A sharp weighted transplantation theorem for Laguerre function expansions

摘要

We find the sharp range of boundedness for transplantation operators associated with Laguerre function expansions in L-p spaces with power weights. Namely, the operators interchanging {L-kappa(alpha)} and {L-kappa(beta)} are bounded in L-p (y(delta p)) if and only if -rho/2 - 1/p < delta < 1 - 1/p + rho/2, where rho = min {alpha, beta}. This improves a previous partial result by Stempak and Trebels, which was only sharp for rho <= 0. Our approach is based on new multiplier estimates for Hermite expansions, weighted inequalities for local singular integrals and a careful analysis of Kanjin's original proof of the unweighted case. As a consequence we obtain new results on multipliers, Riesz transforms and g-functions for Laguerre expansions in L-p (y(delta p)). (c) 2006 Elsevier Inc. All rights reserved.