arXiv subjects

Stefano Pinton

Publications and source records attributed to Stefano Pinton.

At least 19 recordsLinked to original sources

arXiv2026

A variation of the inverse Fueter theorem and the generalized polyanalytic Cauchy-Kovalevskaya extension of order 2

In this paper, we first establish a generalized Cauchy--Kovalevskaya (GCK) extension for axially polyanalytic functions of order $2$. We prove that the extension can be written as a power series involving differential operators acting on two initial functions. We also study the decomposition of the GCK extension in terms of integrals over the sphere 2-sphere $\mathbb{S}$ involving plane-wave type functions. We further establish a connection between the GCK extension for polyanalytic functions of order $2$ and the Fueter theorem. This is one of the most important results in hypercomplex analysis and can be described in two steps. In the first step, starting from holomorphic functions of one complex variable, the application of a suitable operator yields slice hyperholomorphic functions. In the second step, applying the Laplace operator in four real variables (called Fueter map) one obtains axially monogenic functions, i.e. functions in the kernel of the Fueter operator. A suitable factorization of the Fueter map in terms of the Fueter operator and its conjugate gives rise to two intermediate classes of functions between slice hyperholomorphic functions and axially monogenic functions: axially harmonic functions and axially polyanalytic functions of order $2$. Another goal of this paper is to study the invertibility of the factorizations of the Fueter map for harmonic and polyanalytic functions of order 2 on suitable open sets, and to derive integral representation formulas for the inverse of the factorized Fueter map. These integral representations are based on the Cauchy formula for polyanalytic functions of order $2$ and the Poisson integral formula for harmonic functions.

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arXiv2026

On the application of the factorized Fueter-Sce map to the slice hyperholomorphic Cauchy kernel

The Fueter-Sce theorem is one of the most important results in hypercomplex analysis, providing a two-step procedure for constructing axially monogenic functions starting from holomorphic functions of one variable. In the first step, the so-called slice operator is applied to holomorphic functions of one variable, producing the class of slice hyperholomorphic functions. The second step yields the class of axially monogenic functions by applying the pointwise differential operator Delta_{n+1}^{(n-1)/2}, with n odd, known as the Fueter-Sce map. The significance of the Fueter-Sce theorem also lies in the fact that it induces two spectral theories corresponding to the function classes it generates. Over the years, factorizations of the Fueter-Sce map have been studied to identify the intermediate spaces that arise between slice hyperholomorphic functions and axially monogenic functions. Until now, only certain factorizations of the Fueter-Sce map have been considered. In this paper, our goal is to determine the most general factorizations of the Fueter-Sce map and to apply these differential operators to the Cauchy kernel of slice hyperholomorphic functions.

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arXiv2025

Functions and operators of the polyharmonic and polyanalytic Clifford fine structures on the $S$-spectrum

The spectral theory on the $S$-spectrum originated to give quaternionic quantum mechanics a precise mathematical foundation and as a spectral theory for linear operators in vector analysis. This theory has proven to be significantly more general than initially anticipated, naturally extending to fully Clifford operators and revealing unexpected connections with the spectral theory based on the monogenic spectrum, developed over forty years ago by A. McIntosh and collaborators. In recent years, we have combined slice hyperholomorphic functions with the Fueter-Sce mapping theorem, also called Fueter-Sce extension theorem, to broaden the class of functions and operators to which the theory can be applied. This generalization has led to the definition of what we call the {\em fine structures on the $S$-spectrum}, consisting of classes of functions that admit an integral representation and their associated functional calculi. In this paper, we focus on the fine structures within the Clifford algebra setting, particularly addressing polyharmonic functions, polyanalytic functions, holomorphic Cliffordian functions and their associated functional calculi defined via integral representation formulas. Moreover, we demonstrate that the monogenic functional calculus, defined via the monogenic Cauchy formula, and the $F$-functional calculus of the fine structures, defined via the Fueter-Sce mapping theorem in integral form, yield the same operator.

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arXiv2024

An introduction to the fine structures on the $S$-spectrum

Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides a two-step procedure for extending holomorphic functions to hyperholomorphic functions. In the first step, slice hyperholomorphic functions are obtained, and their associated Cauchy formula establishes the $S$-functional calculus for noncommuting operators on the $S$-spectrum. The second step produces axially monogenic functions, which lead to the development of the monogenic functional calculus. In this review paper we discuss the second operator in the Fueter-Sce mapping theorem that takes slice hyperholomorphic to axially monogenic functions. This operator admits several factorizations which generate various function spaces and their corresponding functional calculi, thereby forming the so-called fine structures of spectral theories on the $S$-spectrum.

math.SP↗

arXiv2023

The $H^\infty$-functional calculi for the quaternionic fine structures of Dirac type

In this paper, we utilize various integral representations derived from the Fueter-Sce extension theorem, to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, due to the different factorizations of the Laplace opertor with respect to the Cauchy-Fueter operator and its conjugate, we identify four distinct classes of functions: Slice hyperholomorphic functions (leading to the $S$-functional calculus), axially harmonic functions (leading to the $Q$-functional calculus), axially polyanalytic functions of order $2$ (leading to the $P_2$-functional calculus), and axially monogenic functions (leading to the $F$-functional calculus). By applying the respective product rule, we establish the four different $H^\infty$-versions of these functional calculi.

math.FA↗

arXiv2023

Characterization of continuous homomorphisms on entire slice monogenic functions

This paper is inspired by a class of infinite order differential operators arising in the time evolution of superoscillations. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e., functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra. We introduce the concept of proximate order and establish some fundamental properties of entire hyperholomorphic functions that are crucial for this characterization.

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arXiv2023

The harmonic $H^\infty$-functional calculus based on the S-spectrum

The aim of this paper is to introduce the $H^\infty$-functional calculus for harmonic functions over the quaternions. More precisely, we give meaning to Df(T) for unbounded sectorial operators T and polynomially growing functions of the form Df, where f is a slice hyperholomorphic function and $D=\partial_{q_0}+e_1\partial_{q_1}+e_2\partial_{q_2}+e_3\partial_{q_3}$ is the Cauchy-Fueter operator. The harmonic functional calculus can be viewed as a modification of the well known S-functional calculus f(T), with a different resolvent operator. The harmonic $H^\infty$-functional calculus is defined in two steps: First, for functions with a certain decay property, one can make sense of the bounded operator Df(T) directly via a Cauchy-type formula. In a second step, a regularization procedure is used to extend the functional calculus to polynomially growing functions and consequently unbounded operators Df(T). The harmonic functional calculus is an important functional calculus of the quaternionic fine structures on the S-spectrum, which arise also in the Clifford setting and they encompass a variety of function spaces and the corresponding functional calculi. These function spaces emerge through all possible factorizations of the second map of the Fueter-Sce extension theorem. This field represents an emerging and expanding research area that serves as a bridge connecting operator theory, harmonic analysis, and hypercomplex analysis.

math.FA↗

arXiv2023

Harmonic and polyanalytic functional calculi on the $S$-spectrum for unbounded operators

Harmonic and polyanalytic functional calculi have been recently defined for bounded commuting operators. Their definitions are based on the Cauchy formula of slice hyperholomorphic functions and on the factorization of the Laplace operator in terms of the Cauchy-Fueter operator $\mathcal{D}$ and of its conjugate $\overline{\mathcal{D}}$. Thanks to the Fueter extension theorem when we apply the operator $\mathcal{D}$ to slice hyperholomorphic functions we obtain harmonic functions and via the Cauchy formula of slice hyperholomorphic functions we establish an integral representation for harmonic functions. This integral formula is used to define the harmonic functional calculus on the $S$-spectrum. Another possibility is to apply the conjugate of the Cauchy-Fueter operator to slice hyperholomorphic functions. In this case, with a similar procedure we obtain the class of polyanalytic functions, their integral representation and the associated polyanalytic functional calculus. The aim of this paper is to extend the harmonic and the polyanalytic functional calculi to the case of unbounded operators and to prove some of the most important properties. These two functional calculi belong to so called fine structures on the $S$-spectrum in the quaternionic setting. Fine structures on the $S$-spectrum associated with Clifford algebras constitute a new research area that deeply connects different research fields such as operator theory, harmonic analysis and hypercomplex analysis.

math.SP↗

arXiv2023

The fine structure of the spectral theory on the $S$-spectrum in dimension five

Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define functions of operators. The Fueter-Sce-Qian extension theorem is a two steps procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called $S$-functional calculus for noncommuting operators based on the $S$-spectrum. In the second step this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, contains the Weyl functional calculus as a particular case. In this paper we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allows to define the associated functional calculi based on the $S$-spectrum. The function spaces and the associated functional calculi define the so called {\em fine structure of the spectral theories on the $S$-spectrum}. Among the possible fine structures there are the harmonic and poly-harmonic functions and the associated harmonic and poly-harmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.

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arXiv2023

The general theory of superoscillations and supershifts in several variables

In this paper we describe a general method to generate superoscillatory functions of several variables starting from a superoscillating sequence of one variable. Our results are based on the study of suitable infinite order differential operators on holomorphic functions with growth conditions of exponential type, where additional constraints are required when dealing with infinite order differential operators whose symbol is a function that is holomorphic in some open set, but not necessarily entire. The results proved for the superoscillating sequence in several variables are extended to sequences of supershifts in several variables.

math.FA↗

arXiv2022

Properties of a polyanalytic functional calculus on the $S$-spectrum

The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, i.e., null solutions of the generalized Cauchy-Riemann operator in $\mathbb{R}^4$, denoted by $\mathcal{D}$. This theorem is divided in two steps. In the first step a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for these type of functions is the starting point of the $S$-functional calculus. In the second step a monogenic function is obtained by applying the Laplace operator in four real variables, namely $ Δ$, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of $Δ= \mathcal{D} \mathcal{\overline{D}}$. Instead of applying directly the Laplace operator to a slice hyperholomorphic function we apply first the operator $ \mathcal{\overline{D}}$ and we get a polyanalytic function of order 2, i..e, a function that belongs to the kernel of $ \mathcal{D}^2$. We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on $S$-spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

math.SP↗

arXiv2022

A polyanalytic functional calculus of order 2 on the S-spectrum

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solutions of the Cauchy-Riemann operator in $ \mathbb{R}^4$, denoted by $ \mathcal{D}$. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operator in four real variables ($Δ$) to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. $Δ= \mathcal{\overline{D}} \mathcal{D}$ to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions of $ \mathcal{D}^2$. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on the $S$-spectrum. As far as the authors know, this is the first time that a monogenic polyanalytic functional calculus has been taken into consideration.

math.FA↗

arXiv2022

Axially harmonic functions and the harmonic functional calculus on the S-spectrum

The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S-spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmonic functional calculus is the commutative version of the pseudo S-resolvent operator. This new calculus also appears, in a natural way, in the product rule for the F-functional calculus.

math.SP↗

arXiv2021

Fractional powers of higher order vector operators on bounded and unbounded domains

Using the $H^\infty$-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order $m\geq 1$, acting on the right linear quaternionic Hilbert space $L^2(Ω,\mathbb C\otimes\mathbb H)$. The operators that we consider are of the type $$ T=i^{m-1}\left(a_1(x) e_1\partial_{x_1}^{m}+a_2(x) e_2\partial_{x_2}^{m}+a_3(x) e_3\partial_{x_3}^{m}\right), \ \ \ x=(x_1,\, x_2,\, x_3)\in \overlineΩ, $$ where $\overlineΩ$ is the closure of either a bounded domain $Ω$ with $C^1$ boundary, or an unbounded domain $Ω$ in $\mathbb R^3$ with a sufficiently regular boundary which satisfy the so called property $(R)$, $\{e_1,\, e_2,\, e_3\}$ is an orthonormal basis for the imaginary units of $\mathbb H$, $a_1,\,a_2,\, a_3: \overlineΩ \subset\mathbb{R}^3\to \mathbb{R}$ are the coefficients of $T$. In particular it will be given sufficient conditions on the coefficients of $T$ in order to generate the fractional powers of $T$, denoted by $P_α(T)$ for $α\in(0,1)$, when the components of $T$, i.e. the operators $T_l:=a_l\partial_{x_l}^m$, do not commute among themselves.

math.SP↗

arXiv2021

Testing families of analytic discs in the unit ball

Let $a,b,c\in \mathbb{C}^2$ be three non collinear points such that their mutual joining complex lines do not intersect the unit ball $\mathbb{B}^2$ and such that the line through $a$ and $b$ is tangent to $\mathbb{B}^2$. Then the set of lines concurrent to $a,b$ and $c$ is a testing family for continuous functions on $\mathbb{S}^3$. This improves a result by the authors and solves a case left open in the literature as described by Globevnik.

math.CV↗

arXiv2021

Higher order gradients of monogenic functions

Given a monogenic function on the quaternionic algebra $\mathbb{H}$, the Clifford algebra $\mathbb{R}_n$ or the octonionic algebra $\mathbb{O}$ we prove that $|\nabla^m f|^α$ is subharmonic for some $α>0$ where $\nabla^m f$ is the $m$-th order gradient of $f$. We find also the optimal value of $α$. This is generalization of a result of Calderon and Zygmund.

math.CV↗

arXiv2020

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 $(A_1,...,A_n)$. 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. Here we also discuss how to define the fractional Fourier's law for nonhomogeneous materials, such definition is based on the spectral theory on the $S$-spectrum.

math.SP↗

arXiv2020

The noncommutative fractional Fourier law in bounded and unbounded domains

Using the spectral theory on the $S$-spectrum it is possible to define the fractional powers of a large class of vector operators. This possibility leads to new fractional diffusion and evolution problems that are of particular interest for nonhomogeneous materials where the Fourier law is not simply the negative gradient operator but it is a nonconstant coefficients differential operator of the form $$ T=\sum_{\ell=1}^3e_\ell a_\ell(x)\partial_{x_\ell}, \ \ \ x=(x_1,x_2,x_3)\in \barΩ, $$ where, $Ω$ can be either a bounded or an unbounded domain in $\mathbb{R}^3$ whose boundary $\partialΩ$ is considered suitably regular, $\barΩ$ is the closure of $Ω$ and $e_\ell$, for $\ell=1,2,3$ are the imaginary units of the quaternions $\mathbb{H}$. The operators $T_\ell:=a_\ell(x)\partial_{x_\ell}$, for $\ell=1,2,3$, are called the components of $T$ and $a_1$, $a_2$, $a_3: \barΩ \subset\mathbb{R}^3\to \mathbb{R}$ are the coefficients of $T$. In this paper we study the generation of the fractional powers of $T$, denoted by $P_α(T)$ for $α\in(0,1)$, when the operators $T_\ell$, for $\ell=1,2,3$ do not commute among themselves. To define the fractional powers $P_α(T)$ of $T$ we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo $S$-resolvent operator of $T$. In this paper we consider two different boundary conditions. If $Ω$ is unbounded we consider Dirichlet boundary conditions. If $Ω$ is bounded we consider the natural Robin-type boundary conditions associated with the generation of the fractional powers of $T$.

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