Hyperbolic Extensions of Integral Formulas

Abstract.

The theory of monogenic functions or regular functions is based on Euclidean metric. We consider a function theory in higher dimensions based on hyperbolic metric. The advantage of this theory is that positive and negative powers of hyper complex variables are included to the theory, which is not in the monogenic case. Hence elementary functions can be defined similarly as in classical complex analysis.

Our operator is the modified Dirac operator $M_{k}f = Df +k \frac{Q\prime f}{x_{n}}$, where ′ is the main involution and Qf is given by the decomposition f (x) = Pf(x) + Qf (x) e _n with $Pf (x), Qf (x)\in C\ell_{0,n}$. The functions satisfying M _k f  = 0 are called k-hypermonogenic functions. In case k = n – 1 they are called hypermonogenic functions. The modified Dirac operator may be used to decompose the Laplace Beltrami operator $\Delta f - \frac{k}{x_{n}}\frac{\partial f}{\partial x_{n}}$ connected to the Riemannian metric

$$ds^{2}=x_{n}^{2k/(1-n)} \left(dx_{0}^{2} + \ldots +dx_{n}^{2}\right)$$

.

In this paper we present a new version of the integral formulas for hypermonogenic functions and (1 – n)-hypermonogenic functions valid in whole space. Since the kernels are very natural and the results are valid in the whole space the integral formulas improve the earlier results.