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 ≤ p, q≤∞, for which the potential operators satisfy L p -L q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, 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 Napoli, Drelichman and Duran, and Duoandikoetxea.