [math.CV], 2009 (to appear in Math. J. Okayama Univ.). Roman L´ aviˇcka Mathematical Institute Charles University Sokolovsk´ a 83 186 75 Praha 8 Czech Republic e-mail: [email protected]

Bochner’s Formulae for Dunkl-Harmonics and Dunkl-Monogenics Shanshan Li and Minggang Fei Abstract. We study the counterpart of a direct sum decomposition of L2 (Rd ) into subspaces which are invariant under the classical Fourier transform for the Dunkl transform associated with a family of weight functions hκ which keep invariant under a given finite reflection group. The explicit Dunkl transform formula for each component of a function in L2 (Rd ; h2κ ) is established, which generalizes Bochner’s formulae for spherical harmonics and spherical monogenics in Dunkl case. Mathematics Subject Classification (2010). Primary 30G35, 43A32; Secondary 33C80, 31A05. Keywords. Reflection group, Clifford algebra, Dunkl transform, Dunkl-harmonics, Dunkl-monogenics.

1. Introduction It is well-known that Fourier analysis in Euclidean space is intimately connected with the action of the group of rotations, as well as that of the groups of translations and dilations. This is due to the fact that the Laplace or Dirac operator is invariant under rotations. But in several applications it would be advantageous to have a function theory which is based on reflection groups, particularly on finite reflection groups, instead of rotation groups. However, there exists one major obstacle. While partial derivatives are invariant under rotations this is not the case for reflections. The way out seems to be to consider differential-difference operators ([5],[8]), also called Dunkl operators in the literature. These operators are invariant under reflections and, additionally, are pairwise commuting. Also, they are very important in pure mathematics and physics. Dunkl operators not only provide a useful tool in the study of special functions with root systems ([6],[11]), This work was supported by the grant from Funda¸c˜ ao para Ciˆ encia e a Tecnologia (Portugal) with grant No.: SFRH/BPD/41730/2007.


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but they are closely related to certain representations of degenerated affine Hecke algebras ([2],[12]). Moreover the commutative algebra generated by these operators has been used in the study of certain exactly solvable models of quantum mechanics, namely the Calogero-Moser-Sutherland models, which describe a quantum mechanical system of N identical particles on a circle or line which interacts pairwise through long range potentials of inverse square type ([14],[17]). The classical Fourier transform defined in L1 (Rd ) is isometrically extended 2 to L (Rd ). Moreover, this extension is invariant under rotations. For a family of weight functions hκ invariant under a given finite reflection group W , it is possible to construct a similar transform acting on L2 (Rd ; h2κ ) associated with the Dunkl operators, called Dunkl transform ([7]), which enjoys similar properties to those of the classical Fourier transform. This transform is defined by fˆ(x) = ch K(x, −iy)f (y)h2κ (y)dy, Rd

where the usual complex-valued exponential e−ix,y is replaced by K(x, −iy) = Vκ (e−i·,y )(x) for a specific positive linear operator Vκ (see next section for detail). We remark that, when κ = 0 (and therefore, hκ (x) ≡ 1 and Vκ = id), fˆ is just the classical Fourier transform. Based on the spherical Dunkl-harmonic function theory in [8] it is easy to obtain a direct sum decomposition of L2 (Rd ; h2κ )) into subspaces which keep invariant under the Dunkl transform. However a result on how the Dunkl transform acts on each subspaces is still missing except for some special cases. The task of this paper is to deduce the explicit Dunkl transform formula for each component of a function in L2 (Rd ; h2κ ), which generalizes the Bochner’s formulae for the classical spherical harmonics ([15]) and spherical monogenics ([9]) in the Dunkl case.

2. Clifford Analysis and Dunkl Analysis We will be working with Rd , the real-linear span of e1 , . . . , ed with ei ej + ej ei = −2δij , where δij is the Kronecker symbol. The universal real-valued Clifford algebra R0,d is defined as the 2d -dimensional associative algebra with basis given by e0 = 1 and eA = eh1 · · · ehn , where A = {h1 , . . . , hn } ⊂ {1, . . . , d}, for 1 ≤ h

1 < ··· < hn ≤ d. Hence, each element x ∈ R0,d will be represented by x = A xA eA ,

d ∂ xA ∈ R. We now introduce the Dirac operator ∂ = j=1 ej ∂xj . In particular we have that ∂ 2 = −Δ, where Δ is the d-dimensional Laplacian. A function f : Ω → R0,d is said to be left-monogenic (resp. right-monogenic) if it satisfies the equation ∂f = 0 (resp. f ∂ = 0) for each x ∈ Ω. Basic properties of the Dirac operator and left-monogenic functions can be found in [1], [4] and [10]. For α ∈ Rd \{0}, the reflection σα x of a given vector x ∈ Rd on the hyperplane Hα orthogonal to α is given, in Clifford notation, by σα x := −αxα−1 .

(2.1)

Bochner’s Formulae for Dunkl-Harmonics and -Monogenics

151

4 A finite set R ⊂ Rd \{0} is called a root system if R Rd · α = {α, −α} and σα R = R for all α ∈ R. For a given root system R the reflections σα , α ∈ R, generate a finite group W ⊂ O(d), called the finite reflection group (or Coxeter group) associated with R. All 3 reflections in W correspond to suitable pairs of roots. For a given β ∈ Rd \ α∈R Hα , we fix the positive subsystem R+ = {α ∈ R| α, β > 0}. This means for each α ∈ R either α ∈ R+ or −α ∈ R+ . A function κ : R → R+ on a root system R is called a multiplicity function if it is invariant under the action of

the associated reflection group W . As abbreviation, we introduce the index γκ = α∈R+ κ(α) and μ = 2γκ + d. Hereby μ is called Dunkl-dimension. Associated with the reflection group W and the multiplicity function κ, there is a weight function hκ defined by 5 | x, α|κα , x ∈ Rd . hκ (x) = α∈R+

This is a positive homogeneous function of degree γκ which is invariant under the finite reflection group W . For each fixed positive subsystem R+ and multiplicity function κ we have the differential-difference operators (also called Dunkl operators): ∂ f (x) − f (σα x) Tj f (x) = αj , f (x) + κ(α) j = 1, . . . , d, (2.2) ∂xj

α, x α∈R+

for f ∈ C 1 (Rd ). In the case κ = 0, Tj , j = 1, . . . , d, reduce to the corresponding partial derivatives. This also gives us the justification to think of these differentialdifference operators as the equivalent of partial derivatives in the case of finite reflection groups. In this paper, we will assume throughout that κ ≥ 0 and γκ > 0. More importantly, these operators mutually commute; that is, Ti Tj = Tj Ti . This property allows us to define a Dunkl-Dirac operator in Rd for the corresponding reflection group W by Dh f =

d

ej Tj f.

(2.3)

j=1

In addition, the Dunkl Laplacian Δh on Rd is defined through Δh = −Dh2 =

d 2 j=1 Tj . As usual, functions belonging to the kernel of Dunkl-Dirac operator Dh and Dunkl-Laplacian Δh will be called Dunkl-monogenic functions and Dunklharmonic functions, respectively. The Dunkl intertwining operator Vκ is a positive linear operator defined uniquely by the conditions ([13]) Vκ Pn ⊂ Pn , Vκ 1 = 1, Tj Vκ = Vκ ∂j , 1 ≤ j ≤ d, where Pn denotes the space of homogeneous polynomials of degree n defined in Rd .

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The Dunkl kernel K(x, y) on Rd × Rd which generalizes the usual exponential function exp( x, y), has been introduced in [6] by means of the operator Vκ , as follows. For n ∈ N, N denotes the set of all natural numbers, we put

·, yn )(x), x, y ∈ Rd . Kn (x, y) = Vκ ( n! The Dunkl kernel K is now defined as ∞ K(x, y) = Kn (x, y) = Vκ (exp( ·, y))(x). n=0

If we denote by L (Rd ; h2κ ), p ∈ [1, +∞] the weighted spaces of measurable functions on Rd satisfying

1/p p 2 f κ,p := |f (x)| hκ (x)dx < +∞, if 1 ≤ p < +∞, p

f κ,∞

Rd

:=

ess supx∈Rd |f (x)| < +∞,

the Dunkl kernel above allows us to define the equivalent of the Fourier transform in the Dunkl case. Definition 2.1. The Dunkl transform is given for any f ∈ L1 (Rd ; h2κ ) by ˆ K(x, −iy)f (y)h2κ (y)dy, x ∈ Rd . f (x) = ch

(2.4)

Rd

The detailed study about Dunkl kernel and Dunkl transform can be found in [7], [3] and [16].

3. Bochner’s Formula for Dunkl-Harmonics Definition 3.1. Let n ∈ N. Denote by (i) Hn = Pn ∩ kerΔh and Hn the space consisting of the restrictions to the unit sphere S d−1 of the functions in Hn . The elements belonging to Hn and Hn are called solid spherical Dunkl-harmonics and surface spherical Dunklharmonics of degree n, respectively. (ii) Nn (h2κ ) the space of all linear combinations of functions of the form f (r)P (x), where f ranges over the radial functions and P ∈ Hn . Now we arrive at the following direct sum decomposition theorem of L2 (Rd ; h2κ ): $∞ Theorem 3.2. The direct sum decomposition L2 (Rd ; h2κ ) = n=0 Nn (h2κ ) holds in the sense that (i) each subspace Nn (h2κ ) is closed; (ii) the Nn (h2κ ) are mutually orthogonal; (iii) every element of L2 (Rd ; h2κ )3is a limit of finite linear combinations of elements ∞ which belong to the spaces n=−∞ Nn (h2κ );


153

(iv) the Dunkl transform maps each Nn (h2κ ) into itself. Proof. The results (i) (ii) and (iii) can be proved easily. The key result is the last one. In order to prove (iv), it suffices to consider f ∈ L1 (Rd ; h2κ ) ∩ L2 (Rd ; h2κ ) having the form f (y) = f0 (ρ)P (y) = ρn f0 (ρ)Y (ξ) with Y ∈ Hn , where ρ = y, and y = ρξ. Since the finite linear combinations of such functions are dense in Nn (h2κ ), Nn (h2κ ) will be invariant under the action of the Dunkl transform provided fˆ ∈ Nn (h2κ ), whenever f has the above form. Letting r = x, and x = rω we then obtain fˆ(x) = ch K(y, −ix)f (y)h2κ (y)dy Rd = ch Vκ [e·,−ix ](y)f (y)h2κ (y)dy Rd ∞ = ch f0 (ρ)ρ2γκ +n+d−1 { Vκ [e−irρω,· ](ξ)Y (ξ)h2κ (ξ)dΣ(ξ)}dρ, sd−1

0

where dΣ(ξ) is the usual Lebesgue measure on the unit sphere S d−1 of Rd . From the Funk-Hecke formula in Dunkl case, there exits a function ϕκ (s), s ∈ C, such that Vκ [esω,· ](ξ)Y (ξ)h2κ (ξ)dΣ(ξ) = ϕκ (s)Y (ω), sd−1

where ϕκ (s) =

1 γ +(d−2)/2

Cnκ

(1)

1

−1

est Cnγκ +(d−2)/2 (t)ωγκ +(d−2)/2 (t)dt,

(3.1)

with ωλ denoting the normalized weight function: ωλ (t) = B(λ + 1/2, 1/2)−1(1 − t2 )λ−1/2 , t ∈ [−1, 1], )1 where B(a, b) = 0 ta−1 (1 − t2 )b−1 dt is the Beta function. So, ∞ ˆ f0 (ρ)ϕκ (−irρ)ρ2γκ +n+d−1 dρ}Y (ω) f (x) = {ch 0

and, consequently, fˆ ∈ Nnd (h2κ ).

However, from the proof of Theorem 3.2 we still do not know clearly about how the Dunkl transform acts on each subspace Nn (h2κ ). In fact, when n = 0, i.e. N0 (h2κ ) is a subspace consisting of all radial functions of L2 (Rd ; h2κ ), there is a good answer as follows (Proposition 5.7.8 in [8]), which is a counterpart of the result on Fourier transform of radial functions (Theorem 3.3 in Chapter IV of [15]).

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Theorem 3.3. Suppose that f ∈ L1 (Rd ; h2κ ) is a radial function, then fˆ(x) = F0 (x) is also radial. Furthermore, ∞ f (ρ)Jγκ +(d−2)/2 (ρr)ργκ +d/2 dρ F0 (x) = F (r) = 2(d−2)/2 Γ(d/2)r−[γκ +(d−2)/2] ∞0 (d−2)/2 −[(μ−2)/2] =2 Γ(d/2)r f (ρ)J(μ−2)/2 (ρr)ρμ/2 dρ, 0

where JA (t) is the Poisson representation of Bessel function of index A > −1/2: 1 (t/2)A JA (t) = √ eits (1 − s2 )A−1/2 ds. πΓ(A + 1/2) −1 We now turn our attention to the space Nn (h2κ ), n ≥ 1. From Theorem 3.2, for f (y) = f0 (y)P (y) = ρn f0 (ρ)Y (ξ), where ρ = y, and y = ρξ, if we let ∞ Fn (r) = r−n {ch f0 (ρ)ϕκ (−irρ)ρ2γκ +n+d−1 dρ}, (3.2) 0

which is represented through Gegenbauer polynomials and Beta funtions (see formula (3.1)), then fˆ(x) = Fn (r)rn Y (ω) = Fn (x)P (x), x = rω ∈ Rd . In the following we expect to explicitly express Fn (r) by Bessel functions like the classical Bochner’s formulae in terms of spherical harmonics ([15]). To do so, we need the following lemma which constructs the relationship between Bessel functions and Gegenbauer polynomials in Rd . Lemma 3.4. Let (d − 2)/2 = λ, then

Γ(λ + 1)Γ(n + 2λ) ∞ f0 (ρ)Jn+λ (rρ)ρn+1+λ dρ 2λ i−n r−(n+λ) Γ(n + 1)Γ(2λ) 0 1

∞ −irρt λ 2 λ−1/2 f0 (ρ) e Cn (t)(1 − t ) dt ρn+1+2λ dρ, = 0

(3.3)

−1

where JA is the Poisson representation of Bessel function of index A and Cnλ is the Gegenbauer polynomial of degree n associated with λ. Proof. We use the following definition for the classical Fourier transform F f (x) = e−ix,u f (u)du. Rd

Then, for f (y) = f0 (ρ)P˜ (u) = ρ f0 (ρ)Y˜ (ζ), where ρ = u and u = ρζ, P˜ and Y˜ denote the ordinary solid spherical harmonics and surface spherical harmonics of degree n, respectively, and letting r = x and x = rω, we then obtain ∞ −ix,u n+d−1 F f (x) = e f (u)du = f0 (ρ)ρ { e−irρω,ζ Y˜ (ζ)dΣ(ζ)}dρ. n

Rd

0

S d−1


155

For one thing, from Chapter IV of [15], there exists a function ϕ, defined on [0, ∞], such that e−isω,ζ Y˜ (ζ)dΣ(ζ) = ϕ(s)Y˜ (ω) (3.4) S d−1

and, furthermore F f (x) = Fñ (x)P˜ (x), where Fñ (x) = (2π)d/2 i−n r−[n+(d−2)/2]

∞

f0 (ρ)Jn+(d−2)/2 (rρ)ρn+d/2 dρ.

0

So, we obtain

∞ f0 (ρ)Jn+(d−2)/2 (rρ)ρn+d/2 dρ (2π)d/2 i−n r−(d−2)/2 0 ∞ = f0 (ρ)ϕ(rρ)ρn+d−1 dρ

(3.5)

0

for s ≥ 0. For another, if we use the classical Funk-Hecke formula for the exponential function e−isω,ζ , we have e−isω,ζ Y˜ (ζ)dΣ(ζ) = α(s)Y˜ (ω), (3.6) S d−1

where

α(s) = |S d−1 |

1

−1

e−ist Pnd (t)(1 − t2 )(d−3)/2 dt,

where Pnd (t)(−1 ≤ t ≤ 1) is a function such that zξd,n (η) = cd,n Pnd ( ξ, η), ∀ ξ, η ∈ S d−1 , where zξd,n is the classical zonal harmonics with pole ξ of d variables of degree n, cd,n =

Γ(d/2) (d+2n−2) (d+n−2)! . 2π d/2 (d+n−2) n!(d−2)!

In addition, the relationship between Pnd (t) and Gegenbauer polynomial Cnν in R is as follows. Γ(n + d − 2) P d (t), f or − 1 ≤ t ≤ 1. Cn(d−2)/2 (t) = Γ(n + 1)Γ(d − 2) n d

So, we arrive at α(s) =

2π d/2 Γ(n + 1)Γ(d − 2) Γ(d/2)Γ(n + d − 2)

1

−1

e−ist Cn(d−2)/2 (t)(1 − t2 )(d−3)/2 dt.

(3.7)

From (3.4) and (3.6) we know that ϕ(s) = α(s) for any s ≥ 0. Therefore, formula (3.3) is a consequence of (3.5) by replacing ϕ(rρ) by α(rρ) which is given by (3.7). This proves the lemma.

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Now, invoking (3.1), (3.2) and (3.3), and putting all the constants together, we conclude this section with the following theorem which is a counterpart of the classical Bochner formula in terms of spherical harmonics (Theorem 3.10 in Chapter IV of [15]). Theorem 3.5. Suppose d ≥ 2 and f ∈ L1 (Rd ; h2κ ) ∩ L2 (Rd ; h2κ ) has the form f (y) = f0 (y)P (y), where P (y) is the solid spherical Dunkl-harmonics of degree n, then the Dunkl transform of f has the form fˆ(x) = Fn (x)P (x), where 2(μ−2)/2 ch Γ(μ/2) −n −[(μ+2n−2)/2] ∞ i r f0 (ρ)J(μ+2n−2)/2 (rρ)ρ(μ+2n)/2 dρ. Fn (r) = B((μ − 1)/2, 1/2) 0 This theorem, together with the fact that the space Nn (h2κ ) is spanned by functions of the form f0 (x)P (x), gives the promised description of the action of the Dunkl transform on Nn (h2κ ), n ≥ 1.

4. Bochner’s Formula for Dunkl-Monogenics Definition 4.1. Denote by (i) M+ n the space of homogeneous Dunkl-monogenic polynomials of degree n in Rd . An arbitrary element of M+ n is called an inner Dunkl-monogenic function of degree n. (ii) M− n the space of homogeneous Dunkl-monogenic functions of degree −(n + μ − 1) in Rd \{0}. An arbitrary element of M− n is called an outer Dunklmonogenic function of degree n. − (iii) M+ n and Mn the spaces consisting of the restrictions to the unit sphere d−1 − S of, respectively, the elements belonging to M+ n and Mn . The elements + − of Mn and Mn are called spherical Dunkl-monogenics, or surface spherical Dunkl-monogenics. Thanks to Clifford algebra, for any n ∈ N, there holds Hn = M+ M− n n−1 , where for n = 0 we put by definition M− −1 = {0}. This allows us to consider the counterpart of the Bochner’s formula for monogenics in Dunkl case. To this end, we need the following definition of function spaces. Definition 4.2. Define (i) Ωn (h2κ ), n ≥ 0, to be the space of finite linear combinations ) ∞ of functions of the form f (r)P (x), where f is a radial function satisfying 0 |f (r)|2 r2n+μ−1 dr < ∞ and P ∈ M+ n . Obviously, P ∈ Hn . (ii) Ω−n (h2κ ), n > 0, to be the space of finite linear combinations of functions of the form g(r)Q(x), where g is a radial function satisfying ∞ |g(r)|2 r−(2n+μ−3) dr < ∞ 0


157

and Q ∈ M− n−1 . Using a similar argument in [9] we arrive the following direct sum decomposition theorem of L2 (Rd ; h2κ ) in Dunkl-monogenics: Theorem 4.3. The direct sum decomposition L2 (Rd ; h2κ ) =

∞

Ωn (h2κ )

n=−∞

holds in the sense that (i) each subspace Ωn (h2κ ) is closed; (ii) the Ωn (h2κ ) are mutually orthogonal; (iii) every element of L2 (Rd ; h2κ )3is a limit of finite linear combinations of elements ∞ which belong to the spaces n=−∞ Ωn (h2κ ); (iv) the Dunkl transform maps each Ωn (h2κ ) into itself. This theorem can be proved in the same way as Theorems 1 and 2 in [9], therefore, we omit it here. Below we will concentrate on how the Dunkl transform acts on each subspace Ωn (h2κ ), n ∈ Z, where Z denotes the set of all integers. In fact, when n = 0, it is obvious that Ω0 (h2κ ) = N0 (h2κ ) and this case is well studied in Theorem 3.3. Throughout the rest of this paper, we assume that n is a nonzero integer. Theorem 4.4. Let f ∈ Ωn (h2κ ) and f (y) = h(y)R(y), where R(y) ∈ M+ n if n > 0 and R(y) ∈ M− if n < 0. Then we have |n|−1 fˆ(x) = Hn (x)R(x), and, with cn = (μ + 2|n| − 2)/2, 2(μ−2)/2 ch Γ(μ/2) −|n| − sgn(n)cn i r Hn (r) = B((μ − 1)/2, 1/2)

∞

h(ρ)Jcn (rρ)ρ1+sgn(n)cn dρ, (4.1)

0

where sgn(n) is the signum function that takes value 1 or −1 for n > 0 or n < 0. Proof. When n > 0, it is evident that Ωn (h2κ ) ⊂ Nn (h2κ ). Thus, Bochner’s formula for Dunkl-harmonics (Theorem 3.5) can be used in this case. Therefore, for f (y) = h(y)R(y) with R(y) being either a Dunkl-harmonics of degree n or inner Dunklmonogenics of degree n, there has fˆ(x) = Fn (x)R(x), where 2(μ−2)/2 ch Γ(μ/2) −n −[(μ+2n−2)/2] ∞ i r h(ρ)J(μ+2n−2)/2 (rρ)ρ(μ+2n)/2 dρ Fn (r) = B((μ − 1)/2, 1/2) 0 2(μ−2)/2 ch Γ(μ/2) −n −cn ∞ i r h(ρ)Jcn (rρ)ρ1+cn dρ, (4.2) = B((μ − 1)/2, 1/2) 0 with the constant cn = (μ + 2|n| − 2)/2.

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Below, we assume that n > 0. Let g ∈ Ω−n (h2κ ) of the form g(y) = h(y)R(x) − with R(x) ∈ M− |n|−1 . As a function in M|n|−1 , the function R has the form R(y) =

Hn (y) , yμ+2n−2

where Hn (y) ∈ Hn . Consequently, g(y) = h(y)R(y) = y−(μ+2n−2) h(y)Hn (y) ∈ Nn (h2κ ). So, Bochner’s formula on Dunkl transform of functions in Nn (h2κ ) (Theorem 3.5) gives gˆ(x) = Fn (x)Hn (x) = xμ+2n−2 Fn (x)R(x) = Gn (x)R(x), and Gn (r) = rμ+2n−2 Fn (r)

2(μ−2)/2 ch Γ(μ/2) −n −[(μ+2n−2)/2] ∞ h(ρ)J(μ+2n−2)/2 (rρ)ρ(μ+2n)/2 dρ =r i r B((μ − 1)/2, 1/2) 0 2(μ−2)/2 ch Γ(μ/2) −n (μ+2n−2)/2 ∞ i r h(ρ)J(μ+2n−2)/2 (rρ)ρ−[(μ+2n−4)/2] dρ = B((μ − 1)/2, 1/2) 0 2(μ−2)/2 ch Γ(μ/2) −n cn ∞ i r h(ρ)Jcn (rρ)ρ1−cn dρ. (4.3) = B((μ − 1)/2, 1/2) 0 μ+2n−2

Summarily, the formulas (4.2) and (4.3) together provide a refinement of Bochner’s formula with spherical Dunkl-harmonics replaced by spherical Dunkl-monogenics. Furthermore, by introducing the signum function 1, n > 0 sgn(n) = −1, n < 0, the formulas (4.2) and (4.3) can be unified into one formula, i.e., formula (4.1). This completes the proof.

References [1] F. Brackx, R. Delanghe and F. Sommen, Clifford analysis, Research Notes in Mathematics, Vol. 76, Pitman Advanced Publishing Company, Boston, London, Melbourne, 1982. [2] L. Cherednik, A unification of the Knizhnik-Zamolodchikov equation and Dunkl operators via affine Hecke Algebras, Invent. Math. 106 (1991), 411-432. [3] M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), 147-162. [4] R. Delanghe, F. Sommen and V. Souˇcek, Clifford algebra and spinor valued functions, A function theory for Dirac operator, Kluwer, Dordrecht, 1992. [5] C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.


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[6] C. F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213-1227. [7] C. F. Dunkl, Hankel transforms associated to finite reflection groups, Contemp. Math. 138 (1992), 123-138. [8] C. F. Dunkl and Y. Xu, Orthogonal polynomials of several variables, Cambridge Univ. Press, 2001. [9] M. Fei and T. Qian, Direct sum decomposition of L2 (Rn 1 ) into subspaces invariant under Fourier transformation, J. Fourier Anal. Appl. 12 (2006), 145-155. [10] K. G¨ urlebeck and W. Spr¨ oßig, Quaternionic and Clifford calculus for Engineers and Physicists, John Wiley & Sons, Chichester, 1997. [11] G. J. Heckman, An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), 341-350. [12] E. M. Opdam, Harmonic analysis for certain representations of graded Hecke Algebras, Acta Math. 175 (1995), 75-121. [13] M. R¨ osler, Positivity of Dunkl’s intertwining operator, Duke Math. J. 98 (1999), 445-463. [14] M. R¨ osler, Dunkl Operators: Theory and Applications, Orthogonal polynomials and special functions (Leuven, 2002), 93-135, Lecture Notes in Math. 1817, Springer, Berlin, 2003. [15] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1971. [16] K. Trimèche, Paley-Wiener theorems for the Dunkl transform and Dunkl translation opeators, Intergal Tranfroms and Special Functions, 13 (2002), 17-38. [17] J. F. van Diejen and L. Vinet, Calogero-Moser-Sutherland Models, CRM Series in Mathematical Physics, Springer-Verlag, New York, 2000. Shanshan Li School of Computer Science and Technology Southwest University for Nationalities Chengdu, 610041, P. R. China e-mail: [email protected] Minggang Fei School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu, 610054, P. R. China and Departamento de Matem´ atica Universidade de Aveiro Aveiro, P-3810-193, Portugal e-mail: [email protected]

An Invitation to Split Quaternionic Analysis Matvei Libine Abstract. Six years after William Rowan Hamilton’s discovery of quaternions, in 1849 James Cockle introduced the algebra of split quaternions. (He called them “coquaternions.”) In this paper we define regular functions on split quaternions and prove two different analogues of the Cauchy-Fueter formula for these functions. In the paper “Split quaternionic analysis and the separation of the series for SL(2, R) and SL(2, C)/SL(2, R)” joint with Igor Frenkel we naturally apply the methods and formulas of quaternionic analysis to solve the problems of harmonic analysis on SL(2,R) and the imaginary Lobachevski space SL(2, C)/SL(2, R). Mathematics Subject Classification (2010). Primary 30G35; Secondary 20C15. Keywords. Cauchy-Fueter formula, split quaternions, Minkowski space, harmonic analysis on SL(2, R) and SL(2, C)/SL(2, R), imaginary Lobachevski space, Cayley transform.

1. Introduction Six years after William Rowan Hamilton’s discovery of quaternions, in 1849 James Cockle introduced the algebra of split quaternions [3]. (He called them “coquaternions.”) One way to define the split quaternions HR is by taking the standard generators for the H = R1 ⊕ Ri ⊕ Rj ⊕ Rk and replacing the √ algebra of quaternions √ i and j with ˜ι = −1i and j˜ = − −1j respectively, so that HR = R1⊕R˜ι⊕R˜ j⊕Rk. Another way to realize split quaternions is as real 2 × 2 matrices. And yet another realization is

z11 z12 HR = ; z11 , z12 , z21 , z22 ∈ C, z22 = z11 , z21 = z12 . z21 z22 I. Frenkel initiated development of quaternionic analysis from the point of view of representation theory of the conformal group SL(2, H) and its Lie algebra sl(2, H). This approach has already been proven very fruitful and in our joint work [6] we push further the parallel with complex analysis and develop a rich theory. In I. Sabadini and F. Sommen (eds.), Hypercomplex Analysis and Applications, Trends in Mathematics, DOI 10.1007/978-3-0346-0246-4_12, © Springer Basel AG 2011

161

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M. Libine

particular we show that the quaternionic analogue of the Cauchy integral formula for the second-order pole 6 f (z) dz 1 f (w) = 2πi (z − w)2 are the differential operator Mx f = ∇f ∇ − f + defined on all holomorphic functions of four complex variables f : HC → HC and its integral presentation (note the square of the Fueter kernel) (Z − W )−1 (Z − W )−1 · f (Z) · dZ 4 , (Mx f )(W ) = det(Z − W ) C4 det(Z − W ) where dZ 4 is the volume form, C4 is a four cycle homologous to U (2) = {Z ∈ HC ; Z ∗ Z = 1} sitting in the complexified quaternionic space HC = C ⊗ H. Since the constant functions on C are the holomorphic functions annihilated by the operator d/dz : f (z) → f (z), their quaternionic analogue is the kernel of the operator Mx, which turns out to be the space of solutions of a Euclidean version of the Maxwell equation for the gauge potential. We also identify the Feynman integrals associated to the diagrams

Feynman diagrams with the intertwining operators projecting certain natural unitary representations of su(2, 2) onto their first irreducible components. Then we conjecture that the other Feynman integrals also admit an interpretation via quaternionic analysis and representation theory as the projectors onto the other irreducible components. With this representation theoretic approach it quickly becomes evident that one has to consider the complexifications HC = C ⊗ H and sl(4, C) of H and sl(2, H) and their real forms, such as the Minkowski space M and su(2, 2) or the split quaternions HR and sl(4, R). An important aspect of quaternionic analysis is its ability to compare representation theories of various real forms, and thus produce new results and make previously known results more explicit. Just as (classical) quaternionic analysis is intimately related to the representation theory of SU (2), split quaternionic analysis is related to the representation theory of SL(2, R). The representation theory of SL(2, R) is much richer than that of SU (2) and exhibits most aspects of representations of higher rank real semisimple Lie groups. In particular, the group SL(2, R) exhibits a subtle aspect of representation theory such as the separation of the discrete and continuous series of unitary representations: L2 SL(2, R) " L2discr SL(2, R) ⊕ L2cont SL(2, R) . (1.1)

An Invitation to Split Quaternionic Analysis

163

We study this decomposition from the quaternionic point of view. The denominator of the Cauchy-Fueter kernel det(X − X0 ) determines a region in HC = C ⊗ H Ω = X0 ∈ HC ; det(X − X0 ) = 0 ∀X ∈ HR with det(X) = 1 . Loosely speaking, this region can be broken into several connected components, two of which are open Ol’shanskii semigroups of GL(2, C). Then the Cauchy-Fueter (X−X0 )−1 on HR can be expanded in terms of the K-types of the discrete or kernel det(X−X 0) continuous series of SL(2, R) depending on the choice of the connected component of Ω containing X0 . It follows that the projectors onto the discrete and continuous series of SL(2, R) can be expressed as (X − X0 )−1 1 discr (P · Dz · f (X), )f (X0 ) = 2 2π Cdiscr det(X − X0 ) (X − X0 )−1 1 · Dz · f (X), (Pcont )f (X0 ) = 2 2π Ccont det(X − X0 ) where Cdiscr and Ccont are certain three cycles in Ω. Note that these two integrals are identical to the Cauchy-Fueter formula, except for the choice of contours of integration Cdiscr and Ccont . Moreover, by choosing appropriate cycles we can even get projectors onto the holomorphic discrete series and antiholomorphic discrete series. Such a geometric description of the decomposition (1.1) fits well into the Gelfand-Gindikin program initiated in [10]. This relationship between quaternionic analysis and the separation of the series for SL(2, R) will be the subject of our upcoming paper [7]. In view of how many connections to mathematical physics (classical) quaternionic analysis has (see, for example, [13]), it is almost guaranteed that split quaternionic analysis will have them as well. At this point it is worth mentioning two very recent physics papers [1] and [16] stressing the importance of the (2, 2) signature of the split quaternions HR as opposed to the traditional Lorentzian signature of the Minkowski space M. Once split quaternionic analysis is sufficiently developed, it can be generalized in many different ways. Perhaps the most obvious direction is to extend the new results to higher dimensions. There is a generalization of quaternionic analysis known as Clifford analysis. Let Cl(V ) be a Clifford algebra over a real finitedimensional vector space V with a nondegenerate quadratic form Q(x). Then on R ⊕ V we can introduce Dirac operators D and D + with coefficients in Cl(V ) so that DD+ = D+ D is the wave operator on R ⊕ V with symbol x2 − Q. We define a differentiable function f : R ⊕ V → Cl(V ) to be (left) Clifford analytic (or monogenic) if D+ f = 0. (We think of it as an analogue of the Cauchy-Riemann equations.) Slightly more generally, we can define Clifford analytic functions with values in a Cl(V )-module. When the quadratic form Q is negative definite, there is a well developed theory of Clifford analytic functions called Clifford analysis (see, for example, [2], [5] and [12]). This theory generalizes (classical) quaternionic analysis. Note that in this case DD+ is the Laplacian on R ⊕ V , hence the components of

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Clifford analytic functions are harmonic. Clifford analysis has deep connections with harmonic analysis, representations of spin groups and index theory of Dirac operators. Methods developed in split quaternionic analysis will extend to analysis of Clifford analytic functions associated with quadratic forms of arbitrary signature. Thus we can consider split Clifford analysis as a “real form” of complexified Clifford analysis, which in turn was introduced by Ryan in [17]. On the other hand, the use of the wave equation for the study of harmonic analysis on a hyperboloid goes back to Strichartz [18]. More recently, Kobayashi and Ørsted [14] study representations of O(p + 1, q + 1) in the space of solutions of the wave equation p,q ϕ = 0 on Rp+q . In this light it is natural to expect that this split Clifford analysis will yield new results relating solutions of the wave equation, representation theory and index theory of Dirac operators. In particular, we expect to obtain concrete realizations of representations of O(p + 1, q + 1) in the space of solutions of Clifford analytic functions. Since Clifford analysis is widely used by mathematical physicists, it is very likely that they will find the split version at least as useful as the classical one. In this article we approach the split quaternions HR as a real form of HC , introduce the notion of regular functions and give two different analogues of the Cauchy-Fueter formula valid for different classes of functions. This is done in parallel with (classical) quaternionic analysis. We conclude the paper with an outline of our derivation of the projectors Pdiscr and Pcont onto the discrete and continuous series of SL(2, R). Some contemporary reviews of quaternionic analysis are given in [19] and [4].

2. The Quaternionic Spaces HC , HR and M In this article we use notations established in [6]. In particular, e0 , e1 , e2 , e3 denote the units of the classical quaternions H corresponding to the more familiar 1, i, j, √ k (we reserve the symbol i for −1 ∈ C). Thus H is an algebra over R generated by e0 , e1 , e2 , e3 , and the multiplicative structure is determined by the rules e0 ei = ei e0 = ei ,

(ei )2 = −e0 ,

ei ej = −ei ej ,

1 ≤ i < j ≤ 3,

and the fact that H is a division ring. Next we consider the algebra of complexified quaternions HC = C ⊗ H (also known as biquaternions). We define a complex conjugation on HC with respect to H: Z = z 0 e0 + z 1 e1 + z 2 e2 + z 3 e3

→

Z c = z 0 e0 + z 1 e1 + z 2 e2 + z 3 e3 ,

z 0 , z 1 , z 2 , z 3 ∈ C, so that H = {Z ∈ HC ; Z c = Z}. The quaternionic conjugation on HC is defined by: Z = z 0 e0 + z 1 e1 + z 2 e2 + z 3 e3

→

Z + = z 0 e0 − z 1 e1 − z 2 e2 − z 3 e3 ,

z 0 , z 1 , z 2 , z 3 ∈ C; it is an anti-involution: (ZW )+ = W + Z + ,

∀Z, W ∈ HC .


165

We will also use an involution Z → Z − = −e3 Ze3

(conjugation by e3 ).

Then the complex conjugation, the quaternionic conjugation and the involution Z → Z − commute with each other. In this article we will be primarily interested in the space of split quaternions HR which is a real form of HC defined by HR = {Z ∈ HC ; Z c− = Z} = {R -span of e0 , e˜1 = ie1 , e˜2 = −ie2 , e3 }. We will also consider the Minkowski space M which we regard as another real form of HC : M = {Z ∈ HC ; Z c+ = −Z}, M is spanned over R by e˜0 = −ie0 , e1 , e2 , e3 . On HC we have a quadratic form N defined by N (Z) = ZZ + = Z + Z = (z 0 )2 + (z 1 )2 + (z 2 )2 + (z 3 )2 , hence

Z+ . N (Z) The corresponding symmetric bilinear form on HC is 1 1 Z, W ∈ HC , (2.1)

Z, W = Tr(Z + W ) = Tr(ZW + ), 2 2 where Tr Z = 2z 0 = Z + Z + . When this quadratic form is restricted to H, HR and M, it has signature (4, 0), (2, 2) and (3, 1) respectively. The real forms H and M have been studied in [6], and the signature (1, 3) is equivalent to (3, 1). In this article we study the real form HR realizing the only remaining signature (2, 2). We will use the standard matrix realization of H so that

1 0 0 −i 0 −1 −i 0 , e1 = , e2 = , e3 = , e0 = 0 1 −i 0 1 0 0 i Z −1 =

and H = {Z ∈ HC ; Z c = Z} =

Z=

z11 z21

z12 z22

∈ HC ; z22 = z11 , z21 = −z12 .

Then HC can be identified with the algebra of all complex 2 × 2 matrices:

z12 z HC = Z = 11 ; zij ∈ C , z21 z22 the quadratic form N (Z) becomes det Z and the involution Z → Z − becomes

z11 −z12 1 0 1 0 z11 z12 − = Z → Z = . Z= 0 −1 0 −1 z21 z22 −z21 z22 The split quaternions HR and the Minkowski space M have matrix realizations

z11 z12 HR = Z = ∈ HC ; z22 = z11 , z21 = z12 z21 z22

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and

M=

Z=

z11 z21

z12 z22

∈ HC ; z11 , z22 ∈ iR, z21 = −z12 .

From this realization it is easy to see that the split quaternions form an algebra over R isomorphic to gl(2, R), the invertible elements in HR , denoted by H× R , are nothing else but GL(2, R). We regard SL(2, C) as a quadric {N (Z) = 1} in HC , and we also regard SU (1, 1) " SL(2, R) as the set of real points of this quadric: SU (1, 1) = {Z ∈ HR ; N (Z) = 1}

z11 z12 2 2 = Z= ∈ HR ; det Z = |z11 | − |z12 | = 1 . z12 z11 The algebra of split quaternions HR is spanned

1 0 0 1 0 , e˜1 = , e˜2 = e0 = 0 1 1 0 −i so HR =

x0 e0 + x1 e˜1 + x2 e˜2 + x3 e3 =

0 x − ix3 x1 − ix2

(2.2)

over R by the four matrices

i −i 0 , e3 = , 0 0 i

x1 + ix2 0 1 2 3 ; x , x , x , x ∈ R . x0 + ix3

The quaternionic conjugation in this basis becomes e+ 0 = e0 ,

e˜+ e1 , 1 = −˜

e˜+ e2 , 2 = −˜

e+ 3 = −e3 .

The multiplication rules for HR are: e0 commutes with all elements of HR , e˜1 , e˜2 , e3 anti-commute, e20 = e˜21 = e˜22 = e0 , e23 = −e0 , e˜1 e˜2 = e3 , e˜2 e3 = −˜ e1 , e3 e˜1 = −˜ e2 . The elements e0 , e˜1 , e˜2 , e3 are orthogonal with respect to the bilinear form (2.1) and e0 , e0 = e3 , e3 = 1, ˜ e1 , e˜1 = ˜ e2 , e˜2 = −1. The (classical) quaternions H are oriented so that {e0 , e1 , e2 , e3 } is a positive basis. Let dV = dz 0 ∧ dz 1 ∧ dz 2 ∧ dz 3 be the holomorphic 4-form on HC deter7 mined by dV (e0 , e1 , e2 , e3 ) = 1, then the restriction dV 7H is the Euclidean volume 7 form corresponding to {e0 , e1 , e2 , e3 }. On the other hand, the restriction dV 7H is R also real-valued and hence determines an orientation on HR so that {e0 , e˜1 , e˜2 , e3 } becomes a positively oriented basis. Define a norm on HC by

1 z11 z12 2 2 2 2 ∈ HC , Z = √ |z11 | + |z12 | + |z21 | + |z22 | , Z= z21 z22 2 so that ei = 1, 0 ≤ i ≤ 3. In [6] we defined a holomorphic 3-form on HC with values in HC Dz = e0 dz 1 ∧ dz 2 ∧ dz 3 − e1 dz 0 ∧ dz 2 ∧ dz 3 + e2 dz 0 ∧ dz 1 ∧ dz 3 − e3 dz 0 ∧ dz 1 ∧ dz 2


167

characterized by the property 1 Tr(Z1+ , Dz(Z2 , Z3 , Z4 )) = dV (Z1 , Z2 , Z3 , Z4 ), 2 ∀Z1 , Z2 , Z3 , Z4 ∈ HC . 7 and Dx˜ = Dz 7H .

Z1 , Dz(Z2 , Z3 , Z4 ) = 7 Let Dx = Dz 7H

R

Proposition 2.1. The 3-form Dx takes values in HR . If we write X = x0 e0 +x1 e˜1 + x2 e˜2 + x3 e3 ∈ HR , x0 , x1 , x2 , x3 ∈ R, then Dx is given explicitly by Dx = e0 dx1 ∧ dx2 ∧ dx3 + e˜1 dx0 ∧ dx2 ∧ dx3 − e˜2 dx0 ∧ dx1 ∧ dx3 − e3 dx0 ∧ dx1 ∧ dx2 . (2.3) Remark 2.2. Clearly, the form Dx satisfies the property

X1 , Dx(X2 , X3 , X4 ) =

1 Tr(X1+ , Dx(X2 , X3 , X4 )) = dV (X1 , X2 , X3 , X4 ), 2 ∀X1 , X2 , X3 , X4 ∈ HR ,

which could be used to define it. It is also worth mentioning that in terms of the coordinates Z = on HC , zij ∈ C, Dz =

1 2

−dz11 ∧ dz12 ∧ dz21 dz11 ∧ dz21 ∧ dz22

z11 z21

z12 z22

−dz11 ∧ dz12 ∧ dz22 , dz12 ∧ dz21 ∧ dz22

where we write zij = xij + iyij , xij , yij ∈ R, and dzij = dxij + idyij . Let U ⊂ HR be an open region with piecewise smooth boundary ∂U . We give a canonical orientation to ∂U as follows. The positive orientation of U is determined → be a non-zero vector in by {e0 , e˜1 , e˜2 , e3 }. Pick a smooth point p ∈ ∂U and let n p Tp HR perpendicular to Tp ∂U and pointing outside of U . Then {→ τ1 , → τ2 , → τ3 } ⊂ Tp ∂U → → → → is positively oriented in ∂U if and only if {np , τ1 , τ2 , τ3 } is positively oriented in HR . Lemma 2.3. Let R ∈ R be a constant, then we have the following restriction formulas: 7 7 − 7 7 X X− Dx7 = X dS, Dx7 = X dS = XR dS, {X∈HR ; N (X)=R}

{X∈HR ; X=R}

where the sets {X ∈ HR ; N (X) = R} and {X ∈ HR ; X = R} are oriented as boundaries of the open sets {X ∈ HR ; N (X) < R} and {X ∈ HR ; X < R} respectively, and dS denotes the respective restrictions of the Euclidean measure on HR .

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3. Regular Functions on H and HC Recall that regular functions on H are defined using an analogue of the Cauchy˜ ∈ H as X ˜ =x Riemann equations. Write X ˜ 0 e0 + x ˜1 e1 + x ˜2 e2 + x ˜3 e3 , x ˜0 , x˜1 , x ˜2 , x ˜3 ∈ R, and factor the four-dimensional Laplacian operator on H as a product of two Dirac operators =

∂2 ∂2 ∂2 ∂2 + + + = ∇∇+ = ∇+ ∇, (∂ x ˜0 )2 (∂ x ˜1 )2 (∂ x ˜2 )2 (∂ x ˜3 )2

where ∂ ∂ ∂ ∂ + e1 1 + e2 2 + e3 3 and 0 ∂x ˜ ∂x ˜ ∂x ˜ ∂x ˜ ∂ ∂ ∂ ∂ ∇ = e0 0 − e1 1 − e2 2 − e3 3 . ∂x ˜ ∂x ˜ ∂x ˜ ∂x ˜ The operators ∇+ , ∇ can be applied to functions on the left and on the right. For an open subset U ⊂ H and a differentiable function f on U with values in H or ˜ = 0 for all X ˜ ∈ U , and f is right-regular HC , we say f is left-regular if (∇+ f )(X) + ˜ ˜ if (f ∇ )(X) = 0 for all X ∈ U . ∇+ = e0

Proposition 3.1. For any C 1 -function f on U ⊂ H with values in H or HC , 7 7 d(D˜ x · f ) = −Dx˜ ∧ df = (∇+ f ) dV 7H . d(f · Dx˜) = df ∧ D˜ x = (f ∇+ ) dV 7H , In particular, ∇+ f = 0

⇐⇒

D˜ x ∧ df = 0,

f ∇+ = 0

⇐⇒

df ∧ D˜ x = 0.

Following [6], we say that a differential function f C : U C → HC defined on an open set U C ⊂ HC is holomorphic if it is holomorphic with respect to the complex variables z 0 , z 1 , z 2 , z 3 . Then we define f C to be holomorphic left-regular if it is holomorphic and ∇+ f C = 0. Similarly, f C is defined to be holomorphic right-regular if it is holomorphic and f C ∇+ = 0.

z11 z12 , zij ∈ C, then a If we identify HC with complex 2 × 2 matrices z21 z22 function f C : U C → HC is holomorphic if and only if it is holomorphic with respect to the complex variables zij , 1 ≤ i, j ≤ 2. Let us introduce holomorphic analogues of ∇+ and ∇: ∂ ∂ ∂ ∂ + e1 1 + e2 2 + e3 3 and ∇+ C = e0 ∂z 0 ∂z ∂z ∂z ∂ ∂ ∂ ∂ ∇C = e0 0 − e1 1 − e2 2 − e3 3 . ∂z ∂z ∂z ∂z Then for a holomorphic function f C : U C → HC , the following derivatives are equal:

∂ ∂ ∂ − ∂z∂21 C C C C ∂z22 ∂z11 ∂z21 ∇+ f C = ∇+ f f C, f = 2 , ∇f = ∇ f = 2 C ∂ ∂ ∂ C − ∂z∂12 ∂z11 ∂z12 ∂z22

An Invitation to Split Quaternionic Analysis C

f ∇ =f +

C

∇+ C

= 2f

C

∂ ∂z22 − ∂z∂12

− ∂z∂21 ∂ ∂z11

C

C

169

f ∇ = f ∇C = 2f

,

C

∂ ∂z11 ∂ ∂z12

∂ ∂z21 . ∂ ∂z22

Proposition 3.2. For any holomorphic function f C : U C → HC , C ∇+ Cf = 0

⇐⇒

Dz ∧ df C = 0,

Lemma 3.3. We have: 1 = 0; 1. N (Z) 1 2. ∇C N (Z) = Z −1 N (Z)

1 N (Z) ∇C

−1

f C ∇+ C = 0

⇐⇒

df C ∧ Dz = 0.

+

= −2 NZ(Z) = −2 NZ(Z)2 ;

+

= NZ(Z)2 is a holomorphic left- and right-regular function defined wherever N (Z) = 0; −1 + 4. The form NZ(Z) · Dz = NZ(Z)2 · Dz is a closed holomorphic HC -valued 3-form defined wherever N (Z) = 0. 3.

Lemma 3.4. Let U C ⊂ HC be an open subset. For any differentiable function F : U C → C, we have: ∇ F (Z + ) = (∇+ F )(Z + ), ∇+ F (Z + ) = (∇F )(Z + ), ∇ F (Z −1 ) = −Z −1 · (∇F )(Z −1 ) · Z −1 .

4. Regular Functions on HR We introduce linear differential operators on HR ∂ ∂ ∂ ∂ − e˜1 1 − e˜2 2 + e3 3 and ∂x0 ∂x ∂x ∂x ∂ ∂ ∂ ∂ ∇R = e0 0 + e˜1 1 + e˜2 2 − e3 3 ∂x ∂x ∂x ∂x which may be applied to functions on the left and on the right. Fix an open subset U ⊂ HR and let f be a differentiable function on U with values in HR or HC . ∇+ R = e0

Definition 4.1. The function f is left-regular if it satisfies ∂f ∂f ∂f ∂f (X) − e˜1 1 (X) − e˜2 2 (X) + e3 3 (X) = 0, ∂x0 ∂x ∂x ∂x Similarly, f is right-regular if (∇+ R f )(X) = e0

∀X ∈ U.

∂f ∂f ∂f ∂f (X)e0 − 1 (X)˜ e1 − 2 (X)˜ e2 + 3 (X)e3 = 0, ∀X ∈ U. ∂x0 ∂x ∂x ∂x We denote by 2,2 the ultrahyperbolic wave operator on HR which can be factored as follows: ∂2 ∂2 ∂2 ∂2 + 2,2 = − − + = ∇R ∇+ R = ∇R ∇R . (∂x0 )2 (∂x1 )2 (∂x2 )2 (∂x3 )2 (f ∇+ R )(X) =

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Proposition 4.2. For any C 1 -function f : U → HR (or f : U → HC ), d(f · Dx) = df ∧ Dx = (f ∇+ R ) dV,

d(Dx · f ) = −Dx ∧ df = (∇+ R f ) dV.

In particular, ∇+ Rf = 0

⇐⇒

Dx ∧ df = 0,

f ∇+ R = 0

⇐⇒

df ∧ Dx = 0.

Let U C ⊂ HC be an open set. The restriction relations 7 7 Dz 7H = Dx, Dz 7H = D˜ x R

imply that the restriction of a holomorphic left- or right-regular function to U R = U C ∩ HR produces a left- or right-regular function on U R respectively. And the restriction of a holomorphic left- or right-regular function to UH = U C ∩ H also produces a left- or right-regular function on UH respectively. Conversely, if one starts with, say, a left-regular function on HR , extends it holomorphically to a left-regular function on HC and then restricts the extension to H, the resulting function is left-regular on H. These properties of Dirac operators ∇+ R and ∇R and the notion of regular functions on HR are in complete parallel with the Dirac operators ∇+ M and ∇M and the corresponding notion of regular functions on M introduced in Section 3.2 in [6]. Proposition 4.3. Let f C : U C → HC be a holomorphic function. Then + C C + C ∇+ f C = ∇+ R f = ∇M f = ∇ f ,

∇f C = ∇R f C = ∇M f C = ∇f C ,

C + C + f C ∇+ = f C ∇+ R = f ∇M = f ∇ ,

f C ∇ = f C ∇R = f C ∇M = f C ∇.

+ + Thus, essentially by design, the Dirac operators ∇+ , ∇+ R , ∇M , ∇C and ∇, ∇R , ∇M , ∇C (and hence the notions of regular functions on H, HR , M and holomorphic regular functions on HC ) are all compatible.

5. Fueter Formula for Holomorphic Regular Functions on HR We are interested in extensions of the Cauchy-Fueter formula to functions on HR . First we recall the classical version of the integral formula due to Fueter: Theorem 5.1 (Cauchy-Fueter Formula [8, 9]). Let UH ⊂ H be an open bounded ˜ is left-regular on a subset with piecewise C 1 boundary ∂UH . Suppose that f (X) neighborhood of the closure UH , then ˜ −X ˜ 0 )−1 ˜ 0 ∈ UH ; ˜ 0 ) if X (X 1 f (X ˜ · D˜ x · f (X) = 2 ˜0 ∈ ˜ ˜ 2π ∂UH N (X − X0 ) 0 if X / UH . ˜ is right-regular on a neighborhood of the closure UH , then If g(X) ˜ 0 ∈ UH ; ˜ 0 ) if X ˜ −X ˜ 0 )−1 g(X 1 (X ˜ g(X) · Dx˜ · = 2 ˜0 ∈ ˜ ˜ 2π ∂UH 0 if X / UH . N (X − X0 )


171

Let U ⊂ HR be an open subset, and let f be a C 1 -function defined on a neighborhood of U such that ∇+ R f = 0. In this subsection we extend the Cauchy-Fueter integral formula to left-regular functions which can be extended holomorphically to a neighborhood of U in HC . (In other words, we assume that f is a real-analytic ˜ X ˜ 0 )−1 ˜ function on U .) Observe that the expression in the integral formula (NX− ˜ X ˜ ) · Dx (X− ˜

0 −1

X0 ) is nothing else but the restriction to H of the holomorphic 3-form (Z− ˜ 0 ) · Dz N (Z−X ˜ which is the form from Lemma 3.3 translated by X0 . For this reason we expect an integral formula of the kind

7 7 (Z − X0 )−1 1 · Dz 77 ·f (X), ∀X0 ∈ U. f (X0 ) = 2 2π ∂U N (Z − X0 ) HR

However, the integrand is singular wherever N (Z − X0 ) = 0. We resolve this difficulty by deforming the contour of integration ∂U in such a way that the integral is no longer singular. Fix an ε ∈ R and define an ε-deformation hε : HC → HC , Z → Zε , by: z11 → z11 + iεz11 , z12 → z12 − iεz12 , z21 → z21 − iεz21 , z22 → z22 + iεz22 . Define a quadratic form on HC S(Z) = z11 z22 + z12 z21 . Lemma 5.2. We have the following identities: Zε = Z + iεZ − ,

(Zε )+ = Z + + iεZ +− ,

N (Zε ) = (1 − ε2 )N (Z) + 2iεS(Z), S(X) = X2 ,

∀X ∈ HR .

For Z0 ∈ HC fixed, we use a notation hε,Z0 (Z) = Z0 + hε (Z − Z0 ) = Z + iε(Z − Z0 )− . Theorem 5.3. Let U ⊂ HR be an open bounded subset with piecewise C 1 boundary ∂U , and let f (X) be a C 1 -function defined on a neighborhood of the closure U such that ∇+ R f = 0. Suppose that f extends to a holomorphic left-regular function C C f : V → HC with V C ⊂ HC an open subset containing U , then f (X0 ) if X0 ∈ U ; (Z − X0 )−1 1 C · Dz · f (Z) = − 2 2π (hε,X0 )∗ (∂U) N (Z − X0 ) 0 if X0 ∈ / U, for all ε = 0 sufficiently close to 0. Remark 5.4. For all ε = 0 sufficiently close to 0 the cycle (hε,X0 )∗ (∂U ) lies inside V C and the integrand is non-singular, thus the integrals are well-defined. Moreover, we will see that the value of the integral becomes constant when the parameter ε is sufficiently close to 0. Of course, there is a similar formula for right-regular functions on HR as well.

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Proof. Let M = supX∈∂U X − X0 . Without loss of generality we may assume that V C is the δ-neighborhood of U for some δ > 0. We will show that the integral formula holds for 0 < |ε| < δ/M . Clearly, for this choice of ε the contour of integration (hε,X0 )∗ (∂U ) lies inside V C and, since the integrand is a closed form, the integral stays constant for −δ/M < ε < 0 and 0 < ε < δ/M (a priori the values of the integral may be different on these two intervals). Let Sr = {X ∈ HR ; X − X0 2 = r2 } and Br = {X ∈ HR ; X − X0 2 ≤ r2 } be the sphere and the closed ball of radius r centered at X0 , and choose r > 0 sufficiently small so that Br ⊂ U and r < δ. ˜ ∈ H + X0 ; X ˜ − X0 2 = r2 } be the sphere oriented as Lemma 5.5. Let S˜r = {X the boundary of the open ball, then −S˜r if X0 ∈ U ; (hε,X0 )∗ (∂U ) ∼ 0 if X0 ∈ /U as homology 3-cycles inside {Z ∈ V C ; N (Z − X0 ) = 0}. Proof. We give a proof for ε > 0; the case ε < 0 is similar. As homology cycles in {Z ∈ V C ; N (Z − X0 ) = 0}, (hε,X0 )∗ (∂U )

∼

(hε,X0 )∗ (Sr )

∼

(h1,X0 )∗ (Sr ).

/ U , the cycle (h1,X0 )∗ (Sr ) is homologous to zero. If X0 ∈ Assume now X0 ∈ U . Let PH be the projection HC H defined by y 0 )e0 + (˜ x1 + i˜ y 1 )e1 + (˜ x2 + i˜ y 2 )e2 + (˜ x3 + i˜ y 3 )e3 Z = (˜ x0 + i˜ ˜ =x → X ˜0 e0 + x ˜1 e1 + x ˜2 e2 + x ˜3 e3 , x˜0 , x ˜1 , x ˜2 , x ˜3 , y˜0 , y˜1 , y˜2 , y˜3 ∈ R, and let PH+X0 : HC H + X0 , be the projection PH+X0 (Z) = PH (Z − X0 ) + X0 . We describe the supports of the cycles involved: |Sr | = {X0 + ae0 + b˜ e1 + c˜ e2 + de3 ; a2 + b2 + c2 + d2 = r2 }, e1 + (1 − i)c˜ e2 + (1 + i)de3 ; |(h1,X0 )∗ (Sr )| = {X0 + (1 + i)ae0 + (1 − i)b˜ a2 + b2 + c2 + d2 = r2 }, |(PH+X0 ◦h1,X0 )∗ (Sr )| = {X0 + ae0 + be1 − ce2 + de3 ; a2 + b2 + c2 + d2 = r2 } = |S˜r |. Moreover, (PH+X0 ◦h1,X0 )∗ (Sr ) = −S˜r as homology cycles. It is easy to see that this projection provides a homotopy between (h1,X0 )∗ (Sr ) and −Sr , hence the lemma.

By Stokes’

(hε,X0 )∗ (∂U)

(Z − X0 )−1 ·Dz·f C (Z) = N (Z − X0 )

˜r −S

(Z − X0 )−1 ·Dz·f C (Z) N (Z − X0 )

if X0 ∈ U ,


173

and zero if X0 ∈ / U . Finally, by the Fueter formula for the usual quaternions (Theorem 5.1), the last integral is −2π 2 f (X0 ). (Alternatively, one can let r → 0+ and show directly that the integral remains unchanged and at the same time approaches −2π 2 f (X0 ) in the same way the Cauchy and Cauchy-Fueter formulas are proved.) For a Cauchy-Fueter formula for regular functions on M that extend to holomorphic regular functions on HC see Section 3.3 in [6].

6. Fueter Formula for Regular Functions on HR In this section we prove an analogue of the Cauchy-Fueter formula for smooth left-regular functions on HR which are not necessarily real analytic and do not necessarily have holomorphic extensions. As a “trade-off” for working with “bad” functions the proofs become much more involved. Theorem 6.1. Let U ⊂ HR be a bounded open region with smooth boundary ∂U . Let f : U → HC be a function which extends to a real-differentiable function on an open neighborhood V ⊂ HR of the closure U such that ∇+ R f = 0. Then, for any point X0 ∈ HR such that ∂U intersects the cone {X ∈ HR ; N (X − X0 ) = 0} transversally, we have: −1 (X − X0 )+ f (X0 ) if X0 ∈ U ; lim · Dz · f (X) = ε→0 2π 2 ∂U N (X − X ) + iεX − X 2 2 0 if X0 ∈ / U. 0 0 Proof. The case X0 ∈ / U is easier, so we assume X0 ∈ U . Using Proposition 4.2, we get d

(X − X0 )+

2 · Dz · f (X)

N (X − X0 ) + iεX − X0 2

(X − X0 )+ + = 2 ∇R f (X) dV N (X − X0 ) + iεX − X0 2 = 4iε

X − X0 2 − (X − X0 )+ (X − X0 )− 3 f (X) dV. N (X − X0 ) + iεX − X0 2

(6.1)

In particular, expression (6.1) tends to zero pointwise when ε → 0 except for those X which lie on the cone {X ∈ HR ; N (X −X0 ) = 0}, and we need to be very careful there. By translation we can assume that X0 = 0. Let Sr = {X ∈ HR ; X2 = r2 } and Br = {X ∈ HR ; X2 ≤ r2 } be the sphere and the closed ball of radius r. By

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M. Libine

Stokes’ ∂U

X+ N (X) + iεX2

2 ·Dz ·f (X) =

U \Br

d

+ Sr

X+ N (X) + iεX2

X+ N (X) + iεr2

·Dz ·f (X) 2

2 · Dz · f (X).

We will show that, as ε → 0, the first integral on the right-hand side tends to zero (this part is non-trivial and uses that the cone {N (X − X0 ) = 0} intersects ∂U transversally). On the other hand, as ε → 0 and r → 0+ , the second integral tends to −2π 2 f (0). The proof is essentially a series of integration by parts computations. In regular coordinates (x0 , x1 , x2 , x3 ) we have X = x0 e0 + x1 e˜1 + x2 e˜2 + x3 e3 , and N (X) = (x0 )2 − (x1 )2 − (x2 )2 + (x3 )2 . For computing purposes we replace (x0 , x1 , x2 , x3 ) with spherical coordinates (ρ, θ, ϕ, ψ) so that x0 = ρ cos θ cos ϕ x1 = ρ sin θ sin ψ x2 = ρ sin θ cos ψ x3 = ρ cos θ sin ϕ

ρ ≥ 0, 0 ≤ θ ≤ π/2, 0 ≤ ϕ ≤ 2π, 0 ≤ ψ ≤ 2π.

(6.2)

This is an orientation-preserving change of coordinates, and the vector fields ∂ ∂ ∂ { ∂θ , ∂ϕ , ∂ψ } form a positively-oriented frame on Sr . Then N (X) = ρ2 cos(2θ), N (X) + iεX2 = ρ2 cos(2θ) + iε , and the equation N (X) = 0 becomes θ = π/4. Recall that the function x1n which is singular at x = 0 can be regularized as 1 1 a distribution in two different ways, (x+i0) n and (x−i0)n , so that a test function 8 1 ) g(x) dx 9 8 1 9 g(x) is being sent into (x+i0)n , g(x) = limε→0+ (x+iε)n or (x−i0) = n , g(x) ) g(x) dx limε→0− (x+iε)n . By a similar fashion we have the following lemma: Lemma 6.2. Fix a θ0 ∈ (0, π4 ), and let n be a positive integer, then we have two distributions which send a test function g(θ) into the limits π4 +θ0 π4 +θ0 g(θ) dθ g(θ) dθ n n . lim and lim− π ε→0+ π −θ0 ε→0 cos(2θ) + iε cos(2θ) + iε 4 4 −θ0 Proof. We need to show that the limits exist and depend continuously on the test function g(θ). We do it by induction on n using integration by parts. If n = 1, π4 +θ0 π4 +θ0 g(θ) dθ 2 sin(2θ) g(θ) = · dθ π π cos(2θ) + iε cos(2θ) + iε 2 sin(2θ) 4 −θ0 4 −θ0 7π

π4 +θ0 g(θ) 7 4 +θ0 d g(θ) 7 = dθ−log cos(2θ)+iε · log cos(2θ)+iε · . 7π π dθ 2 sin(2θ) 2 sin(2θ) −θ −θ0 0 4 4

For the purpose of this integration, the complex logarithm function is defined on the complex plane C minus the negative real axis, and the values of the logarithm


175

lie in the strip {z ∈ C; −π < Im z < π}. The function log cos(2θ)+iε is integrable for all values of ε, including ε = 0, hence the limits as ε → 0± exist and depend continuously on g(θ). Now suppose that n > 1, then π4 +θ0 π4 +θ0 g(θ) dθ 2 sin(2θ) g(θ) n = n · dθ π π 2 sin(2θ) cos(2θ) + iε cos(2θ) + iε 4 −θ0 4 −θ0 7 π +θ 1 1 g(θ) 77 4 0 = · n − 1 cos(2θ) + iε n−1 2 sin(2θ) 7 π −θ0 4

π4 +θ0 1 g(θ) 1 d dθ, − · n − 1 π4 −θ0 cos(2θ) + iε n−1 dθ 2 sin(2θ)

and the result follows by induction on n. Lemma 6.3.

lim

ε→0

U \Br

d

X+ N (X) + iεX2

2 · Dz · f (X) = 0.

Proof. We have seen that

X+ d 2 · Dz · f (X) U \Br N (X) + iεX2 X − X0 2 − (X − X0 )+ (X − X0 )− f (X) dV. = 4iε 3 U \Br N (X) + iεX2 Writing the right-hand side integral in the spherical coordinates (6.2) and integrating out the variables ρ, ϕ, ψ we obtain an integral of the type π2 g(θ) dθ 4iε (6.3) 3 , 0 cos(2θ) + iε for some function g(θ). By assumption, the boundary ∂U is smooth, compact and intersects the cone {N (X) = 0} = {θ = π/4} transversally, hence the function g(θ) is smooth at least for θ lying in some interval [ π4 − θ0 , π4 + θ0 ] with θ0 ∈ (0, π4 ). It follows from Lemma 6.2 that the limit of (6.3) as ε → 0 is zero. Lemma 6.4. Sr

X+ N (X) + iεr2

2 · Dz = r

Sr

dS N (X) + iεr2

2 = −

2π 2 . 1 + ε2

Proof. From Lemma 2.3 we see that

X+ N (X) + iεr

7 7 2 · Dz 77 = 2 Sr

X +X − N (X) + iεr2

2 ·

dS . r

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Notice that the involution X → X − preserves the sphere, its orientation, its volume X +X − X +− X + − + form dS and replaces (N (X)+iεr 2 )2 with (N (X)+iεr 2 )2 . Therefore, using X X +− 2 3 X X = 2X , spherical coordinates (6.2) and dS = r sin θ cos θ dψdϕdθ, Sr

X + X − + X +− X dS 1 dS = 2 · · r 2 Sr N (X) + iεr2 2 r N (X) + iεr2 θ=π/2 ϕ=2π ψ=2π r dS sin θ cos θ dψdϕdθ = 2 = 2 θ=0 ϕ=0 ψ=0 Sr N (X) + iεr2 cos(2θ) + iε 7θ=π/2 θ=π/2 7 sin(2θ) dθ π2 2π 2 7 = 2π 2 = =− . 2 7 cos(2θ) + iε θ=0 1 + ε2 θ=0 cos(2θ) + iε X +X −

Lemma 6.5.

lim

r→0+

lim

ε→0

Sr

X + · Dz · f (X) 2 N (X) + iεr2

= −2π 2 f (0).

Proof. By Lemma 2.3 we have: X + · Dz · f (X) X + X − · f (X) dS lim = lim 2 2 ε→0 S ε→0 S r N (X) + iεr2 N (X) + iεr2 r r

(6.4)

If the function f (X) were constant we would be done by previous lemma. However, we cannot argue that since r → 0+ , f (X) is close to f (0) and so may be treated like a constant because there can be derivatives of f (X) involved. So an integration by parts argument will be needed. Writing X = x0 e0 + x1 e˜1 + x2 e˜2 + x3 e3 and using the spherical coordinates (6.2) we compute: X +X −

2(x2 x3 − x0 x1 ) − 2i(x0 x2 + x1 x3 ) (x0 )2 + (x1 )2 + (x2 )2 + (x3 )2 = 2(x2 x3 − x0 x1 ) + 2i(x0 x2 + x1 x3 ) (x0 )2 + (x1 )2 + (x2 )2 + (x3 )2

1 0 0 1 0 −i + ρ2 sin(2θ) sin(ψ − ϕ) + ρ2 sin(2θ) cos(ψ − ϕ) . = ρ2 0 1 1 0 i 0 Thus (6.4) can be rewritten as ' f (X) dS lim r 2 ε→0 Sr N (X) + iεr2 +r Sr

0 −i 0 1 ( + cos(ψ − ϕ) sin(ψ − ϕ) i 0 1 0 · sin(2θ)f dS 2 N (X) + iεr2


177

We have: f (X) dS 1 θ=π/2 ϕ=2π ψ=2π sin(2θ)f dψdϕdθ r 2 = 2 2 θ=0 Sr N (X) + iεr2 ϕ=0 ψ=0 cos(2θ) + iε 7θ=π/2 ϕ=2π ψ=2π

∂f 7 f 1 ∂θ dψdϕdθ 7 . = dψdϕ − 4 ϕ=0 cos(2θ) + iε 7θ=0 cos(2θ) + iε ψ=0 By the chain rule ∂f ∂θ can be written as ρ · g(X) for some smooth function g(X), thus the second integral is r times an expression from Lemma 6.2. Taking limits ε → 0 and then r → 0+ , the first integral tends to −2π 2 f (0) and the second tends to zero. The second term inside the limit is

0 −i 0 1 sin(ψ − ϕ) + cos(ψ − ϕ) i 0 1 0 r · sin(2θ)f dS 2 Sr N (X) + iεr2

0 −i 0 1 sin(ψ − ϕ) + cos(ψ − ϕ) i 0 1 0 1 sin2 (2θ)f dψdϕdθ. (6.5) = 2 2 cos(2θ) + iε But

ψ=2π

ψ=2π

sin(ψ − ϕ)f dψ =

ψ=0 ψ=2π

cos(ψ − ϕ) ψ=0 ψ=2π

cos(ψ − ϕ)f dψ = − ψ=0

∂f f dψ, ∂ψ

sin(ψ − ϕ) ψ=0

∂f f dψ. ∂ψ

By the chain rule can be written as ρ · h(X) for some smooth function h(X), thus the right-hand side of (6.5) is r times an expression from Lemma 6.2. When we take limits first as ε → 0 and then as r → 0+ , integral (6.5) tends to zero. ∂f ∂ψ

This concludes our proof of Theorem 6.1.

7. Separation of the Series for SL(2, R) What makes the representation theory of SL(2, R) more interesting than that of SU (2) is having the separation of the series into discrete and continuous components. Instead of SL(2, R) we prefer to work with SU (1, 1) sitting inside HR , as in (2.2). In this section we outline a relationship between split quaternionic analysis and the decomposition L2 SU (1, 1) " L2discr SU (1, 1) ⊕ L2cont SU (1, 1) . The denominator of the Cauchy-Fueter kernel N (X −X0 ) determines a region in HC Ω = X0 ∈ HC ; N (X − X0 ) = 0 ∀X ∈ HR with N (X) = 1 .

178

M. Libine

This region contains two open Ol’shanskii semigroups of GL(2, C) Γ0 = {Z ∈ HC ; Z ∗ e˜3 Z − e˜3 is positive definite}, Γ0 = (Γ0 )−1 = {Z ∈ HC ; Z ∗ e˜3 Z − e˜3 is negative definite},

1 0 ∈ HC . Following [15] we can show that over Γ0 and Γ0 where e˜3 = ie3 = 0 −1 −1

0) the Cauchy-Fueter kernel (Z−X N (Z−X0 ) can be expanded in terms of the K-types of the discrete series of SU (1, 1). Thus we obtain the following integral formula for the projector onto the discrete series of SU (1, 1)): 1 (Z − X0 )−1 (Pdiscr )f (X0 ) = · Dz · f (Z), (7.1) 2 2π Cdiscr N (Z − X0 )

where Cdiscr is a certain three-cycle in Γ0 ∪ Γ0 . (Strictly speaking, this operator is not a projector because it has eigenvalues ±1 on the discrete series subspace, but its square does give a projection onto the discrete series.) Furthermore, we can + − + decompose the cycle Cdiscr into a sum of two cycles Cdiscr + Cdiscr with Cdiscr and − + 0 0 Cdiscr lying in Γ and Γ respectively. Then integration over Cdiscr (respectively − ) produces a “projector” onto the holomorphic (respectively antiholomorCdiscr phic) discrete series. Our next goal is to obtain a projector onto the continuous series component Pcont . We expect that Pcont will be given by the same formula (7.1) but with a different choice of the cycle of integration, quite possibly supported in Ω\(Γ0 ∪Γ0 ). To get Pcont we use a conformal map γ : HC → HC which sends ˜ M HR →

and

3 SU (1, 1) → ˜ H

(with singularities),

3 is the unit hyperboloid of one sheet in M; we call γ the “Cayley transwhere H 3 can be identified with SL(2, C)/SU (1, 1) and is usuform”. The hyperboloid H ally called the imaginary Lobachevski space. The group SL(2, C) acts naturally 3 ) and decomposes into the discrete and continuous components: on L2 (H 3 ) " L2 (H 3 ) ⊕ L2 (H 3) L2 (H discr cont as representations of SL(2, C) (see, for example, [11]). Then the map γ switches the discrete and continuous components: 3 ), 3 ). L2discr SU (1, 1) " L2cont (H L2cont SU (1, 1) " L2discr (H This explains the purpose of the Cayley transform – it is easier to find the projector onto the discrete component than onto the continuous one! Once the projector onto 3 ) is found we can pull it back to HR to get Pcont . We expect it to have L2discr (H the form (Z − X0 )−1 1 cont (P · Dz · f (Z), )f (X0 ) = 2 2π Ccont N (Z − X0 ) where Ccont is a certain three cycle in Ω.


179

The integral formulas for Pdiscr and Pcont strongly suggest that the separation of the series is a topological phenomenon! Thus there is some underlying homology theory still waiting to be developed. We hope that this geometric analytic realization of the separation of the series will extend to higher rank groups. Such a picture fits well into the Gelfand-Gindikin program initiated in [10] which realizes representations of reductive groups G in function spaces of open domains in GC (complexification of G). This geometric relationship between quaternionic analysis and the separation of the series for SL(2, R) will be the subject of our upcoming paper [7].

References [1] N. Arkani-Hamed, F. Cachazo, C. Cheung, J. Kaplan, The S-Matrix in Twistor Space, arXiv:0903.2110 (2009). [2] F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, Pitman, London, 1982. [3] J. Cockle, On Systems of Algebra Involving More than One Imaginary, Philosophical Magazine (series 3) 35 (1849), 434-435. [4] F. Colombo, I. Sabadini, F. Sommen, D. C. Struppa, Analysis of Dirac systems and computational algebra, Progress in Mathematical Physics, vol. 39, Birkh¨ auser Boston, 2004. [5] R. Delanghe, F. Sommen, V. Souˇcek, Clifford algebra and spinor-valued functions, Kluwer Academic Publishers Group, Dordrecht, 1992. [6] I. Frenkel, M. Libine, Quaternionic analysis, representation theory and physics, Advances in Math 218 (2008), 1806-1877; also arXiv:0711.2699. [7] I. Frenkel, M. Libine, Split quaternionic analysis and the separation of the series for SL(2, R) and SL(2, C)/SL(2, R), submitted, arXiv:1009.2532 (2010). [8] R. Fueter, Die Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Variablen, Comment. Math. Helv. 7 (1934), no. 1, 307-330. ¨ [9] R. Fueter, Uber die analytische Darstellung der regul¨ aren Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 8 (1935), no. 1, 371-378. [10] I. M. Gelfand, S. G. Gindikin, Complex manifolds whose spanning trees are real semisimple Lie groups, and analytic discrete series of representations, Funktsional. Anal. i Prilozhen. 11 (1977), no. 4, 19-27, 96. [11] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Generalized Functions. Vol. 5: Integral Geometry and Representation Theory, translated from the Russian by E. Saletan, Academic Press, New York and London 1966. [12] J. Gilbert, M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Univ. Press, Cambridge, UK, 1991. [13] F. G¨ ursey, C.-H. Tze, On the role of division, Jordan and related algebras in particle physics, World Scientific Publishing Co., 1996. [14] T. Kobayashi, B. Ørsted, Analysis on the minimal representation of O(p, q) I, II, III, Adv. Math. 180 (2003), no. 2, 486-512, 513-550, 551-595. [15] K. Koufany, B. Ørsted, Function spaces on the Ol’shanski˘ı semigroup and the Gel’fand-Gindikin program, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 3, 689-722.

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[16] L. Mason, D. Skinner, Scattering Amplitudes and BCFW Recursion in Twistor Space, arXiv:0903.2083 (2009). [17] J. Ryan, Complexified Clifford Analysis, Complex Variables Theory Appl. 1 (1982/ 83), no. 1, 119-149. [18] R. S. Strichartz, Harmonic analysis on hyperboloids, J. Funct. Anal. 12 (1973), 341383. [19] A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Math. Soc. 85 (1979), 199225. Matvei Libine Department of Mathematics Indiana University Rawles Hall 831 East 3rd St Bloomington, IN 47405 USA e-mail: [email protected]

On the Hyperderivatives of Moisil–Théodoresco Hyperholomorphic Functions M. Elena Luna-Elizarrarás, Marco A. Mac´ıas-Cede˜ no and Michael Shapiro Abstract. Any Moisil-Théodoresco-hyperholomorphic function is also Fueterhyperholomorphic, but its hyperderivative is always zero, so these functions can be thought of as “constants” for the Fueter operator. It turns out that it is possible to give another kind of hyperderivatives “consistent” with the MoisilThéodoresco operator, but there are several of them. We focus in detail on one of these hyperderivatives and develop also the notion of two-dimensional directional hyperderivative along a plane. As in the previous works, an application to the Cliffordian-Cauchy-type integral proves to be instructive. Mathematics Subject Classification (2010). Primary 30G35; Secondary 32A10. Keywords. Hyperderivative, two-dimensional directional hyperderivative, Cauchy-type integrals.

1. Introduction 1.1. We will use the common notation H for the skew-field of real quaternions. The quaternionic imaginary units in the paper are e1 = i, e2 = j, e3 = k, which satisfy the known properties: e1 e2 = −e2 e1 = e3 ; e2 e3 = −e3 e2 = e1 ; e3 e1 = −e1 e3 = e2 ; e21 = e22 = e23 = −1 . The real unit 1 is written, sometimes, as e0 , so

The research of the first and the third named authors was partially supported by CONACYT projects as well as by Instituto Politécnico Nacional in the framework of COFAA and SIP programs.


181

182

M.E. Luna-Elizarrar´ as, M.A. Mac´ıas-Cede˜ no and M. Shapiro

that given a ∈ H we write a = conjugate of a is a := a0 −

3

3

a e with {a0 , a1 , a2 , a3 } ⊂ R. The quaternionic

=0

a e .

=1

1.2. In [10], Shapiro and Vasilevski obtained the correlation: 1 (3) σx DF [f ](x) + σ x(3) DF [f ](x) , d(σx(2) f (x)) = 2

(1.1)

(3)

where σx is the 3-form (3)

σx

:=

dx1 ∧ dx2 ∧ dx3 − i dx0 ∧ dx2 ∧ dx3 + j dx0 ∧ dx1 ∧ dx3 − k dx0 ∧ dx1 ∧ dx2

=:

d% x0 − i d% x1 + j d% x2 − k d% x3 ,

(2)

σx is the 2-form (2)

σx

i dx2 ∧ dx3 − j dx1 ∧ dx3 + k dx1 ∧ dx2

=:

i d% x1,0 − j d% x2,0 + k d% x3,0 ,

∂ ∂ , DF := e are the Fueter operator and its (quater∂x ∂x =0 =0 nionic) conjugate.

and DF :=

3

:=

3

e

The equality (1.1) is a deep structural analog of its complex analysis antecedent ∂g 0 ∂g 0 dg(z 0 ) = (z ) dz + (z ) dz . (1.2) ∂z ∂z In the latter, whence g is a holomorphic function, then its complex derivative ∂g 0

0 g (z ) coincides with its “formal” derivative (z ). Hence the analogy between ∂z (1.1) and (1.2) allows us to conclude that if a quaternionic function f is hyperholo1 morphic (f ∈ ker DF ), then DF [f ](x0 ) is a highly probable candidate for being 2 an “adequate quaternionic hyperderivative” of the hyperholomorphic function f . The paper [8] justifies the idea; it works with the notion of the hyperderivative

f (x0 ) of a Fueter-hyperholomorphic function as the limit of a specific quotient, concluding that 1

f (x0 ) = DF [f ](x0 ) . (1.3) 2 In others words recalling that in the complex case the derivative of a holomorphic function is the proportionality coefficient between the differentials of the

Hyperderivatives of MT-hyperholomorphic Functions

183

function and of the independent variable: dg = g (z 0 ) dz, we conclude that the hyperderivative f (x0 ) is the proportionality coefficient between the two differential forms: 1 (3) (2) d(σx0 f (x0 )) = σx0 D F [f ](x0 ) . 2 1.3. The Moisil-Théodoresco operator DMT :=

3 =1

e

∂ , ∂x

(1.4)

which acts on functions f : Ω ⊂ R3 → H of class C 1 , is related with the Fueter operator as follows: given f as before, define f : Ω × R ⊂ R4 → H such that f(x0 , x1 , x2 , x3 ) := f (x1 , x2 , x3 ) for any x0 . Now, assuming that f is Moisil-Théodoresco-hyperholomorphic (that is, f is a null-solution of the Moisil-Théodoresco operator), one has: 3 ∂ ∂ f DF [f ](x0 , x1 , x2 , x3 ) = (x0 , x1 , x2 , x3 ) + e [f ](x0 , x1 , x2 , x3 ) ∂x0 ∂x =1

=

3 =1

e

∂ [f ](x1 , x2 , x3 ) = DMT [f ](x1 , x2 , x3 ) = 0 , ∂x

thus f is Fueter-hyperholomorphic in this specific domain Ω × R (of course, the cylinder can be of a finite height, and there are another ways of “inflating” the domain Ω). It is common to say that f is Fueter-hyperholomorphic as well, and to write the Fueter operator (for these specific domains and for these specific functions) as 3 ∂ ∂ DF = e = + DMT . ∂x ∂x0 =0

What is more, the left-hyperderivative of f ∈ ker DMT at the point x0 = 0 (x1 , x02 , x03 ) ∈ Ω ⊂ R3 is equal to the left-hyperderivative of f at any point (x0 , x01 , x02 , x03 ) =: (x0 , x0 ) ∈ R4 , and hence, is identically zero in the whole domain Ω: 1

f (x0 , x0 ) = DF [ f](x0 , x0 ) 2 3 ∂ f ∂ f (x0 , x0 ) − e (x0 , x0 ) = ∂x0 ∂x =1

= −DMT [f ](x1 , x2 , x3 ) = 0 . That is, the set of Moisil-Théodoresco-hyperholomorphic functions is a kind of the set of “Fueter-hyperholomorphic constants”, and thus it is obviously not interesting to study the properties of the Fueter-hyperderivative of the MoisilThéodoresco-hyperholomorphic functions.

184


1.4. It is the aim of the paper to show that it is possible to develop another approach to the notion of hyperderivatives of a Moisil-Théodoresco-hyperholomorphic function which is based on the same ideas but in such a way that the new hyperderivative does not vanish identically. We describe also the peculiarities of the situation and we explain why the Moisil-Théodoresco-hyperholomorphic functions need to have several equivalent hyperderivatives. For doing this we need to recall some results on the hyperderivability in Clifford analysis as they are presented in the papers by G¨ urlebeck and Malonek [7] and by Luna-Elizarrarás, Mac´ıas-Cede˜ no and Shapiro [5]. Let C0,m be the real Clifford algebra with imaginary units e1 , e2 , . . . , em m ∂ with negative signature; let DCR := e be the Cauchy-Riemann operator ∂x =0

acting on C 1 (Ω ⊂ Rm+1 ; C0,m ); the (Cliffordian) Cauchy-Riemann-hyperholomorphic functions are null solutions of DCR . In analogy to (1.1) and (1.2), for C0,m valued functions of class C 1 there holds: 1 σ x DCR [f ](x) − σx DCR [f ](x) , (1.5) d(τx f (x)) = 2 where x0 − e1 dˆ x1 + · · · + (−1)m em dˆ xm , σx := dˆ

(1.6)

x1,0 + e2 dˆ x2,0 + · · · + (−1)m em dˆ xm,0 , τx = −e1 dˆ

(1.7)

and

with d% x := dx0 ∧ dx1 ∧ · · · ∧ dx−1 ∧ dx+1 ∧ · · · ∧ dxm , (i.e., d% x is obtained from the differential of volume dV = dx0 ∧· · · ∧dxm , omitting the factor dx ), and d% xs,t is obtained from d% xs omitting also dxt . Similarly to what is written in Subsection 1.2 the Cliffordian hyperderivative is defined as the proportionality coefficient between the differential forms σx and τx , and it also turns out to be the limit of an appropiate quotient; besides, the hyperderivative coincides up to a real factor, as in (1.3), with the conjugate Cauchy-Riemann operator:

1 f (x) = − DCR [f ](x) . 2

In particular for m = 2, the Cauchy-Riemann operator of Clifford analysis for H-valued functions has the form: DCR =

∂ ∂ ∂ + e1 + e2 . ∂x0 ∂x1 ∂x2

(1.8)


185

2. The left-i-hyperderivative 2.1. Since DMT

∂ ∂ ∂ +j +k ∂x1 ∂x2 ∂x3

∂ ∂ ∂ i = i DMT = i −k +j , ∂x1 ∂x2 ∂x3 = i

(2.1)

where ∂ ∂ ∂ −k +j , (2.2) ∂x1 ∂x2 ∂x3 the sets of null-solutions of both operators coincide. At the same time the operator i DMT is a Cauchy-Riemann-type operator, compare with (1.8); as a matter of fact such operators are, for instance, in [9] and [10], where the corresponding functions are called ψ-hyperholomorphic with ψ := {1, −k, j }, and they possess all the usual properties of Clifford analysis. Thus we are going to work with the corresponding hyperderivative and assign it to the Moisil-Théodoresco-hyperholomorphic functions. i := DMT

i 2.2. In particular the operator DMT and its conjugate i

DMT :=

∂ ∂ ∂ +k −j , ∂x1 ∂x2 ∂x3

which act on C 1 (Ω ⊂ R3 , C0,2 ∼ = H) satisfy i

i

i i ◦ DMT = DMT ◦ DMT = ΔR3 ; DMT

the analogs of (1.6) and (1.7) are: σx,i = d% x1 + k d% x2 + j d% x3 = dx2 ∧ dx3 + k dx1 ∧ dx3 + j dx1 ∧ dx2 , and τx,i = −k d% x1,2 − j d% x1,3 = −k dx3 − j dx2 , for which there holds:

1 i i σx,i DMT [f ](x) − σ x,i DMT [f ](x) . (2.3) 2 So we are in a position to define a hyperderivative for Moisil-Théodorescohyperholomorphic functions. d(τx,i f (x)) =

2.3. Definition. Let f ∈ C 1 (Ω ⊂ R3 ; H). The function f is called left-i-hyperderivable in Ω, if for any x ∈ Ω there is a quaternion, denoted by fi (x), such that d(τx,i f (x)) = σx,i fi (x) . (2.4) The quaternion

fi (x) is named the left-i-hyperderivative of f at x.

The next theorem is immediate from (2.3).

186


2.4. Theorem. Let f ∈ C 1 (Ω ⊂ R3 ; H). The function f is Moisil-Théodorescohyperholomorphic in Ω if and only if it is left-i-hyperderivable and for such a function 1 i

fi (x) = D MT [f ](x) , ∀x ∈ Ω . (2.5) 2 2.5. Following the antecedents of this paper (see [8] and [5]), the left-i-hyperderivative already defined is the limit of a quotient of the adequate increments of the function and of the variable. As was proved in [8], the key point here is that the increments have suitable dimensions. So let us consider parallelograms in R3 . Let x0 ∈ R3 and v1 , v2 be two linearly independent vectors in R3 . The (twodimensional) parallelogram Π with vertex x0 and edges v1 , v2 is defined by : 2 0 2 , t v | (t1 , t2 ) ∈ [0, 1] Π := x + =1

and its boundary

0

x +

∂Π :=

2

: t v | (t1 , t2 ) ∈ ∂[0, 1]

2

.

=1

2.6. Theorem. Let f ∈ ker DMT (Ω) and let fi (x) be its left-i-hyperderivative at the point x ∈ Ω ⊂ R3 . Then for any sequence {Πk }∞ k=1 of oriented non-degenerated parallelograms with vertex x0 and with lim diam Πk = 0, there holds: ⎛ lim ⎝

k→∞

⎞−1 ⎛ σx,i ⎠

⎝

Πk

k→∞

⎞

τx,i f (x)⎠ = fi (x0 ) .

(2.6)

∂Πk

The proof follows from (2.3) and from Stokes’ Theorem: first of all, ⎞ ⎛ ⎞−1 ⎛ ⎛ ⎞−1 lim ⎝ σx,i ⎠ ⎝ τx,i f (x)⎠ = lim ⎝ σx,i ⎠ · d(τx,i f (x))

k→∞

k→∞

Πk

Πk

∂Πk

⎛ =

lim ⎝

k→∞

Πk

⎞−1 σx,i ⎠

Πk

· Πk

but also it is direct to prove that

thus

⎛ lim ⎝

k→∞

Πk

fi (x) = fi (x0 ) + o(x − x0 ) , ⎞−1 ⎛ σx,i ⎠

⎝

∂Πk

⎞ τx,i f (x)⎠ = fi (x0 ) .

σx,i fi (x)) ,


187

3. The directional left-i-hyperderivative 3.1. Definition. Let L ⊂ R3 be a hyperplane such that L ∩ Ω = ∅. The function f : Ω ⊂ R3 → H is called left-i-hyperderivable at x0 ∈ L ∩ Ω along the plane L if for any sequence {Πk }∞ k=1 , with Πk ⊂ L and lim diam Πk = 0, of parallelograms k→∞

with vertex x0 , the limit

⎡⎛ ⎞⎤ ⎞−1 ⎛ ⎥ ⎢ σx,i ⎠ ⎝ τx,i · f (x)⎠⎦ lim ⎣⎝

(3.1)

k→∞

Πk

∂Πk

exists and is independent of the sequence {Πk }∞ k=1 . In this case the limit is denoted by fi,L (x0 ) and is called the directional left-i-hyperderivative. 3.2. Note that the limits in (3.1) and (2.6) are quite similar. The crucial difference between them is the fact that the parallelograms considered in (3.1) are “caught” in the plane L. 3.3. Given as before a (two-dimensional) plane L ⊂ R3 , let γ(x) :=

3

n x + d =

=1

0 be its equation, with n % = (n1 , n2 , n3 ) the unitary normal vector, and d ∈ R. Applying formula (2.3) to the function γ, there holds: 1 i i d(τx,i γ(x)) = σx,i D MT [γ](x) − σ x,i DMT [γ](x) 2 =

1 ( σx,i ( n1 + n2 k − n3 j ) − σ x,i (n1 − n2 k + n3 j) ) 2

1 σx,i n ˘ − σ x,i n ˘ , 2 where n ˘ := n1 − n2 k + n3 j. This differential form is identically zero on L, thus =

σx,i n ˘ = σ x,i n ˘

for x ∈ L .

3.4. Let us combine the latter fact with Stokes’ Theorem. Consider a function f which satisfies the conditions in Definition 3.1, then

−1

σx,i τx,i f (x) = Πk

∂Πk

=

1 2

−1 σx,i Πk

i i σx,i DMT [f ](x) − (˘ n)2 DMT [f ](x) .

Πk

This formula implies two facts which one would expect from a suitable notion of directional derivative. The first fact is related with functions of class C 1 (Ω ⊂ R3 ; H) and claims that these functions possess the left-i-hyperderivative along any (two-dimensional) plane that intersects the domain, and it is given in terms of the

188


direction of the corresponding plane. The second fact says that if the function f : Ω ⊂ R3 → H is Moisil-Théodoresco-hyperholomorphic the left-i-hyperderivative does not depend on the direction of the plane. 3.5. Theorem. Let f ∈ C 1 (Ω ⊂ R3 ; H). Then f is left-i-hyperderivable along any (two-dimensional) plane L at every x0 ∈ L∩Ω, and the left-i-hyperderivative along the plane L is given by 2 i 1 i

fi, L (x0 ) = DMT [f ](x0 ) − n ˘ DMT [f ](x0 ) . (3.2) 2 3.6. Corollary. Let f ∈ C 1 (Ω ⊂ R3 ; H). Then f is Moisil-Théodoresco-hyperholomorphic in Ω if and only if, for all x0 ∈ Ω, the hyperderivative fi,L (x0 ) is independent of the hyperplane L, with x0 ∈ L ∩ Ω. Moreover, in this case all the hyperderivatives fi,L (x0 ) are equal to fi (x0 ).

4. The left-i-hyperderivative and the Cauchy-type integral 4.1. As in the previous papers, an application of the hyperderivative to the Clifford-Cauchy-type integral proves to be instructive. In order to compute the left-i-hyperderivative of the Cauchy-type integral, it is necessary to establish the corresponding relations between the Cauchy kernel and the right- and left-CauchyRiemann operators. 4.2. The right-hand side operators. The right-hand side Cauchy-Riemann operator, which acts on functions C 1 (Λ ⊂ Rm+1 ; C0,m ) is given by Dr :=

m =0

where M

e

M e

∂ , ∂x

is the operator of multiplication on the right: M e [f ] := f e .

Analogously the right-hand side Moisil-Théodoresco operator is given as DMT,r := M i

∂ ∂ ∂ + Mj + Mk . ∂x1 ∂x2 ∂x3

It acts on C 1 (Ω ⊂ R3 ; H), and it can be written as

∂ ∂ ∂ . + Mk − Mj DMT,r = M i ∂x1 ∂x2 ∂x3 Define i := DMT,r

∂ ∂ ∂ + Mk − Mj , ∂x1 ∂x2 ∂x3

and its conjugate: i

DMT,r :=

∂ ∂ ∂ − Mk + Mj , ∂x1 ∂x2 ∂x3


189

thus, as it happens with the left-hand side operators, the sets of the null-solutions i of DMT,r and DMT,r , coincide. Moreover, the right-hand side analog of (2.3) is valid: 1 i i d(f (x) τx,i ) = (4.1) DMT,r [f ](x) σ x,i − DMT,r [f ](x) σx,i . 2 The immediate right-hand side analogs of Definition 2.3 and Theorem 2.4 follow: 4.3. Definition. Let f ∈ C 1 (Ω ⊂ R3 ; H). The function f is called right-i-hyperderivable in Ω, if for any x ∈ Ω there is a quaternion denoted by fi (x) such that d(f (x) τx, i ) = fi (x) σ x, i . 4.4. Theorem. Let f ∈ C 1 (Ω ⊂ R3 ; H). The function f is right-Moisil-Théodoresco-hyperholomorphic in Ω if and only if it is right-i-hyperderivable in Ω and in this case for any x ∈ Ω it follows: fi (x) =

1 i D [f ](x) . 2 MT,r

4.5. From (2.3) and (4.1) there holds: 1 i i d(f (x) τx,i g(x)) = DMT,r [f ](x) σ x, i g(x) − DMT,r [f ](x) σx, i g(x) 2 i i + f (x) σx, i DMT [g ](x) − f (x) σ x, i DMT [g](x) .

(4.2)

4.6. Let us recall that E(y − x) =

y−x (y1 − x1 ) i + (y2 − x2 ) j + (y3 − x3 ) k =− , A3 | y − x |3 A3 | y − x |3

with A3 the surface area of the unit sphere S2 ⊂ R3 , is the Cauchy kernel, which is left- and right-Moisil-Théodoresco-hyperholomorphic in R3 \ {x}. Hence from (4.2) one has: 1 i dy (E(y − x) τy, i f (y)) = DMT,r [E](y − x) σ y, i f (y) 2 i

i + E(y − x) σy, i D MT [f ](y) − E(y − x) σ y, i DMT [f ](y) . (4.3)

There are some useful relations between the Cauchy kernel and the operators i i DMT and D MT,r that we shall use: i

i

i

D MT,r,y [E(y − x)] = −DMT,r,x [E(y − x)] = −DMT,x [E(y − x)] .

(4.4)

190


4.7. Let Ω ⊂ R3 and let Γ := y ∈ R3 | (y) = 0 be its smooth boundary. Integrating (4.3) over Γ, on the left side we get: dy (E(y − x) τy, i f (y)) = 0 . Γ

Hence

i DMT,r,y

[E](y − x) σ y, i f (y) = −

Γ

Γ

+

i

E(y − x) σy, i DMT,y [f ](y) i E(y − x) σ y, i DMT,y [f ](y) ;

Γ

using (4.4) one has: i i E(y − x) σ y, i f (y) = − E(y − x) σy, i DMT,y [f ](y) −DMT,x Γ Γ i + E(y − x) σ y, i DMT,y [f ](y) .

(4.5)

Γ

Observe that the function is identically zero on Γ. Combining this fact and formula (2.3) applied to , one concludes that i −1 i i σ y, i = σy, i D MT,y [(y)] DMT,y [(y)] =: σy, i VΓ(y) , i −1 i i where we defined VΓ(y) := DMT,y [(y)] DMT,y [(y)] . Thus (4.5) becomes: i DMT,x E(y − x) σ y, i f (y) Γ i (4.6) i i E(y − x) σy, i DMT,y [f (y)] − VΓ(y) DMT,y [f (y)] . = Γ

Hence the following theorem has been proved. 4.8. Theorem. Let Ω ⊂ R3 be a simply connected domain with boundary Γ := { y ∈ R3 | (y) = 0 }, where ∈ C 1 (R3 , R), grad() | Γ(y) = 0 for all y ∈ Γ and let f ∈ C 1 (Γ, H). Then for all x ∈ / Γ the equality (4.6) holds, that is, the i–hyperderivative of the Cauchy-type integral is also a Cauchy-type integral. An immediate consequence is 4.9. Corollary. Let p ∈ N, f ∈ C p (Γ, H) and ∈ C p (R3 , R). Then for all x ∈ / Γ there follows: i (p) D MT,x E(y − x) σ y, i f (y) Γ

(p) E(y − x) σ y, i f (y)

= = Γ

Γ

i (p) i i E(y − x) σy, i DMT,y − VΓ(y) DMT,y [f (y)] .


191

5. The left j- and k-hyperderivatives In the previous sections the reasoning was concentrated around the imaginary unit i. Let us show that j and k may play a similar role. There arise the corresponding Cauchy-Riemann operators: j DMT := e3

∂ ∂ ∂ + − e1 ∂x1 ∂x2 ∂x3

and k DMT := −e2

∂ ∂ ∂ + e1 + . ∂x1 ∂x2 ∂x3

These operators keep characterizing the Moisil-Théodoresco-hyperholomorphic functions: j k kerDMT = kerDMT = kerDMT ⊂ C 1 (Ω ⊂ R3 ; H ).

Defining the suitable differential forms: σx, j := −e3 dx2 ∧ dx3 + dx1 ∧ dx3 + e1 dx1 ∧ dx2 ; τx, j := e3 dx3 − e1 dx1 ; σx, k := e2 dx2 ∧ dx3 + e1 dx1 ∧ dx3 − dx1 ∧ dx2 ; τx, k := −e2 dx2 − e1 dx1 ; the following crucial correlations are valid for any f ∈ C 1 (Ω ⊂ R3 ; H ): 1 j j σ x, j DMT [f ](x) − σx, j DMT [f ](x) , d(τx, j f (x)) = 2 and d(τx, k f (x)) =

1 k k σ x, k DMT [f ](x) − σx, k DMT [f ](x) . 2

Thus the left j- and k-hyperderivatives are defined in an exact analogy to the previous sections, concluding that:

1 j fj (x) = − DMT [f ](x) , 2

1 k fk (x) = − DMT [f ](x) , 2

for any x ∈ Ω. The rest of definitions and theorems related with the j- and k-hyperderivatives can be given also.

192


6. Comparison with one complex variable case The existence of the three hyperderivatives for the same Moisil-Théodorescohyperholomorphic function may cause an impression that we have a phenomenon with no analogues in the classical one complex variable theory. Let us show briefly that this is not true, i.e., the phenomenon does have its antecedents. Consider a function f = u + i v : Ω ⊂ R2 → C, that is, the function f takes complex values but its domain is not endowed with any complex numbers structure; this mimics the previously considered quaternionic situation. For such functions of class C 1 the Cauchy-Riemann operator is well defined and may be used in order to define the class Hol(Ω) of holomorphic in Ω functions:

1 ∂ ∂ +i [f ] = 0 . (6.1) 2 ∂x ∂y The same class is determined by another operator:

1 ∂ ∂ −i [f ] = 0 , 2 ∂y ∂x

(6.2)

i compare with DMT . But since we assume no complex numbers structure in Ω the derivative as the limit of a special quotient cannot be introduced for f ∈ Hol(Ω). The equation (6.1) corresponds to the complex variable z := x+i y which generates the function f1 : z ∈ Ω1 := {x + i y | (x, y) ∈ Ω } → f (x, y) ∈ C, and equation (6.2) corresponds to the complex variable ζ := y − i x which generates the function f2 : ζ ∈ Ω2 := {ζ = y − i x | (x, y) ∈ Ω } → f (x, y) ∈ C. Thus, to any f ∈ Hol(Ω) we associate two complex functions (that is, both go from C to C), each of them having a derivative in the usual sense: for z ∈ Ω1 and ζ0 = −i z0 ∈ Ω2 there exist the complex numbers f1 (z0 ) and f2 (ζ0 ) but they are different in general; it can be shown that f2 (ζ0 ) = i f1 (z0 ). Both f1 (z0 ) and f2 (ζ0 ) can be equally called “the derivative of f at (x0 , y0 )”, hence we are in exactly the same situation as for Moisil-Théodoresco-hyperholomorphic functions.

References [1] R. Delanghe, F. Sommen, V. Souˇcek. Clifford algebra and spinor-valued functions. Kluwer Academic Publishers. Mathematics and its Applications, 53, 1992. [2] K. G¨ urlebeck, H. R. Malonek. A Hypercomplex Derivative of Monogenic Functions in Rn+1 and its Applications. Complex Variables, 39, 1999, 199-228. [3] K. G¨ urlebeck, W. Spr¨ ossig. Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley and Sons, 1997. [4] V. V. Kravchenko, M. V. Shapiro. Integral representations for spatial models of mathematical physics. Addison-Wesley-Longman, Pitman Research Notes in Mathematics, 351, 1996.


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[5] M.E. Luna-Elzarrar´ as, M.A. Mac´ıas-Cede˜ no, M. Shapiro. Hyperderivatives in Clifford analysis and some applications to the Cliffordian Cauchy-type integrals. In: Hypercomplex Analysis. Trends in Mathematics, 2008, 221-234. [6] H. R. Malonek. Selected Topics in Hypercomplex Function Theory. Clifford Algebras and Potential Theory, S.-L. Eriksson, ed., University of Joensuu. Report Series 7, 2004, 111-150. [7] K. G¨ urlebeck, H.R. Malonek. A Hypercomplex Derivative of Monogenic Functions in Rn+1 and its Applications. Complex Variables, Theory and Applications, Vol. 39, 1999, 199-228. [8] I. M. Mitelman, M. Shapiro. Differentiation of the Martinelli-Bochner Integrals and the Notion of Hyperderivability. Math. Nachr., 172, 1995, 211-238. [9] M. V. Shapiro, N. L. Vasilevski. Quaternionic ψ-Hyperholomorphic Functions, Singular Integral Operators and Boundary Value Problems I. ψ-Hyperholomorphic Function Theory. Complex Variables, Theory and Applications, 27, 1995, 17-46. [10] N. L. Vasilevski, M. V. Shapiro. Some Questions of Hypercomplex Analysis. Complex Analysis and Applications, 87, Sofia, 1989, 523-531. M. Elena Luna-Elizarrar´ as, Marco A. Mac´ıas-Cede˜ no and Michael Shapiro Departamento de Matem´ aticas Escuela Superior de F´ısica y Matem´ aticas Instituto Politécnico Nacional México e-mail: [email protected] [email protected] [email protected]

Deconstructing Dirac Operators. II: Integral Representation Formulas Mircea Martin Abstract. We set up generalized Cauchy-Pompeiu and Bochner-MartinelliKoppelman representation formulas for arbitrary pairs (D, Φ), where D is a first-order homogeneous differential operator on Rn with coefficients in a Banach algebra A, and Φ is a smooth A-valued function on Rn \ {0} homogeneous of degree 1 − n, n ≥ 2. Within our general framework we prove that the integral representation formulas include the expected components, as well as some remainders that are explicitly computed in terms of D and Φ. As a consequence, we obtain necessary and sufficient conditions that ensure the existence of genuine Cauchy-Pompeiu or Bochner-Martinelli-Koppelman formulas for such operator-kernel pairs (D, Φ). Properly interpreted in a Clifford algebra setting these conditions prove valuable in investigating Dirac and Cauchy-Riemann operators. Mathematics Subject Classification (2010). 32A26, 35F05, 47B34, 47F05. Keywords. Dirac operators, Cauchy-Riemann operators, first-order differential operators, Cauchy-Pompeiu formula, Bochner-Martinelli-Koppelman formula, integral representation formulas.

1. Introduction The Dirac operator Deuc = Deuc,n on the Euclidean space Rn , n ≥ 2, and the Cauchy-Riemann operator ∂¯ = ∂¯m on Cm , m ≥ 2, are both natural extensions to two or more real or complex variables of the classical Cauchy-Riemann operator studied in single variable complex analysis. Though rather different from each other, the two operators share several basic properties. To make a first point, we observe that both Deuc and ∂¯ are first-order homogeneous differential operators with constant coefficients, and their coefficients are elements of the real or complex Clifford algebras associated with Rn and Cm , respectively. As yet another common feature, we recall the existence of two types of smooth homogeneous kernels, the Euclidean Cauchy kernel for Deuc , and the Bochner-Martinelli-Koppelman kernel I. Sabadini and F. Sommen (eds.), Hypercomplex Analysis and Applications, Trends in Mathematics, DOI 10.1007/978-3-0346-0246-4_14, © Springer Basel AG 2011

195

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¯ It is well known that the kernels make it possible to set up the Cauchyfor ∂. ¯ Pompeiu formula for Deuc , and the Bochner-Martinelli-Koppelman formula for ∂. Moreover, and this is quite relevant, both kernels come from potential theory, and can be defined by using Newton potentials, that is, fundamental solutions of real Laplace operators. Essential with regard to all these properties are the relationship between Deuc and the real Laplace operator ΔR = ΔR,n on Rn , 2 = −ΔR , Deuc

(1.1)

the relationship between ∂¯ and the complex Laplace operator ΔC = ΔC,m on Cm , (1.2) ∂¯∗ ∂¯ + ∂¯ ∂¯∗ = −ΔC , ∗ ¯ ¯ where ∂ is the formal adjoint of ∂, and the link between the two Laplace operators, ΔR,2m = 2ΔC,m .

(1.3)

We should perhaps indicate that in (1.1), (1.2), and (1.3) above we assume that ΔR,n =

∂2 ∂2 ∂2 , 2 + 2 + ···+ ∂ξ1 ∂ξ2 ∂ξn 2

where ξi , 1 ≤ i ≤ n, are the standard cooordinate functions on Rn , and

∂2 ∂2 ∂2 , + + ···+ ΔC,m = 2 ∂ζ1 ∂ ζ¯1 ∂ζ2 ∂ ζ¯2 ∂ζm ∂ ζ¯m where ζi and ζ¯i , 1 ≤ i ≤ m, are the standard complex cooordinate functions on Cm and their complex conjugates. Of course, to get (1.3) we need to switch from complex coordinates on Cm to real coordinates on R2m . Moreover, since Deuc is formally self-adjoint, that is, ∗ Deuc = Deuc , and because ∂¯2 = 0 and ∂¯∗2 = 0, equations (1.1), (1.2), (1.3) imply √ ∗ ). (1.4) Deuc,2m ∼ = 2 ( ∂¯m + ∂¯m The study of Clifford algebras and Dirac operators in an Euclidean or Hermitian setting has grown into a major research field. The monographs by Anglès [4], Brackx, Delanghe, and Sommen [6], Colombo, Sabadini, Sommen, and Struppa [7], Delanghe, Sommen, and Souˇcek [8], Gilbert and Murray [9], G¨ urlebeck and Sprössig [10], Krausshar [15], Mitrea [24], and Rocha-Chavez, Shapiro, and Sommen [30] are just some of the existing texts that provide insights into the general theory of Clifford algebras and Clifford analysis, including many applications to other branches of mathematics, physics, and related fields. The volumes edited by Ablamowicz [1], Ablamowicz and Fauser [2], Qian, Hempfling, McIntosh, and Sommen [25], Ryan [34], Ryan and Sprössig [36], and Sabadini, Shapiro, and Sommen [37] also provide an excellent illustration of the full scope of the current developments and the work done in this area. For specific contributions related in part to some of the problems addressed in our article we refer to Bernstein [5], Hile [12], Rocha-Chavez, Shapiro, and Sommen [27-29], Ryan [31-33, 35], Shapiro [38], Sommen [41, 42], and Vasilevski and Shapiro [44]. Relevant results regarding the

Deconstructing Dirac Operators. II.

197

Cauchy-Riemann operator and the classical Bochner-Martinelli-Koppelman formula are presented in Aizenberg and Dautov [3], Henkin and Leiterer [11], Krantz [14], and Range [26]. This article complements the investigations reported in Martin [22, 23]. Our goal is essentially the same, namely, to recover and explain properties of Dirac operators by studying first-order differential operators with coefficients in a Banach algebra. We are confident that this line of research will enrich our understanding of Dirac operators, and connect Clifford analysis with new issues of harmonic and complex analysis, or of operator theory. Connections of this kind, that motivated us in developing our approach, are pointed out in Martin [16-21]. In our article we will consider as basic objects triples (A, σ, ϕ) consisting of a real or complex unital Banach algebra A, a real linear embedding σ : Rn → A, and a smooth function ϕ : Rn → A homogeneous of degree 1, where n ≥ 2. By regarding σ as a symbol mapping, we define a first-order differential operator D on Rn with coefficients in A. Using the homogeneous function ϕ, we introduce a new function Φ : Rn0 → A, with Rn0 = Rn \ {0}, by setting Φ(ξ) =

1 |Sn−1 |

·

ϕ(ξ) , |ξ|n

ξ ∈ Rn0 ,

(1.5)

where |Sn−1 | is the total surface area of the unit sphere Sn−1 in Rn . Obviously Φ is a smooth function homogeneous of degree 1 − n, and we will refer to it as a smooth A-valued kernel on Rn of degree 1 − n. Subsequently, each pair (D, Φ) defined in this way will be called an operator-kernel pair on Rn over A. Assume now that (D, Φ) is an operator-kernel pair. The natural domains of D are the spaces C ∞ (Rn , M) of all smooth M-valued functions on Rn , where M is a Banach left or right A-module. Of course, instead of Rn we may take any open subset of Rn . To smooth A-valued functions, so in particular to ϕ or Φ, we can apply D either on the left, or on the right. The resulting smooth functions depend on σ and ϕ, and are denoted by Dϕ and ϕD, or DΦ and ΦD, respectively. As other links between σ and ϕ, or D and Φ, we define the right and left spherical means μR , μL ∈ A of the pair (σ, ϕ) or (D, Φ), given by Φ(ξ) · σ(ξ) darea(ξ), μL = σ(ξ) · Φ(ξ) darea(ξ), μR = Sn−1

Sn−1

where darea is the surface area measure on Sn−1 . Further, let us suppose that Ω ⊂ Rn is a bounded open set with a smooth and oriented boundary Σ. To D, Φ, Ω, and Σ we associate four integral operators, PΩ , RR,Ω , RL,Ω , CΣ : C ∞ (Rn , M) → C ∞ (Rn \ Σ, M). The first three, PΩ , RR,Ω , and RL,Ω are basically the convolutions operators restricted on Ω with kernels Φ, ΦD, and DΦ, respectively. The fourth, CΣ , is an

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integral operator on Σ, with a smooth kernel that depends on both Φ and σ. Explicit equations are given in Section 2. Finally, we define a truncation operator, TΩ : C ∞ (Rn , M) → C ∞ (Rn \ Σ, M), associated with Ω and its boundary Σ by setting, for each u ∈ C ∞ (Rn , M), TΩ u(x) = u(x),

x ∈ Ω,

and TΩ u(x) = 0,

x ∈ Rn \ (Ω ∪ Σ).

We are now in a position to indicate our generalized forms of the Cauchy-Pompeiu and Bochner-Martinelli-Koppelman integral representation formulas for operatorkernel pairs. Suppose that (D, Φ) is an operator-kernel pair on Rn over A, M is a Banach left Amodule, and let Ω ⊂ Rn be a bounded open set with a smooth and oriented boundary Σ. Then, the spherical means μR , μL ∈ A of (D, Φ), the four integral operators PΩ , RR,Ω , RL,Ω , CΣ , and the truncation operator TΩ satisfy the equations μR TΩ = CΣ − PΩ D − RR,Ω ,

(1.6)

(μR + μL )TΩ = CΣ − PΩ D − DPΩ − (RR,Ω + RL,Ω ).

(1.7)

and The operators in (1.6) and (1.7) act from C ∞ (Rn , M) to C ∞ (Rn \ Σ, M). Using (1.6) we get that (D, Φ) has a standard Cauchy-Pompeiu formula if and only if μR = 1, the identity of A, and RR,Ω ≡ 0 for all open sets Ω with smooth and oriented boundaries, and for any Banach left A-module M. Anticipating a result from Section 3, it follows that the existence of a genuine Cauchy-Pompeiu formula is equivalent to Sn−1

ϕ(ξ) · σ(ξ) darea(ξ) = |Sn−1 |,

(1.8)

and |ξ|2 ϕD(ξ) − n ϕ(ξ) · σ(ξ) = 0,

ξ ∈ Rn ,

(1.9)

two simple conditions that only involve the initially selected pair (σ, ϕ). In a similar way, by using equation (1.7) we deduce that the existence of a genuine Bochner-Martinelli-Koppelman formula for arbitrary domains Ω and arbitrary left A-modules M is equivalent to [ϕ(ξ) · σ(ξ) + σ(ξ) · ϕ(ξ)] darea(ξ) = |Sn−1 |, (1.10) Sn−1

and |ξ|2 [ϕD(ξ) + Dϕ(ξ)] − n [ϕ(ξ) · σ(ξ) + σ(ξ) · ϕ(ξ)] = 0,

ξ ∈ Rn ,

once more two easy to check conditions for the original pair (σ, ϕ).

(1.11)


199

Conditions (1, 8), (1.9), and (1.10), (1.11) take simpler forms and have interesting interpretations if we assume that the second component ϕ of the pair (σ, ϕ) is real linear. We will denote ϕ in this case by σ † , and then, by regarding it as a symbol mapping, we introduce a second first-order homogeneous differential operator D† on Rn with coefficients in A. In other words, now we are dealing with pairs (D, D† ) of first-order homogeneous differential operators with symbol mappings (σ, σ † ). Direct calculations show that conditions (1, 8), (1.9) amount to σ † (ξ) · σ(ξ) = |ξ|2 ,

ξ ∈ Rn ,

(1.12)

and conditions (1.10), (1.11) reduce to σ † (ξ) · σ(ξ) + σ(ξ) · σ † (ξ) = |ξ|2 ,

ξ ∈ Rn .

(1.13)

In their turn, equation (1.12) is equivalent to D† D = ΔR,n ,

(1.14)

and equation (1.13) has the equivalent form DD† + D† D = ΔR,n .

(1.15)

A comparison with equations (1.1), (1.2), and (1.3), now explains from a different perspective the existence of representation formulas for Deuc = Deuc,n and ∂¯ = ∂¯m . The significance of property (1.14) in a general setting was first explained by Hile [12], who extended the Cauchy-Pompeiu formula for pairs (D, D† ) related as in (1.14) of first-order diferential operators with constant matrix coefficients. The kernel ΦH involved in the integral representation formula proved by Hile is given by ΦH = D† ER , where ER is the fundamental solution of the real Laplace operator ΔR . As the previous comments point out, our approach is more general, it does not rely on the use of the fundamental solution ER , and it enables us to extend both the Cauchy-Pompeiu and Bochner-Martinelli-Koppelman formulas. As yet another highlight, it works in two directions, in the sense that the existence of such integral representation formulas completely characterizes the pairs (D, D† ) with symbols (σ, σ † ). Moreover, equation (1.5) indicates a simple way of defining both the Euclidean Cauchy and the Bochner-Martinelli-Koppelman kernels, without using potential theory. A detailed analysis of pairs of operators (D, D† ) satisfying (1.14) or (1.15) is presented in Martin [23]. The remainder of the article is organized as follows. In Section 2 we briefly discuss some prerequisites and state the two main results, Theorem A and Theorem B. Section 3 is concerned with several auxiliary results that eventually are used to prove Theorems A and B, and some direct consequences of the main results.

2. Integral Representation Formulas The goal of this section is to set up two types of integral representation formulas for homogeneous first-order differential operators with coefficients in a Banach algebra. We are mainly interested in finding necessary and sufficient conditions for

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the existence of some appropriate generalizations of the classical Cauchy-Pompeiu and Bochner-Martinelli- Koppelman formulas. For general properties of differential operators with constant coefficients we refer to Hörmander [13] and Tarkhanov [43]. 2.1. The Setting Throughout our article, we suppose that A is a real or complex unital Banach algebra and let M be a Banach left A-module. To be specific, we assume that M is a real or complex Banach space and A is realized as a subalgebra of L(M), the algebra of bounded linear operators on M. The space C ∞ (Rn , M) of all smooth M-valued functions on Rn , n ≥ 2, becomes a left A-module by extending the action of A to M-valued functions pointwise. Its A-submodule consisting of compactly supported functions will be denoted by C0∞ (Rn , M). As our first basic object we assume that A is equipped with a real linear embedding σ : Rn → A, n ≥ 2. The coefficients of σ form an n-tuple A = (a1 , a2 , . . . , an ) of elements of A such that σ = sA , where sA : Rn → A is given by sA (ξ) = ξ1 a1 + ξ2 a2 + · · · + ξn an ,

ξ = (ξ1 , ξ2 , . . . , ξn ) ∈ Rn .

(2.1)

Associated with σ, or, equivalently, with the n-tuple A = (a1 , a2 , . . . , an ), we introduce the first-order differential operator D = DA on C ∞ (Rn , M) by setting D A = a1 D 1 + a2 D 2 + · · · + an D n ,

(2.2)

where Di = ∂/∂xi or Di = ∂/∂ξi , 1 ≤ i ≤ n, depending on the specific coordinate functions on Rn that we use, which will be either xi , , 1 ≤ i ≤ n, or ξi , 1 ≤ i ≤ n. The embedding σ is the symbol mapping of D, and obviously σ and D uniquely determine each other. Since M is a left A-module, the action of D on C ∞ (Rn , M) is given by Du = a1 · D1 u + a2 · D2 u + . . . + an · Dn u,

u ∈ C ∞ (Rn , M).

However, if we assume that M is a right A-module, the action of D on C ∞ (Rn , M) will be defined as uD = D1 u · a1 + D2 u · a2 + . . . + Dn u · an ,

u ∈ C ∞ (Rn , M).

In this regard we should notice that whenever M is an A-bimodule, as for instance in the special case when M = A, we do not expect Du and uD to be equal. The second basic object that we need is a smooth A-valued function ϕ : Rn → A, n ≥ 2, which is supposed to be homogeneous of degree 1, that is, ϕ(tξ) = tϕ(ξ),

t ∈ (0, ∞), ξ ∈ Rn .

Associated with ϕ we introduce a function Φ : Rn0 → A, where Rn0 = Rn \ {0}, by setting ϕ(ξ) 1 Φ(ξ) = n−1 · , ξ ∈ Rn0 , (2.3) |S | |ξ|n


201

where |Sn−1 | stands for the total surface area of the unit sphere Sn−1 in Rn . Obviously, Φ is a smooth function homogeneous of degree 1 − n, that is, Φ(tξ) = t1−n Φ(ξ),

t ∈ (0, ∞), ξ ∈ Rn0 .

(2.4)

Such functions Φ will be referred to as smooth A-valued kernels on R of degree 1 − n. Equation (2.3) makes it possible to recover the original function ϕ from its associated kernel Φ. In addition, from (2.3) we also get n

ΦD(ξ) =

1 1 · · [ |ξ|2 ϕD(ξ) − n ϕ(ξ) · σ(ξ) ], |Sn−1 | |ξ|n+2

ξ ∈ Rn0 ,

(2.5)

ξ ∈ Rn0 .

(2.6)

and DΦ(ξ) =

1 |Sn−1 |

·

1 |ξ|n+2

· [ |ξ|2 Dϕ(ξ) − n σ(ξ) · ϕ(ξ) ],

As links between the two basic objects σ and ϕ, or between D and Φ, we next define two elements μR , μL ∈ A given by 1 μR = n−1 ϕ(ξ) · σ(ξ) darea(ξ) = Φ(ξ) · σ(ξ) darea(ξ), (2.7) |S | Sn−1 Sn−1 and μL =

1 |Sn−1 |

Sn−1

σ(ξ) · ϕ(ξ) darea(ξ) =

Sn−1

Φ(ξ) · σ(ξ) darea(ξ),

(2.8)

where darea is the surface area measure on the unit sphere Sn−1 . We will refer to μR and μL as the weighted right and left spherical means of ϕ with weight σ, or just as the right and left spherical means of the pair (D, Φ). We also introduce μ ∈ A given by μ = μR + μL . (2.9) Though subsequently we will be using both D and Φ, we are going to regard D as the primary object, and think of Φ as a secondary object that helps in setting up integral representation formulas for D. The next sections will make the point. For convenience, we will refer to (D, Φ) as an operator-kernel pair on Rn over A. 2.2. Related Integral Operators We now suppose that (D, Φ) is an operator-kernel pair on Rn , n ≥ 2, and let Ω ⊂ Rn be a bounded open set with a smooth and oriented boundary Σ = ∂Ω. To D, Φ, Ω, and Σ we associate the integral operators PΩ , RR,Ω , RL,Ω , CΣ : C ∞ (Rn , M) → C ∞ (Rn \ Σ, M), defined by

PΩ u(x) =

Φ(ξ − x) · u(ξ) dvol(ξ),

(2.10)

Ω

ΦD(ξ − x) · u(ξ) dvol(ξ),

RR,Ω u(x) = p.v. Ω

(2.11)

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M. Martin DΦ(ξ − x) · u(ξ) dvol(ξ),

RL,Ω u(x) = p.v.

(2.12)

Ω

CΣ u(x) =

Φ(ξ − x) · σ(ν(ξ)) · u(ξ) darea(ξ),

(2.13)

Σ

for all u ∈ C ∞ (Rn , M) and x ∈ Rn \ Σ, where dvol is the Lebesgue measure on Ω, p.v. stands for principal value, darea is the surface area measure on Σ and, for each point ξ ∈ Σ, ν(ξ) = (ν1 (ξ), ν2 (ξ), . . . , νn (ξ)) ∈ Rn

(2.14)

denotes the unit outer normal vector to Σ at ξ. The fact that the integral operator PΩ transforms smooth functions on Rn into smooth functions on Rn \ Σ is an easy consequence of (2.4). The operators RR,Ω and RL,Ω have the same property. For more details concerning their behavior we refer to the general Calderon-Zygmund theory as presented, for instance, in Stein [39, 40] or Tarkhanov [43]. We also define a truncation operator associated with Ω and its boundary Σ, TΩ : C ∞ (Rn , M) → C ∞ (Rn \ Σ, M), by setting TΩ u(x) = χΩ (x) · u(x),

u ∈ C ∞ (Rn , M), x ∈ Rn \ Σ,

(2.15)

where χΩ is the characteristic function of Ω. In other words, for u ∈ C ∞ (Rn , M) we set TΩ u(x) = u(x), if x ∈ Ω, and TX u(x) = 0, if x ∈ Rn \ (Ω ∪ Σ). Finally, for compactly supported functions we introduce the integral operators P, RR , RL : C0∞ (Rn , M) → C ∞ (Rn , M), which, when u ∈ C0∞ (Rn , M) and x ∈ Rn , are given by Pu(x) = Φ(ξ − x) · u(ξ) dvol(ξ),

(2.16)

Rn

RR u(x) = p.v. and

Rn

ΦD(ξ − x) · u(ξ) dvol(ξ),

(2.17)

DΦ(ξ − x) · u(ξ) dvol(ξ).

(2.18)

RL u(x) = p.v.

Rn


203

2.3. Main Results We are now in a position to state the two main results of our article. Both theorems are stated for arbitrary pairs (D, Φ) consisting of a first-order homogeneous differential operator D on Rn with coefficients in a Banach algebra A, and a smooth A-valued kernel on Rn of degree 1 − n, n ≥ 2, given by equations (2.2) and (2.3), respectively. The first result deals with a generalized Cauchy-Pompeiu representation formula. Theorem A. Suppose that (D, Φ) as an operator-kernel pair on Rn and let μR be the associated right spherical mean. The following three statements are equivalent: (i) If u ∈ C ∞ (Rn , M) and Ω ⊂ Rn is a bounded open subset of Rn with a smooth oriented boundary Σ, then μR · TΩ u(x) = CΣ u(x) − PΩ Du(x), (ii) If u ∈

C0∞ (Rn , M),

x ∈ Rn \ Σ.

(2.19)

then

μR · u(x) = −P Du(x),

x ∈ Rn .

(2.20)

(iii) The two components of the pair (D, Φ) have the property ΦD(ξ) = 0,

ξ ∈ Rn0 .

(2.21)

The second result provides a generalized form of the Bochner-Martinelli-Koppelman formula in a several real variables setting. Theorem B. Suppose that (D, Φ) as an operator-kernel pair on Rn and let μR and μL be the associated right and left spherical means. The following three statements are equivalent: (i) If u ∈ C ∞ (Rn , M) and Ω ⊂ Rn is a bounded open subset of Rn with a smooth oriented boundary Σ, then (μR + μL ) · TΩ u(x) = CΣ u(x) − PΩ Du(x) − DPΩ u(x), (ii) If u ∈

C0∞ (Rn , M),

x ∈ Rn \ Σ.

(2.22)

then

(μR + μL ) · u(x) = −P Du(x) − DPu(x),

x ∈ Rn .

(2.23)

(iii) The two components of the pair (D, Φ) have the property ΦD(ξ) + DΦ(ξ) = 0,

ξ ∈ Rn0 .

(2.24)

Complete proofs of Theorems A and B will be given in Section 3. Before concluding this section, we want to make a short comment regarding the equations in parts (i) and (ii) of Theorem A. We claim that the assumption that μR needs to be the right spherical mean of (D, Φ) is redundant. Actually, what really matters is the existence of an element μR that makes the two equations true, because from these equations we can prove that such an element must be the right spherical mean. To make a point, let us assume that Ω ⊂ Rn is the open unit ball, so Σ = Sn−1 , let u ∈ C ∞ (Rn , A) be the constant function that equals the identity of A, and select

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x = 0 ∈ Rn . Under such assumptions, equation (2.19) in part (i) in conjunction with equation (2.13) implies μR = CSn−1 u(0) = Φ(ξ) · σ(ξ) darea(ξ), Sn−1

whence, by comparing with (2.7) we get that μR needs to be the right spherical mean. A similar observation can be made for the equations in parts (i) and (ii) of Theorem B.

3. Auxiliary Results and Proofs This section provides some technical results and proofs of Theorems A and B. The setting and the notation are the same as in Section 2. 3.1. An Integral Formula Suppose A = (a1 , a2 , · · · , an ) is the n-tuple that defines D = DA as in equation (2.2). For each 1 ≤ i ≤ n we denote by dξic the (n − 1)-form on Rn defined by dξic = dξ1 ∧ · · · ∧ dξi−1 ∧ dξi+1 ∧ · · · ∧ dξn , where ξi , 1 ≤ i ≤ n, are the standard coordinate functions on Rn , and let ωA be the A-valued form on Rn given by ωA =

n

(−1)i−1 ai dξic .

(3.1)

i=1

Assume now that X ⊂ Rn is an oriented compact smooth submanifold of Rn of dimension n, with a smooth oriented boundary ∂X. Given two smooth functions Θ ∈ C ∞ (X, A) and u ∈ C ∞ (X, M), we introduce on X the M-valued (n − 1)-form Θ · ωA · u, and observe that its exterior derivative equals d(Θ · ωA · u) = (Θ · Du + ΘD · u)dξ, where dξ = dξ1 ∧ · · · ∧ dξi ∧ · · · ∧ dξn is the volume form on X. We next apply Stokes’ Theorem to Θ · ωA · u by using the compact manifold X with boundary ∂X and get Θ · ωA · u = (Θ · Du + ΘD · u)dξ. (3.2) ∂X

X

Both sides of (3.2), which are integrals of M-valued differential forms, can be expressed as integrals of M-valued functions, by taking the surface area measure darea on ∂X, and the volume measure dvol on X. To be specific, for each ξ ∈ ∂X we let ν(ξ) = (ν1 (ξ), ν2 (ξ), · · · , νn (ξ)) ∈ Rn be the unit outer normal vector to ∂X at ξ, and recall that on ∂X we have (−1)i−1 dξic |∂X = νi (·)darea,

1 ≤ i ≤ n,


205

whence, by (3.1) and (2.1), we get that equation (3.2) amounts to Θ(ξ) · σ(ν(ξ)) · u(ξ)darea(ξ) = [ Θ(ξ) · Du(ξ) + ΘD(ξ) · u(ξ) ]dvol(ξ), (3.3) ∂X

X

an equation that provides an alternative definition of the differential operator D. 3.2. Integral Representation Formulas with Remainders The next technical results, combined into a single lemma, point out relationships between the integral operators, the truncation operator, and the spherical means introduced in Section 2. They could be regarded as general integral representation formulas with remainders, and will prove quite useful in completing the proofs of Theorems A and B. The notation and the assumptions are as in Section 2. Main Lemma. Suppose that (D, Φ) is an operator-kernel pair on Rn , n ≥ 2, and let μR and be μL the associated right and left spherical means. (i) If Ω ⊂ Rn is a bounded open set with a smooth oriented boundary Σ, then μR TΩ = CΣ − PΩ D − RR,Ω ,

(3.4)

and μL TΩ = −DPΩ − RL,Ω , (3.5) ∞ n ∞ n with both sides regarded as operators from C (R , M) to C (R \ Σ, M). (ii) Moreover, μR = −P D − RR , (3.6) and (3.7) μL = −DP − RL , ∞ n ∞ n with both sides regarded as operators from C0 (R , M) to C (R , M). Proof. In order to prove (3.4) we need to show that μR · TΩ u(x) = CΣ u(x) − PΩ Du(x) − RR,Ω u(x), ∞

(3.8)

for all u ∈ C (R , M) and each x ∈ R \ Σ = [R \ (Ω ∪ Σ)] ∪ Ω. n

n

n

We first assume that x ∈ Rn \ (Ω ∪ Σ), set X = Ω ∪ Σ, and let Θ ∈ C ∞ (X, A) be the function given by Θ(ξ) = Φ(ξ − x), ξ ∈ X. (3.9) Since the boundary ∂X of X equals Σ, equation (3.3) becomes Φ(ξ − x)·σ(ν(ξ))·u(ξ)darea(ξ) = [ Φ(ξ − x)·Du(ξ)+ ΦD(ξ − x)·u(ξ) ]dvol(ξ). Σ

Ω

Using equations (2.10), (2.11), (2.13), and the definition (2.15) of the truncation operator, we notice that the last equation reduces to (3.8). Let us next suppose that x ∈ Ω. We choose ε > 0 such that Bn (x, ε) ⊂ Ω, where Bn (x, ε) is the closed ball of center x and radius ε in Rn , and define the compact manifold X as the closure of the bounded open set Ω \ Bn (x, ε). Its boundary ∂X consists of Σ, with the standard orientation, and the sphere Sn−1 (x, ε) of center x

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and radius ε, with the opposite orientation. We define Θ(ξ) for ξ ∈ X as in (3.9) and then, by applying (3.3) we get Φ(ξ − x) · σ(ν(ξ)) · u(ξ)darea(ξ) − Φ(ξ − x) · σ(ν(ξ)) · u(ξ)darea(ξ) Sn−1 (x,ε)

Σ

Φ(ξ − x) · Du(ξ)dvol(ξ) +

= Ω\Bn (x,ε)

ΦD(ξ − x) · u(ξ)dvol(ξ). (3.10) Ω\Bn (x,ε)

Further, we observe that the second integral in the left-hand side of (3.10) can be changed using the transformation ξ = x + ε η,

η ∈ Sn−1 .

Based on some simple calculations we have Φ(ξ − x) · σ(ν(ξ)) · u(ξ)darea(ξ) = Sn−1 (x,ε)

Sn−1

Φ(η) · σ(η) · u(x + ε η)darea(η).

Therefore, by using (2.7) we get lim Φ(ξ − x) · σ(ν(ξ)) · u(ξ)darea(ξ) = μR · u(x). ε↓0

Sn−1 (x,ε)

Equation (3.8) now follows, since obviously lim Φ(ξ − x) · Du(ξ)dvol(ξ) = PΩ Du(x), ε↓0

Ω\Bn (x,ε)

and

ΦD(ξ − x) · u(ξ)dvol(ξ) = RR,Ω u(x).

lim ε↓0

Ω\Bn (x,ε)

The proof of equation (3.4) is complete. The proof of equation (3.5) amounts to showing that μL · TΩ u(x) = −DPΩ u(x) − RL,Ω u(x),

(3.11)

for any u ∈ C ∞ (Rn , M) and x ∈ Rn \ Σ. Equation (3.11) can be proved based on a reasoning similar to the proof of (3.8) above, or, as we would prefer to do, by taking advantage of an integral formula proved in Tarkhanov [43, Section 2.1.7]. In our specific setting that formula becomes Di PΩ u(x) = −p.v. Di Φ(ξ − x) · u(ξ) dvol(ξ) − μi · TΩ u(x), 1 ≤ i ≤ n, (3.12) Ω

where μi =

1 |Sn−1 |

Sn−1

ξi · ϕ(ξ) darea(ξ) =

Sn−1

ξi · Φ(ξ) darea(ξ).

(3.13)

To get (3.11), we only need to multiply both sides of (3.12) by ai , 1 ≤ i ≤ n, take the sum of the resulting equations, and recall the definitions of D, RL,Ω , and μL .


207

The two equations in part (ii) are direct consequences of part (i). Specifically, let us assume that u ∈ C0∞ (Rn , M) is a given compactly supported function and take as Ω ⊂ Rn an open ball that contains the support of u. Using next (3.8) and (3.11) we get μR · u(x) = −P Du(x) − RR u(x), (3.14) and μL · u(x) = −DPu(x) − RL u(x),

(3.15)

for all x ∈ Rn . We conclude the proof by observing that (3.14) and (3.15) are just explicit forms of (3.6) and (3.7). 3.3. Proofs of Theorems A and B Proof of Theorem A. We first show that (i) and (iii) are equivalent. To this end, we compare equation (2.19) in Theorem A with (3.4) in Main Lemma, and deduce that statement (i) in Theorem A is equivalent to RR,Ω ≡ 0,

(3.16)

for any open and bounded set Ω ⊂ Rn with a smooth oriented boundary and any Banach left A-module M . It remains to observe that according to (2.11), the definition of RR,Ω , (3.16) takes place for all sets Ω and Banach modules M with the required properties if and only if the pair (D, Φ) satisfies equation (2.21) in part (iii) . The equivalence of (ii) and (iii) is proved in a similar way. Specifically, a comparison of (2.20) and (3.6) shows that statement (ii) in Theorem A is equivalent to RR ≡ 0,

(3.17)

a property that, in its turn, is also equivalent to equation (2.21) in part (iii), as one can easily infer from (2.17), the definition of RR . The Proof of Theorem A is complete.

Proof of Theorem B. We first observe that by combining (3.4) with (3.5), and then (3.6) with (3.7), from our Main Lemma we get (μR + μL )TΩ = CΣ − PΩ D − D PΩ − (RR,Ω + RL,Ω ),

(3.18)

and, respectively, μR + μL = −P D − D P − (RR + RL ).

(3.19)

Due to (3.18) we deduce that statement (i) in Theorem B is equivalent to RR,Ω + RL,Ω ≡ 0,

(3.20)

for any open and bounded set Ω ⊂ Rn with a smooth oriented boundary and any Banach left A-module M. Based on (3.19) we next get that statement (ii) is equivalent to RR + RL ≡ 0.

(3.21)

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Finally, a quick inspection of the definitions of RR,Ω , RL,Ω , RR , and RL in Section 2 suffices in concluding that property (3.20) for all adequate open sets Ω and modules M, as well as (3.21), is each equivalent to DΦ(ξ) + ΦD(ξ) = 0,

ξ ∈ Rn0 ,

the same equation as in part (iii) of Theorem B.

The Proof of Theorem B is complete.

3.4. Concluding Remarks We end our investigations with some direct consequences of Theorems A and B stated and proved above. For instance, if, as for the Dirac operator Deuc , we assume that D = DA is elliptic and let Φ be its fundamental solution satisfying ΦD(ξ) = DΦ(ξ) = 0,

ξ ∈ Rn0 ,

then Theorem A is true and μR = 1, so we get a genuine Cauchy-Pompeiu representation formula. Moreover, in this case the first term CΣ u in formula (2.19) has the property DCΣ u(x) = 0, x ∈ Rn \ Σ. Returning to the general setting, we next want to point out that according to equation (2.5) from Section 2, condition (2.21) in Theorem A amounts to equation (1.9), as we already mentioned in Section 1. Similarly, based on (2.6) we get that condition (2.24) in Theorem B is equivalent to (1.11). We could now make the stronger hypothesis ϕ = σ † , where σ † : Rn → A, n ≥ 2 is a real linear mapping whose coefficients form the n-tuple A† = (a†1 , a†2 , . . . , a†n ) of elements of A, that is, σ † = sA† , with sA† (ξ) = ξ1 a†1 + ξ2 a†2 + · · · + ξn a†n ,

ξ = (ξ1 , ξ2 , . . . , ξn ) ∈ Rn .

The definitions of the two spherical means associated with ϕ = σ † and σ imply μR =

1 † 1 (a1 a1 + a†2 a2 + · · · + a†n an ) = · ϕD(ξ), n n

ξ ∈ Rn ,

and

1 1 (a1 a†1 + a2 a†2 + · · · + an a†n ) = · Dϕ(ξ), ξ ∈ Rn . n n Using the last two equations we conclude that (1.9) and (1.11) are equivalent to μR =

σ † (ξ) · σ(ξ) = μR |ξ|2 ,

ξ ∈ Rn ,

(3.22)

and [σ † (ξ) · σ(ξ) + σ(ξ) · σ † (ξ)] = (μR + μL )|ξ|2 ,

ξ ∈ Rn .

(3.23)

In their turn, equations (3.22) and (3.23) have the equivalent forms D† D = μR · ΔR,n ,

(3.24)

DD† + D† D = (μR + μL ) · ΔR,n ,

(3.25)

and


209

where D† = DA† is the differential operator with symbol mapping σ † . For further investigations of pairs of differential operators that satisfy (3.24) or (3.25) in either an Euclidean or a Hermitian setting we refer to Martin [23].

References [1] Ablamowicz, R. (Ed.), Clifford Algebras Applications to Mathematics, Physics, and Engineering, Progress in Mathematical Physics Series, 34, Birkh¨ auser Verlag, 2004. [2] Ablamowicz, R. and Fauser, B., (Eds.), Clifford Algebras and their Applications in Mathematical Physics, Volume 1: Algebra and Physics, Progress in Mathematical Physics, 18, Birkh¨ auser Verlag, 2000. [3] Aizenberg, L. A. and Dautov, Sh. A., Differential Forms Orthogonal to Holomorphic Functions or Forms, and Their Properties, Transl. Math. Monographs, 56, Amer. Math. Soc., 1983. [4] Anglès, P., Conformal Groups in Geometry and Spin Structures, Progress in Mathematical Physics Series, 50, Birkh¨ auser Verlag, 2008. [5] Bernstein, S., A Borel-Pompeiu formula in Cn and its applications to inverse scattering theory, in Progress in Mathematical Physics Series, 19: Clifford Algebras and Their Applications in Mathematical Physics, Birkh¨ auser Verlag, 2000, pp. 117–185. [6] Brackx, F., Delanghe, R., and Sommen, F., Clifford Analysis, Pitman Research Notes in Mathematics Series, 76, 1982. [7] Colombo, F., Sabadini,I., Sommen, F., and Struppa, D.C., Analysis of Dirac Systems and Computational Algebra, Progress in Mathematical Physics Series, 39, Birkh¨ auser Verlag, 2004. [8] Delanghe, R., Sommen, F., and Souˇcek, V., Clifford Algebra and Spinor-Valued Functions, Kluwer Academic Publishers, 1992. [9] Gilbert, J. E. and Murray, M. A. M., Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics, 26, Cambridge University Press, 1991. [10] G¨ urlebeck, K. and Spr¨ ossig, W., Quaternionic and Clifford Calculus for Physicists and Engineers, John Wiley & Sons, 1997. [11] Henkin, G. M. and Leiterer, J., .Theory of Functions on Complex Manifolds, Birkh¨ auser Verlag, 1984. [12] Hile, G. N., Representations of solutions of a special class of first order systems, Journal of Differential Equations, 25 (1977), 410–424. [13] H¨ ormander, L., The Analysis of Linear Partial Differential Operators,Volume II: Differential Operators with Constant Coefficients, Springer Verlag, 1983. [14] Krantz, S. G.,Function Theory of Several Complex Variables, John Wiley & Sons, 1982. [15] Krausshar, R. S., Generalized Analytic Automorphic Forms in Hypercomplex Spaces Frontiers in Mathematics Series, 15, Birkh¨ auser Verlag, 2004. [16] Martin, M., Higher-dimensional Ahlfors-Beurling inequalities in Clifford analysis, Proc. Amer. Math. Soc., 126 (1998), 2863–2871.

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[17] Martin, M., Convolution and maximal operator inequalities, in Progress in Mathematical Physics Series, 19: Clifford Algebras and Their Applications in Mathematical Physics, Birkh¨ auser Verlag, 2000, pp. 83–100. [18] Martin, M., Self-commutator inequalities in higher dimension, Proc. Amer. Math. Soc., 130 (2002), 2971–2983. [19] Martin, M., Spin geometry, Clifford analysis, and joint seminormality, in Trends in Mathematics Series, 1: Advances in Analysis and Geeometry, Birkh¨ auser Verlag, 2004, pp. 227–255. [20] Martin, M., Uniform approximation by solutions of elliptic equations and seminormality in higher dimensions, Operator Theory: Advances and Applications, 149, Birkhuser Verlag, 2004, pp. 387-406. [21] Martin, M., Uniform approximation by closed forms in several complex variables, Advances in Applied Clifford Algebras, 19 (3-4) (2009), 777-792. [22] Martin, M., Deconstructing Dirac operators. I: Quantitative Hartogs-Rosenthal theorems, in More Progress in Analysis, Proceedings of the Fifth International Society for Analysis, Its Applications and Computation Congress, ISAAC 2005, World Scientific, 2009, pp. 1065–1074. [23] Martin, M., Deconstructing Dirac operators. III: Dirac and semi-Dirac pairs , Operator Theory: Advances and Applications, 203, Birkhuser Verlag, 2009, pp. 347-362. [24] Mitrea, M., Singular Integrals, Hardy Spaces, and Clifford Wavelets, Lecture Notes in Mathematics, 1575, Springer Verlag, 1994. [25] Qian, T., Hempfling, Th., McIntosh, A., and Sommen, F. (Eds.), Advances in Analysis and Geometry. New Developments Using Clifford Algebras, Trends in Mathematics Series, 14, Birkh¨ auser Verlag, 2004. [26] Range, R. M., Holomorphic Functions and Integral Representations in Several Complex Variables, Springer Verlag, 1986. [27] Rocha-Chavez, R., Shapiro M., and Sommen, F., On the singular Bochner-Martinelli integral, Integral Equations Operator Theory, 32 (1998), 354–365. [28] Rocha-Chavez, R., Shapiro M., and Sommen, F., Analysis of functions and differential forms in Cm , in Proceedings of the Second International Society for Analysis, Its Applications and Computation Congress, ISAAC 1999, Kluwer, 2000, pp. 1457–1506. [29] Rocha-Chavez, R., Shapiro M., and Sommen, F., Integral theorems for solutions of the complex Hodge-Dolbeaut system , in Proceedings of the Second International Society for Analysis, Its Applications and Computation Congress, ISAAC 1999, Kluwer, 2000, pp. 1507–1514. [30] Rocha-Chavez, R., Shapiro M., and Sommen, F., Integral Theorems for Functions and Differential Forms in Cm , Research Notes in Mathematics, 428, Chapman & Hall, 2002. [31] Ryan, J., Applications of complex Clifford analysis to the study of solutions to generalized Dirac and Klein-Gordon equations, with holomorphic potential, J. Diff. Eq. 67 (1987), 295–329. [32] Ryan, J., Cells of harmonicity and generalized Cauchy integral formulae, Proc. London Math. Soc., 60 (1990), 295–318. [33] Ryan, J., Plemelj formulae and transformations associated to plane wave decompositions in complex Clifford analysis, Proc. London Math. Soc., 64 (1991), 70–94.


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[34] Ryan, J. (Ed.), Clifford Algebras in Analysis and Related Topics, CRC Press, Boca Raton, FL, 1995. [35] Ryan, J., Intrinsic Dirac operators in Cn , Advances in Mathematics, 118 (1996), 99–133. [36] Ryan, J. and Spr¨ ossig, W. (Eds.), Clifford Algebras and Their Applications in Mathematical Physics, Volume 2: Clifford Analysis, Progress in Mathematical Physics, 19, Birkh¨ auser Verlag, 2000. [37] Sabadini, I., Shapiro, M., and Sommen, F. (Eds.), Hypercomplex Analysis, Trends in Mathematics Series, 6,Birkh¨ auser Verlag, 2009. [38] Shapiro, M., Some remarks on generalizations of the one-dimensional complex analysis: hypercomplex approach, in Functional Analytic Methods in Complex Analysis and Applications to Partial Differential Equations, World Sci., 1995, pp. 379–401. [39] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [40] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993. [41] Sommen, F., Martinelli-Bochner formulae in complex Clifford analysis, Zeitschrift f¨ ur Analysis und ihre Anwendungen, 6 (1987), 75–82. [42] Sommen, F., Defining a q-deformed version of Clifford analysis, Complex Variables: Theory and Applications, 34 (1997), 247–265. [43] Tarkhanov, N. N., The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, 1995. [44] Vasilevski, N. and Shapiro, M., Some questions of hypercomplex analysis, in Complex Analysis and Applications, Sofia, Bulgaria, 1987, 1989, pp. 523–531. Mircea Martin Department of Mathematics Baker University Baldwin City, 66006 Kansas USA e-mail: [email protected]

A Differential Form Approach to Dirac Operators on Surfaces Heikki Orelma and Frank Sommen Abstract. In this paper we study what is a suitable method to restrict the classical Dirac operator ∂x , defined on Rm , to a k-surface S ⊂ Rm . The fundamental result is that for each k-surface S there exists (at least locally) a first order linear differential operator Dk satisfying d(dxk−1 f )|S = (−1)k−1 (dxk Dk f )|S . If S = Rm , then Dm = ∂x is the classical Dirac operator. Mathematics Subject Classification (2010). Primary 58G03; Secondary 30G35. Keywords. Dirac operator, tangential Dirac operator, restricted Dirac operator, surface geometry, surface monogenics.

1. Introduction Clifford analysis offers a function theory related with the Dirac operator which is a higher dimensional generalization of classical complex analysis in the complex plane to m-dimensional Euclidean space. In the beginning of the 1990s the second author introduced an extension to differential forms of classical Clifford analysis in his paper [12]. The key idea was that a Clifford algebra-valued differential form can also be a solution of the Dirac equation. This is obtained if the Dirac operator is defined as a Lie derivative. In this paper, we shall restrict these results to k-surfaces in Rm . In Chapter 2 we consider Clifford algebras, differential forms and their tensor products. In Chapter 3, we consider some geometric tools for k-dimensional The first author was partially supported by the Vilho, Yrj¨ o and Kalle V¨ ais¨ al¨ a foundation and the Magnus Ehrnrooth foundation.


213

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surfaces in Rm . One of the main tools will be the Clifford algebra-valued surface element: dxk = k! eA dxA . |A|=k

If F is a k-form on R , then its restriction to a k-surface S is of the form F |S = gω where ω is a fixed k-form on S. The main result of the section is that the restriction is of the form F |S = (dxk f )|S . m

In Chapter 4 we define the restricted Dirac operators on a k-surface S. For example, the left restricted Dirac operator is d(dxk−1 f )|S . We shall also consider some of its basic properties. Then we will prove that the restricted Dirac operator can be represented as a Lie derivative: L∂x (F |S ) = −kd(dxk−1 f )|S , where F |S = (dxk f )|S . Then we define the tangential Dirac operator D by means of the formula d(dxk−1 f )|S = (−1)k−1 (dxk Df )|S . Chapter 5 gives an example. We will study tangential Dirac operators on the paraboloid.

2. Basic Language 2.1. Clifford Algebra In this subsection we will consider the classical theory of Clifford algebras. Historical remarks and more comprehensive introduction to the topic may be found in, e.g., [4], [6], [7], or [9]. The Clifford algebra Rm is the free associative algebra generated by symbols e1 , . . . , em together with the defining relations ei ej + ej ei = −2δij . As a real vector space Rm has dimension 2m . A canonical basis of Rm is given by eA = ea1 · · · eak , A = {a1 , . . . , ak } ⊂ M = {1, . . . , m} and 1 ≤ a1 < · · · < ak ≤ m. Especially e∅ = 1 and e{j} = ej . The pseudoscalar is the element eM = e1 · · · em . The space of k-vectors is defined by Rkm = span{eA : |A| = k} and thus any a ∈ Rm admits the following multivector decomposition, a = [a]0 + [a]1 + · · · + [a]m with [a]k ∈

Rkm .

There exists the canonical embedding Rm → Rm defined by (x1 , . . . , xm ) → x =

m j=1

ej xj .

Differential Form Approach to Dirac Operators on Surfaces

215

Hence the set of 1-vectors R1m is usually identified with Rm . For a 1-vector x ∈ Rm and a k-vector a ∈ Rkm , their product xa splits into a (k − 1)-vector and a (k + 1)vector, namely xa = [xa]k−1 + [xa]k+1 , where [xa]k−1 =

1 (xa − (−1)k ax) 2

and 1 (xa + (−1)k ax). 2 The (k + 1)-part of the product is denoted by x ∧ a = [xa]k+1 . [xa]k+1 =

The conjugation is an algebra endomorphism a → a defined by relations ej = −ej and ab = ba for a, b ∈ Rm . We define the norm in the Clifford algebra Rm by a2A . |a|2 = [aa]0 = A

0, it is well known (see, e.g., [7]) that there exists a unique If x ∈ R and |x| = 1-vector x−1 satisfying x−1 x = xx−1 = 1. The 1-vector x−1 is called the inverse of x. m

2.2. Differential Forms Differential forms are a powerful tool for research in modern differential geometry. In this subsection we only recall some basic facts. There is a vast literature on differential forms. We mention a few: [8], [11] or [14]. We shall introduce differential forms axiomatically as an extension of the algebra of polynomials Alg{x1 , . . . , xm } as follows: For each variable xj we introduce a new generator dxj leading to the extended associative algebra Φm = Alg{x1 , . . . , xm } ⊗R Alg{dx1 , . . . , dxm } where Alg{dx1 , . . . , dxm } is the free associative algebra generated by dx1 , . . . , dxm and the map d : xj → dxj extends to an endomorphism of this algebra satisfying d(xj F ) = dxj F + xj dF, d(dxj F ) = −dxj dF. First note that these axioms also imply that dxj dxk

= −dxk dxj , i.e., Alg{dx1 , . . . , 2 dxm } is the Grassmann algebra. Also we have d = m j=1 dxj ∂xj and d = 0. A canonical basis of Alg{dx1 , . . . , dxm } is given by dxA = dxa1 · · · dxak , A = {a1 , . . . , ak } ⊂ M = {1, . . . , m} and 1 ≤ a1 < · · · < ak ≤ m. Especially dx∅ = 1 and dx{j} = dxj . The element dxM = dx1 · · · dxm corresponds the volume element in Euclidean space. The space of k-forms is defined by Φkm = Alg{x1 , . . . , xm } ⊗R span{dxA : |A| = k}.

216


Now we shall consider some important algebra endomorphisms on Φm . The basic contraction operator F → ∂xj F is determined by the relations ∂xj (xk F ) = xk ∂xj F, ∂xj (dxk F ) = δjk F − dxk ∂xj F. Then, the basic endomorphisms on Φm are F → xj F,

F → ∂xj F,

F → dxj F,

F → ∂xj F.

2.3. Clifford Algebra-valued Differential Forms As shown in the previous subsections, there are many similarities between algebra of differential forms and the Clifford algebra (see also [3]). Despite similarities, the dxj and ej are different calculus objects. We will see that in next section of this paper. Now we shall consider the Clifford extension of differential forms

the the the the

Φm ⊗R Rm = Alg{x1 , . . . , xm ; dx1 , . . . , dxm ; e1 , . . . , em }. Each ω ∈ Φm ⊗R Rm admits the representation ωA,B eA dxB ω= A,B

where eA and dxB are elements of the canonical basis of Rm and Φm respectively. Let us consider the following example. The differential form m dx = ej dxj j=1

is called the vector differential. The vector differential plays a fundamental role in the theory of monogenic forms, see [4] and [13]. Moreover (cf. [5] pp. 6-7) dxk = k! eA dxA . |A|=k

2.4. Monogenic Differential Calculus We consider smooth functions defined on an open subset Ω of Rm with values in the algebra of Clifford algebra-valued differential forms, that is, we consider sections of Φm ⊗R Rm defined on Ω. We denote the set of smooth sections by E(Ω) = Γ(Ω, Φm ⊗R Rm ). The set of Clifford algebra-valued k-sections is denoted by E k (Ω). Hence every F ∈ E(Ω) admits the representation FA,B (x)eA dxB , F (x) = A,B

where FA,B : Ω → R is a smooth real function or F (x) = FA (x)dxA A


217

where

m FA : Ω → Rm is a smooth Clifford algebra-valued function. Let v = j=1 vj ∂xj be a Clifford algebra-valued vector field. The contraction is given by vj FA (x)(∂xj dxA ). vF (x) = j,A

The Lie derivative of F ∈ E (Ω) with respect to v is defined by k

Lv F (x) = d(vF (x)) + v(dF (x)). The above identity is known as Cartan’s magic formula. The Dirac operator is the vector field m ∂x = ej ∂xj . j=1

Let k = 0, . . . , m and let F ∈ E k (Ω) such that F (x) = we put ∂x F (x) = ∂x FA (x)dxA .

|A|=k

FA (x)dxA . Then

|A|=k

One may verify (cf. [4] or [12]) that ∂x F (x) = d(∂x F (x)) + ∂x (dF (x)), that is ∂x F (x) = L∂x F (x). An Rm -valued k-form F ∈ E k (Ω) is called left (right) monogenic if ∂x F (x) = 0 (resp. F (x)∂x = 0). The equation L∂x F (x) = 0 is called the Dirac equation. The case k = 0 leads us to the classical theory of Clifford analysis, see [2], [4], [6] or [7].

3. Clifford Algebraic Tools for Surfaces A k-dimensional embedded submanifold S of Rm is called a k-surface. Consider a k-surface S with equations xj = xj (u1 , . . . , uk ), j = 1, . . . , m or briefly x = x(u). Let ι : S → Rm be the embedding, i.e., ι(u) = x. Let F (x) = FA (x)eA dxA . |A|=k

We may pull global objects back to objects on a surface by ι∗ (FA )(u) = FA (ι(u)), ι∗ (eA ) = eA , ι∗ (dxi ) =

k ∂xi duj . ∂u j j=1

218


The restriction of a function F to the surface S is then F |S (u) = ι∗ (F )(u). Next we shall study the geometry of k-surfaces. The structure constants are defined by ∂xa1 ∂xak cA (u) = sgn(π) ··· . ∂uπ(1) ∂uπ(k) π∈Sk

Hence dxA |S = cA (u)du1 · · · duk , and we may prove the following lemma. Lemma 3.1. Let S ⊂ Rm be a k-surface and let Ω be an open subset of Rm satisfying S ⊂ Ω. For each F ∈ E k (Ω) there exists an Rm -valued function f satisfying F |S (u) = f (u)du1 · · · duk , where

f (u) =

FA |S (u)cA (u).

|A|=k

Proof. Assume F (x) =

FA (x)dxA .

|A|=k

Then

F |S (u) =

ι∗ (FA )(u)ι∗ (dxA )

|A|=k

and we obtain F |S (u) =

FA |S (u)cA (u) du1 · · · duk .

|A|=k

Using the above lemma we obtain dxk |S = k!c(u)du1 · · · duk , where c(u) =

eA cA (u).

|A|=k

Next we shall study what is a meaning of the k-vector c(u) in the surface geometry. We recall the following famous factorization formula. Let x1 , . . . , xk be vectors in Rm . Then 1 x1 ∧ · · · ∧ xk = sgn(π)xπ(1) · · · xπ(k) . k! π∈Sk

Next we prove that c(u) admits the following surprisingly handy representation.


219

Proposition 3.2. On a k-surface S with the coordinates x = x(u) the k-vector c(u) admits the representation c(u) =

∂x ∂x ∧ ···∧ . ∂u1 ∂uk

Proof. First we compute k

dxk |S = dx|S · · · dx|S =

j1 ,...,jk

∂x ∂x ··· duj1 · · · dujk . ∂u ∂u j1 jk =1

This implies that dxk |S =

sgn(π)

π∈Sk

∂x ∂x ··· du1 · · · duk . ∂uπ(1) ∂uπ(k)

Using the factorization formula we have dxk |S = k!

∂x ∂x ∧ ···∧ du1 · · · duk , ∂u1 ∂uk

which completes the proof.

Thus we infer that c(u) is a k-blade tangent to S at the point u. Thus c(u) is called the tangent k-blade on S. Obviously |c(u)|2 = c(u)c(u) = 0 and we may c(u) define the unit tangent k-blade by |c(u)| . Using the tangent k-blade we obtain the following proposition. Proposition 3.3. Let S be a k-surface and Ω be an open subset of Rm satisfying S ⊂ Ω. For each F ∈ E k (Ω) there exists a smooth function f satisfying F |S = (dxk f )|S . Proof. Let g be the function satisfying F |S (u) = g(u)du1 · · · duk and c be the tangent k-blade such that dxk |S = k!c(u)du1 · · · duk . Assume u ∈ S is fixed. Consider the equation k!c(u)f (u) = g(u). Since c(u)c(u) = 0, the solution of the above equation is f (u) = for each u ∈ S and the proof is complete.

1 −2 c(u)g(u) k! |c(u)|

4. Surface Monogenics In this section we will define the tangential Dirac operators on surfaces. Also we will study their basic properties.

220


4.1. Restricted Dirac Operator Recall ι∗ ◦ d = d ◦ ι∗ (see [8]), i.e., the exterior derivative and the restriction commute. It is hard to try to restrict the classical Dirac operator ∂x directly to a k-surface S. We recall the following example. Assume S = Rm . Then one may verify that d(dxm−1 f ) = (−1)m−1 dxm (∂x f ). In the left-hand side of the above example appears the term what one may restrict to a surface. This motivates us to give the following definition. Definition 4.1. Assume S ⊂ Rm is a k-surface. A function f is left surface monogenic on a k-chain C ⊂ S if d(dxk−1 f )|C = 0 and right surface monogenic if d(f dxk−1 )|C = 0. The operator d(dxk−1 f )|C is called the restricted left-Dirac operator on a chain C and the operator d(f dxk−1 )|C is called the restricted right-Dirac operator on a chain C. Let F ∈ E k−1 (Ω) and C ⊂ Rm be a k-chain such that C ⊂ Ω. The well-known integral formula F = dF ∂C

C

is called Stokes’ formula (see [8]). Using Lemma IV.1.7 of [4] we obtain ∂x dxk = −k(m − k + 1)dxk−1 . Since f is a 0-form we obtain ∂x (dxk f ) = (∂x dxk )f. Hence we may prove the following theorem. Theorem 4.2 (Cauchy’s Theorem). Let f be a left surface monogenic function on a chain C ⊂ S and let F = dxk f . Then ∂x F = 0. ∂C

Proof. Let f be a left surface monogenic function on a k-chain C ⊂ S, i.e., d(dxk−1 f )|C = 0. Hence ∂x F = ∂x dxk f ∂C ∂C = −k(m − k + 1) dxk−1 f ∂C = −k(m − k + 1) d(dxk−1 f ) = 0. C


221

The proof is complete. The formula d(f dxk−1 g) = d(f dxk−1 )g + f d(dxk−1 g)

is called the basic identity. The basic identity and Stokes’ formula implies the following proposition. Proposition 4.3. Let f and g be smooth function on a k-chain C. Then d(f dxk−1 )g + f d(dxk−1 g) . f dxk−1 g = ∂C

C

Moreover, if f is right surface monogenic and g is left surface monogenic, then f dxk−1 g = 0. ∂C

Next we express the equation d(dxk−1 f )|S = 0 in terms of the embedding of S. First we prove the following more general result. Proposition 4.4. Let (U, ϕ) be a chart in Rm such that S ⊂ U and S = {x ∈ Rm : ϕi (x) = 0, i = k + 1, . . . , m}. Then F |S = 0 if and only if

m

F =

(ϕj Gj + dϕj Hj )

j=k+1

for some Gj and Hj . Proof. Let F be a Clifford algebra-valued function. Then F |S = 0 if and only if F =

m

ϕj Gj

j=k+1

for some Gi . Let F =

FA dϕA

|A|=k

and let N = {k + 1, . . . , m}. Hence F = FA dϕA + FA dϕA A∩N =∅

A∩N =∅

If A ∩ N = ∅, then for each A there exists j ∈ N such that A∩N =∅

FA dϕA =

m j=k+1

dϕj Hj

222


and the last sum is obviously zero on S. Hence F |S = FA |S dϕA |S = 0 A∩N =∅

if and only if FA |S = 0 for each A such that A ∩ N = ∅. Since FA is a Clifford algebra-valued function it is of the form m FA = ϕj GA,j . j=k+1

Then

m

FA dϕA =

ϕj

j=k+1

A∩N =∅

GA,j dϕA

A∩N =∅

which completes the proof.

Next we will see how the restricted Dirac operator depends on the embedding. Corollary 4.5. The equation d(dxk−1 f )|S = 0 holds if and only if d(dxk−1 f ) =

m

(ϕj Gj + dϕj Hj )

j=k+1

for some Gj and Hj . 4.2. Connection with Lie Derivatives If S is a k-surface in Rm , then we may always choose an orthonormal local frame w1 , . . . , wm in a neighborhood of x ∈ S in Rm such that w1 , . . . , wk is a basis of the tangent space of S at a point x. Lemma 4.6. If w1 , . . . , wm is a local frame, then m wj wj , ∂x , ∂x = j=1

where ·, · : Rm × Rm → R is the Euclidean inner product.

m

m Proof. Let ∂x = k=1 ek ∂xk and wj = i=1 wj , ei ei . Thus m

wj , ∂x = Since ek =

wj , ei ei , ek ∂xk =

j=1 wj , ek wj ,

∂x =

m j=1


wj , ei ∂xi .

i=1

i,k=1

m

m

we compute

wj

m k=1

wj , ek ∂xk =

m

wj wj , ∂x .

j=1


223

The above lemma allows us to define the parallel part of the Dirac operator by ∂x =

k

wj wj , ∂x

j=1

and the perpendicular part by m

∂x⊥ =

wj wj , ∂x .

j=k+1

Hence ∂x = ∂x + ∂x⊥ . The Lie derivative of F |S with respect to ∂x admits the following representation. Lemma 4.7. Let F be a smooth k-form on Ω ⊂ Rm and S ⊂ Ω be a k-surface. Then L∂x (F |S ) = d(∂x (F |S )). Proof. Since dF is a (k + 1)-form, then dF |S = 0 and L∂x (F |S ) = d(∂x F |S ) + ∂x (dF |S ) = d(∂x F |S ).


We may always choose (see [8]) the slice coordinates on the k-surface S at least locally, i.e., in the neighborhood U of the point x ∈ S there exists local coordinates ϕ = (ω1 , . . . , ωm ) such that ϕ(y) = (ω1 (y), . . . , ωk (y), 0, . . . , 0) for each y ∈ S ∩ U . Moreover there exists vectors vj orthonormal at a point x such that v1 , . . . , vk is a basis of the tangent space of S at a point x. Then we obtain ∂x =

k

vj ∂ωj ,

∂x⊥ =

j=1

m j=k+1

Since the restriction of F is F |S = f dω1 · · · dωk , then it is clear that ∂x (F |S ) = (∂x F )|S . Similarly the vector differential is of the form dx =

m

vj dωj

j=1

and we define its parallel and perpendicular parts by

dx =

k j=1

vj dωj

vj ∂ωj .

224


and dx⊥ =

m

vj dωj .

j=k+1

Hence, in a given point x ∈ S we have the decomposition dx = dx + dx⊥ . Then ∂x dx = −k, ∂x⊥ dx⊥ = −(m − k) and ∂x dx⊥ = ∂x⊥ dx = 0. These rules allows us to prove the following lemma. Lemma 4.8. On a k-surface S, (∂x⊥ dxk )|S = −k(m − k)dxk−1 |S . Proof. Let x ∈ S be a fixed point. The lemma will be proved if we can show (∂x⊥ dxp )|S = −p(m − k)dxp−1 |S for p = 1, 2, . . . , m. First we compute m vj ∂ωj (dxp−1 dx) ∂x⊥ dxp = =

j=k+1 m

vj (∂ωj dxp−1 )dx + (−1)p−1 dxp−1 (∂ωj dx)

j=k+1

= (∂x⊥ dxp−1 )dx + (−1)p−1

m

vj dxp−1 vj .

j=k+1

In a given point x ∈ S we may express dx = dx + dx⊥ . Hence m m vj dxp−1 vj = (−1)p−1 vj (dx )p−1 vj + P ERP (−1)p−1 j=k+1

j=k+1

=

m

vj2 (dx )p−1 + P ERP

j=k+1

= −(m − k)(dx )p−1 + P ERP, where P ERP denotes the perpendicular part of the form. Hence ∂x⊥ dxp = (∂x⊥ dxp−1 )dx − (m − k)(dx )p−1 + P ERP, that is, (∂x⊥ dxp )|S = (∂x⊥ dxp−1 )|S dx|S − (m − k)dxp−1 |S . Since (∂x⊥ dx)|S = −(m − k), the proof follows inductively. Since x ∈ S is an arbitrary point, we obtain the result. Lemma 4.9. Let S be a k-surface. Then m−k (∂x dxk )|S . (∂x⊥ dxk )|S = m−k+1


225

Proof. Recall ∂x dxk = −k(m − k + 1)dxk−1 . Especially (∂x dxk )|S = −k(m − k + 1)dxk−1 |S . The above lemma implies (∂x⊥ dxk )|S = −k(m − k)dxk−1 |S

and the proof follows.

Theorem 4.10. Let F ∈ E (Ω) and S be a k-surface such that S ⊂ Ω. If f is a function satisfying F |S = (dxk f )|S , then k

L∂x (F |S ) = −kd(dxk−1 f )|S . Proof. Using Lemma 4.7 we obtain L∂x (F |S ) = d(∂x (dxk f )|S ) = d(∂x (dxk f ))|S . Consider the equation (∂x (dxk f ))|S = (∂x (dxk f ))|S − (∂x⊥ (dxk f ))|S where the last term is equal to (∂x⊥ (dxk f ))|S = (∂x⊥ dxk )|S f |S + ∂˙x⊥ dxk f˙|S . Since f is a 0-form, we obtain that ∂˙x⊥ dxk f˙|S = 0 and using the above lemma we obtain m−k (∂x⊥ (dxk f ))|S = (∂x⊥ dxk )|S f |S = (∂x (dxk f ))|S . m−k+1 Then m−k ∂x (dxk f ))|S = (∂x (dxk f ))|S − (∂x (dxk f ))|S m−k+1 1 (∂x (dxk f ))|S . = m−k+1 Thus we may write the Lie derivative in the form 1 d(∂x (dxk f ))|S . L∂x (F |S ) = m−k+1 Since f is a 0-form, we have ∂x (dxk f ) = ∂x dxk f = −k(m − k + 1)dxk−1 f and then we obtain L∂x (F |S ) = −kd(dxk−1 f )|S . The proof is complete.

As a corollary we obtain the following result. Corollary 4.11. Let F ∈ E k (Ω) and S be a k-surface such that S ⊂ Ω. If f is a function satisfying F |S = (dxk f )|S , then f is a left surface monogenic function on S if and only if L∂x (F |S ) = 0.

226


4.3. Tangential Dirac Operator In this chapter of the paper we will show that for each restricted Dirac operator there exists a unique linear first-order differential operator such that it is an operator on the set of Clifford algebra-valued functions to itself. Also we see that their null-solutions are the same. Theorem 4.12. Let ι : S → Rm be the embedding and let C ⊂ S be a k-chain. Then there exists a unique differential operator D satisfying d(f dxk−1 g)|C = (f D)dxk g |C + (−1)k−1 f dxk Dg |C and (a) D(f + ga) = Df + (Dg)a and (f + ag)D = (f D) + a(gD), where f, g are smooth Rm -valued functions and a ∈ Rm . (b) D(f g) = (f g)D = (Df )g + f (Dg) where f, g are smooth real functions. Proof. Using Lemma 4.3 there exists the function h such, that (dxk−1 dg)|C = (dxk h)|C . Let us denote Dg := h. Consider the equation (dxk Df )|C = 0. Since dxk |C : f → (dxk f )|C is an isomorphism, we infer that Df = 0. Since Df = 0 for all f , we have D ≡ 0. Hence the operator D is unique. Assume f and g are smooth Rm -valued functions and a ∈ Rm . Then (dxk D(f + ga))|C = d(dxk−1 (f + ga))|C = (dxk (Df + Dga))|C . The uniqueness of the operator implies D(f + ga) = Df + (Dg)a. The right part of the proof is similar and thus omitted.

Definition 4.13. The operator D is called the tangential Dirac operator. Obviously the kernel of D and the kernel of the operator d(dxk−1 g)|S are the same. The above theorem and Stokes’ formula implies the following integral formula. Proposition 4.14. Let f and g be smooth functions on a k-chain C ⊂ S. Then (f D)dxk g + (−1)k−1 f dxk Dg . f dxk−1 g = ∂C

C

5. Clifford Analysis on the Paraboloid In this section we give an example how to use the preceding tools. We will study the paraboloid and derive the tangential Dirac operator D on it. Also we will study solutions of the equation Df = 0.


227

5.1. The Tangential Dirac Operator on the Paraboloid The paraboloid is an m-surface S in Rm+1 governed by the relation x0 = x21 + · · · + x2m . If u = x1 e1 + · · · + xm em , then x0 = u, u and x(u) = u, ue0 + u is a point on the paraboloid in Rm+1 . We choose u to be a local coordinate in S. The restriction of the vector differential is dx|S = du + 2 u, due0 . Let N = {1, . . . , m}. Then duk = k!

eA dxA .

|A|=k A⊂N

Next we deduce some technical tools. Lemma 5.1. (a) u, du2 = 0, (b) du u, du = − u, dudu, (c) d u, du = 0, (d) dum ej = −mdum−1 dxj for j = 1, . . . , m, (e) dum ei ej = −m(m − 1)dum−2 dxi dxj for i < j. In the above, (d) and (e) are called the first and the second transformation rule respectively. Proof. The proofs of (a), (b) and (c) are just short computations and thus omitted. xj = dx1 · · · dxj−1 dxj+1 · · · dxm . We (d) Put e%j = e1 · · · ej−1 ej+1 · · · em and d% compute −mdum−1 dxj = −m!

m

e%k d% xk dxj = −m!% ej d% xj dxj .

k=1

Since e%j = −e2j e%j = −(−1)m+j eN ej and d% xj dxj = (−1)j+m dxN we obtain −mdum−1 dxj = m!eN dxN ej = dum ej . (e) Assume i < j. Put e%ij = e1 · · · ei−1 ei+1 · · · ej−1 ej+1 · · · em and d% xij = dx1 · · · dxi−1 dxi+1 · · · dxj−1 dxj+1 · · · dxm . Hence −m(m − 1)dum−2 dxi dxj = −m! e%lk d% xlk dxi dxj = −m!% eij d% xij dxi dxj . l 0} of Rp+1,q+1 . Consequently, the symbols X A and xA represent the same object – the identity vector field on Rp+1,q+1 . Let’s explicitly define the mapping of the coordinate change: ⎛ ⎞ ⎛ 0⎞ t y ⎠ = ⎝ ya ⎠ . txa Φ(t, xa , ρ) = ⎝ a y∞ t(ρ − x 2xa )

254

V. Tuˇcek

We see that φ(xa ) = Φ(1, xa , 0) and the identity (2.1) simplifies on the image of φ to δA B = XA Z B + ZA X B + YAc YcB . This identity will be of a great use later on. Lemma 2.4. The Euler operator in the new coordinates is equal to E = t∂t . Proof. For f (y A ) ∈ C ∞ (Rn ) we have ∂f ∂f xb xb ∂f ) + xa a + (ρ − ) 0 ∂y ∂y 2 ∂y ∞ ∂f y a ∂f y ∞ ∂f ) = y0( 0 + 0 a + 0 ∂y y ∂y y ∂y ∞ = (y A ∂A f ) ◦ Φ(t, xa , ρ)

t∂t f (y A (t, xa , ρ)) = t(

Since we want to deal with differential operators we need to incorporate smooth functions on Rp,q into the picture. Suppose that f is a smooth function defined on the neighbourhood of origin in Rp,q . Then for any w ∈ C, f˜(Φ(t, xa , 0)) = tw f (xa )

(2.3)

defines a smooth function on a ‘conical neighbourhood’ of (1, 0, 0) inside the null cone N . Moreover it is a homogeneous function of degree w because f˜(λy A ) = λw f˜(y A ) for λ > 0. Conversely f may be recovered from f˜ by setting t = 1. In order to be able to apply ambient differential operators to f˜ we need to extend it from the null cone to the whole space or at least to some open (in Rp+1,q+1 ) neighbourhood of (1, 0, 0). We will call any such extension ambient extension. There are infinitely many choices for such an extension even if we stick to the homogeneous ones. Nevertheless, any two such extensions will differ by a very convenient factor. Lemma 2.5. Let f˜ and fˆ be two smooth w-homogeneous extensions of f on some open neighbourhood of (1, 0, 0) not containing zero. Then there exist a smooth (w − 2)-homogeneous function h such that (f˜ − fˆ)(y A ) = r(y A )h(y A ) where r is the defining quadric of the null cone. Proof. For any smooth function k on Rp+1,q+1 it holds 1 1 d ∂k a a a a k(t, x , sρ)ds = k(t, x , 0) + ρ (t, xa , sρ)ds. k(t, x , ρ) = k(t, x , 0) + ds ∂ρ 0 0 According to the first equation of (2.2) we have ρ = r/2t2 . For k = (f˜ − fˆ) ◦ Φ we have k(t, xa , 0) = 0 and the result follows by substitution with Φ−1 . Remark 2.6. The classical chain rule formula, with regard to (2.3), gives ∂a f = ∂a (f˜ ◦ φ) = (∂a φB )(∂B f˜) ◦ φ = (YaB ∂B f˜) ◦ φ

(2.4)

Construction of Conformally Invariant Differential Operators

255

for any ambient extension f˜ of f because ∂a φB equals YaB on the image of φ. Using this expression for f = YbA ◦ φ we get ∂c (YbA ◦ φ) = (YcD ∂D YbA ) ◦ φ = (−tgcb Z A ) ◦ φ.

(2.5)

3. Construction of conformally invariant differential operators Let (V1 , 1 ), (V2 , 2 ) be two representations of a Lie group G and let G have a smooth action on a manifold M . One defines the induced action of G on smooth functions C ∞ (M, Vi ) by (g · f )(x) = i (g)(f (g −1 )x). The G-invariant differential operators are defined as those differential operators which are equivariant with respect to the induced action of G. In our case, the Lie group in question is SO(p + 1, q + 1) and the underlying manifold is N . As a consequence, we may find the conformally invariant operators ˜ induces among the orthogonally invariant ones. An ambient differential operator D n ˜ ˜ an operator on R if and only if the value of (D f ) ◦ φ does not depend on the ˜ f˜ = D ˜ fˆ). For a linear operator, this condition ambient extension f˜ of f (i.e., D ˜ f˜ − fˆ) = D(rh) ˜ is equivalent to D( = 0. Of course such a condition can hold only for some weights w of the extension and not for the other weights. The conclusion is that a conformally invariant linear differential operator is induced by ˜ for which there exists a weight w ∈ C an SO(p + 1, q + 1)-invariant operator D such that ˜ r]f˜ = Dr ˜ f˜ − rD ˜ f˜ = 0. [D, ˜ with leading term S A1 ···Ai (xA )∂A1 ···Ai . Suppose we are given such an operator D In order to find the coefficients of the induced operator we use the adapted frame ˜ = S A1 ···Ai (xA )δA1 B1 · · · δAi Bi ∂B1 · · · ∂Bi + LOT which on the (2.1) and write D image of φ simplifies to S A1 ···Ai (xa )(XA1 Z B1 + ZA1 X B1 + YAc1 YcB1 ) · · · · · · (XAi Z Bi + ZAi X Bi + YAci YcBi )∂B1 ···Bi , since we already assume that the result doesn’t depend on the ambient extension and hence we can drop all the terms containing . Also the terms containing Z Bi can be omitted as well, because they represent derivatives in the direction transversal to the null cone and the result doesn’t depend on the ambient extension. Let us compute the expression for the operator induced by the ambient Laplace operator as an illustration of this method, g AB ∂A ∂B f˜ ◦ φ = g AB δA C δB D ∂C ∂D f˜ ◦ φ 1 = g AB 2 (XA Z C + ZA X C + YAq YqC − ρUA C ) t 1 D D D r D ˜ · 2 (XB Z + ZB X + YB Yr − ρUB )∂C ∂D f ◦ φ t

256

V. Tuˇcek = [(Z C X D + X C Z D )∂C ∂D f + g qr YqC YrD ∂C ∂D f˜] ◦ φ.

by (2.2)

Because we have [∂A , ∂B ] = 0, the first term in the last expression equals 2Z C X D ∂C ∂D = 2 Z C ∂C X D ∂D − Z C (∂C X D )∂D = 2 Z C ∂C X D ∂D − Z C (δC D )∂D = 2Z C ∂C (E − 1) applied to f˜ and evaluated on the image of φ. The second term is qr C D g Yq Yr ∂C ∂D f˜ ◦ φ = g qr [YqC ∂C YrD ∂D − YqC (∂C YrD )∂D ]f˜ ◦ φ = g qr YqC ∂C (YrD ∂D f˜) ◦ φ + g qr gqr Z D ∂D f˜ ◦ φ by (2.5) = Δf + n(Z D ∂D f˜) ◦ φ. f˜) ◦ φ = Δf + Z D ∂D (n + 2E − 2)f˜ ◦ φ (Δ

Hence

and we see that, for w = 1 − n/2, the ambient Laplace operator on Rp+1,q+1 induces the Laplace operator on Rp,q . We see that the computation of the coefficients boils down to two parts – calculation of the contractions with the symbol of the operator and calculation of the contractions with the differentials. For the latter part we can record here the following lemma. Lemma 3.1. Let D(k, s) be an operator defined as s k+1 D(k, s) = X D1 · · · X Dk YcD · · · YcD ∂D1 · · · ∂Ds . s k+1

Then modulo the terms depending on the ambient extension we have for w-homogeneous functions, / k 0 5 ˜ (D(k, s)f ) ◦ φ = (w − s + i) ∂c · · · ∂c f. k+1

s

i=1

Proof. Let T (k) denote the operator X D1 · · · X Dk ∂D1 · · · ∂Dk . We have T (k) = X D1 · · · X Dk−2 X Dk−1 X Dk ∂D1 · · · ∂Dk = X D1 · · · X Dk−2 X Dk−1 (∂D1 X Dk − δD1 Dk )∂D2 · · · ∂Dk = X D1 · · · X Dk−2 X Dk−1 ∂D1 X Dk ∂D2 · · · ∂Dk f − T (k − 1) = X D1 · · · X Dk−2 (∂D1 X Dk−1 − δD1 Dk−1 )X Dk ∂D2 · · · ∂Dk − T (k − 1) = X D1 · · · X Dk−2 ∂D1 X Dk−1 X Dk ∂D2 · · · ∂Dk − 2T (k − 1) .. . = X D1 ∂D1 X D2 · · · X Dk ∂D2 · · · ∂Dk − (k − 1)T (k − 1) = E T (k − 1) − (k − 1)T (k − 1) = (E −k + 1)T (k − 1)


257

Since T (1) = X D1 ∂D1 = E we see that T (k) = X D1 · · · X Dk ∂D1 · · · ∂Dk = (E −k + 1)(E −k + 2) · · · (E −1) E . We can view YbA as a 1-homogeneous function on Rn with values in Rn+2 because the homogeneity in the standard coordinates translates to homogeneity in t by Lemma (2.4). Therefore the Euler operator acts as the identity on YcD which i implies [E, YcD ] = 0. Using this fact we can write i s k+1 · · · YcD ∂Dk+1 ···Ds . D(k, s) = T (k)YcD s k+1

Iterating the formula (2.4) we get 1 2 k ∂c1 · · · ∂ck f = (YcD ∂D1 (YcD ∂D2 (· · · ∂Dk−1 (YcD ∂Dk f˜)) · · · ) ◦ φ. 1 2 k

Since YcD ∂D YbA = −tgcb Z A by (2.5), we see that the difference between the above 1 k · · · YcD ∂c1 · · · ∂ck yields a differentiation of f in expression and the formula YcD 1 k the direction transversal to the embedding φ that clearly depends on the choice of an ambient extension. Since each differentiation lowers the homogeneity by one, T (k) acts in the expression for (D(k, s)f˜) ◦ φ on w − (s − k)-homogeneous function and the result follows.

4. Symmetry operators of the Laplace equation As an application of the just presented method we compute the so-called higher symmetry operators for the Laplace equation. We say that a linear differential operator D is symmetry operator of the Laplace equation if there exists another linear differential operator δ such that ΔD = δΔ. It is easy to see that these operators preserve the space of harmonic functions. It was shown in [6] that, modulo the trivial symmetry operators of the form DΔ, all the symmetry operators are induced from the ambient operators of the form D V := V A1 B1 ···As Bs XA1 · · · XAs ∂B1 · · · ∂Bs where D2s n+2 R V A1 B1 ···As Bs ∈ is a tensor that is skew in each pair of indices Ai Bi , is totally trace-free, and such that skewing over any three indices gives zero. It follows that V A1 B1 ···As Bs is symmetric with respect to changes of the form Ai Bi ↔ Aj Bj and that symmetrising over any s + 1 indices gives zero. These symmetries can be expressed by a Young tableau ··· ··· trace-free part. E FG H s boxes in each row

From these symmetry properties it is easy to prove that DV commutes with Since it also preserves homogeneity it follows that the operator r and with Δ. induced on (1 − n)/2-homogeneous functions is a symmetry operator with δ being the operator induced on (−3 − n)/2-homogeneous functions.

258

V. Tuˇcek

Theorem 4.1. Let V A1 B1 ···As Bs be a tensor with the aforementioned symmetries and let V c1 ···cs = V A1 B1 ···As Bs XA1 · · · XAs YBc11 · · · YBcss ◦ φ. Let f be a smooth function on Rn and let f˜ be its w-homogeneous extension on some open neighbourhood of φ(Rn ). The operator on Rn defined by Dw V f = (D V f˜) ◦ φ equals to s Ik k s c1 ···cs w−s+i Dw f = (−1) )∂ck+1 · · · ∂cs f. (4.1) V i=1 n+2s−1−i (∂c1 · · · ∂ck V k k=0

Proof. We apply the method described in the previous section and discard the terms containing and Z Di . We arrive at the following equality on the image of φ ⎛ ⎞ 5 c 5 j⎠ ⎝ ZBi X Di DV f˜ = V A1 B1 ···As Bs XA1 · · · XAs YBjj YcD ∂D1 ···Ds f˜. j I∪J={1,...s} I∩J=∅

i∈I

j∈J

Because the tensor XA1 · · · XAs is symmetric and V A1 B1 ···As Bs is symmetric in pairs Ai Bi , we can write ⎛ ⎞ 5 c 5 j⎠ YBjj YcD V A1 B1 ···As Bs XA1 · · · XAs ⎝ ZBi X Di j i∈I

j∈J

⎛

= V A1 B1 ···As Bs XA1 · · · XAs ⎝

k 5

i=1

ZBi X Di

s 5

⎞ c j⎠ YBjj YcD j

j=k+1

for any two disjoint subsets I, J of {1, . . . , s} whose union is the whole set and s where I has cardinality k. For brevity we introduce the symbols XA1 ···As , YBc11 ···c ···Bs , ZB1 ···Bs as shorthands for XA1 · · · XAs etc. So far, we have got the following expression for DV on the image of φ, s s c ···cs Dk+1 ···Ds ˜ V A1 B1 ···As Bs XA1 ···As ZB1 ···Bk YBk+1 X D1 ···Dk Yck+1 ···cs ∂D1 · · · ∂Ds f , k+1 ···Bs k k=0

where it is understood that for k = 0 the term under the sum equals s k+1 · · · YcD ∂D1 · · · ∂Ds f V A1 B1 ···As Bs XA1 ···As YBc11 · · · YBcss X D1 · · · X Dk YcD s k+1

and analogously there are only ‘Z terms’ for k = s. c ···cs Let S(k) = V A1 B1 ···As Bs XA1 ···As ZB1 ···Bk YBk+1 be the symbol part of the k+1 ···Bs operator. Using the chain rule and the Leibniz rule we get ∂ck+1 (S(k) ◦ φ) = (YcD ∂ S(k)) ◦ φ k+1 D c ···cs = V A1 B1 ···As Bs YcD ∂D (XA1 ···As )ZB1 ···Bk YBk+1 k+1 k+1 ···Bs c ···cs + XA1 ···As ZB1 ···Bk ∂D (YBk+1 ) ◦ φ, ···B s k+1


259

because YaD ∂D ZB equals zero by a straightforward computation. ∂D XAi = Yc1 Ai and the Leibniz rule again we write Using the identity YcD 1 the first summand as s ck+1 ···cs D (V A1 B1 ···As Bs XA1 ···A i ···As (Yc1 ∂D XAi )ZB1 ···Bk YBk+1 ···Bs ) ◦ φ. i=1

∂D XAi = Yc1 Ai we can simplify it to Plugging in the identity YcD 1 s ck+1 ···cs V A1 B1 ···As Bs XA1 ···A ◦ φ, (g − X Z − Z X )Z Y A B A B A B B ···B i ···As i 1 i 1 i 1 1 k Bk+1 ···Bs i=1

because Yc1 Ai YBc11 equals gAi B1 − XAi ZB1 − ZAi XB1 on the image of φ due to the (2.1). Using trace-freeness of V A1 B1 ···As Bs and its antisymmetry in A1 B1 , we see that for i = 1 the only nontrivial contraction is with the term XAi ZB1 ; for i = 1 we contract V with the tensor field ZA1 XB1 + XA1 ZB1 which is symmetric in its indices. Therefore the first summand equals −(s − 1)S(k + 1). With the help of the identity (2.5), the second summand yields s c ci cs ci J V A1 B1 ···As Bs XA1 ···As ZB1 ···Bk YBk+1 · · · Y · · · Y (−δ Z ) ◦ φ. c B 1 i B B i s k+1 i=k+1

For i = 1 the contraction results in −nS(k + 1) whereas for i = 1 we get −S(k + 1) because of the symmetry in pairs A1 B1 ↔ Ai Bi . Thus the second summand equals −(n + s − k − 1)S(k + 1). Putting it all together we arrive at ∂ck+1 (S(k) ◦ φ) = −(n + 2s − 2 − k)S(k + 1) ◦ φ. Since S(0) ◦ φ = V c1 ···cs , we see that S(k + 1) =

(−1)k+1 ∂c · · · ∂c1 V c1 ···cs (n + 2s − 2) · · · (n + 2s − 2 − k) k+1

and using the lemma 3.1 we finally conclude ˜ Dw V f = (D V f ) ◦ φ ( ' s s S(k)D(k, s)f˜ ◦ φ = k k=0 s s S(k) ◦ φ · (D(k, s)f˜) ◦ φ = k k=0 s s A(k, s, w)(∂ck · · · ∂c1 V c1 ···ck )∂ck+1 · · · ∂cs f = k k=0

where A(k, s, w) = (−1)k

(w − s + 1) · · · (w − s + k) . (n + 2s − 2) · · · (n + 2s − 1 − k)

260

V. Tuˇcek

Remark 4.2. The formula (4.1) agrees with the formula on page 1659 of [6], where the author uses a rather sophisticated notion of naturality in order to obtain the result.

References [1] Baston, R. J., Conformally Invariant Operators: Singular Cases, Bull. London Math. Soc., 23 (1991), 153-159 [2] Baston, R. J., Eastwood, M. G., Invariant operators, Twistors in mathematics and physics, 129–163, London Math. Soc. Lecture Note Ser., 156, Cambridge Univ. Press, Cambridge, 1990. [3] Calderbank, D., Diemmer, T., Differential invariants and curved Bernstein-GelfandGelfand sequences, J. Reine Angew. Math. 537 (2001), 67–103. ˇ [4] Cap, A., Slov´ ak, J., Souˇcek, V., Bernstein-Gelfand-Gelfand sequences, Ann.Math., 154 (2001), 97–113. ˇ [5] Cap, A., Gover, A. R., Standard tractors and the conformal ambient metric construction, Ann. Global Anal. Geom. 24 (2003), no. 3, 231–259. [6] Eastwood, M. G., Higher symmetries of the Laplacian, Annals Math. 161, 1645 (2005) [arXiv:hep-th/0206233]. [7] Eastwood, M. G., Rice, J. W., Conformally invariant differential operators on Minkowski space and their curved analogues, Comm. Math. Phys. Volume 109, Number 2 (1987), 207-228. [8] Eastwood, M. G., Slov´ ak, J., Semiholonomic Verma modules. Journal of Algebra, vol. 197 (1997), no. 2, 424–448. [9] Franek, P., Dirac operator in two variables from the viewpoint of parabolic geometry, Adv. Appl. Clifford Algebr. 17 (2007), no. 3, 469–480. [10] Franek, P., Generalized Dolbeault sequences in parabolic geometry, J. Lie Theory 18 (2008), no. 4, 757–774. [11] Kr´ ysl, S., Classification of 1st order symplectic spinor operators over contact projective geometries, Differential Geom. Appl. 26 (2008), no. 5, 553–565. [12] Kr´ ysl, S., Decomposition of a tensor product of a higher symplectic spinor module and the defining representation of sp(2n, C), J. Lie Theory 17 (2007), no. 1, 63–72. [13] Slov´ ak, J. , Invariant operators on conformal manifolds, Research Lecture Notes, University of Vienna, 138 pages, 1992. (extended and revised version submitted as habilitation thesis at Masaryk University in Brno, 1993) available online at http://www.math.muni.cz/~ slovak/ftp/papers/vienna.ps V´ıt Tuˇcek ´ MUUK Sokolovsk´ a 49/83 Praha 186 00 Czech Republic e-mail: [email protected]

Remarks on Holomorphicity in Three Settings: Complex, Quaternionic, and Bicomplex Daniele C. Struppa, Adrian Vajiac and Mihaela B. Vajiac Abstract. We compare holomorphicity in three distinct settings: complex, quaternionic, and bicomplex variables. We show how the analysis in the complex case can be extended in substantially different directions when dealing with quaternions or bicomplex numbers. In particular, we highlight the different dimensionality of hyperfunctions in the three settings. Mathematics Subject Classification (2010). 30G35. Keywords. Bicomplex numbers, hyperfunction theory, quaternions.

1. Introduction We have recently devoted some attention [8], [9] to the study of holomorphicity in the setting of bicomplex numbers. This topic is not new, and in fact bicomplex numbers, and the analysis of functions defined on bicomplex numbers, have been studied at irregular intervals in the last decades. An important seminal work was the one of Ryan, [29], but it was only with the book of Price, [25], that a full foundation of the theory of bicomplex numbers was given. Afterwards, a few isolated works analyzed either the properties of bicomplex numbers, or the properties of holomorphic functions defined on bicomplex numbers. In this respect, we should highlight the work of Rochon, Shapiro, and their collaborators, [4], [27], and [28]. Our recent articles are devoted mostly to the study of singularities of holomorphic functions of bicomplex variables, and to the development of a systematic approach to a hyperfunction theory in the bicomplex setting. It is worth pointing out that different authors are often led to similar research interests on the basis of different starting points. For example, Ryan’s interest in bicomplex numbers stems from his work on Clifford analysis, while Price’s work is clearly in the tradition of complex analysis. As for us, as we point out in [8], we were led to bicomplex numbers because of our long-standing interest and work in the theory of analyticity for functions of quaternionic variables. I. Sabadini and F. Sommen (eds.), Hypercomplex Analysis and Applications, Trends in Mathematics, DOI 10.1007/978-3-0346-0246-4_18, © Springer Basel AG 2011

261

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D.C. Struppa, A. Vajiac and M.B. Vajiac

Thus, we think it may be of some interest to collect here some reflections on both commonalities and differences that arise when working in the complex, the quaternionic, and the bicomplex setting. We believe that by paying attention to these three settings, we are likely to identify new areas for research, and make better sense of existing results. We will, therefore, attempt to highlight the most important results from classical complex analysis, and see how they can be interpreted in the quaternionic and the bicomplex case. What makes the comparison even more intriguing is the fact that analyticity in the quaternionic setting can be interpreted in many different ways. For the sake of simplicity, we will restrict our analysis to the case of CauchyFueter regular functions [14], [15], and to the more recent study of slice regular functions, [11], [17], [18], and [19]. Acknowledgements. The authors are grateful to Professor M. Shapiro for the many stimulating conversations on the topics of bicomplex numbers and of hyperholomorphic functions. The authors also want to express their gratitude to the anonymous referee for his/her helpful comments.

2. Algebraic Definitions

7 Complex numbers are defined as the set C = {x + iy 7 x, y ∈ R}, where i is an imaginary unit, i.e., i2 = −1, which commutes with the real variables x and y. The algebra of complex numbers is well known, therefore we will not spend any additional time here. Note however that C is isomorphic to the real Clifford algebra R0,1 (see, e.g., [12, page 213]). Since in the rest of the paper we will use several different complex spaces, with different imaginary units, we will adopt the convention of indicating the imaginary unit as an index, so that the set defined above will also be denoted by Ci . The real associative algebra of quaternions H is also well known. One considers three imaginary units I, J and K which anti-commute with each other and such that I2 = J2 = K2 = −1, IJ = −JI = K, JK = −KJ = I, KI = −IK = J. Then a real basis for H is given by (1, I, J, K), so every quaternion q ∈ H is uniquely written as q = x0 + Ix1 + Jx2 + Kx3 , where xi ∈ R are real variables. It is immediate to see that q can also be written as q = z + Jw = z + wJ where z = x0 + Ix1 and w = x2 + Ix3 are complex numbers in CI , and the conjugate of w is the usual complex conjugate. In this notation, the conjugate of q ∈ H is defined by q = z − Jw = z − wJ = x0 − Ix1 − Jx2 − Kx3 , and its (Euclidean) norm is qH = qq =

x20 + x21 + x22 + x23 .

Remarks on Holomorphicity

263

It is immediate to see that H can be identified with the real Clifford algebra R0,2 (see again [12, page 213]), but is also a skew field, in which the inverse of any element q = 0 is given by: q q −1 = . q2 We further notice that every non-real quaternion q can be uniquely expressed as q = x + yI for some x, y ∈ R, y > 0 and some I in the set of imaginary units 7 S = {q ∈ H 7 q 2 = −1}, which is a 2-sphere in the 3-space of purely imaginary quaternions. In other words: H= (R + IR), I∈S

where the complex lines LI := R + IR are all isomorphic to the complex field CI and intersect at R. This decomposition is the central observation in the new theory of slice regular functions on H (see, e.g., [17] or, for a more complete treatment, [16]). Finally, the bicomplex space BC [25, 27] is the product of two copies of the complex spaces Ci over the space of complex numbers Cj , i.e., 7 BC = {z + jw 7 z, w ∈ Ci }, where i and j are commuting imaginary units ij = ji = k, i2 = j2 = −1, and k2 = 1. Explicitly, we write a bicomplex number as Z = x0 + ix1 + jx2 + kx3 = z + jw, where z, w ∈ Ci . The algebra of BC is not a division algebra since, for example, (1 + k)(1 − k) = 0. If we consider the following two idempotents elements of BC, 1+k 1−k , e2 = e1 = 2 2 we immediately verify that every Z = z + jw ∈ BC, can be uniquely written in the so-called idempotent representation as Z = (z − iw)e1 + (z + iw)e2 . Let Z = x0 + ix1 + jx2 + kx3 = z + jw ∈ BC. Because of the many imaginary units that appear in this definition, there are several natural ways to define a notion of conjugation in BC. The various definitions are all interesting, and lead to different ways of regarding the algebra of bicomplex numbers. The Z ∗ -conjugate is defined as: Z ∗ = z − jw, where we took the conjugates in the two complex spaces Ci corresponding to the is defined variables z and w, respectively, and the conjugate in Cj . The Z-conjugate as: = z + jw, Z

264


where we took only the conjugates in the two complex spaces Ci corresponding to the variables z and w, respectively. Finally the Z † -conjugate is defined as follows: Z † = z − jw, where we took only the conjugate in Cj . Note that Z · Z ∗ = |z|2i + |w|2i + 2k(x0 x3 − x1 x2 ) is an element of the real Clifford algebra R1,0 , see [12]; this algebra is not a complex space because k2 = 1, and it is also known as the set of hyperbolic numbers or duplex numbers. Similarly, Z · Z = |z|2i − |w|2i + 2j(x0 x2 + x1 x3 ) ∈ Cj , and Z · Z † = z 2 + w 2 ∈ Ci . Since z 2 +w2 ∈ Ci has an inverse whenever different from zero, the bicomplex number Z† Z −1 = 2 (1) z + w2 is the inverse of Z in BC. For this reason we say that the bicomplex number Z = z + jw is nonsingular (or invertible) if Z · Z † = z 2 + w2 = 0. In terms of the ideals I1 = e1 and I2 = e2 , the set of nonsingular bicomplex numbers is BC \ (I1 ∪ I2 ). A simple computation shows that, in terms of the idempotent representation, the inverse of a bicomplex number Z = z + jw ∈ / I1 ∪ I2 is given by: Z −1 = (z − iw)−1 e1 + (z + iw)−1 e2 , As we close this section, we will mention one more “variation on the theme”, namely the space of biquaternions BH defined as the associative algebra of complexified quaternions: BH = H ⊗R Cj (see, e.g., [12, page 278]). A biquaternion Z ∈ BH is written as Z = z0 + Iz1 + Jz2 + Kz3 = q + jq ,

(2)

where zμ = xμ + jyμ ∈ Cj , for μ = 0, 1, 2, 3, and q = x0 + Ix1 + Jx2 + Kx3 and similarly q are in H. Here j is an imaginary complex unit which commutes with the quaternionic imaginary units I, J, K. The biquaternion algebra is not a division algebra: the zero divisors occur when ZBH = z02 − z12 − z22 − z32 = 0, similarly to what happens in the case of bicomplex numbers. In this setup, the bicomplex space BC can be viewed as a slice Lj = CI + jCI of BH, by considering, for example, z2 = z3 = 0 in (2). A theory of holomorphic functions for this space (and its significance in electromagnetism) was studied in [6], but its analysis goes beyond the scope of this paper.


265

3. Differentiability and Regularity Let U be an open set in C and let f : U → C. Its complex derivative at a point z is defined as the limit (if it exists) f (z) := lim

h→0

f (z + h) − f (z) , h

where h is allowed to take values in C. A function is said to be holomorphic in an open set if it admits derivative in every point of the set, and it turns out that if one defines the so-called Cauchy-Riemann operator ∂ ∂ ∂ = +i , ∂z ∂x ∂y then a function is holomorphic if and only if ∂f = 0. ∂z In the quaternionic case, one can analogously define the derivative of a quaternion-valued function at a point q by requiring the existence of the limit f (q + h) − f (q) , h where now h is allowed to take values in H. Unfortunately it is easy to show that the existence of f (q) implies f (q) = aq + b, for some a, b ∈ H, and so the most natural generalization of holomorphicity to the quaternionic domain does not yield a viable theory. However, Fueter showed that the theory of one complex variable can more or less be extended to the quaternionic case if we generalize the Cauchy-Riemann operator to the quaternionic case. Two such generalizations can be given, known as the left and the right Cauchy-Fueter operators f (q) := lim

h→0

∂l ∂q ∂r ∂q

= =

∂ ∂ ∂ ∂ +I +J +K ∂x0 ∂x1 ∂x2 ∂x3 ∂ ∂ ∂ ∂ + I+ J+ K. ∂x0 ∂x1 ∂x2 ∂x3

and holomorphic functions on H are then defined as solutions of such operators. This gives two (absolutely equivalent) theories to which the literature refers to as left and right (Cauchy-Fueter) regular functions. Even though the theory of Cauchy-Fueter regular functions is indeed very rich (see, e.g., [12]), there are aspects of the theory, which are problematic, such as the fact that even simple polynomials are not regular in the sense of CauchyFueter, or the fact that a simple power series expansion is not available for such functions. It was because of these concerns, that a new definition of holomorphicity on H was given in [11, 16, 17, 18].

266


Let Ω be 7a domain in H and let f : Ω → H. For each I ∈ S, let ΩI = Ω ∩ LI and let fI = f 7ΩI be the restriction of f to the complex line LI . The restriction fI is called holomorphic if it has continuous partial derivatives and

∂ 1 ∂ +I fI (x + yI) ≡ 0. ∂ I f (x + yI) = 2 ∂x ∂y The function f is called slice regular if, for all I ∈ S, fI is holomorphic. Note that even though slice regular functions are defined as solutions to a differential operator, this operator cannot be written as a constant coefficients ∂ depends on the point at which operator, since the I in front of the derivative ∂y the function is being computed. This point is a delicate one, which makes the study of slice regular functions distinctively different from the study of other forms of holomorphicity (which usually arise by considering linear constant coefficients differential operators). It is interesting also to note that if a function f is slice regular, then it is possible to define a notion of derivative for it. Specifically, for each I ∈ S, the I-derivative of f is defined as

∂ 1 ∂ ∂I f (x + yI) = −I fI (x + yI) 2 ∂x ∂y on ΩI . The slice derivative of f is the function ∂s f = f : Ω → H defined by ∂I f on ΩI , for all I ∈ S. Thus, just like in the complex case, the notions of differentiability and holomorphicity are strongly intertwined. We now turn to the study of a bicomplex function F : BC → BC which can be always written as F = u + jv, where u, v : BC → Ci . Following [25] we define bicomplex holomorphicity in the same way one defines classical holomorphicity in one complex variable: Definition 1. A function is said to be bicomplex holomorphic in an open set U ⊆ BC if it admits a bicomplex derivative at each point, i.e., if the limit lim H→0

H −1 (F (Z + H) − F (Z))

H invertible

exists and is finite for any Z in U . This limit will be called the derivative of F and denoted by F (Z). We know that holomorphic functions in one complex variable are solutions of the Cauchy-Riemann system. Something similar occurs in this case, [25]: Theorem 2. Let U be an open set in BC, and let F : U → BC be such that F = u + jv ∈ C 1 (U ). Then F is bicomplex holomorphic if and only if: 1. u and v are complex holomorphic in z and w ∂v ∂v ∂u ∂u = and =− on U . 2. ∂z ∂w ∂z ∂w ∂u ∂v ∂v ∂u 1 ∂F Moreover, F = = +j = −j . 2 ∂Z ∂z ∂z ∂w ∂w


267

The next result is the foundation for the theory of bicomplex holomorphic functions, and for the understanding of their behavior, and it was first proved in [28]: Theorem 3. Let U ⊆ BC be an open set and F : U → BC such that F = u + jv ∈ C 1 (U ). Then F is holomorphic in BC if and only if F satisfies the following three systems of differential equations: ∂F ∂F ∂F = = = 0. ∗ † ∂Z ∂Z ∂ Z˜

(3)

This result shows that holomorphic functions on the bicomplex space are once again solutions of a system of linear constant coefficients differential equations. The key difference from the case of one complex or one quaternionic variable lies in the fact that the system we use now is an overdetermined system, see [13], and the algebraic consequences of this fact are rather far reaching. Specifically, if we regard F as a vector F = (u, v), then Theorem 3 can be restated by saying that bicomplex holomorphic functions are the solutions to the system ⎡ ⎢ ⎢ ⎢ P (D)F = ⎢ ⎢ ⎢ ⎣

∂z ∂w ∂z ∂w ∂z ∂w

−∂w ∂z ∂w −∂z −∂w ∂z

⎤ ⎥ ⎥ ⎥ ⎥ F = 0. ⎥ ⎥ ⎦

(4)

∂ where for simplicity we have replaced the differential operator with the short∂z hand ∂z , and so on. An immediate consequence of the Ehrenpreis’ fundamental principle (see [13]) is that the bicomplex holomorphic functions form a sheaf of rings H on the space BC. This is analogous to what happens for holomorphic functions in C, and for Cauchy-Fueter regular functions in H, but substantially different from what happens for slice regular functions. As we noted before, these latter functions are not solutions of constant coefficients differential operators, and they do not form a sheaf. The consequence of this observation is central, because it allows us to use the methods described in [12] to study the algebraic properties of bicomplex holomorphic functions. In particular, and without offering any details for which we refer the reader to [12] and to [8], we obtain the following removability of singularities theorem: Theorem 4. Let K ⊂ BC be a compact convex set and U a neighborhood of K. Then any distribution solution to the homogeneous system P (D)f = 0 on U \ K can be uniquely extended to a solution of the the system on U . This result gives a striking difference between the theory of one complex (or quaternionic) variable, and the theory of one bicomplex variable. As it will become clear in the next section, this result really tells us that, at least in some of

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its features, the theory of holomorphic functions in one bicomplex variable really behaves like a theory of several variables.

4. Bicomplex Hyperfunctions in One and Several Variables One of the great developments of the theory of holomorphic functions is the introduction, by the Japanese school of Sato, Kawai, and Kashiwara, of hyperfunctions (see, e.g., [20] for a simple introduction to the topic). Roughly speaking, hyperfunctions are generalized functions defined on R or Rn , as suitable boundary values of holomorphic functions. The definition in one real variable is quite simple. If U is an open set on the real line, the space of hyperfunctions B(U ) is defined by O(C \ U ) . O(C) This definition is easily extended to the quaternionic case; if we denote by R the sheaf of Cauchy-Fueter left regular functions on H, and if U is now an open set in {q ∈ H : x0 = 0}, then the space BH (U ) of quaternionic hyperfunctions on U can be defined by B(U ) =

R(H \ U ) . R(H) In order to understand how such a definition may be extended to the bicomplex case, it is however necessary to explore a different way to understand hyperfunctions. This way is not as intuitive as the one described above, and actually requires that we look first at how to extend the notion of hyperfunctions to several variables. In the complex case, for example, the appropriate definition for the sheaf of hyperfunctions on Cn involves the use of relative sheaf cohomology of dimension n. The reason for the specific dimensionality is too elaborate to be discussed here, but relies on some important algebraic properties of the sheaf of holomorphic functions in several variables. We will therefore take a short detour on this topic. As is well known, the theory of holomorphic functions in several complex variables began as a somewhat obvious extension of the theory in one variable. Functions are said to be holomorphic in several variables if they are holomorphic in each variable independently (there was of course a famous theorem of Hartogs to show that the original request that the function be continuous was superfluous). Because of this definition, some of the early results on holomorphic functions of several complex variables were predictable, for example the Cauchy formula for polydisks, and did not provide any new insight in the structure of holomorphic functions. It was only in 1906, with the discovery by Hartogs of the famous “removability of compact singularities” phenomenon that now carries his name, that mathematicians fully understood that the theory of several complex variables was BH (U ) =


269

fundamentally different from the theory in one variable (in this context, see Theorem 4 above). The nature of this difference eluded mathematicians for many years, but gave rise to some impressive achievements, like (for example) the BochnerMartinelli formula, which shed further light on the peculiarity of several complex variables. Incidentally, but relevant here, it is well known that both Bochner and Martinelli were led to the now famous formula through their readings of the work of Fueter, who was trying to use quaternions to find integral representations for holomorphic functions in two complex variables (for a completely modern take on this approach, one should read the work of Shapiro and some of his collaborators, for example [22], [26]). It was however when sheaf theory became an important instrument for analysts, that the actual nature of the theory of several complex variables became apparent, and that an algebraic approach to the theory of holomorphic functions in Cn emerged. The reader interested in a modern look at several complex variables, and willing to further explore some of the connections we have quickly mentioned, should probably read Krantz’s excellent [21]. As we mentioned above, it is through the algebraic properties of the sheaf O of holomorphic functions, that one can fully develop a hyperfunction theory. For example, one can prove that almost all the relative cohomology groups of the form H j (Cn , Cn \ Rn ; O) vanish. In fact they all vanish, with the exception of the case in which j = n. The technical term for this property usually says that Rn is purely n-codimensional in Cn with respect to O, but what it means is that Rn is an initial surface for the Cauchy problem for the Cauchy-Riemann system in Cn , or (in yet other terms) that every real analytic function on Rn extends uniquely to a holomorphic function in a neighborhood of Rn . This ultimately implies that one can define a sheaf of hyperfunctions on Rn by setting, on every open set U ∈ Rn , B(U ) := H n (Cn , Cn \ U ; O). Note that when n = 1, this definition is equivalent to the one we gave at the beginning of the section. A somewhat similar evolution, but on a shortened time-scale, occurred for the generalization of the notion of Cauchy-Fueter regularity from one to several variables. While the original work of Fueter dates back to the 1930s, there was no real attempt to extend his ideas to Hn , until Pertici’s work [24], where he proved an analog of the Bochner-Martinelli formula in the quaternionic setting, as well as a version of the Hartogs’ phenomenon of removability of compact singularities. Pertici’s work, however, did not lead to the development of a full theory, which only occurred with [1], [2], [3] and the other works discussed in [12], where an algebraic approach is taken, in the same vein as the one that had been successful in the complex case. By noticing that Cauchy-Fueter regular functions are solutions of overdetermined systems of differential equations (the Cauchy-Fueter system), just like holomorphic functions are solutions of the Cauchy-Riemann system, one can

270


systematically study the algebraic structure of the Cauchy-Fueter system and show that if S is a (2n + 1)-dimensional initial variety for the Cauchy problem for the Cauchy-Fueter system, then all the relative cohomology groups of the form H j (Hn , Hn \ S; R) vanish, with the exception of the case in which j = 2n − 1. This once again allows us to define a sheaf of quaternionic hyperfunctions on S by setting, on every open set U ∈ S, BH (U ) := H 2n−1 (Hn , Hn \ U ; R). Once again, when n = 1 this definition is equivalent to the one we gave at the beginning of the section. The nature of varieties S that are initial for the Cauchy-Fueter system is rather delicate, and we refer the reader to [23, 10]. It is worth pointing out that both the complex and quaternionic hyperfunctions in one variable are essentially defined as 1-cohomology classes, and that the reason for that is the fact that the initial varieties on which they are defined are codimension 1 varieties. One can then use a simple cohomological argument to show that these 1-cohomology classes can always be represented by holomorphic (or regular) functions. Thus, the cohomological definition coincides with the more natural boundary value definition. One may then ask what happens in the case of bicomplex functions. In this case, one can show that the initial variety for the system that defines holomorphic functions of one bicomplex variable is indeed R, just like in the complex case. Remark 5. We take this opportunity to correct an imprecision in [9], where we stated, incorrectly, that every real analytic function from R to BC extends to a holomorphic function on all of BC (Theorem 12 in [9]). This is clearly not possible, since even in the complex case it is not true that a real analytic function from R to C extends to a holomorphic function on all of C (consider, for example, f (x) = 1 1 1+x2 , which extends to f (z) = 1+z 2 , which is holomorphic in a neighborhood of the real axis but certainly not entire). Thus, our statement in [9] needs to say that every real analytic function from R to BC extends uniquely to a holomorphic function defined in a bicomplex neighborhood of R in BC. The proof of Theorem 12 in [9] does not require any modification. Since now R has codimension 3 in BC, one should expect that hyperfunctions in the bicomplex setting should be defined by means of 3-cohomology classes. That this is indeed the case is demonstrated in [9], where moreover a cohomological argument is established to show that: Theorem 6. If U is any open convex set in BC, then, with obvious meaning of the notations, and with H the sheaf of holomorphic functions on BC, BBC (U ∩ R) ∼ = H 2 (U \ R; H). This result is obtained through a novel construction of a Dolbeault-like resolution for the sheaf of C ∞ functions by means of differential forms which represent the three operators defining holomorphic functions in the bicomplex setting. We


271

believe this result to be very interesting, because it shows once again that the theory of holomorphicity in the bicomplex setting is essentially different from the theory of holomorphicity in one complex or one quaternionic variable, and it is by its own nature a several variables kind of theory. We had a first glimpse of this with Theorem 4, and we now have a confirmation with Theorem 6. From this point of view, the step from one to several variables is more natural for the bicomplex case, than it was in the standard complex and quaternionic settings. We begin by giving a natural definition for holomorphicity in several bicomplex variables. Denote by (Z1 , . . . , Zn ) a variable in BCn . For each Zi = zi + jwi we have the conjugates Zi∗ , Z˜i , Zi† . The notion of bicomplex hyperholomorphicity in several variables can be introduced by requiring bicomplex hyperholomorphicity in each variable as in the following definition. Definition 7. Let U be an open set in BCn and let F be a differentiable function from U to BC. Then F = u + jv is bicomplex holomorphic if and only if: 1. u and v are complex holomorphic in zi and wi for every i = 1, . . . , n 2. ∂zi u = ∂wi v and ∂zi v = −∂wi u on U . We immediately have: Proposition 8. Let U ⊆ BCn be an open set and let F = u + jv be a differentiable function from U to BC. Then F is bicomplex holomorphic in (Z1 , . . . , Zn ) if and only if F is Zi∗ , Z˜i , and Zi† -regular for all i = 1, . . . , n. The generalization in several variables of theorem 3 holds in an obvious way, and therefore we can conclude that holomorphic functions in several bicomplex variables form a sheaf, and that (like in the one variable case) they do not allow isolated (or compact) singularities. If one now defines the sheaf of bicomplex hyperfunctions on Rn to be, for any open set U ∈ Rn , BBC (U ) := H 3n (BCn , BCn \ U ; H). then one obtains a perfectly consistent theory, that allows the reconstruction of most natural results for classical hyperfunctions. Since hyperfunctions (in all three settings) are sheaves, one has a well-defined notion of support. In both the complex and the quaternionic case there is then a beautiful characterization of compactly supported hyperfunctions in terms of a duality theorem which, in its early original (and one-dimensional) formulation, was due to Köthe, and in the most general (and n-dimensional) formulation is known as the K¨ othe-Martineau-Grothendieck duality theorem for several complex variables. The beauty of the result in the complex case rests on the fact that one can represent hyperfunctions supported by a compact set K as analytic functionals on that same compact set. In other words, the n-th relative cohomology group of O with support in K is isomorphic to the dual of the space of germs of holomorphic functions on K. In symbols:

272


Theorem 9. If K is a compact set in Cn , then H n (Cn ; O) ∼ = (O(K)) . K

The quaternionic case introduces an interesting variation, in that the theorem cannot be stated in this same elegant form, because instead of germs of regular functions, one has to resort to germs of functions in a different sheaf (a sort of dual sheaf of R which we usually denote by S), which only coincides with R when n = 1, 2. The reasons for this discrepancy are quite deep and related to the nature of the version of the Dolbeault resolution that one needs to employ in the quaternionic case. This is quite beyond the scope of this paper, and we refer the reader to [12] as well as to [30]. The situation in the bicomplex case returns to normality, in the sense that one can still employ the same sheaf both in the definition of hyperfunctions, and in the characterization of those which have compact support. Specifically, one can prove the following result, [9]: Theorem 10. If K is a compact set in BCn , then H 3n (BCn ; H) ∼ = (H(K)) . K

The situation we have described can therefore be summed up nicely in the following table. Cn n

Hn Codimension of the Initial Variety 2n − 1 Duality

n HK (Cn ; O) ∼ = (O(K))

2n−1 HK (Hn ; R) ∼ = (S(K))

BCn 3n

3n HK (BCn ; H) ∼ = (H(K))

References [1] W.W. Adams, C.A. Berenstein, P. Loustaunau, I. Sabadini, D.C. Struppa, Regular functions of several quaternionic variables and the Cauchy–Fueter complex, J. Geom. Anal. 9 no.1 (1999), 1-15. [2] W.W. Adams, P. Loustaunau, V.P. Palamodov, D.C. Struppa, Hartogs’ phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring, Ann. Inst. Fourier 47 (1997), 623-640. [3] C.A. Berenstein, I. Sabadini, D.C. Struppa, Boundary values of regular functions of quaternionic variables, Pitman Res. Notes in Math., Vol. 347 (1996), 220-232. [4] K.S. Charak, D. Rochon, N. Sharma, Normal Families of Bicomplex Holomorphic Functions, Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society, Vol: 17, Issue: 3 (2009), pp. 257–268. [5] F. Colombo, G. Gentili, I. Sabadini, A Cauchy kernel for slice regular functions, Ann. Global Anal. Geom. 37 (2010), 361-378.


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[6] F. Colombo, P. Loustaunau, I. Sabadini, D.C. Struppa, Regular functions of biquaternionic variables and Maxwell’s equations, J. Geom. Phys., 26 (1998), 183-201. [7] F. Colombo, I. Sabadini, A structure formula for slice-monogenic functions and some of its consequences, Trends in Math., Birkh¨ auser, Basel, 2009, 101-114. [8] F. Colombo, I. Sabadini, D.C. Struppa, A. Vajiac, M. Vajiac, Singularities of functions of one and several bicomplex variables, to appear in Ark. f¨ ur Mat., 2011. [9] F. Colombo, I. Sabadini, D.C. Struppa, A. Vajiac, M. Vajiac, Bicomplex Hyperfunction Theory, to appear in Ann. Mat. Pura Appl., 2010. [10] F. Colombo, A. Damiano, I. Sabadini, D.C. Struppa, Quaternionic hyperfunctions on five-dimensional varieties in H2 , J. Geom. Anal. 17, no.3 (2007), 435-454. [11] F. Colombo, G. Gentili, I. Sabadini, D.C. Struppa, Extension results for slice regular functions of a quaternionic variable, Adv. Math. 222, 1793–1808, 2009. [12] F. Colombo, I. Sabadini, F. Sommen, D.C. Struppa, Analysis of Dirac Systems and Computational Algebra, Birkh¨ auser (2004). [13] L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley-Interscience, New York (1970). ¨ [14] R. Fueter, Uber Hartogs’schen Satz, Comm. Math. Helv. 12 (1939), 75-80. ¨ [15] R. Fueter, Uber einen Hartogs’schen Satz in der Theorie der analytischer Funktionen von n Komplexen Variables, Comm. Math. Helv. 14 (1942), 394-400. [16] G. Gentili, C. Stoppato, D.C. Struppa, Regular functions of a quaternionic variable, book, draft. [17] G. Gentili, D.C. Struppa, A new approach to Cullen-regular functions of a quaternionic variable, C.R. Math. Acad. Sci. Paris, 342(10), 741–744, 2006. [18] G. Gentili, D.C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math. 216(1), 279–301, 2007. [19] R. Ghiloni, A. Perotti, Slice regular functions on real associative algebras, Preprint, 2009. [20] G. Kato, D.C. Struppa, Fundamentals of Algebraic Microlocal Analysis, Marcel Dekker, New York, 1999. [21] S. Krantz, Theory of Several Complex Variables, Wadsworth and Brooks/Cole Advanced Books and Softward, Belmont, California, 1992. [22] V.V. Kravchenko, M.V. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, Pitman Res. Notes in Math. Series, Vol. 351, 1996. [23] D. Napoletani, D.C. Struppa, On a large class of supports for quaternionic hyperfunctions in one variable, Pitman Res. Notes in Math. Series, Vol. 394 (1999), 170-175. [24] D. Pertici, Funzioni regolari di piu’ variabili quaternioniche, Ann. Mat. Pura e Appl. serie IV, CLI (1988), 39-65. [25] G.B. Price, An Introduction to Multicomplex Spaces and Functions, Marcel Dekker, New York, 1991. [26] R. Rocha-Chavez, M.V. Shapiro, F. Sommen, Integral Theorems for Functions and Differential Forms in Cm , Chapman and Hall/CRC Res. Notes in Math. Series, Vol. 428 (2002).

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[27] D. Rochon, M.V. Shapiro, On algebraic properties of bicomplex and hyperbolilc numbers, Anal. Univ. Oradea, fasc. math., vol. 11 (2004). [28] D. Rochon, On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schr¨ odinger equation, Complex Variables and Elliptic Equations, Vol. 53, Issue 6 (June 2008), pp. 501–521. [29] J. Ryan, Complexified Clifford analysis, Complex Variables and Elliptic Equations, Vol. 1, Issue 1 (September 1982), pp. 119–149. [30] I. Sabadini, F. Sommen, D.C. Struppa, The Dirac complex on abstract vector variables: megaforms, Exp. Math. 12 (2003), 351-364. [31] B. Schuler, Zur Theorie der regul¨ aren Funktionen einer Quaternionen-variablen, Comm. Math. Helv. (10), 327–342, 1937. Daniele C. Struppa, Adrian Vajiac and Mihaela B. Vajiac Chapman University Schmid College of Science Orange, CA 92866 USA e-mail: [email protected] [email protected] [email protected]

The Gauss-Lucas Theorem for Regular Quaternionic Polynomials Fabio Vlacci Abstract. In this paper we extend the classical Gauss-Lucas Theorem from the setting of complex polynomials to the case of regular polynomials of quaternionic variable. Mathematics Subject Classification (2010). 30G35, 26C10, 30C15. Keywords. Zeroes and critical points of quaternionic polynomials.

1. Introduction After Hamilton’s discovery of quaternions, the strange new phenomena which take place in this non-commutative algebra attracted the interests of many mathematicians. For instance, in the study of quaternions from the algebraic point of view, it immediately turned out that, in general, the Fundamental Theorem of Algebra fails to be valid for quaternionic polynomials, as shown in the following example: Example 1.1. For any n ∈ N and for any quaternion q, the polynomial aq n − q n a + 1 (with coefficient the quaternion a) has real part identically equal to 1. From the analytic point of view, the richness of the theory of holomorphic functions of one complex variable, along with motivations from physics, aroused interest in a theory of quaternion-valued functions of a quaternionic variable. In fact, several interesting theories have been introduced in the last century. The most famous is the one due to Fueter (see [5, 6]) of the mid 1930s; the basic results of this theory are accurately summarized in [22]. Recent work on Fueter-regularity includes [2, 15] and references therein. In the same years (see [17, 18]) there are the This work is partially supported by G.N.S.A.G.A. of the I.N.D.A.M. and by M.I.U.R..


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first attempts to find techniques to calculate roots of a class of quaternionic polynomials whereas an extensive description of the algebraic properties of polynomials in a non commutative setting can be found in [16]. A different theory of quaternion-valued functions of one quaternionic variable has been recently proposed by G. Gentili and D. C. Struppa (see [11, 12]). The theory is based on a definition of regularity for quaternionic functions inspired by C. G. Cullen (see [4]). Several basic results of the theory are proven in [12], including the Cullen-regularity of quaternion power series and some nice properties of their zeroes. The study of the zero-sets of regular funcions has been further developed in [9, 24]; in this paper we apply specific techniques from the theory of regular functions and establish a generalization of the Gauss-Lucas Theorem for quaternionic polynomials.

2. Basic preliminary results for complex polynomials In this section we will shortly review well-known results1 for polynomials whose coefficients belong to C (the field of complex numbers) usually called complex polynomials. If p is a complex polynomial, the set of all zeroes of p will be denoted by Zp and also called the zero set of p. If p is considered as the holomorphic function z → p(z), then we define p to be the (complex) derivative of p. We recall that, by the celebrated Fundamental Theorem of Algebra, any non-constant complex polynomial has a root in C and that any zero of the (complex) polynomial p is called critical point of p. The following geometric definition turns out to be very useful. Definition 2.1. Given a subset U of C, we denote by K(U ) the intersection of all convex sets which contain U . Clearly the set K(U ) is convex. It is also called the convex hull of U . A geometrical description of critical points in terms of roots of a complex polynomial p is contained in the famous Theorem 2.2 (Gauss–Lucas). Let p be a complex polynomial. The convex hull K(Zp ) of the zero-set of p also contains the critical points of p; in particular, K(Zp ) ⊆ K(Zp ). The assertion of the previous theorem is equivalent to Proposition 2.3. Every critical point ξ of the complex polynomial p is a convex k I (z − zj )mj , then linear combination of all zeroes of p, namely, if p(z) = c j=1

ξ=

k

λj zj

j=1 1 We refer the interested reader to [20] for a more detailed introduction to the subject and for the proofs of the results stated in this section.

Gauss-Lucas Theorem for Regular Quaternionic Polynomials with λj ∈ R, λj ≥ 0 ∀j = 1, . . . , k such that

k

277

λj = 1.

j=1

One of the advantages of Proposition 2.3 (compared to Theorem 2.2) is that it enlights the role played by the zeroes of p for the determination of critical points of p.

3. The Gauss–Lucas Theorem for regular polynomials in H Denote by S the two-dimensional sphere of quaternion imaginary units: S = {q ∈ H : q 2 = −1}. For any imaginary unit I ∈ S, let LI = R + IR be the complex line through 0, 1 and I. Any quaternion q which is not real uniquely determines two real numbers x and y (with y > 0) and an imaginary unit I such that q = x + Iy. Definition 3.1. Let Ω ⊆ H be a domain in H; we say that Ω is • a slice domain if Ω ∩ R is non-empty and if LI ∩ Ω is a domain in LI for all I ∈ S; • an axially symmetric domain if, for all x + Iy ∈ Ω, the whole 2-sphere x + Sy is contained in Ω. If V is a subset of H, then the set V = x + Sy x+Iy∈V

is called the (axially) symmetric completion of V . Clearly the unit ball B ⊂ H is the first example of an axially symmetric slice domain and, in general, the axially symmetric slice domains turn out to be the natural domains of definition for an interesting class of functions which are currently under investigation and known as slice-regular or simply regular functions (see [3, 12]). Definition 3.2. Let Ω be a domain in H. Given f : Ω → H, then f is said to be slice-regular if, for all I ∈ S, its restriction fI along LI has continuous partial derivatives and the function ∂¯I f : Ω ∩ LI → H defined by

∂ 1 ∂ ¯ +I fI (x + Iy) ∂I f (x + Iy) = (1) 2 ∂x ∂y vanishes identically. With the notations ΩI = Ω ∩ LI and fI = f|ΩI , we may refer to the vanishing of ∂¯I f by saying that the restriction fI is holomorphic on ΩI . From now on we will refer to these functions

simply just as regular functions. As observed in [12], a quaternion power series q n an with an ∈ H defines a regular function in its n∈N

domain of convergence, which proves to be an open ball B(0, R) = {q ∈ H : |q| < R}. In the same paper, it is also proven that

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Theorem 3.3. A function f : B = B(0, R) → H is regular if and only if there exist quaternions an ∈ H such that q n an (2) f (q) = n∈N

for all q ∈ B. In particular, if f is regular, then f ∈ C ∞ (B). The zero-set of a regular function is a very interesting geometric object (see [3, 9, 12]). Theorem 3.4 (Structure of the Zero Set). Let f : B(0, R) → H be a regular function and suppose f does not vanish identically. Then the zero set of f consists of isolated points or isolated 2-spheres of the form S = x + yS, for x, y ∈ R. In particular, the set of zeroes of a regular function whose coefficients are all in LI are either sphericals or they are isolated points contained in LI . The fact that a regular function f has all the coefficients in LI has the important geometrical meaning that f maps LI into itself; if the same happens for at least two different slices LI and LJ , then necessarily all the coefficients of f are real and hence f maps any slice LI into itself (see [12]). Definition 3.5. Let f (q) =

+∞

q n an and g(q) =

n=0

+∞

q n bn be given quaternionic

n=0

power series with radii of convergence greater than R. We define the regular prod+∞ n

n

q cn , whose coefficients cn = ak bn−k uct of f and g as the series f ∗ g(q) = n=0

k=0

are obtained by discrete convolution from the coefficients of f and g. The regular product of f and g has a series expansion with radius of convergence greater than R. It can be easily proven that the regular multiplication ∗ is an associative, non-commutative operation and that (see [9]) f ∗ g(q) = f (q)g(f (q)−1 qf (q)). Theorem 3.6. Let f (q) =

+∞

(3)

q n an be a given quaternionic power series with radius

n=0

of convergence R and let α ∈ B(0, R). Then f (α) = 0 if and only if there exists a quaternionic power series g with radius of convergence R such that f (q) = (q − α) ∗ g(q).

(4)

This result (whose proof can be found in [9]) would of course be uninteresting if the other zeroes of f did not depend on the zeroes of g. Fortunately, this is not the case: the zeroes of a regular product f ∗ g are strongly related with those of f and g, as shown by the following (see [9]) Theorem 3.7 (Zeroes of a regular product). Let f, g be given quaternionic power series with radii of convergence greater than R and let α ∈ B(0, R). Then f ∗g(α) = 0 if and only if f (α) = 0 or f (α) = 0 and g(f (α)−1 αf (α)) = 0.

Gauss-Lucas Theorem for Regular Quaternionic Polynomials Definition 3.8. Let f (q) =

+∞

279

q n an be a given quaternionic power series with

n=0

radius of convergence R. We define the regular conjugate of f as the series f c (q) = +∞

n q a ¯n . n=0

We remark that f c also has radius of convergence R and, in general, if h = f ∗ g, then hc = g c ∗ f c . If we define f s = f ∗ f c = f c ∗ f , then f s also has radius R. Notice furthermore that the coefficients of f s are all real and that if the coefficients of f are all real, then simply f s = f 2 . Something more precise can be actualy proven about the zeroes of f , of f c and of f s (see [9]). Proposition 3.9. Let f be a given quaternionic power series with radius of convergence R and let x, y ∈ R be such that S = x + yS ⊆ B(0, R). The zeroes of f in S are in one-to-one correspondence with those of f c . Proposition 3.10. Let f (q) =

+∞

q n an be a given quaternionic power series with

n=0

radius of convergence R. If α = x0 + I0 y0 (with x0 , y0 ∈ R, I0 ∈ S) is such that f (α) = f (x0 + I0 y0 ) = 0, then f s (x0 + Ly0 ) = 0 for all L ∈ S. Symmetrization allows us to transform any zero into a “spherical” zero and these zeroes can not accumulate unless the regular function is constant: indeed if the spherical zeroes accumulate, then the same happens in each complex line LI and this is impossible for the Identity Principle, unless the regular function is constant. From now on we will focus our attention to the case of regular polynomials of quaternionic variable, i.e., according to Theorem 3.3, polynomials with coefficients in H of the form P (q) = q n an + q n−1 an−1 + . . . + qa1 + a0 . First of all, for regular polynomials the Fundamental Theorem of Algebra holds true (see [14, 17, 18]); furthermore, the following result is proved in [13]. Theorem 3.11. Let P be a regular polynomial of degree m. Then there exist p, m1 , . . . , mp ∈ N, and w1 , . . . , wp ∈ H, generators of the spherical roots of P , so that P (q) = (q 2 − 2qRe(w1 ) + |w1 |2 )m1 · · · (q 2 − 2qRe(wp ) + |wp |2 )mp Q(q),

(5)

where Re(wi ) denotes the real part of wi and Q is a regular polynomial with coefficients in H having only non spherical zeroes. Moreover, if n = m−2(m1 +· · ·+mp ) there exist a constant c ∈ H, t distinct 2-spheres S1 = x1 + y1 S, . . . , St = xt + yt S, t integers n1 , . . . , nt with n1 + · · ·+ nt = n, and (for any i = 1, . . . , t) ni quaternions αij ∈ Si , j = 1, . . . , ni , such that ni t 5 5 Q(q) = [ ∗ ∗ (q − αij )]c i=1 j=1

(6)

280

F. Vlacci

I I where ∗ is the analogous of in the case of ∗-product. Notice that, even though αij ∈ Si is not necessarily a root of the polynomial P , the polynomial P certainly has an isolated zero on the sphere Si = xi + yi S. Definition 3.12. Let P be a regular polynomial with quaternionic coefficients. If x + yS is a spherical root of P , its multiplicity is defined as two times the largest integer m for which it is possible to factor (q 2 − 2qx + (x2 + y 2 ))m from P . On the other hand, we say that α ∈ H has multiplicity k as an (isolated) root for P if, in the factorization (6), there are exactly k quaternions αij which lie on the sphere Sα . Remark 3.13. Since the coefficients of P s are real, then P s |LI is a complex polynomial and maps LI into itself; therefore the Gauss–Lucas Theorem can be applied to the restriction along LI of the symmetrization P s of a polynomial P . Actually, this can be done for any I ∈ S. Notice furthermore that, according to Proposition 3.4 and the definition of symmetrized polynomial, if deg P = n, then the zero-set of P s – when restricted along LI – is a set of (at most) 2n points; these points, when non-real, are symmetric in pairs, with respect to the real axis. And this geometric configuration of the zero-set along LI is exactly the same for the restriction of P s along any LI with I ∈ S. Thus, recalling Proposition 3.10, we have Proposition 3.14. Any critical point of a regular polynomial P with quaternionic coefficients is in (the axially symmetric completion of ) the convex hull K(ZP s ) of the zero set of P s . Proof. Consider P (q) = (q − 2qRe(w1 ) + |w1 | ) 2

2 m1

· · · (q − 2qRe(wp ) + |wp | ) 2

2 mp

ni t 5 5 [ ∗ ∗ (q − αij )]c; i=1 j=1

then, after rearranging indexes, the zero-set of the restriction of P s along LI can be written as ZP s |LI = {w1I , . . . , wpI , w1I , . . . , wpI } ∪ {αI1 , . . . , αIt1 −1 , αI1 , . . . , αIt1 −1 } ∪ {αIt1 , . . . , αIt } where αIs is real for t1 ≤ s ≤ t. Thus, according to Proposition 2.3 applied to P s |LI , any critical point ξ of P s |LI can be written in the following way, ξ=

p t t 1 −1 (λj wjI + λ j wjI ) + (μk αIk + μ k αIk ) + τs αIs j=1

s=t1

k=1

≥ 0 ∀j = 1, . . . , p, μk ≥ 0, μ k ≥ 0 t1

−1 k

(λj + λ j ) + (μk + ∀k = 1, . . . , t1 − 1, τs ≥ 0 ∀s = t1 , . . . , t and such that

with

λj , λ j , μk , μ k , τs

μ k ) +

t

s=t1

∈ R, λj ≥ 0,

λ j

j=1

τs = 1.

k=1

Gauss-Lucas Theorem for Regular Quaternionic Polynomials

281

We can now provide a different, more direct, proof of the previous result in the case of quaternionic polynomials with non-spherical zeroes only. Given such a regular polynomial P with quaternionic coefficients and degree d, then there exist d quaternions αj and a constant c such that P (q) = (q − α1 ) ∗ (q − α2 ) ∗ . . . (q − αd )c; consider the associated polynomial #2 ) ∗ . . . (q − α #d ) P (q) = (q − α1 ) ∗ (q − α where if αk = xk + Ik yk , the d − 1 quaternions α #2 , . . . , α #d are uniquely determined by the conditions α #j = xj + I1 yj

∀j, such that 2 ≤ j ≤ d.

%P of the zero-set ZP of P clearly coincides The axially symmetric completion Z % of the zero-set of P . But since the with the axially symmetric completion Z P regular polynomial P has all the coefficients in LIα1 , the zero-set of P is contained in LIα1 – we recall that P (and hence P) has no spherical zeroes. Thus the set ZP of critical points of P is contained in the convex hull K(ZP ) of the zero-set ZP of P and hence the set ZP of critical points of P is contained in the axially symmetric completion K(Z ) of the convex hull of the zero-set of P . P

References [1] L. Ahlfors, Complex analysis, McGraw-Hill, New York (1979). [2] F. Colombo, I. Sabadini, F. Sommen, D.C. Struppa, Analysis of Dirac Systems and Computational Algebra, Birkh¨ auser, 2004. [3] F. Colombo, G. Gentili, I. Sabadini, D. C. Struppa, Extension results for slice regular functions of a quaternionic variable. Adv. Math. 222 (2009), 1793–1808. [4] C. G. Cullen, An integral theorem for analytic intrinsic functions on quaternions. Duke Math. J. 32 (1965), 139–148. [5] R. Fueter, Die Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Variablen. Comment. Math. Helv. 7 (1934/5), 307–330. ¨ [6] R. Fueter, Uber die analytische Darstellung der regul¨ aren Funktionen einer Quaternionenvariablen. Comment. Math. Helv. 8 (1935/6), 371–378. [7] G. Gentili, C. Stoppato, Power series and analyticity over the quaternions, to appear in Math. Ann. [8] G. Gentili, C. Stoppato, The open mapping theorem for quaternionic regular functions. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5 (2009), 805–815. [9] G. Gentili, C. Stoppato, Zeros of regular functions and polynomials of a quaternionic variable. Michigan Math. J. 56 (2008), 655–667. [10] G. Gentili, C. Stoppato, D. C. Struppa, F. Vlacci, Recent developments for regular functions of a hypercomplex variable. Hypercomplex Analysis, Trends in Mathematics, 165-186, Birkhauser, Basel (2009).

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[11] G. Gentili, D. C. Struppa, A new approach to Cullen-regular functions of a quaternionic variable. C. R. Math. Acad. Sci. Paris 342 (2006), 741–744. [12] G. Gentili, D. C. Struppa, A new theory of regular functions of a quaternionic variable. Adv. Math. 216 (2007), 279–301. [13] G. Gentili, D. C. Struppa, On the multiplicity of the zeroes of polynomials with quaternionic coefficients, Milan J. Math., 76, (2008), 15–25. [14] G. Gentili – D. C. Struppa – F. Vlacci, The Fundamental Theorem of Algebra for Hamilton and Cayley numbers, Math. Z., 259, (2008), 895–902. [15] V.V. Kravchenko, M.V. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, Pitman Research Notes in Mathematics Series, 351, Longman, Harlow, 1996. [16] T. Y. Lam, A first course in noncommutative rings. Graduate Texts in Mathematics, 123, Springer-Verlag, New York, 1991. [17] J. Niven Equations in quaternions, Amer. Math. Monthly 48, (1941), 654–661. [18] J. Niven The roots of a quaternion,Amer. Math. Monthly 49, (1942), 386–388. [19] A. Pogorui, M. V. Shapiro, On the structure of the set of zeros of quaternionic polynomials. Complex Variables Theory Appl. 49 (2004), no. 6, 379–389. [20] Q.I. Rahman, G. Schmeisser, Analytic Theory of Polynomials Claredon Press, Oxford, (2002). [21] C. Stoppato, Poles of regular quaternionic functions, Complex Var. Elliptic Equ. 54 (2009), 1001–1018. [22] A. Sudbery, Quaternionic analysis. Math. Proc. Camb. Phil. Soc. 85 (1979), 199–225. [23] I. Vignozzi, Funzioni Intere sui Quaternioni: Fattorizzazioni degli Zeri, Tesi di Laurea, Universit` a di Firenze (2008). [24] F. Vlacci, The Argument Principle for Quaternionic Regular Functions, to appear in Michigan Math. J. Fabio Vlacci Dipartimento di Matematica “U. Dini” Universit` a di Firenze Viale Morgagni 67/A 50134 Firenze Italy e-mail: [email protected]

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