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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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