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

Abstract

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 \( L_{q,s,\alpha}(0,\infty) \), \( 1 < p < q < \infty \), \( 1 \le r \le s \le \infty \). We obtain boundedness conditions in the limiting cases \( p = 1 \) and \( p = \frac{2\alpha+2}{\beta} \). Finally, for the limiting case \( p = \frac{2\alpha+2}{\beta} \) we prove an analogue of the Adams theorem on exponential integrability of \( I_{\beta,\alpha} \) in \( L_{\frac{2\alpha+2}{\beta},r,\alpha}(0,\infty) \).