The harmonic H^∞-functional calculus based on the S-spectrum

The aim of this paper is to introduce the H1-functional calculus for harmonic functions over the quaternions. More precisely, we give meanring to Df .T / for unbounded sectorial operators T and polynomially growing functions of the form Df , where f is a slice hyperholomorphic function and D = @q0 + e1 @q1 + e2@q2 + e3@q3 is the Cauchy-Fueter operator. The harmonic functional calculus can be viewed as a modification of the well-known S-functional calculus f .T /, with a different resolvent operator. The harmonic H1-functional calculus is defined in two steps. First, for functions with a certain decay property, one can make sense of the bounded operator Df .T / directly via a Cauchy-type formula. In a second step, a regularization procedure is used to extend the functional calculus to polynomially growing functions and consequently unbounded operators Df .T /. The harmonic functional calculus is an important functional calculus of the quaternionic fine structures on the S-spectrum, which arise also in the Clifford setting and they encompass a variety of function spaces and the corresponding functional calculi. These function spaces emerge through all possible factorizations of the second map of the Fueter-Sce extension theorem. This field represents an emerging and expanding research area that serves as a bridge connecting operator theory, harmonic analysis, and hypercomplex analysis.