Potential operators associated with Hankel and Hankel-Dunkl transforms

Abstract

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those $1\le p,q \le \infty$, for which the potential operators satisfy $L^p-L^q$ estimates. In case of the Riesz potentials, we also characterize those $1\le p,q \le \infty$, for which two-weight $L^p-L^q$ estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight $L^p-L^q$ bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De N\'apoli, Drelichman and Dur\'an, and Duoandikoetxea.