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The resolvent equations for the Harmonic and bi-Harmonic functional calculi in dimension five

Fabrizio Colombo, Antonino De Martino, Joao Marques Da Costa

Abstract

The fine structures on the $S$-spectrum constitute a new research area that includes a class of functional calculi based on the $S$-spectrum and on integral transforms determined by the Fueter--Sce mapping theorem and the Cauchy formula for slice hyperholomorphic functions. This strategy, based on integral transforms, allows us to construct functional calculi that include harmonic and polyharmonic functional calculi. The resolvent operators in this setting do not arise directly from a Cauchy kernel, but rather from suitable manipulations of it. For this reason the corresponding resolvent equations differ substantially from those associated with the classical Cauchy kernel. In this paper, we investigate the harmonic and biharmonic resolvent equations in dimension five, as well as the corresponding product rules and Riesz projectors for these functional calculi.

The resolvent equations for the Harmonic and bi-Harmonic functional calculi in dimension five

Abstract

The fine structures on the -spectrum constitute a new research area that includes a class of functional calculi based on the -spectrum and on integral transforms determined by the Fueter--Sce mapping theorem and the Cauchy formula for slice hyperholomorphic functions. This strategy, based on integral transforms, allows us to construct functional calculi that include harmonic and polyharmonic functional calculi. The resolvent operators in this setting do not arise directly from a Cauchy kernel, but rather from suitable manipulations of it. For this reason the corresponding resolvent equations differ substantially from those associated with the classical Cauchy kernel. In this paper, we investigate the harmonic and biharmonic resolvent equations in dimension five, as well as the corresponding product rules and Riesz projectors for these functional calculi.
Paper Structure (4 sections, 16 theorems, 98 equations)

This paper contains 4 sections, 16 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The resolvent equation and the product rule for the bi-harmonic functional calculus
  4. The harmonic case associate with the $\Delta D$ operator

Key Result

Theorem 2.6

Let $U \subset \mathbb{R}^{n+1}$ be a bounded slice Cauchy domain, let $I \in \mathbb{S}$, and set $ds_I = ds(-I)$. If $f$ is a left (resp. right) slice hyperholomorphic function on a domain containing $\overline{U}$ and $x \notin [s]$, then where $S^{-1}_L(s,x)$ (resp. $S^{-1}_R(s,x)$) denotes the left (resp. right) slice hyperholomorphic Cauchy kernel. Moreover, the integrals in integrals are i

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Slice hyperholomorphic functions
  • Definition 2.4: Slice Cauchy domain
  • Definition 2.5: Slice hyperholomorphic Cauchy kernels
  • Theorem 2.6: Cauchy-formulas for slice hyperholomorphic functions
  • Definition 2.7
  • Remark 2.8
  • Theorem 2.9: Fueter-Sce mapping theorem
  • Definition 2.10: Dirac fine structures on the $S$-spectrum
  • ...and 37 more