A hyperbolic Dirac operator and its kernels

作者：Eriksson Sirkka Liisa^*

来源：Complex Variables and Elliptic Equations, 2013, 58(6): 767-781.

DOI：10.1080/17476933.2011.620096

• monogenic
• hypermonogenic
• Dirac operator
• hyperbolic metric
• modified Dirac operator
• 30G35
• 30A05
• 30F45

We consider the hyperbolic generalization of the classical complex analysis to higher dimensions based on the hyperbolic metric . The field of complex numbers is generalized by the associative Clifford algebra Cl-0,Cl-n generated by the anti-commutating elements e(i) satisfying . H. Leutwiler has noticed around 1992 that the power x(0)+x(1)e(1)+%26lt;bold%26gt;%26lt;/bold%26gt;x(n)e(n))(m) is the conjugate gradient of a hyperbolic harmonic function. He started to study these types of functions that include positive and negative powers and elementary functions, defined similarly as in classical complex analysis. Their total Clifford algebra valued generalizations, called hypermonogenic functions, are defined in terms of the modified Dirac operator introduced by H. Leutwiler and the author in 2000. The integral formula for hypermonogenic functions has been proved by the author. In this article we compute the same kernels using the hyperbolic distance. The kernel is surprisingly the shifted Euclidean Cauchy kernel. Using this we are able to prove mean-value properties for hypermonogenic functions and hyperbolic harmonic functions.

• 出版日期2013-6-1