  Holomorphic functions play a crucial role in operator theory and the Cauchy
formula is a very important tool to define functions of operators. The
Fueter-Sce-Qian extension theorem is a two steps procedure to extend
holomorphic functions to the hyperholomorphic setting. The first step gives the
class of slice hyperholomorphic functions; their Cauchy formula allows to
define the so-called $S$-functional calculus for noncommuting operators based
on the $S$-spectrum. In the second step this extension procedure generates
monogenic functions; the related monogenic functional calculus, based on the
monogenic spectrum, contains the Weyl functional calculus as a particular case.
In this paper we show that the extension operator from slice hyperholomorphic
functions to monogenic functions admits various possible factorizations that
induce different function spaces. The integral representations in such spaces
allows to define the associated functional calculi based on the $S$-spectrum.
The function spaces and the associated functional calculi define the so called
{\em fine structure of the spectral theories on the $S$-spectrum}. Among the
possible fine structures there are the harmonic and poly-harmonic functions and
the associated harmonic and poly-harmonic functional calculi. The study of the
fine structures depends on the dimension considered and in this paper we study
in detail the case of dimension five, and we describe all of them. The
five-dimensional case is of crucial importance because it allows to determine
almost all the function spaces will also appear in dimension greater than five,
but with different orders.
