  We study potential operators (Riesz and Bessel potentials) associated with
classical Jacobi and Fourier-Bessel expansions. We prove sharp estimates for
the corresponding potential kernels. Then we characterize those $1 \le p,q \le
\infty$, for which the potential operators are of strong type $(p,q)$, of weak
type $(p,q)$ and of restricted weak type $(p,q)$. These results may be thought
of as analogues of the celebrated Hardy-Littlewood-Sobolev fractional
integration theorem in the Jacobi and Fourier-Bessel settings. As an ingredient
of our line of reasoning, we also obtain sharp estimates of the Poisson kernel
related to Fourier-Bessel expansions.
