The Fractional Two-Sided Quaternionic Dunkl Transform and Heisenberg-Type Inequalities

Abstract

This report investigates the main definitions and fundamental properties of the fractional two-sided quaternionic Dunkl transform in two dimensions. We present key results concerning its structure and emphasize its connections to classical harmonic analysis. Special attention is given to inversion, boundedness, spectral behavior, and explicit formulas for structured functions such as radial or harmonic functions. Within this framework, we establish a generalized form of the classical Heisenberg-type uncertainty principle. Building on this foundation, we further extend the result by proving a higher-order Heisenberg-type inequality valid for arbitrary moments $p \geq 1$, with sharp constants characterized through generalized Hermite functions. Finally, by analyzing the interplay between the two-sided fractional quaternionic Dunkl transform and the two-sided fractional quaternionic Fourier transform, we derive a corresponding uncertainty principle for the latter.