Negative powers of Laguerre operators

Abstract

We study negative powers of Laguerre differential operators in $\mathbb{R}$, $d\ge 1$. 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 $\mathbb{R}$ and isomorphic to ${\mathbb{Z}}_2^d$. 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.

30 pages

References in corpus (1)

• Imaginary Powers of the Dunkl Harmonic Oscillator

Cited by in corpus (7)

• Vector-valued extensions for fractional integrals of Laguerre expansions
• Sharp estimates of the potential kernel for the harmonic oscillator with applications
• Potential operators associated with Jacobi and Fourier-Bessel expansions
• On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings
• The Riesz transform for the harmonic oscillator in spherical coordinates
• Sharp estimates for potential operators associated with Laguerre and Dunkl-Laguerre expansions
• Multipliers of Laplace transform type in certain Dunkl and Laguerre settings