Beckner logarithmic uncertainty principle for the Riemann-Liouville operator (Q2855491)

The Riemann-Liouville operator \(\mathcal{R}_\alpha\) is defined by NEWLINE\[NEWLINE\mathcal{R}_\alpha(f)(r, x) = \frac{\alpha}{\pi} \int_{-1}^{1} \int_{-1}^{1} f(rs {\sqrt{1-t^2}}, x+rt)~(1-t^2)^{\alpha - \frac{1}{2}} (1-s^2)^{\alpha-1}~dt ~dsNEWLINE\]NEWLINE if \(\alpha > 0\) and NEWLINE\[NEWLINE \frac{1}{\pi} \int_{-1}^1 f(r \sqrt{1-t^2}, x+rt)~\frac{dt}{\sqrt{1-t^2}}NEWLINE\]NEWLINE if \(\alpha = 0.\) The Fourier transform \(\mathcal{F}_\alpha\) associated with \(\mathcal{R}_\alpha\) is defined by NEWLINE\[NEWLINE\forall (\mu, \lambda) \in \Upsilon, ~\mathcal{F}_\alpha(f) = \int_0^\infty~\int_{\mathbb R}~f(r, x) \mathcal{R}_\alpha (\cos(\mu.)e^{-i\lambda.})(r,x)~d\nu_\alpha(r,x),NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \Upsilon = \mathbb R^2 \cup \{(i\mu, \lambda); (\mu, \lambda) \in \mathbb R^2, |\mu| \leq |\lambda| \},NEWLINE\]NEWLINE and NEWLINE\[NEWLINEd\nu_\alpha(r, x) = \frac{r^{2\alpha+1}dr}{2^\alpha \Gamma(\alpha+1)} \otimes \frac{dx}{\sqrt{2\pi}}.NEWLINE\]NEWLINENEWLINENEWLINEThis paper establishes the Stein-Weiss inequality for the B-Riesz potential generated by the Riemann-Liouville operator. Pitt's and Beckner logarithmic inequalities for the associated Fourier transform are also proved.