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This article is cited in 4 scientific papers (total in 4 papers)
On Harmonic Analysis Operators in Laguerre–Dunkl and Laguerre-Symmetrized Settings
Adam Nowaka, Krzysztof Stempakb, Tomasz Z. Szareka a Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00–656 Warszawa, Poland
b Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50–370 Wrocław, Poland
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.
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.
Received: May 25, 2016; in final form September 23, 2016; Published online September 29, 2016
Citation:
Adam Nowak, Krzysztof Stempak, Tomasz Z. Szarek, “On Harmonic Analysis Operators in Laguerre–Dunkl and Laguerre-Symmetrized Settings”, SIGMA, 12 (2016), 096, 39 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1178 https://www.mathnet.ru/eng/sigma/v12/p96
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