Hi,
have you got already introduced Hardy inequality for the quaternion Fourier transform.?
Yes Indeed, one of our first challenges was to extend the validity of Hardy's Theorem, which is one of the most important uncertainty principles of classical harmonic analysis, to the Quaternionic domain, and it was established: See Theorem 5.3 in
https://link.springer.com/article/10.1007/s00009-017-1024-5
Furthermore, a generalized version of this theorem has been demonstrated using Beurling's theorem, I refer you to Corollary 5.1 in
https://link.springer.com/article/10.1007/s11868-019-00281-7
Have a look at http://repository.ias.ac.in/12063/1/333.pdf
https://link.springer.com/content/pdf/10.1155/JIA.2005.495.pdf
Article An improved Hardy-Sobolev inequality in W1,p and its applica...
https://www.tandfonline.com/doi/abs/10.1080/00036811.2014.1001381?journalCode=gapa20
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