Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators

Abstract

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Mixed norm estimates for the Riesz transforms on the Heisenberg group

“…The key point is to use an idea of Rubio de Francia [27] along with a lemma of Herz and Riviere [18]. The same technique was successfully employed to prove mixed norm estimates for Riesz transforms associated to the Dunkl harmonic oscillator [24] and also for Riesz transforms on compact Lie groups [26]. In this article we demonstrate that mixed norm estimates can also be proved for Riesz transforms on the Heisenberg groups H d .…”

“…We use the same idea used in Sect. 4.2 of [24] to study the Hermite-Riesz transforms. First, we prove the weighted mixed norm inequality for S j := S j (−1).…”

“…But the mixed norm inequalities for these operators are not known. Note that the mixed norm estimates for the Riesz transforms associated with Dunkl-Hermite operators and the Riesz transforms on the group SU (2) have been proved in the articles [24,26] respectively.…”

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

“…For basic concepts of this theory see Dunkl's pioneering work [16] and, for instance, the survey article by Rösler [35]. Harmonic analysis related to the Dunkl harmonic oscillator (DHO in short) has been intensively studied in recent years by the authors [29,30,31,43,44] and many other mathematicians, see, e.g., [3,6,7,12,13,26,46] and references therein. A commonly appearing assumption in this literature is that the underlying multiplicity function is non-negative.…”

“…It is worth mentioning that recently unweighted L p -boundedness of first order Riesz transforms and imaginary powers associated with the DHO and an arbitrary finite group of reflections was obtained by Amri [3] and Amri and Tayari [6], respectively. See [12,13] for more results in the general DHO setting. Our present analysis is a natural, but by no means trivial, first step towards conjecturing and proving further results for the DHO and a general reflection group, possibly with non-positive multiplicity functions admitted.…”