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On Harmonic Analysis Operators in Laguerre–Dunkl and Laguerre-Symmetrized Settings
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: 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; Calderón–Zygmund operator. 
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     author = {Adam Nowak and Krzysztof Stempak and Tomasz Z. Szarek},
     title = {On {Harmonic} {Analysis} {Operators} in {Laguerre{\textendash}Dunkl} {and~Laguerre-Symmetrized} {Settings}},
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}
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Adam Nowak; Krzysztof Stempak; Tomasz Z. Szarek. On Harmonic Analysis Operators in Laguerre–Dunkl and Laguerre-Symmetrized Settings. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a95/

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