<p>In this paper, we establish several qualitative uncertainty principles for the two-sided quaternionic Dunkl transform (QDT) on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7991_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. This transform combines the non-commutative structure of quaternions with the reflection symmetry and weighted analysis inherent in Dunkl theory. Our main results include quaternionic analogues of classical theorems due to Beurling, Hardy, Cowling-Price, Morgan, and Gelfand-Shilov, adapted to the QDT setting. The proofs rely on refined decay estimates, properties of the Dunkl kernel, and the use of the intertwining operator to transfer results from the quaternionic Fourier domain. These results extend known uncertainty principles to a broader, quaternionic Dunkl framework and lay the groundwork for further applications in quaternionic harmonic analysis.</p>

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UNCERTAINTY PRINCIPLE FOR THE TWO-SIDED QUATERNIONIC DUNKL TRANSFORM

  • A. Achak,
  • R. Daher,
  • A. Elhyat,
  • El. Loualid

摘要

In this paper, we establish several qualitative uncertainty principles for the two-sided quaternionic Dunkl transform (QDT) on \(\mathbb {R}^2\) R 2 . This transform combines the non-commutative structure of quaternions with the reflection symmetry and weighted analysis inherent in Dunkl theory. Our main results include quaternionic analogues of classical theorems due to Beurling, Hardy, Cowling-Price, Morgan, and Gelfand-Shilov, adapted to the QDT setting. The proofs rely on refined decay estimates, properties of the Dunkl kernel, and the use of the intertwining operator to transfer results from the quaternionic Fourier domain. These results extend known uncertainty principles to a broader, quaternionic Dunkl framework and lay the groundwork for further applications in quaternionic harmonic analysis.