On the Application of the Factorized Fueter-Sce Map to the Slice Hyperholomorphic Cauchy Kernel

Abstract The Fueter-Sce theorem is one of the most important results in hypercomplex analysis, providing a two-step procedure for constructing axially monogenic functions starting from holomorphic functions of one variable. In the first step, the so-called slice operator is applied to holomorphic functions of one variable, producing the class of slice hyperholomorphic functions. The second step yields the class of axially monogenic functions by applying the pointwise differential operator $$\Delta _{n+1}^{\frac{n-1}{2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:msubsup> </mml:math> , with n odd, known as the Fueter-Sce map. The significance of the Fueter–Sce theorem also lies in the fact that it induces two spectral theories corresponding to the function classes it generates. Over the years, factorizations of the Fueter-Sce map have been studied to identify the intermediate spaces that arise between slice hyperholomorphic functions and axially monogenic functions. Until now, only certain factorizations of the Fueter–Sce map have been considered. In this paper, our goal is to determine the most general factorizations of the Fueter–Sce map and to apply these differential operators to the Cauchy kernel of slice hyperholomorphic functions.