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.


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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