  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.
