Heinz Leutwiler

Modified clifford analysis

The following generalization of the Cauchy-Riemann system is being considered . This system, denoted by (H), is closely related to the hyperbolic metric on R n+1 +. Embedding R n+1 + into the Clifford algebra Cn, the powers xk(k∈N) yield solutions of (H). Also, compositions of solution with Mobius transformations in R n+1 + lead to solutions of (H). General rules associating new solutions to given ones are finally being investigated.

Modified Quaternionic Analysis in ℝ 4

This paper deals with classes of solutions of the generalized Cauchy-Riemann system

Hypermonogenic functions and Möbius transformations

\left\{ {\matrix{ {s({{\partial u} \over {\partial x}} - {{\partial v} \over {\partial y}} - {{\partial w} \over {\partial t}} - {{\partial r} \over {\partial s}}) + 2r = 0,} \cr {{{\partial u} \over {\partial y}} = - {{\partial v} \over {\partial x}},{{\partial u} \over {\partial t}} = - {{\partial w} \over {\partial x}},{{\partial u} \over {\partial s}} = - {{\partial r} \over {\partial x}},} \cr {{{\partial v} \over {\partial t}} = {{\partial w} \over {\partial y}},{{\partial v} \over {\partial s}} = {{\partial r} \over {\partial y}},{{\partial w} \over {\partial s}} = {{\partial r} \over {\partial t}},} \cr } } \right.

Contributions to the theory of hypermonogenic functions

where f = u + iv + j w + kr is a C 2-function of the quaternionic variable z = x + iy + jt + ks, defined on an open set Ω ℝ4. The system (1) has been examined in the more general, n-dimensional, setting in [7] and in the 3-dimensional case in [8] and [9].

On the Invariance of the Solutions of the Weinstein Equation under Möbius Transformations

Let Cl n be the (universal) Clifford algebra generated by e 1,…, e n satisfying e i e j +e j ,e i =−2δ ij , i,j=1,…,n. The Dirac operator in Cl n is defined by \( D = \sum\nolimits_{i = 0}^n {{e_i}\frac{\partial }{{\partial {x_i}}}} \), where e0=1. The modified Dirac operator is introduced for \(k \in \mathbb{R}\) By \( {M_K}f = Df + k\frac{{Qf}}{{{x_n}}}\), where ′ is the main involution and Qf is given by the decomposition f(x)=Pf(x)+Qf(x)e n with P f (x), Q f (x) ∈Cl nℒ1. A k+1-times continuously differentiable function f: Ω→Cl n , is called k-hypermonogenic in an open subsetΩof \( {\mathbb{R}^{n + 1}}\), if M k f (x) = 0 outside the hyperplane x n = 0. Note that 0-hypermonogenic functions are monogenic and n−1-hypermonogenic functions are hypermono-genic as defined by the authors in 15. The power function x m is hypermono-genic. The set of k-hypermonogenic functions is a right Cl n−1-module. We state a Cauchy type theorem for k-hypermonogenic functions. We also prove an integral formula for the P-part of an hypermogenic function.

On a quasilinear elliptic equation and a Riemannian metric invariant under Möbius transformations

LetCln be the (universal) Clifford algebra generated bye1, …,en satisfyingeiej+ejei=−2σij,i, j=1, …,n. The Dirac operator inCln is defined by {ie67-1}, wheree0=1. The second author started to develop the theory of functionsf=Σi=0nfiei satisfying the modified Cauchy-Riemann systemxnDf+(n −1)fn=0, calledH-solutions. The power function xm, x=x0+x1e1+…+xnen, in contrast to classical Clifford analysis, is anH-solution. We study an extension ofH-solutions named hypermonogenic functions. They are solutions of the equationxnDf+(n − 1)Q′f=0, where denotes the main involution of the Clifford algebra andQf is given by the decompositionf (x)=Pf (x)+Qf (x)en withPf (x) ,Qf (x) ∈Cln−1. Note that the values of hypermonogenic functions are in the full Clifford algebraCln. Hypermonogenic functions form a rightClnn−1-module. They are closely related to hyperbolic geometry. Locally any hypermonogenic function may be represented as {ie67-2} in terms of some hyperbolic harmonic functionH with values inCln−1. We prove that compositions of hypermonogenic functions with Möbius transformations lead to hypermonogenic functions and discuss related matters.

On Quaternionic Analysis and its Modifications

We study a generalization of the Cauchy-Riemann system in ℝ3, which is based on the hyperbolic metric, and has the property that classical holomorphic functions naturally extend to solutions having their values in ℍ, the algebra of quaternions.

On the Appell Transformation

Let Cℓ n be the (universal) Clifford algebra generated by e 1, …, en , satisfying . The Dirac operator in Cℓ n is defined by , where e 0=1. The modified Dirac operator is introduced in (Eriksson-Bique and Leutwiler 2000, Hypermonogenic functions. In: Clifford Algebras and their Applications in Mathematical Physics, Vol. 2 (Boston: Birkhäuser), pp. 287–302) by Mf=Df+k(Q′f/x n ), where ′ denotes the main involution in Cℓ n and Qf is given by the decomposition with . A continuously differentiable function f:Ω→ Cℓ n is called hypermonogenic in an open subset Ω of , if , for all x ∈Ω. Paravector-valued hypermonogenic functions are called H-solutions, see (Leutwiler, 1992, Modified Clifford analysis, Complex Variables, 17, 153–171). The power function is an H-solution. We give a Cauchy-type formula for H-solutions. Furthermore we derive the equation for the restriction g of the hypermonogenic function f to the hyperplane xn =0. This equation has been considered by Laville, Lehman and Ramadanoff, see (2004, Analytic Cliffordian functions, Annales Academiae Scientiarium Fennicae. Mathematica, 29(2), 251–268; 1998, Holomorphic Cliffordian functions, Advances in Applied Clifford Algebras, 8(2), 323–340) and in a different context by J. Ryan, see (1990, Iterated Dirac operators in Cn , Zeitschrift fur Analysis und ihre Anwendungen, 9(5), 385–401), and others. §Dedicated to Richard Delanghe on the occasion of his 65th birthday.