Riccardo Ghiloni, Alessandro Perotti, Caterina Stoppato
This work looks at the theory of octonionic slice regular functions through the lens of differential topology. It proves a full-fledged version of the Open Mapping Theorem for octonionic slice regular functions. Moreover, it opens the path for a possible use of slice regular functions in the study of almost-complex structures in eight dimensions.
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We study the $L^p$ norm of the orthogonal projection from the space of quaternion valued $L^2$ functions to the closed subspace of slice $L^2$ functions.
In this paper we prove two Bloch type theorems for quaternionic slice regular functions. We first discuss the injective and covering properties of some classes of slice regular functions from slice regular Bloch spaces and slice regular Bergman spaces, respectively. And then we show that there exits a universal ball contained in the image of the open unit ball $\mathbb{B}$ in quaternions $\mathbb{H}$ through the slice regular rotation $\widetilde{f}_{u}$ of each slice regular function $f:\overline{\mathbb{B}}\rightarrow \mathbb{H}$ with $f'(0)=1$ for some $u\in \partial\mathbb{B}$.
The functions studied in the paper are quaternion-valued functions of a quaternionic variable. It is show that the left slice regular functions and right slice regular functions are related by a particular involution. The relation between left slice regular functions, right slice regular functions and intrinsic regular functions is revealed. The classical Laplace transform can be naturally generalized to quaternions in two different ways, which transform a quaternion-valued function of a real variable to a left or right slice regular quaternion-valued function of a quaternionic variable. The u…
We study global properties of quaternionic slice regular functions (also called s-regular) defined on symmetric slice domains. In particular, thanks to new techniques and points of view, we can characterize the property of being one-slice preserving in terms of the projectivization of the vectorial part of the function. We also define a "Hermitian" product on slice regular functions which gives us the possibility to express the $*$-product of two s-regular functions in terms of the scalar product of suitable functions constructed starting from $f$ and $g$. Afterwards we are able to determine, …
Domains of existence of slice regular functions in one quaternionic variable
本文完整刻画了四元数空间中切片正则函数的存在域，证明切片开集为存在域、切片正则域、切片正则凸集与2-路径对称集四者等价，并证明了一个插值定理。
In this paper, we are concerned with the S-polyregularity the regular dot product of slice regular functions. We prove that the product of a slice regular function and right quaternionic polynomial function is a S-polyregular function and we determinate its exact order. The general case of the product of any two slice regular functions is also discussed. In fact, we provide sufficient and necessary conditions to the product of slice regular functions be a S-polyregular function. The obtained results are then extended to the product of S-polyregular functions and remain valid for a special dot …
Two-sided slice regular functions
本文提出两类双侧切片正则函数，证明其在轴对称域上等价，研究其基本性质，定义保持正则性的星积，并建立柯西积分公式。
Growth theorems for slice Dirac-regular mappings over Clifford algebras
本文研究了基于克莱因代数的切Dirac-正则映射，证明了其在特定条件下的消失性质，并建立了相关映射的增长定理。
Paley-Wiener Theorems For Slice Regular Functions
该研究在切片正则环境下证明了两个佩利-维纳定理，据此可计算切片正则佩利-维纳空间的再生核并得到相关抽样定理，为该领域分析提供新工具。
On slice regular Bergman spaces and fiber bundles
本文将纤维丛理论应用于切片正则函数研究，证明了四元数切片正则Bergman空间是坐标球丛的底空间，并从纤维丛视角探讨了其相关性质。
In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato, and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (called \textit{spherical expansion}), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relat…
Zeroes of weakly slice regular functions of several quaternionic variables on non-axially symmetric domains
本研究聚焦非轴对称域下多四元数变量弱切片正则函数的零点，提出路径切片茎函数与新型$*$-乘积，证明其零点构成路径切片解析集，推进了四元数分析相关理论。
We generalize the representation formula from slice-domains of regularity to general Riemann slice-domains. This result allows us to extend the $*$-product of slice regular functions on axially symmetric domains to certain Riemann slice-domains by introducing holomorphic stem systems and tensor holomorphic functions. In particular, we construct a power series expansions of slice regular functions on certain Riemann slice-domains, which implies a relation between stem holomorphic, tensor holomorphic and slice regular functions.
Algebra of slice regular functions on non-symmetric domains in several quaternionic variables
本文针对多四元数变量弱切片正则函数空间，引入保持路径切片性质的*-乘积，定义路径开邻域以确立茎函数全纯性，证明两弱切片正则函数的*-乘积仍具弱切片正则性，建立了该类函数的代数框架。
Resultants of slice regular polynomials in two quaternionic variables
本文为两个四元数变量的切片正则多项式定义了基于Dieudonné行列式的非交换结式，并借此研究多项式公共零点的存在性。
In this paper we study Hankel operators in the quaternionic setting. In particular we prove that they can be exploited to measure the $L^{\infty}$ distance of a slice $L^{\infty}$ function from the space of bounded slice regular functions.
In the article the class of slice regular functions is shown to be closed under a new regular composition. The new regular composition turns out to be globally defined in contrast to the locally defined version by Vlacci. Its advantage over Vlacci's is demonstrated by its associated theory of composition operators and dynamical systems for slice regular functions. Especially, the corresponding Littlewood subordination principle and the Denjoy-Wolff type theorem can be established.
On slice regular fractal-fractional Dirichlet type spaces
该文研究基于切片上截断四元数指数函数的广义分形-分数阶导数诱导的四元数切片正则函数右模族，证明Bergman模与Dirichlet模属于其中一类模族，拓展了四元数分析中的函数空间理论。
Some slice regular functions in several variables and fiber bundles
本文通过纤维丛理论将多复变量全纯函数与多四元数变量切片正则函数联系起来，并利用纤维丛理论论证了多变量切片正则函数的定义域。
Quaternionic slice regularity beyond slice domains
本文通过引入矛形域和铰链域扩展切片域，并利用全纯茎函数族与诱导切片正则函数族的新方法，建立了切片正则函数的半全局与全局延拓定理及表示公式。
Slice-regular functions of a quaternionic variable have been studied extensively in the last 12 years, resulting, in many ways, quite close to classical holomorphic functions of a complex variable; indeed, there is a correspondence between slice-regular functions and a certain family of holomorphic maps from the complex plane to $\mathbb{C}^4$, as noted by Ghiloni and Perotti. However, such a construction does not seem to offer any insight on the behaviour of slice-regular functions, due to the lack of a connection between the values of the holomorphic map and the values of the associated slic…
In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator $\mathcal S_a$ with the decomposition process $$f=e_a\langle f, e_a\rangle+B_{a}*\mathcal S_a f,$$ where $e_a$ denotes the slice normalized Szegö kernel and $ B_a $ the slice Blaschke factor. Iterating the above decomposition process, a corresponding maximal selection principle gives rise to the slice adaptive Fourier decomposition. This leads to a adaptive slice Takenaka-Malmquist orthonormal system.
In this paper we prove a new representation formula for slice regular functions, which shows that the value of a slice regular function $f$ at a point $q=x+yI$ can be recovered by the values of $f$ at the points $q+yJ$ and $q+yK$ for any choice of imaginary units $I, J, K.$ This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy fo…
The purpose of this paper is twofold. One is to enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the octonionic space. As applications, we obtain two Landau-Toeplitz type theorems for slice regular functions with respect to regular diameter and slice diameter respectively, together with a Cauchy type estimate. Along with these results, we introduce some new and useful ideas, which also allow to prove the minimum principle and one version of the open mapping …
Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra $\mathbb{O}$, recently introduced by M. Jin, G. Ren and I. Sabadini. A function $f:Ω_D\subset\mathbb{O}\to\mathbb{O}$ is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra $\mathbb{H}_\mathbb{I}$ of $\mathbb{O}$ generated by a pair $\mathbb{I}=(I,J)$ of orthogonal imaginary units $I$ and $J$ ($\mathbb{H}_\mathbb{I}$ is a `quaternionic slice' of $\mathbb{O}$), the restriction of $f$ to $Ω_D\cap\mathbb{H}_\mathbb{I}…
In this paper we prove the Bohr Theorem for slice regular functions. Following the historical path that led to the proof of the classical Bohr Theorem, we also extend the Borel-Carathéodory Theorem to the new setting.
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.
Due to the invalidation of the Gauss-Lucas type result for quaternionic polynomials, we first give in this paper an alternative proof of the Bernstein inequality in $L^{p} (1\leq p \leq+\infty)$ for slice regular polynomials by the Fejér kernel and the Minkowski inequality. Secondly, we extend a result of Ankeny-Rivlin to the quaternionic setting via the Hopf lemma. By the way, some Turan inequalities are established for slice regular polynomials.
Given a quaternionic slice regular function $f$, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of $f$ with those of its slice derivative $\partial_{c}f$ obtaining a countable family of differential equations satisfied by any slice regular function. The results are proved in all details and are accompanied to several examples. For some of the results, we also give alternative proofs.
Octonionic analysis is becoming eminent due to the role of octonions in the theory of G2 manifold. In this article, a new slice theory is introduced as a generalization of the holomorphic theory of several complex variables to the noncommutative or nonassociative realm. The Bochner-Martinelli formula is established for slice functions of several octonionic variables as well as several quaternionic variables. In this setting, we find the Hartogs phenomena for slice regular functions
A local Cauchy integral formula for slice-regular functions
本文证明了无需轴向对称的四元数空间开子集边界上的切片正则函数柯西型积分公式，并给出局部形式，通过分解为两个轴向单演函数实现证明。
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 funct…
In this paper we show how to construct a regular, non commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results.
Partial slice regularity and Fueter's Theorem in several quaternionic variables
本文扩展了多四元数变量切片分析理论，引入部分球面值与导数概念，并证明了Fueter定理的推广，连接了切片正则函数与轴向单演函数。
Slice analysis is a generalization of the theory of holomorphic functions of one complex variable to quaternions. Among the new phenomena which appear in this context, there is the fact that the convergence domain of $f(q)=Σ_{n\in\mathbb{N}}(q-p)^{*n} a_n$, given by a $σ$-ball $Σ(p,r)$, is not open in $\mathbb{H}$ unless $p\in\mathbb{R}$. This motivates us to investigate, in this article, what is a natural topology for slice regular functions. It turns out that the natural topology is the so-called slice topology, which is different from the Euclidean topology and nicely adapts to the slice st…
Weak slice regular functions on the $n$-dimensional quadratic cone of octonions
本文在八元数 $n$ 维二次锥上采用复平面全纯限制的方法定义了弱切片正则函数，证明了其与基于茎函数的方法在非轴对称域上的差异，并研究了其泰勒展开等性质。
Slice regular functions have been extensively studied over the past decade, but much less is known about their boundary behavior. In this paper, we initiate the study of Julia theory for slice regular functions. More specifically, we establish the quaternionic versions of the Julia lemma, the Julia-Carathéodory theorem, the boundary Schwarz lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball $\mathbb B$ and of the right half-space $\mathbb H^+$. Our quaternionic boundary Schwarz lemma involves a Lie bracket reflecting the non-commutativity of quat…
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…
Bohr phenomena for slice regular functions over Quaternions
本文针对四元数上的切片正则函数，建立了切片星形、近凸函数的玻尔不等式，给出其推广与改进版本，并提供满足实部条件的切片正则函数玻尔不等式精细化形式，且所有结果最优。
Almansi-type decomposition and Fueter-Sce theorem for generalized partial-slice regular functions
本文针对 Clifford 代数上的广义偏切片正则函数，提出了 Almansi 分解的两个类比版本，并获得了 Fueter-Sce 定理的两个增强版本。
In this paper, we define a class of slice mappings of several Clifford variables, and the corresponding slice regular mappings. Furthermore, we establish the growth theorem for slice regular starlike or convex mappings on the unit ball of several slice Clifford variables, as well as on the bounded slice domain which is slice starlike and slice circular.
After their introduction in 2006, quaternionic slice regular functions have mostly been studied over domains that are symmetric with respect to the real axis. This choice was motivated by some foundational results published in 2009, such as the Representation Formula for axially symmetric domains. The present work studies slice regular functions over domains that are not axially symmetric, partly correcting the hypotheses of some previously published results. In particular, this work includes a Local Representation Formula valid without the symmetry hypothesis. Moreover, it determines a class …
We construct a counterexample to a well-known extension theorem for slice regular functions, which motivates us to develop a theory of Riemann slice-domains by introducing a new topology on quaternions. By some paths describing axial symmetry in Riemann slice-domains, we rectify the classical extension formula in the theory of slice regular functions and prove a representation formula over slice-domains of regularity. This proof involves an intertwining relation between imaginary units of quaternions and a fixed matrix corresponding to a complex structure.
A slice regular analogue of the Malmquist-Takenaka system is investigated. It is proved that they form a complete orthonormal system in the quaternionic Hardy spaces of the unit ball. The properties of associated projection operator are studied. Key Words and Phrases: Functions of hypercomplex variables, quaternionic Hardy spaces, series expansions, interpolation, approximation by rational functions, quaternionic Malmquist-Takenaka system
The theory of slice regular functions of a quaternionic variable extends the notion of holomorphic function to the quaternionic setting. This theory, already rich of results, is sometimes surprisingly different from the theory of holomorphic functions of a complex variable. However, several fundamental results in the two environments are similar, even if their proofs for the case of quaternions need new technical tools. In this paper we prove the Landau-Toeplitz Theorem for slice regular functions, in a formulation that involves an appropriate notion of regular $2$-diameter. We then show that …
Along with the development of the theory of slice regular functions over the real algebra of quaternions $\mathbb{H}$ during the last decade, some natural questions arose about slice regular functions on the open unit ball $\mathbb{B}$ in $\mathbb{H}$. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of $\mathbb{B}$ fixing the origin, it establishes two variants of the quaternionic Schwarz-Pick lemma, specialized to maps $\mathbb{B}\to\mathbb{B}$ that are not injective. These results allow a full generalization to quaternio…
In this paper a quaternionic sharp version of the Carathéodory theorem is established for slice regular functions with positive real part, which strengthes a weaken version recently established by D. Alpay et. al. using the Herglotz integral formula. Moreover, the restriction of positive real part can be relaxed so that the theorem becomes the quaternionic version of the Borel-Carathéodory theorem. It turns out that the two theorems are equivalent.
Slice Fueter-regular functions on arbitrary domains in octonions
本文研究八元数任意领域上的一类广义切Fueter正则函数，推广最大模原理，并通过新工具发现茎向量的条件唯一性，揭示切Fueter正则函数与Riemann领域理论的自然联系。
Dunkl approach to slice regular functions
本文通过Dunkl分析与Clifford代数的联系，提出了一种基于全纯函数构造单生成函数的新方法，揭示了切正则函数与Dunkl算子核之间的关系。
Improved and Refined Bohr-Type Inequalities for Slice Regular Functions over Octonions
本文研究八元数上切片正则函数的 Bohr 型不等式，给出其推广、改进与精细化版本，并证明所有结果都是精确的。
Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice regular setting, a framework which includes polynomials and power series of the quaternionic variable. In the first chapters of this work we introduce and discuss the algebra and the analysis of slice regular functions. In addition to offering a self-contained introduction to the theory of slice-regular functions, these chapters also contain a few new results (…
Almansi-type decomposition for slice regular functions of several quaternionic variables
本文提出多四元数变量切片正则函数的Almansi型分解，产生2^n个唯一分解，并由此证明Fueter定理、双调和性及均值与Poisson公式。
The aim of this paper is to prove that a large class of quaternionic slice regular functions result to be (ramified) covering maps. By means of the topological implications of this fact and by providing further topological structures, we are able to give suitable natural conditions for the existence of $k$-th $\star$-roots of a slice regular function. Moreover, we are also able to compute all the solutions which, quite surprisingly, in the most general case, are in number of $k^2$. The last part is devoted to compute the monodromy and to present a technique to compute all the $k^2$ roots start…
A representation formula for slice regular functions over slice-cones in several variables
本文将切片分析推广至偶数维实向量空间余维情形，通过在 $[End(\mathbb{R}^{2n})]^d$ 中定义锥并推广切片拓扑，证明了切片正则函数的表示公式，且适用于四元数、八元数等多种代数。
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 $ψ$ the Borel-Pompieu formula induced by the operator ${}^ψG$ and its consequences in the slice regular function …
The main purpose of this paper is to prove some density results of polynomials in Fock spaces of slice regular functions. The spaces can be of two different kinds since they are equipped with different inner products and contain different functions. We treat both the cases, providing several results, some of them based on constructive methods which make use of the Taylor expansion and of the convolution polynomials. We also prove quantitative estimates in terms of higher order moduli of smoothness and in terms of the best approximation quantities.
According to [5] we define the $*$-exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for $\exp_*(f)$ are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the $*$-exponential of a function is either slice-preserving or $\mathbb{C}_J$-preserving for some $J\in\mathbb{S}$ and show that $\exp_*(f)$ is never-vanishing. Sharp necessary and sufficient conditions are given in order that $\exp_*(f+g)=\exp_*(f)*\exp_*(g)$, finding an exceptional and unexpe…
The celebrated 100-year old Phragmen-Lindelof principle is a far reaching extension of the maximum modulus theorem for holomorphic functions of one complex variable. In some recent papers there has been a resurgence of interest in principles of this type for functions of a hypercomplex variable and for solutions of suitable partial differential equations. In the present article we obtain a Phragmen-Lindelof principle for slice regular functions, a class of quaternion-valued functions of a quaternionic variable which has been recently introduced.
The theory of slice regular functions over the quaternions, introduced by Gentili and Struppa in [5], was born on domains that intersect the real axis. This hypothesis can be overcome using the theory of stem functions introduced by Ghiloni and Perotti ([6]), in the context of real alternative algebras. In this paper I will recall the notion and the main properties of stem functions. After that I will introduce the class of slice regular functions induced by stem functions and, in this set, I will extend the identity principle, the maximum and minimum modulus principles and the open mapping th…
A promising theory of quaternion-valued functions of one quaternionic variable, now called slice regular functions, has been introduced in 2006. The basic examples of slice regular functions are power series centered at 0 on their balls of convergence. Conversely, if f is a slice regular function then it admits at each point of its domain an expansion into power series, where the powers are taken with respect to an appropriately defined multiplication *. However, the information provided by such an expansion is somewhat limited by a fact: if the center p of the series does not lie on the real …
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