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

Abstract

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 Z d 2 . 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 < ∞, 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.

Lusin area integrals related to Jacobi expansions

“…We will need a slightly more general definition of the standard kernel, or rather standard estimates, than the one used in [11,13]. More precisely, we will allow slightly weaker smoothness estimates as indicated below, see for instance [5,15,21] where the Lusin area integrals were treated in some other orthogonal expansions settings.…”

“…Nevertheless, to treat Lusin area integrals, which have more complex nature than the operators considered in the above mentioned papers, some generalization of this technique is needed. The latter is of independent interest and is inspired by similar tools elaborated recently in other settings of orthogonal expansions, see [5,15,21].…”

“…These operators, sometimes called conical square functions, were also studied in some contexts of orthogonal expansions, see e.g. [1,2,3,5,8,10,15,16,21]. In the classical situation these objects turn out to be not only interesting on their own right, but also have significant applications.…”

Potential kernels for radial Dunkl Laplacians

“…The rank one direct product case Z J 2 is an important case of Dunkl analysis, developing intensely in recent years, see e.g. [7,13,26,27,28,34,36]. In these papers, the root systems B 1 × • • • B 1 are commonly considered.…”

Laguerre expansions on conic domains

“…If f (x, t) = f 0 (t) with f 0 : R + → R, then the orthogonal expansions of f on the conic domain becomes the classical Laguerre expansions on R + , which has been studied extensively in the literature; see [1,7,8,9,10,14,17,18] and, for some more recent works, see [2,11,12,15] and the references therein. As in the study of classical Laguerre expansions, our study relies on intrinsic properties of orthogonal polynomials that hold only for particular weight functions and special domains.…”