The inverse Fueter mapping theorem

In a recent paper the authors have shown how to give an integral representation of the Fueter mapping theoremusing the Cauchy formula for slice monogenic functions. Specifically, given a slice monogenic function f of the form f=α+ωβ(where α, β satisfy the Cauchy-Riemann equations) we represent in integral formthe axially monogenic function f̄=A+ωB (where $A,B$ satisfy the Vekua's system) given by f̄(x)=Δn-1/2f(x) where Δ is the Laplace operator in dimension $n+1$. In this paper we solve the inverse problem: given an axially monogenic function f̄ determine a slice monogenic function f (called Fueter's primitive of f̄ such that f̄=Δn-1/2f(x). We prove an integral representation theorem for f in terms of f̄ 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 Δn-1/2f(x)=f̄ (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.