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

Abstract

In this paper we study weighted mixed norm estimates for Riesz transforms associated to Dunkl harmonic oscillators. The idea is to show that the required inequalities are equivalent to certain vector valued inequalities for operator defined in terms of Laguerre expansions. In certain cases the main result can be deduced from the corresponding result for Hermite Riesz transforms.

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

“…On the other hand, in [20] the derivatives are as general as in our present result, but neither weights are allowed nor the case p = 1 is treated there (the main objective of [20] are dimension free L p estimates). Finally, we note that recently Riesz transforms associated with the DHO and an arbitrary group of reflections were studied by Amri [3] and Boggarapu and Thangavelu [13], in both cases with only non-negative multiplicity functions admitted. More precisely, in [3] unweighted L p -boundedness, 1 < p < ∞, and weak type (1, 1) for Riesz-Dunkl transforms of order 1 (defined by means of counterparts of D i , but not D * i ) were obtained.…”

“…More precisely, in [3] unweighted L p -boundedness, 1 < p < ∞, and weak type (1, 1) for Riesz-Dunkl transforms of order 1 (defined by means of counterparts of D i , but not D * i ) were obtained. In [13] the authors prove mixed norm estimates (weighted L p,2 -boundedness, 1 < p < ∞) for Riesz-Dunkl transforms of order 1 defined via the counterparts of both D i and D * i . Our Theorem 3.1 suggests that the results of [3,13] can be substantially generalized.…”

“…In [13] the authors prove mixed norm estimates (weighted L p,2 -boundedness, 1 < p < ∞) for Riesz-Dunkl transforms of order 1 defined via the counterparts of both D i and D * i . Our Theorem 3.1 suggests that the results of [3,13] can be substantially generalized.…”

“…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.…”