Sharp estimates for potential operators associated with Laguerre and Dunkl-Laguerre expansions

Abstract

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 $1 \le p,q \le \infty$, 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.