A Mean-Value Theorem for Some Eigenfunctions of the Laplace- Beltrami Operator on the Upper-Half Space

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1 Tampereen teknillinen yliopisto. Matematiikan laitos. Tutkimusraportti 97 Tampere University of Technology. Department of Mathematics. Research Report 97 Sirkka-Liisa Eriksson & Heikki Orelma A Mean-Value Theorem for Some Eigenfunctions of the Laplace- Beltrami Operator on the Upper-Half Space Tampereen teknillinen yliopisto. Matematiikan laitos Tampere 2

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3 A Mean-Value Theorem for Some Eigenfunctions of the Laplace-Beltrami Operator on the Upper-Half Space Sirkka-Liisa Eriksson and Heikki Orelma February 6, 2 Tampere University of Technology, Department of Mathematics, P.O. Box 553, 33 Tampere, Finland Abstract In this paper we study a mean-value property for solutions of the eigenvalue equation of the Laplace-Beltrami operator lb h = (n )h with respect to the volume and the surface integral on the Poincaré upper-half space R n+ + = {(x,..., x n ) R n+ : x n > } with the Riemannian metric ds 2 = dx2 +dx2 + +dx2 n. x 2 n Mathematics Subject Classication (2). Primary 3A5, Secondary 3A5, 3F45 Keywords. Laplace-Beltrami operator, mean-value theorem, hypermonogenic function, Möbius transformation, hyperbolic harmonic function Möbius Transformations in R n+ A Möbius transformation of the complex plane is a fractional transformation of one complex variable represented in the form f(z) = az + b cz + d

4 where z is a complex number and the coecients a, b, c, d are complex numbers satisfying ad bc. The theory of Möbius transformations plays an essential role in the geometric function theory. In this section we shall study corresponding transformations in the (n+)-space R n+. More information and results with complete proofs are available in the sources [], [3], [4], [5], [6], [9], [5], [2] and [23]. One of our further interests will be to study Möbius transformations on the hyperbolic upper-half space, which is a direct generalization of the upper-half plane. Following Ahlfors [6] we consider the reection with respect the unit sphere, dened by J : x x x 2 where x 2 = x 2 + x2 + + x2 n. We extend the reection to innity, by dening J() = and J( ) =. The full Möbius group M(R n+ ) is the group of similarities, i.e., the group generated by J and by the mappings x λax + b, where b R n+, λ > and A belongs to the orthogonal group O(n+). One may continue the theory by to studying the preceding mappings. But this way is some sense very complicated and technical since the ring structure does not exists (ch. [6]). We shall proceed more algebraically. An essential tool in the theory of Möbius transformations on the complex plane is the eld structure of the complex plane. Unfortunately corresponding algebraic structure is not available in the (n + )-space R n+ for n 2. Around 9 K. Th. Vahlen noticed that Möbius transformations in higher dimensions can be studied by extending the vector space R n+ to an associative algebra with unity (cf. [22]). The algebra studied by Vahlen is the Cliord algebra. Next we shall briey recall a few basic properties of the Cliord algebra in R n+. The Cliord algebra Cl,n is the free associative algebra generated by the symbols e,..., e n together with the dening relations e i e j + e j e i = 2δ ij for i, j =,..., m. As a vector space the dimension of the Cliord algebra Cl,n is 2 n. A canonical basis is given by e A = e a e ak, 2

5 where A = {a,..., a k } {,..., n} and a <... < a k. In particular e = and e {j} = e j. The (n + )-dimensional Euclidean space R n+ is a subspace of Cl,n under the canonical embedding (x, x,..., x n ) x + n x j e j and thus we may assume that R n+ Cl,n. In some connections it is convenient to dene e =. An element a Cl,n is called a Cliord number and often the algebra Cl,n is called the algebra of Cliord numbers. Elements u = n j= u je j Cl,n are called vectors. Thus we see that an element x of R n+ is x = x + x j= with x = x e + + x n e n and called a paravector. If n = we have the following simple example. The Cliord algebra Cl, is spanned by the element and e where e 2 =. Thus Cl, = C and complex numbers z = x + iy are actually paravectors in the Cliord approach. The Cliord algebra has several involutions which generalize complex conjugation. Let a, b Cl,n. The main involution is dened by (ab) = a b and e j = e j. The reversion is dened by (ab) = b a and e j = e j. The conjugation is dened by a = (a ) = (a ). For vectors x = x = x and x = x. Also, x 2 = xx = x 2 x2 n. Thus we may compute xx = (x + x)(x x) = x 2 + x x 2 n for x R n+. The Euclidean norm is just x 2 = xx = xx. The major diculty dealing with Cliord numbers is that Cl,n is not a division algebra. This diculty can be avoided by restricting much of the attention to the multiplicative subgroup Γ n generated by non-zero paravectors. The group Γ n is founded by Rudolf Lipschitz and it is sometimes called the Lipschitz group or the Cliord group. Following Vahlen (cf. [3] or [22]) we give the next denition. Denition. A matrix ( ) a b g = c d is a Vahlen matrix if it satises the following conditions: 3

6 (a) a, b, c, d Γ n {}, (b) ad bc R, (c) ab, cd, c a, d b R n+. The expression (g) = ad bc is known as a pseudo-determinant of a Vahlen matrix g. As long it is real it has the multiplicative property (g g 2 ) = (g ) (g 2 ). Next we study the connection brterrn Vahlen matrices and Möbius transformations. Theorem.2 (Vahlen-Maaÿ, [3]) The Vahlen matrices form a group GL 2 (Γ n ) under matrix multiplication. Each g GL 2 (Γ n ) induces a orientation preserving Möbius transformation T g M(R n+ ) given by with the property T g (x) = (ac + b)(cx + d) T g g 2 = T g T g2 for each g, g 2 GL 2 (Γ n ). Conversely, every T M(R n+ ) is induced by a pair of Vahlen matrices ±g. Hence we see that the Cliord formalism with the Vahlen matrices give us a really handy way to study Möbius transformation in higher dimensions. Next we study the following important special cases. Lemma.3 ([23]) A Möbius transformation T M(R n+ ) is a composition of the following transformations: () a translation T g : x x + µ, µ R n+ induced by the Vahlen matrix ( ) µ g = (2) an inversion T g2 : x x induced by the Vahlen matrix ( ) g 2 = (3) a dilatation T g3 : x λ 2 x, for λ > induced by the Vahlen matrix ( ) λ g 3 = /λ (4) an identity transformation T g4 : x x induced by the Vahlen matrix ( ) g 4 = 4

7 (5) a (special) orthogonal transformation T g5 : x axa, a Γ b and a = induced by the Vahlen matrix ( ) a g 5 = a (6) reection T g6 : x x induced by the Vahlen matrix ( ) g 6 = As we see from the precedin, the higher dimensional generalization of Möbius transformations in higher dimensions have many similar properties than the special case on the complex plane. We may always obtain the case choosing n = in above discussion. Thus there are many mapping properties which are not written in above, but are available in the references in above. In the next section we shall consider Möbius transformations in the Poincaré half-space. 2 The Laplace-Beltrami Operator on the Poincaré Half-Space In this section we study function theoretical questions concerned with the Laplace-Beltrami operator, that is a direct generalization from the classical case on the Poincaré upper half-plane (cf. [2]). Especially we are interested to study some eigenvalue problem for it. First we recall briey what is the Poincaré upper-half space, dene the Laplace- Beltrami operator and hyperbolic harmonic functions. Then we discuss what is the relation between the Poincare upper-half space, Möbius transformations and hyperbolic harmonic functions. Let us denote R n+ + = {x = x + x e + + x n e n : x n > }. The Poincare half-space is a Riemannian manifold (R n+ +, ds2 ) where the Riemannian metric is ds 2 = dx2 + dx2 + + dx2 n x 2. n The Laplace-Beltrami operator on the Poincare upper-half space is the operator (details are available for example in [6]) lb f = x 2 n f (n )x n f x n where f : Ω R is a smooth enough function dened on an open subset Ω of R n+ + and = The null solutions of the x 2 x 2 n 5

8 Laplace-Beltrami operator are called hyperbolic harmonic functions. We shall study Möbius transformations in the Poincare upper-half space. The only problem is that if we take a Möbius transformation T g and an element x R n+ + but does T g (x) R n+ +? The answer is given in the following theorem. Theorem 2. ([3], Corollary 4.2) Every g SL 2 (Γ n ) induces a Möbius transformation T g mapping R n+ + onto itself. Another important property is the following observation that hyperbolic harmonic functions are invariant under Möbius transformations. Theorem 2.2 ([7]) Let Ω be an open subset of the Poincare upperhalf space R n+ +. If f : Ω R is hyperbolic harmonic then f T is hyperbolic harmonic on T (Ω) for each Möbius transformation T. We start to consider Cliord algebra-valued functions. Cliord algebravalued functions are related to hyperbolic harmonic functions in the similar manner as holomorphic functions are related to harmonic functions. The relation is initiated and studied by Heinz Leutwiler in the series of articles [7], [8] and [2] and further studied by the rst author and Leutwiler. Let us consider the Cliord algebra valued function f : Ω Cl,n where Ω R n+ + is an open subset. Since the Cliord algebra Cl,n is generated by the symbols e,..., e n we obtain that then the Cliord algebra Cl,n is generated by the symbols e,..., e n. Thus each a Cl,n may be represented in the form a = b + ce n where b, c Cl,n. We abbreviate P a = b and Qa = c and Q a = (Qa) and P a = (P a). Then we dene the modied Dirac operator by Mf = Df + n Q f x n where D = x + e x + + e n x n is the Dirac operator on R n+. The theory of null-solutions of the modied Dirac operator is called hyperbolic function theory see, e.g., []. The function f : Ω Cl,n is called a hypermonogenic if Mf(x) = for each x Ω. 6

9 Hypermonogenic functions have many nice function theoretic properties for example they have Cauchy-type integral formulas. Also the function x x k, where k Z, is hypermonogenic. Many properties and more references can be found from the review article []. The conjugate of the modied Dirac operator is dened by where D = x e x e n Mf = Df n Q f x n x n. In the hyperbolic function theory we dene hyperbolic harmonic functions f : Ω Cl,n as a solutions of the equation for x Ω. MMf(x) = The next theorem give us a connection between hypermonogenic functions and hyperbolic harmonic functions. Also we may see that the equation in above is really a good generalization for real-valued hyperbolic harmonic functions. Theorem 2.3 ([2]) Let Ω R n+ + be an open subset and let f : Ω Cl,n be a twice dierentiable function. Then P (MMf) = P f n P f x n x n and Q(MMf) = Qf n Qf + (n ) Qf x n x n x 2 n If f is hypermonogenic, then P f satises the equation and Qf satises the equation P f n P f = x n x n Qf n Qf + (n ) Qf x n x n x 2 n =. Thus we see that the Q-part of a hypermonogenic function is a solution of the following eigenvalue problem lb h = (n )h. Similarly than in the case of null solutions we obtain the following theorem.. 7

10 Theorem 2.4 ([7]) Let Ω be an open subset of the Poincare upperhalf space R n+ +. If g : Ω R is a solution of the equation lb h = (n )h then g T is also so a solution on T (Ω) for each Möbius transformation T mapping R n+ + onto itself. In the next section we shall study more detail the structure of the above eigenfunctions. Also we see that the P -part of a hypermonogenic function is a direct generalization of a real-valued hyperbolic harmonic function. For a Cl,n -valued function, especially for the P -part of a hypermonogic function, we obtain the following structure theorem. Theorem 2.5 ([]) Let Ω R n+ + be open and g : Ω Cl,n be a dierentiable function. The following properties are equivalent. (a) g is a solution of the equation g n x n (b) g is smooth and g(a) = ω n+ sinh n R h g =. x n g(x)dσ h (x) for all B h (a, R h ) Ω. In the formula ω n+ denotes the surface area of the (n )-dimensional sphere. (c) g is smooth and g(a) = g(x)dx h (x) V (B h (a, R h )) B h (a,r h ) for all B h (a, R h ) Ω where V (B h (a, R h )) = σ n Rh sinh n tdt is the hyperbolic volume of the ball B h (a, R h ). In the previous theorem B h (a, R h ) is the hyperbolic ball with the center a and the radius R h. In the next section we shall give more detailed description for it. Since R is a canonical subset of Cl,n we obtain the following obvious corollary. Corollary 2.6 The preceding theorem is true also for functions g : Ω R. In the next section we shall give and prove similar theorem for preceding eigenfunctions. 8

11 3 A Mean-Value Theorem for some Eigenfunctions of the Laplace-Beltrami Operator Our aim is the give a detailed proof in the following structure theorem for the eigenfunctions represented in the previous section. First we recall a few basic facts from the geometry. More detailed survey to the topic is available in [4]. In [4] it is shown that the hyperbolic ball with the radius R h and the center a is the euclidean ball with the center c(a, R h ) and the radius R e (a, R h ) where and The n-form B h (a, R n ) = {x R n+ + : x c(a, R h) < R e (a, R h )} c(a, R h ) = a + a e + + a n e n + a n e n cosh R h R e (a, R h ) = a n sinh R h. dσ = n ( ) j e j d x j j= is often very usefull vector valued dierential form on R n+ +, where d x j = dx dx j dx j+ dx n. Let K be an (n + )-dimensional manifold-with-boundary. On the boundary K the form dσ admits the representation dσ = νds where ν is a unit normal vector eld and ds a scalar n-form. The corresponding surface form on the hyperbolic space is dσ h and if dx is the volume form on the Euclidean space then the corresponding hyperbolic form is dx h = dx. More x n+ n detailed introduction to integration and certain dierential forms in the Poincare upper-half space can be found from [4]. = dσ x n n Theorem 3. Let Ω R n+ + be an open subset and let h : Ω Cl,n be a smooth function. The following properties are equivalent. () h is an eigenfunction, i.e, is a solution of lb h(x) = (n )h(x) (2) for x Ω. h(x)dσ h (x) ω n+ ψ(r h ) 9

12 (3) where whenever B(a, R h ) Ω. Rh ψ(r h ) = sinh R h sinh n 2 (t)dt n h(x)dx h ω n+ φ(r h ) B h (a,r h ) where ω n+ is the surface area of the (n + )-unit sphere and Rh φ(r h ) = (n ) cosh R h sinh n 2 (t)dt sinh n R h. whenever B(a, R h ) Ω. The corresponding result in the case n = 2 is already known. Heinz Leutwiler proved the theorem in his paper [9] but he used a little bit dierent methods in his proof. The rst consequence is the following remark. Corollary 3.2 The preceding theorem is true also for functions h : Ω R. The proof of the theorem is based on a sequence of lemmata. First we recall the Cauchy's formula for the Q-part of a hypermonogenic function and other useful results. Proposition 3.3 ([8]) If f is a hypermonogenic function on Ω and K Ω is an oriented (n + )-dimensional manifold-with-boundary. Then for each a K we have Qf(a) = 2n a n n ω n+ K Q(q(x, a)ν(x)f(x))ds(x) where ds is the scalar surface element, ν is the outer unit normal vector eld, and q(x, a) = 2(n ) D x a n x â n = (x a) + (x â) 2 x a n x â n. The kernel in the above integral admits the following expression.

13 Theorem 3.4 ([4]) where q(x, a) = (x τ(a, x)) cosh d h(x, a) a n sinh 2 d h (x, a)e n (2a n x n ) n sinh n+ d h (x, a) τ(a, x) = a + a e + + a n e n + a n cosh d h (x, a)e n. and d h is the distance function with respect to the hyperbolic metric. Also we need the following integration result. We dene rst the modied version of the modied Dirac operator by M n f = Df n x n Q f. Theorem 3.5 ([3]) Let Ω be an open subset of R n+ +. If K Ω is an oriented (n + )-dimensional manifold-with-boundary and g is a smooth Cliord algebra-valued function on Ω, then K P (ν(x)g(x)) ds(x) x n n = K M n g(x) dx x n. The theorem in above allow us to prove the following lemma. Lemma 3.6 Assume that f is hypermonogenic on Ω and B h (a, R h ) Ω. Then ( en ν(x)f(x) ) ds(x) = Qf(x)dx h. B h (a,r h ) Q x n n Proof. Since Q(e n ν(x)f(x)) = P (ν(x)f(x)) we obtain ( en ν(x)f(x) ) P (ν(x)f(x)) Q x n ds(x) = x n ds(x). Using Theorem 3.5 we have ( en ν(x)f(x) Q Since x n n ) ds(x) = B h (a,r h ) Mf(x) = Df(x) + n x n Q f(x) = we obtain M n f(x) = Mf(x) + Q f(x) x n Thus the proof is complete. Also we shall need the following result. P (M n f(x)) dx x n. = Q f(x) x n.

14 Lemma 3.7 ([]) If f is a twise continuously dierentiable function from Ω R n+ + into Cl,n, and B h (e n, R h ) Ω we obtain d ( ) dr h sinh n fdσ h = R h sinh n h fdx h. R h B h (a,r h ) Moreover we need the following theorem, which tell us that if g is an eigenvector then there exists a hypermonogenic function such that g = Qf. The theorem is formulated only for a ball but similar theorem holds also for more general star-shaped sets (cf. [2]). Theorem 3.8 ([2]) Let h : B h (a, R) Cl,n be a solution of the equation lb h(x) = (n )h(x). There exists a hypermonogenic function f : B h (a, R) Cl,n satisfying h = Qf on B h (a, R). Now we may start to give the proof for the Theorem 3.. First we show that the statement () implies (2). Lemma 3.9 Let h : Ω Cl,n be a solution of on Ω and let B h (a, R h ) Ω. Then lb h(x) = (n )h(x) (n )ω n+ sinh R h Rh sinh n 2 (t)dt h(x)dσ h (x) Proof. Let f be a hypermonogenic function satisfying Qf = h on B h (a, R h ). Applying Proposition 3.3 and Theorem 3.4 we obtain ω n+ Qf(a) = 2 n = a n n Q(q(x, a)ν(x)f(x))ds(x) ( (x τ(a, x)) cosh dh (x, a) a n sinh 2 d h (x, a)e ) n Q a n x n sinh n+ ν(x)f(x) ds(x) d h (x, a) Since on the ball B h (a, R h ) the unit normal eld is given by we infer ν(x) = x τ(a, x) R e (a, R h ) ω n+ Qf(a) = ( Re (a, R h )ν(x) cosh d h (x, a) a n sinh 2 d h (x, a)e ) n Q a n x n sinh n+ ν(x)f(x) ds(x) d h (x, a) 2

15 Since ν(x)ν(x) = we obtain ω n+ Qf(a) = R e(a, R h ) cosh R h a n sinh n+ R h sinh n R h Qf(x)dσ h ( en ν(x)f(x) ) Q ds(x). Since R e (a, R h ) = a n sinh R h, by virtue of Lemma 3.6 we have ω n+ Qf(a) = cosh R h sinh n Qf(x)dσ h R h sinh n R h Using Lemma 3.7 and the assumption we have ω n+ Qf(a) = cosh R h sinh n Qf(x)dσ h R h sinh R h d ( n dr h sinh n R h x n n The equation in above give us the dierential equation sinh(r h )g (R h ) + (n ) cosh(r h )g(r h ) = C where C = (n )Qf(a) and g(r h ) = ω n+ sinh n R h The general solution of this equation is g(r h ) = C R h sinh n 2 tdt + C sinh n. (R h ) Since g is a continuous function we have lim g(r h) = Qf(a) R h + and then C =. The proof is complete. We show next that the statement (2) implies (3). Lemma 3. Assume Then where ω n+ sinh R h Rh sinh n 2 (t)dt Qf(x)dσ h. n Qf(x)dx h ω n+ φ(r h ) B h (a,r h ) B h (a,r h ) Qf(x)dσ h ). h(x)dσ h (x). Rh φ(r h ) = (n ) cosh R h sinh n 2 (t)dt sinh n R h. Qf(x)dx h. 3

16 Proof. Using Lemma 3.7 we have n sinh n h(x)dx h = d ( R h dr h sinh n R h B h (a,r h ) By the assumptions n ω n+ sinh n h(x)dx h R h B h (a,r h ) = d ( Rh sinh n 2 tdt) dr h sinh n h(a) R h ( = (n ) cosh R Rh h sinh n 2 tdt) sinh R h sinh n h(a) R h Then n ω n+ (n ) cosh R h Rh sinh n 2 (t)dt sinh n R h and the proof is complete. h(x)dσ h ) B h (a,r h ) We show next that (3) implies (2). First we need to prove the following lemma. Lemma 3. Let T : B h (e n, R h ) B h (a, R h ) be the mapping T (x) = a n x + P a, where a R n+ +. Then T is dieomorphism, and the following transformation rules holds: (a) f(y)dσ h(y) = B h (e n,r h ) f T (x)dσ h (x), (b) f T (x)dσ h(x) = B h (e n,r h ) f(y)dσ h(y), (c) B h (a,r h ) h(y)dy h = B h (e n,r h ) h T (x)dx h, (d) B h (a,r h ) h T (x)dx h = B h (e n,r h ) h(y)dy h. Proof. Let f : B h (a, r) Cl n. In the canonical coordinates h(x)dx h dσ h (y) = dσ(y) y n n = x n n n ( ) j e j dˇy j. j= The j'th coordinate of T is { a n x j + a j, forj =,..., n, y j = a n x n, forj = n. 4

17 Hence the pull-back of dσ h with respect to T is T (fdσ h )(x) = dσ(y) y n n = f T (x) x n n n ( ) j e j dˇx j = f T (x)dσ h (x). j= The second formula follows from the basic properties of pull-back mappins. The proof is complete. That allows us to prove the following lemma. Lemma 3.2 Assume Then n h(x)dx h. ω n+ φ(r h ) B h (a,r h ) ω n+ sinh R h Rh sinh n 2 (t)dt Proof. Using the previous proposition we infer n h(y)dy h ω n+ φ(r h ) B h (a,r h ) n = h T h(x)dx h. ω n+ φ(r h ) B h (e n,r h ) h(x)dσ h (x). Using the polar coordinates we have (cf. SPIVAK, KYYRLEPECK ETAL, SLE-HEINZ implisiiitisesti) Then Since n Rh h T (x)dσ h (x)dt ω n+ φ(r h ) B h (e n,t) ω n+ φ(r h ) (n ) Rh B h (e n,t) h T (x)dσ h (x)dt. Rh φ (R h ) = (n ) sinh R h sinh n 2 (t)dt Using Proposition 3.7 we have ω n+ sinh R h Rh sinh n 2 (t)dt B h (e n,r h ) h T (x)dσ h (x). 5

18 Then using the (a)-part of the preceding proposition we have Rh h(y)dσ ω n+ sinh R h sinh n 2 h (y). (t)dt The proof is complete. Lastly we deduce that (2) implies (). Lemma 3.3 Assume Then for x B h (a, R h ). Proof. Recall Since ω n+ sinh R h Rh sinh n 2 (t)dt lb h(x) = (n )h(x) h(x)dσ h (x). Rh φ(r h ) = (n ) cosh R h sinh n 2 (t)dt sinh n R h. d dr h sinh n R h Rh sinh n 2 (t)dt = sinh n 2 R h φ(r h ) ( Rh sinh n 2 (t)dt using Lemma 3.7 we obtain = d ( sinh n R h dr Rh h sinh n 2 (t)dt ) sinh n h(x)dσ h (x) R h B h (e nr h ) = sinhn 2 R h φ(r h ) ( ) Rh 2 sinh n 2 sinh (t)dt n h(x)dσ h (x) R h sinh n R h + Rh sinh n 2 (t)dt sinh n lb h(x)dx h R h B h (a,r h ) Since (2) and (3) are equivalent we obtain the formula h(x)dσ h (x) = (n ) sinh R Rh h sinh n 2 (t)dt φ(r h ) Then (n )φ(r h ) = ( ) Rh 2 sinh n 2 (t)dt + Rh sinh n 2 (t)dt sinh R h Rh sinh R h sinh 2 R h B h (a,r h ) 6 ) 2 sinh n 2 (t)dt φ(r h ) lb h(x)dx h, B h (a,r h ) B h (a,r h ) h(x)dx h. h(x)dx h

19 that is, sinh R h Rh sinh n 2 (t)dt B h (a,r h ) Since R h is arbitrary we obtain that The proof is complete. References lb h(a) + (n ). ( lb h(x) + (n )h(x))dx h =. [] Ahlfors, L., Cliord numbers and Möbius transformations in R n, Cliord algebras and their applications in mathematical physics (Canterbury, 985), 6775, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 83, Reidel, Dordrecht, 986. [2] Ahlfors, L., Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 973. [3] Ahlfors, L., Möbius transformations in R n expressed through 2 2 matrices of Cliord numbers, Complex Variables Theory Appl. 5 (986), no. 2-4, [4] Ahlfors, L., Möbius transformations and Cliord numbers, Differential geometry and complex analysis, 6573, Springer, Berlin, 985. [5] Ahlfors, L., Old and new in Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (984), 935. [6] Ahlfors, L. Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics. University of Minnesota, School of Mathematics, Minneapolis, Minn., 98. [7] Akin, Ö., Leutwiler, H., On the invariance of the solutions of the Weinstein equation under Möbius transformations, Classical and modern potential theory and applications (Chateau de Bonas, 993), 929, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 43, Kluwer Acad. Publ., Dordrecht, 994. [8] Eriksson, S.-L., Integral formulas for hypermonogenic functions, Bull. Bel. Math. Soc. (24), [9] Eriksson, S.-L., Möbius transformations in several function classes, Cliord algebras and potential theory, 23226, Univ. Joensuu Dept. Math. Rep. Ser., 7, Univ. Joensuu, Joensuu, 24. 7

20 [] Eriksson, S.-L., Leutwiler, H., Introduction to hyperbolic function theory, Cliord Algebras and Inverse Problems (Tampere 28), Tampere Univ. of Tech. Institute of Math. Research Report No. 9 (29), pp. 28. [] Eriksson, S.-L, Leutwiler, H., Hyperbolic harmonic functions and their function theory, Potential Theory and Stochastics in Albac, (29), 85-. [2] Eriksson, S.-L., Leutwiler, H., Hypermonogenic functions, Cliord algebras and their Applications in Mathematical Physics 2, pp Birkhäuser, Boston, 2. [3] Eriksson, S.-L., Leutwiler, H., On hyperbolic function theory, Adv. Appl. Cliord Algebr. 8 (28), no. 3-4, [4] Eriksson, S.-L., Orelma, H., A mean-value theorem for hyperbolic harmonic functions, submitted [5] Gürlebeck, K., Habetha, K., Spröÿig, W., Holomorphic functions in the plane and n-dimensional space, Birkhäuser Verlag, Basel, 28. [6] Leutwiler, H., Appendix: Lecture notes of the course Hyperbolic harmonic functions and their function theory, Cliord algebras and potential theory, 859, Univ. Joensuu Dept. Math. Rep. Ser., 7, Univ. Joensuu, Joensuu, 24. [7] Leutwiler, H., Best constants in the Harnack inequality for the Weinstein equation, Aequationes Math. 34 (987), no. 2-3, [8] Leutwiler, H., Modied Cliord analysis, Complex Variables Theory Appl. 7 (992), no. 3-4, 537. [9] Leutwiler, H., Quaternionic analysis in R 3 versus its hyperbolic modication In F. Brackx et al.(eds.) Cliord Analysis and its Applications, Kluwer, Dordrecht 2, 932. [2] Leutwiler, H., Remarks on modied Cliord analysis, Potential theoryicpt 94 (Kouty, 994), , de Gruyter, Berlin, 996. [2] Lounesto, P., Cliord algebras and spinors, Second edition. London Mathematical Society Lecture Note Series, 286. Cambridge University Press, Cambridge, 2. [22] Vahlen, K., Ueber Bewegungen und complexe Zahlen, Math. Ann. 55 (92), no. 4, [23] Waterman, P., Möbius transformations in several dimensions, Adv. Math. (993), no.,