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Modified Clifford analysis. (English) Zbl 0758.30037
Let \(\Omega\subset\mathbb R^{n+1}\) be an open set, \((x_ 0,\ldots,x_ n)\in\Omega\) and the vector \((u_ 0,\ldots,u_ n)\) a solution of the following generalized Cauchy-Riemann system \[ x_ n\left({\partial u_ 0 \over \partial x_ 0} -{\partial u_ 1 \over \partial x_ 1}-\ldots- {\partial u_ n \over \partial x_ n}\right)+(n-1)u_ n=0, \]
\[ {\partial u_ j \over \partial x_ k} = {\partial u_ k \over \partial x_ j}\text{ for }1\leq j,k\leq n \quad\text{and} \quad {\partial u_ 0 \over \partial x_ k} =- {\partial u_ k \over \partial x_ 0}\text{ for } 1\leq k\leq n. \] This system denoted by \((H_ n)\), is a non-Euclidean version of the well known Riesz system.
The author studies the system \((H_ n)\) in the terminology of Clifford analysis (\(\mathbb R^{n+1}\) is embedded into the Clifford algebra \(C_ n\)) and gives some facts for the solutions of \((H_ n)\) associated to the classical holomorphic functions. He considers compositions of solutions of \((H_ n)\) with Möbius transformations of \((H_ n)\), with Möbius transformations in \(\mathbb R^{n+1}\) and the vector fields mapping the set of solutions of \((H_ n)\), \(n>1\), into itself.
A characterization of the solutions of \((H_ n)\) in analogy to the complex case, where a complex-valued function \(f\) is holomorphic iff the functions \(f\) and \(zf\) are harmonic, is given. Finally, an analogue of the Weierstrass convergence theorem is obtained.

MSC:
30G35 Functions of hypercomplex variables and generalized variables
53A35 Non-Euclidean differential geometry
31C12 Potential theory on Riemannian manifolds and other spaces
58A14 Hodge theory in global analysis
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