On Some Uncertainty Principles for the Fractional Dunkl Transform

This paper establishes several fundamental uncertainty principles associated with the fractional Dunkl transform (FrDT), a natural generalization of the Dunkl transform. In particular, we prove Hardy’s theorem, an L p –L q version of Miyachi’s theorem, and local uncertainty principles in the context of the FrDT. Unlike classical approaches that focus on the relationship between a function and its transform, our results characterize the interplay between two distinct fractional Dunkl transforms. As a consequence, we recover the standard uncertainty principles involving a function, its FrDT, and its Dunkl transform as special cases. These findings enrich the existing theory of uncertainty principles and extend the framework from Fourier and Dunkl analysis to their fractional analogue, thereby providing new tools for harmonic analysis with potential applications in mathematical physics and signal processing.