  In this paper, we utilize various integral representations derived from the
Fueter-Sce extension theorem, to introduce novel functional calculi tailored
for quaternionic operators of sectorial type. Specifically, due to the
different factorizations of the Laplace opertor with respect to the
Cauchy-Fueter operator and its conjugate, we identify four distinct classes of
functions: Slice hyperholomorphic functions (leading to the $S$-functional
calculus), axially harmonic functions (leading to the $Q$-functional calculus),
axially polyanalytic functions of order $2$ (leading to the $P_2$-functional
calculus), and axially monogenic functions (leading to the $F$-functional
calculus). By applying the respective product rule, we establish the four
different $H^\infty$-versions of these functional calculi.
