The O’Neil inequality for the Hankel convolution operator and some applications - Aykol, Canay; Serbetci, Ayhan; Guliyev, Vagif S. (Eurasian Mathematical Journal, 2013)
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 \( I_{β,α} \), associated with the Hankel transform in the Lorentz-Hankel spaces \( L_{p,r,α}(0,∞) \). We establish necessary and sufficient conditions for the boundedness of \( I_{β,α} \) from the Lorentz-Hankel spaces \( L_{p,r,α}(0,∞) \) to \( L_{q,s,α}(0,∞) \), \( 1 < p < q < ∞ \), \( 1 \le r \le s \le ∞ \). We obtain boundedness conditions in the limiting cases \( p = 1 \) and \( p = \frac{2α+2}{β} \). Finally, for the limiting case \( p = \frac{2α+2}{β} \) we prove an analogue of the Adams theorem on exponential integrability of \( I_{β,α} \) in \( L_{\frac{2α+2}{β},r,α}(0,∞) \).