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

[摘要] 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.

[关键词] Dunkl harmonic oscillator;generalized Hermite functions;negative multiplicity function;Laguerre expansions of convolution type;Bessel harmonic oscillator;Laguerre– Dunkl expansions;Laguerre-symmetrized expansions;heat semigroup;Poisson semigroup;maximal operator;Riesz transform;g-function;spectral multiplier;area integral;Calder´on– Zygmund operator