A polyanalytic functional calculus of order 2 on the S-spectrum

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solutions of the Cauchy-Riemann operator in $\mathbb{R}^4$, denoted by $\mathcal{D}$. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operator in four real variables ($\Delta$) to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. $\Delta= \mathcal{\overline{D}} \mathcal{D}$ to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions of $\mathcal{D}^2$. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on the $S$-spectrum. As far as the authors know, this is the first time that a monogenic polyanalytic functional calculus has been taken into consideration.Comment: arXiv admin note: text overlap with arXiv:2205.0816

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

The aim of this paper is to introduce the $H^\infty$-functional calculus for harmonic functions over the quaternions. More precisely, we give meaning to Df(T) for unbounded sectorial operators T and polynomially growing functions of the form Df, where f is a slice hyperholomorphic function and $D=\partial_{q_0}+e_1\partial_{q_1}+e_2\partial_{q_2}+e_3\partial_{q_3}$ 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 $H^\infty$-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