Title: Riesz transforms and g-function for Laguerre expansions

Authors: E. Harboure, B. Viviani and J. L. Torrea

Issue: Volume 55 (2006), Issue 3, 999-1014

Abstract: We analyze boundedness properties of some operators related to the heat-diffusion semigroup associated to Laguerre functions systems. In particular, for any $\alpha$ greater than $-1$, we introduce appropriate Laguerre Riesz transforms and we obtain power-weighted $L^p$ inequalities, $1$ less than $p$ less than $\infty$. We achieve this result by taking advantage of the existing classical relationship between $n$-variable Hermite polynomials and Laguerre polynomials on the half line of type $\alpha = n/2 - 1$. Such connection allows us to transfer known boundedness properties for Hermite operators to Laguerre operators corresponding to those specific values of $\alpha$. To extend the results to any $\alpha$ greater than $-1$, we make use of transplantation and some weighted inequalities that we obtain in the Hermite setting (which we believe of independent interest).