The Two-Sided Clifford Dunkl Transform and Miyachi's Theorem

Abstract

Recent advances have extended the Dunkl transform to the setting of Clifford algebras. In particular, the two-sided quaternionic Dunkl transform has been introduced as a Dunkl analogue of the two-dimensional quaternionic Fourier transform. In this paper, we develop the two-sided Clifford Dunkl transform, defined using two square roots of -1 in Cl_{p,q}. We establish its fundamental properties, including the inversion and Plancherel formulas, and provide two explicit expressions for the associated translation operator. Moreover, we prove an analogue of Miyachi's theorem for this transform, thereby extending a classical result in harmonic analysis to the Clifford-Dunkl framework.