Riesz transforms and g-function for Laguerre expansions E. HarboureJose TorreaB. Viviani 42A4542B1542B2042B2542C10Laguerre semigroupRiesz transformsLittlewood-Paley theory 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). Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2650 10.1512/iumj.2006.55.2650 en Indiana Univ. Math. J. 55 (2006) 999 - 1014 state-of-the-art mathematics http://iumj.org/access/