Inequalities in the Setting of Clifford Analysis

Beurling's theorem for the Clifford‐Fourier transform

“…Corollary (El Kamel and Jday 12, Corollary 5.5 ) Let $$$ m>2 $$$ be even. Suppose that $$$ f(x)={f}_0\left({\left\Vert x\right\Vert}_c\right) $$$ is a real‐valued radial function in $$$ B\left({R}^m\right) $$$ and $$$ g\in B\left({\mathbb{R}}^m\right)\otimes \mathcal{C}{l}_{0,m} $$$.…”

“…Let $$$ 1\le p,q<\infty $$$ and $$$ r\ge 1 $$$ with $$$ 1/p+1/q=1+1/r $$$. (El Kamel and Jday 12, Theorem 5.2 ) For $$$ m=2 $$$, if $$$ f\in {L}^p\left({\mathbb{R}}^2\right)\otimes \mathcal{C}{l}_{0,2} $$$ and $$$ g\in {L}^q\left({\mathbb{R}}^2\right)\otimes \mathcal{C}{l}_{0,2} $$$, then $$$ f{\ast}_{Cl}g\in {L}^r\left({\mathbb{R}}^2\right)\otimes \mathcal{C}{l}_{0,2} $$$. (El Kamel and Jday 12, Theorem 5.4 )For $$$ m>2 $$$, if $$$ f(x)={f}_0\left({\left\Vert x\right\Vert}_c\right) $$$ is a real‐valued radial function in $$$ {L}^p\left({\mathbb{R}}^m\right) $$$…”

“…In this context, many uncertainty principles for the CFT were proved, such as the Hardy theorem and the Heisenberg inequality. [11][12][13] However, the Cowling and Price theorem still needs to be shown in the Clifford analysis setting, particularly in quaternionic analysis. Furthermore, Beurling's theorem, which is the most relevant one since it yields various other uncertainty principle theorems, is not provided.…”

On the Clifford short-time Fourier transform and its properties

“…Since (20) and ( 21) are very different we do not have any commutative relations between the generalised translation and modulation operators.…”

“…where the kernel K ± (x, y) is given by an explicit expression. For this kind of Clifford-Fourier transform many generalizations were found, see [11,12,4,13,7] and some important properties such as the uncertainty principle and Riemann-Lebesgue lemma were proved [19,20].…”