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Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field $\mathbb H$. In this work we deals with a well-known $(\theta, u)-$hyperholomorphic $\mathbb H-$valued functions class related to elements of the kernel of the Helmholtz operator with a parameter $u \in\mathbb H$, just in the same way as the usual quaternionic analysis is related to the set of the harmonic functions. Given a domain $\Omega\subset\mathbb H\cong \mathbb C^2$, we define and study a Bergman spaces theory for $(\theta, u)-$ hyperholomorphic quaternion-valued functions introduced as elements of the kernel of ${}^{\theta}_{u}\mathcal D [f]= {}^{\theta} \mathcal D [f] + u f$ with $u\in \mathbb H$ defined in $C^1(\Omega, \mathbb H)$, where \[ {}^\theta\mathcal D:= \frac{\partial}{\partial \bar z_1} + ie^{i\theta}\frac{\partial}{\partial z_2}j = \frac{\partial}{\partial \bar z_1} + ie^{i\theta}j\frac{\partial}{\partial \bar z_2},\hspace{0.5cm} \theta\in[0,2\pi). \] Using as a guiding fact that $(\theta, u)-$hyperholomorphic functions includes, as a proper subset, all complex valued holomorphic functions of two complex variables we obtain some assertions for the theory of Bergman spaces and Bergman operators in domains of $\mathbb C^2$, in particular, existence of a reproducing kernel, its projection and their covariant and invariant properties of certain objects.

It is well known the connection between the growth of the Schwarzian with both the univalence [see Beardon and Gehring (Comment Math Helv 55: 50–64, 1980), Nehari (Bull Am Math Soc 55:545–551, 1949), Ovesea (Novi Sad J Math 26(1):69–76, 1996)] and the quasiconformal extension of the function [see Ahlfors and Weill (Proc Am Math Soc 13:975–978, 1962), Osgood (Old and new on the Schwarzian derivative, Quasiconformal mappings and analysis. Springer, New York, 1998)]. This work shows that previous relationships have geometrical interpretations when the Schwarzian is applied on the Bloch space and on the Dirichlet space. These interpretations are given in terms of a family of three-dimensional cones. Even more, these function spaces allow us to obtain Mobius invariant properties related to the norm induced by the Schwarzian among other consequences.

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-...

The quaternionic valued functions of a quaternionic variable, often referred to as slice regular functions has been studied extensively due to the large number of generali\-zed results of the theory of one complex variable, see \cite{cgs,CSS,GSC,GS2,gssbook,gp,gpr,GS} and the references given there. Recently, several global properties of these functions has been found of the study of a differential operator, see \cite{GlobalOp,GP_2, G, GG1,GG2}. Particularly, given a structural set $\psi$ the Borel-Pompieu formula induced by the operator ${}^{\psi}G$ and its consequences in the slice regular function theory were studied in \cite{GG1}. The aim of this paper is to present some global and local properties of a kind of quaternionic generalized slice regular functions. We shall see that the global properties are consequences of the study of the perturbed global-type operator: \begin{align*} {}^{\psi}G_v [f] := {}^{\psi} G [f] -\frac{{\bf x}_{\psi}}{ 2} ({\bf x}_{\psi} v + v {\bf x}_{\psi} ) f , \end{align*} where $v$ is a quaternionic constant and $f$ is a quaternionic-valued continuously differentiable function with domain in $\mathbb H$ since our generalized slice regular function space coincides with $\textrm{Ker} {}^{\psi_{\textrm{st}}}G_v$ associated to an axially symmetric s-domain, where the $\psi_{\textrm{st}}$ is standard structural set. Among the local properties studied in this work are the versions of Splitting Lemma and Representation Theorem that show us a deep relationship between this generalized slice regular function space with a complex generalized analytic function space on each slice.

Several topological methods have been used successfully in the study of the hypercomplex analysis; for example, in the theory of functions of several complex variables (Grauert et al., in: Encyclopaedia of mathematical science, vol. 74, Springer, 1991, Hirzebruch, in: Topological methods in algebraic geometry, Classics in Mathematics. Reprint of the 1978 Edition. Springer, Berlin, 1995, Krantz, in Function theory of several complex variables, 2nd edn. American Mathematical Society, Providence, 2001), in the Clifford analysis (Sabadini et al. in Adv. Appl. Clifford Algebras 24:1131–1143, 2014), and in the theory of slice regular functions (Colombo et al. in Math Nachr 285:949–958, 2012). Particularly, the fiber bundle is one of these topological subjects that had an intensive development in a number of papers (Bernstein and Philips in Sci Am 245(1):122–137, 1981, Bleecker, in: Guage Theory and Variational Principles. Dover Books on physics Dover Books on mathematics. Courier Corporation, North Chelmsford, 2005, Bredon in Topology and geometry, Springer, Berlin, 1913, Cohen in The topology of fiber bundles, Stanford University, Stanford, 1998, Hatcher in Algebraic-Topology, Cambridge University Press, Cambridge, 2002, Husemoller in fibre bundles, 3rd edn, Springer, Berlin, 1993, Steenrod in The topology of fibre bundles, Princeton University Press, Princeton, 1951, Walschap in Metric structures in differential geometry, Springer, New York, 2004, Weatherall in Synthese 193:2389–2425, 2016). The aim of this work is to show how the Splitting Lemma and the Representation Formula intrinsically determine a fiber bundle over the space of quaternionic slice regular functions and as a consequence, several properties of this function space are interpreted in terms of sections, pullbacks and isomorphism of fiber bundles.

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.

The papers \cite{O1,O2} are the first works to apply the theory of fiber bundles in the study of the quaternionic slice regular functions. The main goal of the present work is to extend the results given in \cite{O1}, where the quaternionic right linear space of quaternionic slice regular functions was presented as the base space of a fiber bundle. When the quaternionic right linear space of quaternionic slice regular functions is associated to certain domains then this paper shows that the elements of total space, given in \cite{O1}, are defined from a pair of harmonic functions and a pair of orthogonal vectors. Simplifying the computations presented in \cite{O1}, where each element of the total space is formed by two pair of conjugate harmonic functions and a pair of orthogonal unit vectors. This work also gives some interpretations of the behavior of the zero sets of some quaternionic slice regular polynomials in terms of the theory of fiber bundles.

Some important global properties of the slice regular functions have been obtained from the global operator $$\begin{aligned}G:= \Vert \mathbf {x}\Vert ^2 \partial _0 + \mathbf {x} \sum _{i=1}^3 x_i \partial _i , \end{aligned}$$ such as a global characterization, a global Cauchy integral theorem and a global Borel–Pompeiu formula, see Colombo et al. (Trans Am Math Soc 365:303–318, 2013), Gonzalez Cervantes (Complex Anal Oper Theory 13:2527–2539, 2019) and Gonzalez Cervantes and Gonzalez-Campos (Complex Var Elliptic Equ 65:1–10, 2020, https://doi.org/10.1080/17476933.2020.1738410) [4, 13, 14], respectively. The aim of this work is to show: some relationships between G and the composition operator with the conformal mappings, a conformal covariance property of G along with its interpretations in terms of a covariant functor, all consequences of these facts for the slice regular functions, a Leibnitz rule associated to the operator G and a characterization of the real Components of slice regular functions in terms of a Non-constant Coefficient second order differential equation.

The history of Bergman spaces goes back to the book [4] in the early fifties by S. Bergman, where the first systematic treatment of the subject was given, and since then there have been a lot of papers devoted to this area. Some standard works here are [5] , [8] , [15] , [28] and the references therein, which contain a broad summary and historical notes of the subject, that frees us from referring to missing details. In their recent works Gonzalez-Cervantes, Luna-Elizarraras and Shapiro [11] , [12] , laid the foundations for the generalization of the theory of Bergman spaces induced by Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields by taking advantage on the intimate connections between harmonic vector fields theory and quaternionic analysis for the Moisil-Theodorescu operator (MT-operator for short). A deeper discussion of the last mentioned relation can be found in [1] . On the setting of general bounded domains in R 3 , we extend the aforementioned study in a very natural way to the case of an introduced v-MT-operator for v ∈ R 3 , proving several properties of induced Bergman spaces and some relative results about Stokes and Borel-Pompieu formulas for v-MT-hyperholomorphic functions, i.e., functions which belong to kernel of the v-MT-operator. In addition, we show that this v-MT operator satisfies a conformal co-variant property. Application of all the above allows to study of Bergman type spaces induced by v-Laplacian vector fields theory, which represents the main goal of this paper.

The aim of this paper is to introduce, in the context of slice regular functions, three notions of the quaternionic Schwarz derivative with their the basic properties: the characterization of zeros, the conformal covariant property and the conformal invariant property. These concepts mimic the structure of the complex Schwarz derivative which is a differential operator with several properties and applications. Particularly, one of these operators is curiously related to the generalization of the schwarzian derivative over vector spaces given by Ryan (Ann Pol Math 57:29–44, 1992).

In this paper 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 paper we 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 suitable weights are taken into account. Finally, we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane., Comment: to appear in Advances in Geometry

The goal of this paper is study the univalence property of slice regular functions of quaternion variable. We prove a number of results, among which several sufficient conditions for univalence, an area-type Theorem, and a Bieberbach–de Branges Theorem for a subclass of slice regular functions. We also discuss some geometric and algebraic interpretations of our results in terms of maps from to itself.

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.

This paper has a twofold purpose: on one hand we deepen the study of slice regular functions by studying their behavior with respect to the so-called C-property and anti-C-property. We show that, for any fixed basis of the algebra of quaternions $\mathbb{H}$ any slice regular function decomposes into the sum of four slice regular components each of them satisfying the C-property. Then, we will use these results to show a reproducing property of the Bergman kernels of the second kind.

In this paper we continue the study of important Banach spaces of slice hyperholomorphic functions on the quaternionic unit ball by investigating the BMO- and VMO-spaces of slice hyperholomorphic functions. We discuss in particular conformal invariance and a refined characterization of these spaces in terms of Carleson measures. Finally we show the relations with the Bloch and Dirichlet space and the duality relation with the Hardy space $\mathcal{H}^1(\mathbb{D})$. The importance of these spaces in the classical theory is well known. It is therefore worthwhile to study their slice hyperholomorphic counterparts, in particular because slice hyperholomorphic functions were found to have several applications in operator theory and Schur analysis.

Slice monogenic functions have had a rapid development in the past few years. One of the main properties of such functions is that they allow the definition of a functional calculus, called S S -functional calculus, for (bounded or unbounded) noncommuting operators. In the literature there exist two different definitions of slice monogenic functions that turn out to be equivalent under suitable conditions on the domains on which they are defined. Both the existing definitions are based on the validity of the Cauchy-Riemann equations in a suitable sense. The aim of this paper is to prove that slice monogenic functions belong to the kernel of the global operator defined by G ( x ) := | x _ | 2 ∂ ∂ x 0 + x _ ∑ j = 1 n x j ∂ ∂ x j , G(x):=|\underline {x}|^2\frac {\partial }{\partial x_0} \ + \ \underline {x} \ \sum _{j=1}^n x_j\frac {\partial }{\partial x_j}, where x _ \underline {x} is the 1-vector part of the paravector x = x 0 + x _ x=x_0+\underline {x} and n ∈ N n\in \mathbb {N} . Despite the fact that G G has nonconstant coefficients, we are able to prove that a subclass of functions in the kernel of G G have a Cauchy formula. Moreover, we will study some relations among the three classes of functions and we show that the kernel of the operator G G strictly contains the functions given by the other two definitions.

In this article we deepen the study of the Bergman theory in the slice regular setting by providing a description of the Bergman projection and by considering a subclass of slice regular functions which generate isometric isomorphisms of Bergman spaces. As a tool to show our result, we further study the multiplication of slice regular functions and their composition.

This is a continuation of our work (Gonzalez-Cervantes et al. in On the Bergman theory for solenoidal and irrotational vector fields. I. General theory. Operator theory: advances and applications. Birkhauser, accepted) where for solenoidal and irrotational vector fields theory as well as for the Moisil–Theodoresco quaternionic analysis we introduced the notions of the Bergman space and the Bergman reproducing kernel and studied their main properties. In particular, we described the behavior of the Bergman theory for a given domain whenever the domain is transformed by a conformal map. The formulas obtained hint that the corresponding objects (spaces, operators, etc.) can be characterized as conformally covariant or invariant, and in the present paper we construct a series of categories and functors which allow us to give such characterizations in precise terms.

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 Mobius maps by using a suitable notion of composition.

The goal of this paper is to introduce and study some geometric properties of slice regular functions of quaternion variable like univalence, subordination, starlikeness, convexity and spirallikeness in the unit ball. We prove a number of results, among which an Area-type Theorem, Rogosinski inequality, and a Bieberbach-de Branges Theorem for a subclass of slice regular functions. We also discuss some geometric and algebraic interpretations of our results in terms of maps from $\mathbb R^4$ to itself. As a tool for subordination we define a suitable notion of composition of slice regular functions which is of independent interest.

In this paper we show that the classical Bergman theory admits two possible settings for the class of slice regular functions. Let Ω be a suitable open subset of the space of quaternions ℍ that intersects the real line and let \(\mathbb{S}^{2}\) be the unit sphere of purely imaginary quaternions. Slice regular functions are those functions f:Ω→ℍ whose restriction to the complex planes ℂ(i), for every \(\mathbf{i}\in \mathbb{S}^{2}\), are holomorphic maps. One of their crucial properties is that from the knowledge of the values of f on Ω∩ℂ(i) for some \(\mathbf{i}\in \mathbb{S}^{2}\), one can reconstruct f on the whole Ω by the so called Representation Formula. We will define the so-called slice regular Bergman theory of the first kind. By the Riesz representation theorem we provide a Bergman kernel which is defined on Ω and is a reproducing kernel. In the slice regular Bergman theory of the second kind we use the Representation Formula to define another Bergman kernel; this time the kernel is still defined on Ω but the integral representation of f requires the calculation of the integral only on Ω∩ℂ(i) and the integral does not depend on \(\mathbf{i}\in \mathbb{S}^{2}\).

We discuss the notion of slice monogenic function as well as its variation and we compare the various definitions among them.