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 \( L_q,s,α(0,∞) \), \( 1 < p < q < ∞ \), \( 1 r s ∞ \). We obtain boundedness conditions in the limiting cases \( p = 1 \) and \( p = 2α+2β \). Finally, for the limiting case \( p = 2α+2β \) we prove an analogue of the Adams theorem on exponential integrability of \( I_β,α \) in \( L_2α+2β,r,α(0,∞) \).