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On the harmonic generalized Cauchy-Kovalevskaya extension and its connection with the Fueter-Sce theorem

Antonino De Martino, Ali Guzmán Adán

TL;DR

This work develops a generalized Cauchy–Kovalevskaya extension for axially harmonic functions, expressing the extension HGCK[A_0,A_1] as a Bessel-operator series acting on two initial data functions and providing a sphere-plane-wave representation; it then connects this harmonic extension to the Fueter–Sce theorem, clarifying how the Fueter–Sce map factors through CK-extensions to place axially harmonic functions between slice monogenic and axially monogenic classes. The authors introduce harmonic polynomials $oldsymbol{ ext{P}}_k^m(x)$ and Clifford–Appell polynomials, show their harmonicity and relations to Gegenbauer polynomials and the Riesz potential, and construct a basis for the space of axially harmonic functions via scalar and 1-vector parts. The results yield explicit formulas, decomposition formulas, and a structured diagram linking CK-extension and Fueter–Sce theory in Clifford analysis, with an eye toward extensions to polyharmonic settings.

Abstract

One of the primary objectives of this paper is to establish a generalized Cauchy-Kovalevskaya extension for axially harmonic functions. We demonstrate that the result can be expressed as a power series involving Bessel-type functions of specific differential operators acting on two initial functions. Additionally, we analyze the decomposition of the harmonic CK extension in terms of integrals over the sphere $ \mathbb{S}^{m-1} $ involving functions of plane wave type. Another key goal of this paper is to explore the relationship between the harmonic Cauchy-Kovalevskaya extension and the Fueter-Sce theorem. The Fueter-Sce theorem outlines a two-step process for constructing axially monogenic functions in $ \mathbb{R}^{m+1}$ starting from holomorphic functions in one complex variable. The first step generates the class of slice monogenic functions, while the second step produces axially monogenic functions by applying the pointwise differential operator $ Δ_{\mathbb{R}{^{m+1}}}^{\frac{m-1}{2}} $ with $m$ being odd, known as the Fueter-Sce map, to a slice monogenic function. By suitably factorizing the Fueter-Sce map, we introduce the set of axially harmonic functions, which serves as an intermediate class between slice monogenic and axially monogenic functions. In this paper, we establish a connection between the harmonic CK extension and the factorization of the Fueter-Sce map. This connection leads to a new notion of harmonic polynomials, which we show to form a basis for the Riesz potential. Finally, we also construct a basis for the space of axially harmonic functions.

On the harmonic generalized Cauchy-Kovalevskaya extension and its connection with the Fueter-Sce theorem

TL;DR

This work develops a generalized Cauchy–Kovalevskaya extension for axially harmonic functions, expressing the extension HGCK[A_0,A_1] as a Bessel-operator series acting on two initial data functions and providing a sphere-plane-wave representation; it then connects this harmonic extension to the Fueter–Sce theorem, clarifying how the Fueter–Sce map factors through CK-extensions to place axially harmonic functions between slice monogenic and axially monogenic classes. The authors introduce harmonic polynomials and Clifford–Appell polynomials, show their harmonicity and relations to Gegenbauer polynomials and the Riesz potential, and construct a basis for the space of axially harmonic functions via scalar and 1-vector parts. The results yield explicit formulas, decomposition formulas, and a structured diagram linking CK-extension and Fueter–Sce theory in Clifford analysis, with an eye toward extensions to polyharmonic settings.

Abstract

One of the primary objectives of this paper is to establish a generalized Cauchy-Kovalevskaya extension for axially harmonic functions. We demonstrate that the result can be expressed as a power series involving Bessel-type functions of specific differential operators acting on two initial functions. Additionally, we analyze the decomposition of the harmonic CK extension in terms of integrals over the sphere involving functions of plane wave type. Another key goal of this paper is to explore the relationship between the harmonic Cauchy-Kovalevskaya extension and the Fueter-Sce theorem. The Fueter-Sce theorem outlines a two-step process for constructing axially monogenic functions in starting from holomorphic functions in one complex variable. The first step generates the class of slice monogenic functions, while the second step produces axially monogenic functions by applying the pointwise differential operator with being odd, known as the Fueter-Sce map, to a slice monogenic function. By suitably factorizing the Fueter-Sce map, we introduce the set of axially harmonic functions, which serves as an intermediate class between slice monogenic and axially monogenic functions. In this paper, we establish a connection between the harmonic CK extension and the factorization of the Fueter-Sce map. This connection leads to a new notion of harmonic polynomials, which we show to form a basis for the Riesz potential. Finally, we also construct a basis for the space of axially harmonic functions.
Paper Structure (7 sections, 36 theorems, 182 equations, 1 figure)

This paper contains 7 sections, 36 theorems, 182 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Cauchy-Kovalevskaya extension for axially harmonic functions
  4. Plane waves decomposition
  5. Connection between the Fueter-Sce's theorem and the generalized CK-extension for axially harmonic functions
  6. A sequence of harmonic polynomials related to the Fueter-Sce's theorem
  7. Concluding remarks

Key Result

Theorem 2.3

Let $\Omega \subset \mathbb{R}^{m+1}$ be a an axially symmetric slice domain, and let $\Omega_2 \subset \mathbb{C}$ be the intrinsic complex domain such that $\Omega= \Omega_2 \times \mathbb{S}^{m-1}$. Then any function $f \in \mathcal{SM}(\Omega)$ can be written as where the functions $\alpha$, $\beta$ are Clifford-valued real functions defined in $\Omega_2$, such that and moreover satisfy the

Figures (1)

  • Figure :

Theorems & Definitions (77)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Representation theorem CSS3
  • Definition 2.4
  • Theorem 2.5: Slice extension map
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Definition 2.9: Axially monogenic function
  • ...and 67 more