Multipliers of Laplace Transform Type for Laguerre and Hermite Expansions

TL;DR

This paper introduces a new criterion for the boundedness of multiplier operators related to Laguerre and Hermite expansions, simplifying and unifying existing results on fractional integrals through Laplace-Stieltjes transforms.

Contribution

It provides a novel criterion for weighted boundedness of multiplier operators for Laguerre and Hermite expansions using Laplace-Stieltjes transforms, unifying previous results.

Findings

Established a new criterion for weighted $L^p-L^q$ boundedness.

Unified approach simplifies proofs of known fractional integral estimates.

Extended results to a broader class of multiplier operators.

Abstract

We present a new criterion for the weighted $L^p-L^q$ boundedness of multiplier operators for Laguerre and Hermite expansions that arise from a Laplace-Stieltjes transform. As a special case, we recover known results on weighted estimates for Laguerre and Hermite fractional integrals with a unified and simpler approach.