On quaternionic analysis and a certain generalized fractal-fractional $ψ$-Fueter operator

TL;DR

This work extends quaternionic analysis to a fractional-fractal setting by introducing a $\psi$-Fueter operator adapted to fractal measures and proportional fractional derivatives. It defines a generalized fractal-fractional derivative with a $\beta$-proportional structure and constructs a quaternionic $\beta$-proportional fractal Fueter operator $^{\psi}\mathcal{D}^{\sigma,\beta}_{\nu}$, along with a truncated exponential fractal measure variant, to develop a function theory in $\mathbb{H}$ that accommodates fractal media. The authors derive quaternionic Stokes and Borel-Pompeiu formulas in this framework and establish decomposition formulas that separate principal, error, and cross terms, illustrating dualities and kernel representations $K_{\psi}$ within fractal contexts. Special cases and limiting forms (e.g., $\beta=(1,1,1,1)$, $k=(1,1,1,1)$, or $k=\infty$) are analyzed to connect with classical results and to show how the new operator reduces to known fractal or fractional variants. Overall, the paper provides a comprehensive algebraic-analytic toolkit for fractal-quaternionic function theory with potential applications in physics and engineering where fractal structures appear.

Abstract

This paper introduce a fractional-fractal $ψ$-Fueter operator in the quaternionic context inspired in the concepts of proportional fractional derivative and Hausdorff derivative of a function with respect to a fractal measure. Moreover, we establish the corresponding Stokes and Borel-Pompeiu formulas associated to this generalized fractional-fractal $ψ$-Fueter operator.

Theorems & Definitions (24)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Definition 4.1
• Proposition 4.2
• proof
• Definition 4.3