The O'Neil inequality for the Hankel convolution operator and some applications

In this paper we prove the O'Neil inequality for the Hankel (Fourier–Bessel) convolution operator and consider some of its applications. By using the O'Neil inequality we study the boundedness of the Riesz–Hankel potential operator $I_{\beta,\alpha}$, associated with the Hankel transform in the Lorentz–Hankel spaces $L_{p,r,\alpha}(0,\infty)$. We establish necessary and sufficient conditions for the boundedness of $I_{\beta,\alpha}$, from the Lorentz–Hankel spaces $L_{p,r,\alpha}(0,\infty)$ to $L_{q,s,\alpha}(0,\infty)$, $1<p<q<\infty$, $\le r\le s\le\infty$. We obtain boundedness conditions in the limiting cases $p=1$ and $p=(2\alpha+2)/\beta$. Finally, for the limiting case $p=(2\alpha+2)/\beta$ we prove an analogue of the Adams theorem on exponential integrability of $I_{\beta,\alpha}$, in $L_{(2\alpha+2)/\beta,r,\alpha}(0,\infty)$.