K–HYPERMONOGENIC FUNCTIONS

Let Cℓ_n be the (universal) Clifford algebra generated by e₁,…,e_n satisfying e_ie_j + e_je_i = -2δ_ij, i, j = 1,…,n. The Dirac operator is defined by , where e₀ = 1. The modified Dirac operator is introduced by , 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) ∈ Cℓ_n-1. A k + 1-times continuously differentiate function f : Ω → Cℓ_n is called k-hypermonogenic in an open subset Ω of ℝⁿ⁺¹, if M_kf(x) = 0 outside the hyperplane x_n = 0. Note that 0-hypermonogenic are monogenic and n - 1-hypermonogenic functions are hypermonogenic defined by H. Leutwiler and the author. The power function x^m is hypermonogenic. A function f : Ω → Cℓ_n is be called k-hyperbolic harmonic if , where . In the real-valued case, f is k–hyperbolic harmonic, if and only if it satisfies the Laplace-Beltrami equation associated with the hyperbolic metric. We show that f is k-hypermonogenic if and only if both f and xf are k–hyperbolic harmonic. We also prove that a function g is monogenic in the case n odd if and only if locally for a hypermonogenic function f. We prove an integral formula for hypermonogenic functions and also compare hypermonogenic with the holomorphic Cliffordian functions investigated by G. Laville, I. Ramadanoff and L. Pernas.