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In this paper, we prove some splitting results for holomorphic functions of a complex variable and for slice regular functions of a quaternionic variable, in both cases on open domains having some specific properties of symmetry. We use these results to show an application to the slice regular Bergman space. [ABSTRACT FROM AUTHOR]

Given a structural set ψ ⊂ H , some papers [M. Shapiro, V. Vasilevski, (Complex Var Theory Appl. 1995;27:17–46 and 1995;27:67–96)] study the quaternionic ψ-Fueter operator calculus, where the Fueter-type operator is ψ D = ∑ i = 0 3 ψ i ∂ ∂ x i . The aim of this paper is to present the function theory and the Bergman theory induced by the ψ-Fueter-type operator: α ψ D [ f ] = α f + ψ D [ f ] , where a is a quaternionic constant and f is a continuously differentiable function. Particularly, this work shows the Stokes and Borel–Pompieu formulas associated to α ψ D and presents several results of the Bergman theory such as the existence of the Bergman kernel, Bergman projection and their conformal covariant properties. [ABSTRACT FROM AUTHOR]

In the hypercomplex analysis, it is well known that the quaternionic slice regular functions are defined on axially symmetric slice domains and that each quaternion is represented in the form x + i y , where x , y ∈ R and i ∈ S 2 but the mapping ((x , y) , i) ↦ x + i y allows to see that any axially symmetric slice domain is the base space of a trivial sphere bundle. Remember that a sphere bundle is a useful concept in topology and it is a fiber bundle in which the fibers are spheres of dimension n. The previous idea is the natural motivation of this work whose purpose is to present a definition of the quaternionic slice regular functions on the total space of some sphere bundles, show two non-trivial examples of these function spaces and establish some important results of this function theory. [ABSTRACT FROM AUTHOR]

This paper presents the global Borel-Pompieu- and the global Cauchy-type integral formulas for the quaternionic slice regular functions using the relationship between this function space and a non-constant coefficient differential operator given by G := ∥ x → ∥ 2 ∂ 0 + x → ∑ k = 1 3 x k ∂ k , according to [González-Cervantes JO. On cauchy integral theorem for quaternionic slice regular functions. Complex Anal Oper Theory. 2019;13(6):2527–2539; Colombo F, González-Cervantes JO, Sabadini I. A non-constant coefficients differential operator associated to slice monogenic functions. Trans Am Math Soc. 2013;365:303–318]. This association allows to show a behavior of the theory of slice regular functions similar to the well known theories of the hypercomplex analysis. [ABSTRACT FROM AUTHOR]

The aim of this work is to show that the operator G, which has been introduced in Colombo et al. (Trans Am Math Soc 365:303–318, 2013) and whose kernel kerG coincides with the set of quaternionic slice regular functions, is a member of a family of operators with similar properties, such that all the members possess the respective versions of Stokes and Cauchy–type integral theorems. As direct consequences, these theorems are obtained for slice regular functions. [ABSTRACT FROM AUTHOR]

We continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paperwe mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitableweights are taken into account. Finally,we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane. [ABSTRACT FROM AUTHOR]

In this paper we begin the study of some important Banach spaces of slice hyperholomorphic functions, namely the Bloch, Besov and weighted Bergman spaces, and we also consider the Dirichlet space, which is a Hilbert space. The importance of these spaces is well known, and thus their study in the framework of slice hyperholomorphic functions is relevant, especially in view of the fact that this class of functions has recently found several applications in operator theory and in Schur analysis. We also discuss the property of invariance of these function spaces with respect to Möbius maps by using a suitable notion of composition. [ABSTRACT FROM AUTHOR]