Slice monogenic functions of a Clifford variable via the 𝑆-functional calculus

In this paper we define a new function theory of slice monogenic functions of a Clifford variable using the S S -functional calculus for Clifford numbers. Previous attempts of such a function theory were obstructed by the fact that Clifford algebras, of sufficiently high order, have zero divisors. The fact that Clifford algebras have zero divisors does not pose any difficulty whatsoever with respect to our approach. The new class of functions introduced in this paper will be called the class of slice monogenic Clifford functions to stress the fact that they are defined on open sets of the Clifford algebra R n \mathbb {R}_n . The methodology can be generalized, for example, to handle the case of noncommuting matrix variables.

A New Approach to Slice Analysis Via Slice Topology

In this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following Dou et al. (A representation formula for slice regular functions over slice-cones in several variables, arXiv:2011.13770, 2020), how this setting allows us to generalize slice analysis to the general case of functions with values in a real left alternative algebra, which includes the case of slice monogenic functions with values in Clifford algebra. Moreover, we further extend slice analysis, in one and several variables, to functions with values in a Euclidean space of even dimension. In this framework, we study the domains of slice regularity, we prove some extension properties and the validity of a Taylor expansion for a slice regular function.

Octonionic Kerzman–Stein Operators

In this paper we consider generalized Hardy spaces in the octonionic setting associated to arbitrary Lipschitz domains where the unit normal field exists almost everywhere. First we discuss some basic properties and explain structural differences to the associative Clifford analysis setting. The non-associativity requires special attention in the definition of an appropriate inner product and hence in the definition of a generalized Szegö projection. Whenever we want to apply classical theorems from reproducing kernel Hilbert spaces we first need to switch to the consideration of real-valued inner products where the Riesz representation theorem holds. Then we introduce a generalization of the dual Cauchy transform for octonionic monogenic functions which represents the adjoint transform with respect to the real-valued inner product $$\langle \cdot , \cdot \rangle _0$$ ⟨ · , · ⟩ 0 together with an associated octonionic Kerzman–Stein operator and related kernel functions. Also in the octonionic setting, the Kerzman–Stein operator that we introduce turns out to be a compact operator. A motivation behind this approach is to find an approximative method to compute the Szegö projection of octonionic monogenic functions offering a possibility to tackle BVP in the octonions without the explicit knowledge of the octonionic Szegö kernel which is extremely difficult to determine in general. We also discuss the particular cases of the octonionic unit ball and the half-space. Finally, we relate our octonionic Kerzman–Stein operator to the Hilbert transform and particularly to the Hilbert–Riesz transform in the half-space case.

An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators

The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators $$(A_1,\ldots ,A_n)$$ ( A 1 , … , A n ) . A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.