Pitt's inequalities and uncertainty principle for generalized Fourier transform

Abstract

We study the two-parameter family of unitary operators \[ \mathcal{F}_{k,a}=\exp\Bigl(\frac{iπ}{2a}\,(2\langle k\rangle+{d}+a-2 )\Bigr) \exp\Bigl(\frac{iπ}{2a}\,Δ_{k,a}\Bigr), \] which are called $(k,a)$-generalized Fourier transforms and defined by the $a$-deformed Dunkl harmonic oscillator ${\Delta }_{k,a}=|x{|}^{2-a}{\Delta }_k-|x{|}^a$, $a>0$, where ${\Delta }_k$ is the Dunkl Laplacian. Particular cases of such operators are the Fourier and Dunkl transforms. The restriction of ${\mathcal{F}}_{k,a}$ to radial functions is given by the $a$-deformed Hankel transform $H_{\lambda ,a}$. We obtain necessary and sufficient conditions for the weighted $(L^p,L^q)$ Pitt inequalities to hold for the $a$-deformed Hankel transform. Moreover, we prove two-sided Boas--Sagher type estimates for the general monotone functions. We also prove sharp Pitt's inequality for ${\mathcal{F}}_{k,a}$ transform in $L^2({\mathbb{R}}^d)$ with the corresponding weights. Finally, we establish the logarithmic uncertainty principle for ${\mathcal{F}}_{k,a}$.

arXiv comment16 pages