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

Symmetry Integrability and Geometry-Methods and Applications

article

Adam Nowak¹  Krzysztof Stempak²  Tomasz Z. Szarek¹ 

[1] Institute of Mathematics, Polish Academy of Sciences;Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology

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;   

DOI  :  10.3842/SIGMA.2016.096

来源: National Academy of Science of Ukraine

【 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.

【 授权许可】

Unknown