The inverse Fueter mapping theorem for axially monogenic functions of degree k

Abstract

In this paper we first prove an important formula for the fractional Laplacian, and then we use it to invert the Fueter mapping theorem for axially monogenic functions of degree k. In fact, we prove that for every axially monogenic function of degree kf(x)=[A(x0,|x_|)+x_|x_|B(x0,|x_|)]Pk(x_),x∈Rn+1, there exists a holomorphic intrinsic function fk in C such thatf(x)=τk(fk)(x):=(−Δ)k+(n−1)/2(f→k(x)Pk(x_)), where n can be any positive integer, k can be any non-negative integer, f→k is the slice monogenic function induced by fk, and Pk(x_) is an inner spherical monogenic polynomial of degree k. Using the maps τk, k=0,1,2,…, we obtain a decomposition of a monogenic function for any value of the dimension n.