THE INVERSE FUETER MAPPING THEOREM

In a recent paper the authors have shown how to give an integral representation of the Fueter mapping theorem using the Cauchy formula for slice monogenic functions. Specifically, given a slice monogenic function f of the form f = alpha + (omega) under bar beta (where alpha, beta satisfy the Cauchy-Riemann equations) we represent in integral form the axially monogenic function f = A + (omega) under barB (where A, B satisfy the Vekua's system) given by f(x) = Delta n-1/2 f (x) where Delta is the Laplace operator in dimension n + 1. In this paper we solve the inverse problem: given an axially monogenic function f determine a slice monogenic function f (called Fueter's primitive of f) such that f = Delta n-1/2 f (x). We prove an integral representation theorem for f in terms of f which we call the inverse Fueter mapping theorem (in integral form). Such a result is obtained also for regular functions of a quaternionic variable of axial type. The solution f of the equation Delta n-1/2 f(x) = f(x) in the Clifford analysis setting, i.e. the inversion of the classical Fueter mapping theorem, is new in the literature and has some consequences that are now under investigation.