Sharp Estimates for Potential Operators Associated with Laguerre and Dunkl-Laguerre Expansions

We study potential operators associated with Laguerre function expansions of convolution and Hermite types, and with Dunkl-Laguerre expansions. We prove qualitatively sharp estimates of the corresponding potential kernels. Then we characterize those 1 a parts per thousand currency sign p,q a parts per thousand currency sign 8, for which the potential operators are L (p) - L (q) bounded. These results are sharp analogues of the classical Hardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and Dunkl-Laguerre settings.