Pitt's inequalities and uncertainty principle for generalized Fourier transform

We study the two-parameter family of unitary operators $${\mathcal{F}}_{k,a}=\exp (\frac{i\pi }{2a}(2⟨k⟩+d+a-2))\exp \left(\frac{i\pi }{2a}{\mathrm{\Delta }}_{k,a}\right),$$ which are called $(k,a)$-generalized Fourier transforms and defined by the $a$-deformed Dunkl harmonic oscillator ${\mathrm{\Delta }}_{k,a}=\mathrm{\mid }x{\mathrm{\mid }}^{2-a}{\mathrm{\Delta }}_k-\mathrm{\mid }x{\mathrm{\mid }}^a$, $a>0$, where ${\mathrm{\Delta }}_k$ 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 ${\mathcal{F}}_{k,a}$.