Negative powers of Laguerre operators

TL;DR

This paper investigates negative powers of Laguerre differential operators, establishing two-weight $L^p-L^q$ estimates across various settings, including Hermite-type and convolution-type expansions, and extends results to Dunkl harmonic oscillators.

Contribution

It introduces a unified approach to derive two-weight estimates for negative powers of Laguerre operators, utilizing kernel properties, interpolation, and the convexity principle, and extends findings to Dunkl harmonic oscillators.

Findings

Established two-weight $L^p-L^q$ estimates for Laguerre operators.

Applied kernel monotonicity properties to Hermite-type Laguerre expansions.

Extended results to Dunkl harmonic oscillator setting.

Abstract

We study negative powers of Laguerre differential operators in $\R$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates, with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices $\alpha$ we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the {\sl convexity principle} to cover the continuous range of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\R$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.