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## The Cauchy Transform, Potential Theory and Conformal Mapping

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The natural projection on R of a will be called the real part of a, written as R a. Since the Clifford algebra for n 1 is not commutative it must be remarked that in the expression arb the inverse is meant to be put on the right hand side, i.

Moreover a periodicity theorem is needed: let R pq 1, qq1 be the algebra over R pq 1, qq1 with the orthonormal basis e1 ,. In the sequel both the matrix notation and the classical notation for an element of R pq 1, qq1 will be used. An explicit calculation gives the explicit formulae for the automor- phism and the antiautomorphisms, e. As we shall work projectively, this is sufficient for our purposes without making explicit c. For this it is necessary first to describe spheres in R p, q in terms of the projective space over R pq 1, qq1.

It is identified with the ray in R pq 1, qq1 of the form r m ym2 y r 2. Points can be looked upon as spheres with zero radius, and so are identified with rays in the light cone of R pq 1, qq1. A characterisation of the matrices in the Spin group was given in w7, 4x, based on the work of Vahlen w11x for the case p s 0. Notice that all such matrices have pseudodetermi- nant "1.

An important consequence of this characterisation is that cx q d always is a product of vectors, and so, in the Euclidean case, it is either zero or in the Lipschitz group. The same holds for the expression yyc q a. In the sequel we shall only consider the Euclidean case p s 0. Hilbert Modules.

A right module of functions H is a vector space of R 0, n-valued functions on some set which is closed under right pointwise multiplication with any Clifford number i. This inner product induces a real valued inner product R w? If this inner product is positive definite, and H is closed for the induced norm, then H is called a Hilbert module over R 0, n. Notice that multiplication is continuous. Let now H be a Hilbert module of functions on a set V, with inner product w? V The module of monogenic functions in V for which w f, f x V exists, together with this inner product, is the Bergman module for V.

## The Cauchy Transform, Potential Theory and Conformal Mapping (2nd ed.)

It has a reproducing kernel, which is called the Bergman kernel for V. Fix x. This proves the theorem for this case. For any l in the Clifford algebra l a s 0 implies l c s 0. Notice that w F, F x s bab s bc is self-adjoint. Moreover F is the solution of the minimum problem. There is however only one function in H satisfying this equation. For functions f and g which are monogenic in V and continu- ous in V, g f and g g have analogous properties in g V.

Take v fixed and let u s g v.

go According to the expression given in Theorem 1. This leads to a simplification of the numerator. For the denominator it should be remarked that yye n q 1, 1 q xv and ye n u q 1 are in the Lipschitz group or zero. This proves the theorem. As a result the components of the reproducing kernels can be used to define conjugate harmonic functions in the sense of Stein and Weiss.

Indeed, let h be a harmonic function in the appropriate Hilbert space, and apply the kernel for the corresponding module of monogenic functions. The resulting function has as real part h, and its other components are conjugate harmonic functions.

Axler, P. Bourdon, and W. V, Amer. Brackx, R.