This paper establishes new real Paley–Wiener theorems for the two-sided quaternionic Dunkl transform (QDT) on \(\mathbb {R}^2\) . The QDT unifies quaternion-valued harmonic analysis with Dunkl-type differential-difference operators, which incorporate reflection symmetries and weighted measures. Our main results characterize the support of the QDT in terms of growth conditions on sequences of Dunkl-transformed functions and their derivatives. In particular, we prove \(L^2\) -type real Paley–Wiener theorems, spectral-gap theorems for tempered distributions, and a quaternionic version of Roe’s theorem adapted to the Dunkl setting. The proofs rely on the properties of the Dunkl kernel, the Plancherel formula for the QDT, and the intertwining operator that links the QDT with the classical quaternion Fourier transform. These results extend known Paley–Wiener and Roe-type theorems to a broader, symmetry-aware, non-commutative framework, paving the way for future work in quaternionic signal processing and multidimensional harmonic analysis.
