Pitt's inequalities and uncertainty principle for generalized Fourier transform

Abstract

We study the two-parameter family of unitary operators \[ \mathcal{F}_{k,a}=\exp\Bigl(\frac{i\pi}{2a}\,(2\langle k\rangle+{d}+a-2 )\Bigr) \exp\Bigl(\frac{i\pi}{2a}\,\Delta_{k,a}\Bigr), \] 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 $\mathcal{F}_{k,a}$.