Higher order Riesz transforms for Laguerre expansions

Abstract

In this paper, we investigate $L^p$-boundedness properties for the one-dimensional higher order Riesz transforms associated with Laguerre operators. We also prove that the $k$-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when $k$ is even). To establish our results, we exploit a new estimate connecting Riesz transforms in the Hermite and Laguerre settings in dimension one.