The $$H^\infty $$ -Functional Calculi for the Quaternionic Fine Structures of Dirac Type

In recent works, various integral representations have been proposed for specific sets of functions. These representations are derived from the Fueter–Sce extension theorem, considering all possible factorizations of the Laplace operator in relation to both the Cauchy–Fueter operator (often referred to as the Dirac operator) and its conjugate. The collection of these function spaces, along with their corresponding functional calculi, are called the quaternionic fine structures within the context of the S-spectrum. In this paper, we utilize these integral representations of functions to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, by leveraging the aforementioned factorization of the Laplace operator, 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.