Modified Quaternionic Analysis on the Ball

A natural generalization of the classical complex analysis to higher dimensions is the theory of monogenic functions (see, e.g. Brackx et al. in Clifford analysis, Pitman, Boston, 1982). Unfortunately the powers of the underlying variable are not in this system. But they are in the modified system introduced by the first author in Leutwiler (Complex Var Theory Appl 20:19–51, 1992). Since this modified system lives in the upper half space R+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}_{+}^{3}$$\end{document} it is natural to transplant it—with the so-called Cayley mapping—to the unit ball B(0, 1) in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{3}$$\end{document}. It is this transplanted system which we are going to investigate in this article. It represents the counterpart to the theory of (H)-solutions studied in Leutwiler (Complex Var Theory Appl 20:19–51, 1992).


Introduction
In his paper "Modified Quaternionic Analysis in R 3 " (see [8]) the first author introduced a modification of the classical quaternionic analysis with the aim that the powers Communicated by Roman Lavicka.This article is part of the topical collection "Higher Dimensional Geometric Function Theory and Hypercomplex Analysis" edited by Irene Sabadini, Michael Shapiro and Daniele Struppa.
of the underlying quaternionic variable stay in the system.This new "modified system" is living on the upper half space R 3 + = {(x, y, t) : t > 0} and is defined as follows: For a function f = u + iv + jw we require that Solutions f of this system are called (H )-solutions in [8] and in what follows (see also [4,9,10]).
It is now quite natural to ask: What happens with this system if we transplant it with the Cayley transform from the upper half-space R 3  + to the unit ball B(0, 1) = {(x, y, t) : x 2 + y 2 + t 2 < 1}, i.e. if we apply the mapping The transformation of the system (H ) to the unit ball B(0, 1) -with this mappingwill be achieved as follows: Assign to the function f = u + iv + jw the first order differential form ω = udx − vdy − wdt and check that f satisfies (H ) if and only if ω is harmonic in the sense of W. V. D. Hodge, i.e. dω = 0 and δω = 0, where the invariant operators d and δ (see, e.g., [12] for their definition) are computet with respect to the hyperbolic metric, defined by the Riemannian differential Obviously the last three equations of (H ) are fulfilled if and only if the outer derivative dω vanishes.So only in the first equation of (H ) the metric ds 2 appears.And here in the form δ = * d * , where * denotes the famous Hodge Star (see [12]), which transforms k-forms to (3 − k)-forms, and depends on the underlying metric.
In order to transform the system (H ) it now suffices to transform the metric.But ds 2 from (1.1) transformed with (C) leads to the metric Computing now the operator δ with respect to this metric, the system dω = 0 and δω = 0 reads as follows: It is this system which we are going to study in this article.

Transformation of (H)-Solutions to (W)-Solutions
Let us first recall that the Laplace-Beltrami operator associated with the conformal metric reads as follows (see, e.g., [1], page 55): In case of the metric (1.1), where ρ = 1 t , we thus have L B h = 0, if and only if 3) The Laplace-Beltrami equation in case of the metric (1.2), where ρ = 2 1−r 2 , reads as follows: The last three equations in the system (H ) being the "integrability conditions" there is, at least locally, a real function h such that It is now easy to check that f satisfies (H ), if and only if, h satisfies (2.3).Similarly, if F is a solution of the system (W ), then, locally, there is a real function H such that where H is a solution of (2.4).The two hyperbolic metrics (1.1) and (1.2) being transformed into each other by the Cayley transform (C), the corresponding Laplace-Beltrami Eqs.(2.3) and (2.4) can be viewed as the Laplace-Beltrami equations of the same harmonic differential form with respect to different parameters.There results: Lemma 2.1 Let h be a solution of (2.3).Then H = h • C is a solution of (2.4).Conversely, if H is a solution of (2.4), then h = H • C −1 is a solution of (2.3).Hereby C −1 denotes the inverse mapping of the Cayley transform C.
At this stage it is profitable to make use of the quaternions in order to represent the Cayley transform C. (See the appendix for the definition of the set H of quaternions.) We then have The inverse mapping C −1 then reads as follows: The following theorem now shows how one can transform the (H )-solutions into (W )-solutions.

Möbius Transformations on the Ball
Nowadays Möbius transformations in R 3 are well studied in the context of quaternions (see, e.g., [1,2] or [6]), but we shall mainly rely on Ahlfors [1].For the convenience of the reader we shall however repeat the basic definitions in the appendix.
Our goal in this section is to find the action of the Möbius transformations on the (W ) -solutions.
We start with rotations, i.e. orthogonal mappings R with determinant 1.As is well known (see, e.g., [2]) R can be written elegantly with the help of quaternions as follows: and z = x +iy + jt.Note that the definition of the involution * is given in the appendix.We now have Proof Let F be a (W )-solution.Then F is, as we know, locally representable in the form where H satisfies (2.4).Set Besides the rotations the following mapping, the Möbius transformation (see, Ahlfors [1, p. 25]), is essential in what follows.We remark that the definition of the involution is given in the appendix.
The action of T a on the (W )-solutions is the following one: Lemma 3.2 Let F be a (W )-solution on B(0, 1) and T a the mapping (3.2).Then also Proof We already know that F admits the representation F = grad H , where H is a solution of (2.4).
A straightforward calculation then shows that with H also h(z) = H (T a (z)) is a solution of (2.4).
On the other hand, as a tedious differentiation shows, Hence the right hand side, also without the factor 1 − |a| 2 , is a (W )-solution.
The following result is helpful: Every Möbius transformations T which maps B(0, 1) onto itself admits the representation

respectively
(2) T = T a • R, for another a ∈ B(0, 1), and another R, where R denotes a rotation.
Proof To (1): Set a = T −1 (0) and observe that Therefore T • T −1 a is a Möbius transformation which keeps the origin 0 fixed.Thus, by Ahlfors ( On the other hand we have Theorem 3.5 If the Möbius transformation T has the form (3.3), then T maps B(0, 1) onto itself.
Consequently a ∈ B(0, 1).On the other hand B = −a A and thus B = −a A. This implies that is a (W )-solution on B(0, 1).

Proof Set c = A
|A| and a = −A −1 B. Then |c| = 1 and we claim that a ∈ B(0, 1).Indeed, from AB * = B A * we conclude that AB * (A −1 ) * = B A * (A −1 ) * = B and thus |A| 2 > 0 we get |a| < 1 and therefore a ∈ B(0, 1).Now a and c inserted in Assume now that F is a (W )-solution on B(0, 1).By lemma 3.1, P(z) = c * F(czc * )c is a (W )-solution and thus, by lemma 3.2, so is the function There remains to be seen To (1): To (2): By ( 1) Hereby we used the fact that a = a * and z = z * , due to a ∈ R 3 and z ∈ R 3 .This completes the proof of Theorem 3.6, since the factor |A| 2 doesn't matter.
Remark.This theorem is the counterpart to Theorem 5.1 in the appendix.

Examples
A simple solution of the Laplace-Beltrami Eq. ( 2.3) is the function h(x, y, t) = t 2 .Transplanted to B(0, 1) by the Cayley mapping C one gets the solution of the Eq.(2.4).
Applying now a suitable rotation R one obtains the following positive solution of (2.4): In ( [7], Theorem 3.2) it then has been shown: Every positive solution H of (2.4) admits the unique representation where μ denotes a positive measure on the unit sphere S of B(0, 1).In order to find further solutions of the Eq.(2.4) we set where p k is a homogeneous harmonic polynomial of degree k (in the classical sense).Then Inserted in (2.4) we thus find that, setting x = r 2 , the hypergeometric differential equation with the regular solution F(k, − 1 2 , 2k+3 2 ; x) (see, e.g., [5], page 1045).
Consequently the function where p k is an arbitrary homogeneous harmonic polynomial of degree k, is a solution of the Laplace-Beltrami Eq. (2.4).A basis for the vector space of all homogeneous harmonic polynomials of degree k is given, e.g., in [3].
The singular solution of the Eq.(4.2) is the function (see [5], p. 1046) we have x n .
According to ([5], p. 1042) All together we have: The singular function where p k denotes an arbitrary homogeneous harmonic polynomial of degree k, is a solution of the Laplace-Beltrami Eq. (2.4), which vanishes on the unit sphere S.
The computation of the regular solution F(k, − 1 2 , 2k+3 2 ; x) of (4.2) is more difficult.Let us compute the solution in case k = 1.Starting with multiplying both sides with x and integrating, then multiplying again both sides with x and integrating yields Thus and therefore where a, b, c are arbitrary real numbers.It is not difficult to verify that for k ≥ 2 r 2n , but the summation seems to be a problem.Another set of solutions of the Eq.(2.4) is the following one: Let q k be a homogeneous polynomial of degree k which solves the Eq.(2.3).Then the transformed function is a solution of (2.4).
A basis for the vector space of all homogeneous polynomials of degree k which solve the Eq.(2.3) is given in ([11], Theorem 2.1).
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Appendix: Quaternions and Möbius Transforms
As mentioned in Sect. 3 we mainly rely on Ahlfors Lecture Note [1].
The skew-field H of quaternions q = x + iy + jt + ks, with x, y, t, s ∈ R, (5.1) is defined by the relations The term x is called the real part of q.As usual, the real field R will be identified with the real parts of q.The complex field C ∼ = R 2 will be identified with the set {x + iy ∈ H, x, y ∈ R}.
A dominant role in this paper is played by the quaternions of the form called reduced quaternions.They will be identified with the points of R 3 .We thus look at (5.2) as an element of H or R 3 , as the case may be.
The algebra H has three important involutions, similar to complex conjugation.The first, or main involution, is the isomorphism q → q , defined by q = x −iy − jt + ks.We have (q 1 + q 2 ) = q 1 + q 2 and (q 1 q 2 ) = q 1 q 2 .
The second involution, or reversion, is the anti-isomorphism q → q * , defined by q * = x+iy+ jt−ks.It satisfies the relation (q 1 q 2 ) * = q * 2 q * 1 .The third involution is the composition of the two others and we shall write q = (q ) * = (q * ) = x −iy − jt −ks.It is again an anti-isomorphism q 1 q 2 = q 2 .q 1 .
The second involution yields a convenient mean to check whether or not a quaternion is reduced.Indeed q is reduced ⇔ q * = q.
Note that q • q = |q| 2 , where |q| 2 = x 2 + y 2 + t 2 + s 2 .It is then clear that every non-zero quaternion q is invertible with inverse The quaternions yield an elegant way to represent the Möbius group M(R 3 + ).Recall that by definition (see, e.g., [1]) the Möbius group M(R 3 ) is the group of sensepreserving transformations on R + onto itself.Since, conversely, every element of M(R 3 + ) is representable in this way (in terms of some T ∈ SL 2 (C)), it will be easy to handle these transformations.

Theorem 3 . 4
a rotation and thus T = T a • R. Next we present a characterization of the Möbius transformations which map the unit ball B(0, 1) onto itself.Every Möbius transformation T which maps B(0, 1) onto itself admits the representation T (z) = (Az + B)(B z + A ) −1 , with A, B ∈ H, AB * = B A * and |A| 2 − |B| 2 = 1.(3.3) Proof According to lemma 3.3, T = R • T a , where R is a rotation and T a is the mapping (3.2).By (3.1), R(z) = czc * , where c ∈ H with |c| = 1.From c • c * = |c| 2 = 1 we then get

Theorem 3 . 6
is a rotation, by(3.1), and thus we have T = T a • R. Since T is a composition of two Möbius transformations which map B(0, 1) onto itself it has the same property and that is what we want to prove.The action of the Möbius transformation (3.3) on the (W )-solutions is as follows: Let T be a Möbius transformation on B(0, 1), written in the form (3.3), and F be a (W )-solution.Then also

3 = R 3 ∪
{∞}, generated by the similarities and the reflectionJ : R 3 → R 3 : z → z |z| 2 (J (0) = ∞, J (∞) = 0).By a similarity we mean a transformation of the form z → mz + c, where c ∈ R 3 and m is a matrix of the form m = λU , with λ a positive number and U an orthogonal 3×3 matrix.The group M(R 3 + ) is by definition the subgroup of all those transformations in M(R 3 ) which map the upper half space R 3 + onto itself.From complex analysis one knows that the class M(C) of all Möbius transformations in C ∼ = R 2 is in a one-to-one correspondence with SL 2 (C)\{±E}, where E denotes the identity matrix.Denoting, as above, the points in R 3 byz = x + iy + jt, (x, y, t ∈ R)the action of T ∈ SL 2 (C) can be lifted to R 3 settingT (z) = (az + b)(cz + d) −1 (5.3)where a, b, c, d ∈ C with ad − bc = 1.The fact that, with z an arbitrary reduced quaternion with positive third component, T (z) also has this property, is evident, once it has been checked thatT (z) = [a(x + iy) + b][c(x − iy) + d] + act 2 + jt |c(x + iy) + d| 2 + |c| 2 t 2 ,for all z = x + iy + jt ∈ R 3 + .Thus every T ∈ SL 2 (C) defines a Möbius transformation mapping R 3