A sharp weighted transplantation theorem for Laguerre function expansions

Abstract

We find the sharp range of boundedness for transplantation operators associated with Laguerre function expansions in Lp spaces with power weights. Namely, the operators interchanging {Lkα} and {Lkβ} are bounded in Lp (yδ p) if and only if - frac(ρ, 2) - frac(1, p) < δ < 1 - frac(1, p) + frac(ρ, 2), where ρ = min {α, β}. This improves a previous partial result by Stempak and Trebels, which was only sharp for ρ ≤ 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 Lp (yδ p).