arXiv subjects

Caterina Stoppato

Publications and source records attributed to Caterina Stoppato.

At least 19 recordsLinked to original sources

arXiv2026

A manifold Fueter-Sce phenomenon in one hypercomplex variable

Fueter's theorem states, in modern terms, that the Laplacian maps slice-regular quaternionic functions into Fueter-regular functions with axial symmetry. This phenomenon is also present in the Clifford setting, where both slice-monogenic functions and generalized partial-slice monogenic functions are mapped by the Laplacian into monogenic functions with axial symmetry. These results are due, respectively, to Sce and Qian and to Xu and Sabadini. The present work puts the Fueter-Sce phenomenon into context for the wider class of strongly $T$-regular functions. It shows that the phenomenon appears over general associative $*$-algebras. Moreover, the symmetry considered here is multi-axial in a sense introduced by Eelbode. Additionally, but more surprisingly, the phenomenon studied by Fueter, Sce, Xu and Sabadini turns out to be the last step in a multi-step process. A new phenomenon in one hypercomplex variable is therefore discovered.

math.CV↗

arXiv2026

A unified theory of regular functions of a hypercomplex variable

This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises Fueter-regular functions, slice-regular functions and a recently-discovered function class. In the special case of Clifford-valued functions of one paravector variable, it encompasses monogenic functions, slice-monogenic functions, generalized partial-slice monogenic functions, and a variety of function classes not yet considered in literature. For $T$-regular functions over an associative $*$-algebra, this work provides integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle and a Representation Formula. It also proves some foundational results about $T$-regular functions over an alternative but nonassociative $*$-algebra, such as the real algebra of octonions.

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arXiv2025

On a higher-dimensional worm domain and its geometric properties

We construct new $3$-dimensional variants of the classical Diederich-Fornaess worm domain. We show that they are smoothly bounded, pseudoconvex, and have nontrivial Nebenhülle. We also show that their Bergman projections do not preserve the Sobolev space for sufficiently large Sobolev indices.

math.CV↗

arXiv2024

Which Fueter-regular functions are holomorphic?

We provide a classification of Fueter-regular quaternionic functions $f$ in terms of the degree of complex linearity of their real differentials $df$. Quaternionic imaginary units define orthogonal almost-complex structures on the tangent bundle of the quaternionic space by left or right multiplication. Every map of two complex variables that is holomorphic with respect to one of these structures defines a Fueter-regular function. We classify the differential $df$ of a Fueter-regular function $f$, roughly speaking, in terms of how many choices of complex structures make $df$ complex linear. It turns out that, generically, $f$ is not holomorphic with respect to any choice of almost-complex structures. In the special case when it is indeed holomorphic, generically there is a unique choice of almost-complex structures making it holomorphic. The case of holomorphy with respect to several choices of almost-complex structures is limited to conformal real affine transformations or constants.

math.CV↗

arXiv2023

Quaternionic slice regularity beyond slice domains

After Gentili and Struppa introduced in 2006 the theory of quaternionic slice regular function, the theory has focused on functions on the so-called slice domains. The present work defines the class of speared domains, which is a rather broad extension of the class of slice domains, and proves that the theory is extremely interesting on speared domains. A Semi-global Extension Theorem and a Semi-global Representation Formula are proven for slice regular functions on speared domains: they generalize and strengthen some known local properties of slice regular functions on slice domains. A proper subclass of speared domains, called hinged domains, is defined and studied in detail. For slice regular functions on a hinged domain, a Global Extension Theorem and a Global Representation Formula are proven. The new results are based on a novel approach: one can associate to each slice regular function $f:Ω\to\mathbb{H}$ a family of holomorphic stem functions and a family of induced slice regular functions. As we tighten the hypotheses on $Ω$ (from an arbitrary quaternionic domain to a speared domain, to a hinged domain), these families represent $f$ better and better and allow to prove increasingly stronger results.

math.CV↗

arXiv2023

A unified notion of regularity in one hypercomplex variable

We define a very general notion of regularity for functions taking values in an alternative real $*$-algebra. Over Clifford numbers, this notion subsumes the well-established notions of monogenic function and slice-monogenic function. Over quaternions, in addition to subsuming the notions of Fueter-regular function and of slice-regular function, it gives rise to an entirely new theory, which we develop in some detail.

math.CV↗

arXiv2023

Irregularity of the Bergman projection on smooth unbounded worm domains

In this work we consider smooth unbounded worm domains $\mathcal Z_λ$ in $\mathbb C^2$ and show that the Bergman projection, densely defined on the Sobolev spaces $H^{s,p}(\mathcal Z_λ)$, $p\in(1,\infty)$, $s\ge0$, does not extend to a bounded operator $P_λ:H^{s,p}(\mathcal Z_λ)\to H^{s,p}(\mathcal Z_λ)$ when $s>0$ or $p\neq2$. The same irregularity was known in the case of the non-smooth unbounded worm. This improved result shows that the irregularity of the projection is not a consequence of the irregularity of the boundary but instead of the infinite windings of the worm domain.

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arXiv2022

Slice regular functions and orthogonal complex structures over $\mathbb{R}^8$

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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arXiv2020

Geometric function theory over quaternionic slice domains

The theory of quaternionic slice regular functions was introduced in 2006 and successfully developed for about a decade over symmetric slice domains, which appeared to be the natural setting for their study. Some recent articles paved the way for a further development of the theory: namely, the study of slice regular functions on slice domains that are not necessarily symmetric. The present work is a panorama of geometric function theory in this new context, where new phenomena appear. For instance, the nature of the zero sets can be drastically different than in the symmetric case. The work includes differential, algebraic, topological properties, as well as integral and series representations, of slice regular functions over slice domains.

math.CV↗

arXiv2020

Zeros of slice functions and polynomials over dual quaternions

This work studies the zeros of slice functions over the algebra of dual quaternions and it comprises applications to the problem of factorizing motion polynomials. The class of slice functions over an alternative $*$-algebra $A$ was defined by Ghiloni and Perotti in 2011, extending the class of slice regular functions introduced by Gentili and Struppa in 2006. Both classes strictly include the polynomials over $A$. We focus on the case when $A$ is the algebra of dual quaternions $\mathbb{DH}$. The specific properties of this algebra allow a full characterization of the zero sets, which is not available over general alternative $*$-algebras. This characterization sheds some light on the study of motion polynomials over $\mathbb{DH}$, introduced by Hegedüs, Schicho, and Schröcker in 2013 for their relevance in mechanism science.

math.CV↗

arXiv2020

A local representation formula for quaternionic slice regular functions

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 of domains, called simple, having the following property: every slice regular function on a simple domain can be uniquely extended to the symmetric completion of its domain.

math.CV↗

arXiv2020

Bergman kernel and projection on the unbounded worm domain

In this paper we study the Bergman kernel and projection on the unbounded worm domain $$ \mathcal{W}_\infty = \big\{(z_1,z_2)\in\mathbb{C}^2 : \big|z_1-e^{i\log|z_2|^2}\big|^2<1, z_2\neq0\big\}. $$ We first show that the Bergman space of $\mathcal{W}_\infty$ is infinite dimensional. Then we study Bergman kernel $K$ and Bergman projection $\mathcal{P}$ for $\mathcal{W}_\infty$. We prove that $K(z,w)$ extends holomorphically in $z$ (and antiholomorphically in $w$) near each point of the boundary except for a specific subset that we study in detail. By means of an appropriate asymptotic expansion for $K$, we prove that the Bergman projection $\mathcal{P}:W^s\not\to W^s$ if $s>0$ and $\mathcal{P}:L^p\not\to L^p$ if $p\neq2$, where $W^s$ denotes the classic Sobolev space, and $L^p$ the Lebesgue space, respectively, on $\mathcal{W}_\infty$.

math.CV↗

arXiv2020

Completeness on the worm domain and the Müntz-Szász problem for the Bergman space

In this paper we are concerned with the problem of completeness in the Bergman space of the worm domain $\mathcal{W}_μ$ and its truncated version $\mathcal{W}'_μ$. We determine some orthogonal systems and show that they are not complete, while showing that the union of two particular of such systems is complete. In order to prove our completeness result we introduce the Muentz-Szasz problem for the 1-dimensional Bergman space of the disk $\{ζ: |ζ-1|<1\}$ and find a sufficient condition for its solution.

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arXiv2019

Division algebras of slice functions

This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.

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arXiv2019

Minkowski products of unit quaternion sets

The Minkowski product of unit quaternion sets is introduced and analyzed, motivated by the desire to characterize the overall variation of compounded spatial rotations that result from individual rotations subject to known uncertainties in their rotation axes and angles. For a special type of unit quaternion set, the spherical caps of the 3-sphere $S^3$ in $\mathbb{R}^4$, closure under the Minkowski product is achieved. Products of sets characterized by fixing either the rotation axis or rotation angle, and allowing the other to vary over a given domain, are also analyzed. Two methods for visualizing unit quaternion sets and their Minkowski products in $\mathbb{R}^3$ are also discussed, based on stereographic projection and the Lie algebra formulation. Finally, some general principles for identifying Minkowski product boundary points are discussed in the case of full-dimension set operands.

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arXiv2018

Some notions of subharmonicity over the quaternions

This works introduces several notions of subharmonicity for real-valued functions of one quaternionic variable. These notions are related to the theory of slice regular quaternionic functions introduced by Gentili and Struppa in 2006. The interesting properties of these new classes of functions are studied and applied to construct the analogs of Green's functions.

math.CV↗

arXiv2017

Landau's theorem for slice regular functions on the quaternionic unit ball

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 quaternions of two theorems proven by Landau for holomorphic self-maps $f$ of the complex unit disk with $f(0)=0$. Landau had computed, in terms of $a:=|f'(0)|$, a radius $ρ$ such that $f$ is injective at least in the disk $Δ(0,ρ)$ and such that the inclusion $f(Δ(0,ρ))\supseteqΔ(0,ρ^2)$ holds. The analogous result proven here for slice regular functions $\mathbb{B}\to\mathbb{B}$ allows a new approach to the study of Bloch-Landau-type properties of slice regular functions $\mathbb{B}\to\mathbb{H}$.

math.CV↗

arXiv2016

Singularities of slice regular functions over real alternative *-algebras

The main goal of this work is classifying the singularities of slice regular functions over a real alternative *-algebra A. This function theory has been introduced in 2011 as a higher-dimensional generalization of the classical theory of holomorphic complex functions, of the theory of slice regular quaternionic functions launched by Gentili and Struppa in 2006 and of the theory of slice monogenic functions constructed by Colombo, Sabadini and Struppa since 2009. Along with this generalization step, the larger class of slice functions over A has been defined. We introduce here a new type of series expansion near each singularity of a slice regular function. This instrument, which is new even in the quaternionic case, leads to a complete classification of singularities. This classification also relies on some recent developments of the theory, concerning the algebraic structure and the zero sets of slice functions. Peculiar phenomena arise, which were not present in the complex or quaternionic case, and they are studied by means of new results on the topology of the zero sets of slice functions. The analogs of meromorphic functions, called (slice) semiregular functions, are introduced and studied.

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