The $$H^\infty $$ -Functional Calculi for the Quaternionic Fine Structures of Dirac Type

IF 1.2 3区 数学 Q1 MATHEMATICS
Fabrizio Colombo, Stefano Pinton, Peter Schlosser
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引用次数: 0

Abstract

In recent works, various integral representations have been proposed for specific sets of functions. These representations are derived from the Fueter–Sce extension theorem, considering all possible factorizations of the Laplace operator in relation to both the Cauchy–Fueter operator (often referred to as the Dirac operator) and its conjugate. The collection of these function spaces, along with their corresponding functional calculi, are called the quaternionic fine structures within the context of the S-spectrum. In this paper, we utilize these integral representations of functions to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, by leveraging the aforementioned factorization of the Laplace operator, 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.

Abstract Image

狄拉克型四元精细结构的 $$H^\infty $$ 函数计算
在最近的研究中,针对特定函数集提出了各种积分表示法。这些表示法源自 Fueter-Sce 扩展定理,考虑了拉普拉斯算子与考希-富特算子(通常称为狄拉克算子)及其共轭算子的所有可能因式分解。这些函数空间的集合,连同其相应的函数计算,被称为 S 谱背景下的四元精细结构。在本文中,我们利用这些函数的积分表示,引入了为扇形四元数算子量身定制的新型函数计算。具体来说,通过利用上述拉普拉斯算子的因式分解,我们确定了四类不同的函数:片超全态函数(引出 S 函数计算)、轴谐函数(引出 Q 函数计算)、2 阶轴多解析函数(引出 \(P_2\)-functional calculus)和轴单成函数(引出 F 函数计算)。通过应用各自的乘积规则,我们建立了这些函数计算的四个不同的 \(H^\infty \)-versions。
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来源期刊
CiteScore
2.60
自引率
0.00%
发文量
23
审稿时长
>12 weeks
期刊介绍: Milan Journal of Mathematics (MJM) publishes high quality articles from all areas of Mathematics and the Mathematical Sciences. The authors are invited to submit "articles with background", presenting a problem of current research with its history and its developments, the current state and possible future directions. The presentation should render the article of interest to a wider audience than just specialists. Many of the articles will be "invited contributions" from speakers in the "Seminario Matematico e Fisico di Milano". However, also other authors are welcome to submit articles which are in line with the "Aims and Scope" of the journal.
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