Linear and Bilinear Multiplier Operators for the Dunkl Transform

Abstract

The main purpose of this article is to study the L p -boundedness of linear and bilinear multiplier operators for the Dunkl transform in the one dimensional case.Mathematics Subject Classification (2010). Primary 42B15, 42B20; Secondary 46F12, 46E30.

Riesz transforms for Dunkl transform

“…In [2] this result is improved by removing 2γ k ∈ N, where Riesz transform is called Hilbert transform. If γ k = 0 (k = 0), this operator coincides with the usual Riesz transform R j given by (1.1).…”

On fundamental harmonic analysis operators in certain Dunkl and Bessel settings

“…Riesz transforms in the Dunkl setting have also drawn considerable attention. L p mapping properties of the first order Riesz transforms were investigated in the one-dimensional situation by Thangavelu and Xu [23] and then by Amri, Gasmi and Sifi in [2]. Later on Amri and Sifi [3] developed a variant of Calderón-Zygmund theory, which turned out to be well suited to the general Dunkl framework and, in particular, allowed them to obtain unweighted L p bounds for the first order Riesz transforms.…”

“…Our considerations fit into a recent line of research connected with analysis for "low" values of type parameters, which was developed in the Bessel [5], Jacobi [11] and Laguerre [16] settings. Furthermore, in comparison with [2,3,22,23] we obtain weighted L p estimates with a large class of weights admitted.…”

Multivariate spectral multipliers for the Dunkl transform and the Dunkl harmonic oscillator

“…In [1,Theorem 3.1] Amri, Gasmi and Sifi obtained a one-dimensional version of Theorem 3.1, assuming m satisfies a certain Hörmander's type condition of integer order greater than˛C 1=2. It is not hard to verify that, for d D 1, their condition implies condition (3.1) of our Theorem 3.1.…”