Potential operators associated with Jacobi and Fourier-Bessel expansions

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≤p,q≤∞, 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.