A new integral formula for the inverse Fueter mapping theorem

In this paper we provide an alternative method to construct the Fueter primitive of an axial monogenic function of degree $k$, which is complementary to the one used in [F. Colombo, I. Sabadini and F. Sommen, The inverse Fueter mapping theorem in integral form using spherical monogenics, Israel Journal of Mathematics, 2012]. As a byproduct, we obtain an explicit description of the kernel of the Fueter mapping. We also apply our method to obtain the Fueter primitives of the Cauchy kernels with singularities on the unit sphere.