On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

Abstract

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to ${\mathbb{Z}}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calderón-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1<p<\infty$, and from weighted $L^1$ to weighted weak $L^1$. We also obtain similar results for analogous set of operators in the closely related multi-dimensional Laguerre-symmetrized framework. The latter emerges from a symmetrization procedure proposed recently by the first two authors. As a by-product of the main developments we get some new results in the multi-dimensional Laguerre function setting of convolution type.

References in corpus (8)

• Imaginary Powers of the Dunkl Harmonic Oscillator
• Sharp estimates for potential operators associated with Laguerre and Dunkl-Laguerre expansions
• Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators
• Potential and Sobolev Spaces Related to Symmetrized Jacobi Expansions
• A Perturbation of the Dunkl Harmonic Oscillator on the Line
• Mixed norm estimates for the Cesàro means associated with Dunkl--Hermite expansions
• Hilbert Transforms Associated with Dunkl-Hermite Polynomials
• Area Littlewood-Paley functions associated with Hermite and Laguerre operators

Cited by in corpus (2)

• Conical square functions associated with Bessel, Laguerre and Schrödinger operators in UMD Banach spaces
• Laguerre expansions on conic domains