Lp-Fourier multipliers for the Dunkl operator on the real line

Abstract

We consider Fourier multipliers for L p associated with the Dunkl operator on R and establish a version of Ho¨rmander's multiplier theorem. In applying this version, we come up with some results regarding the oscillating multipliers, partial sum operators and generalized Bessel potentials. r 2004 Elsevier Inc. All rights reserved. MSC: primary 43A62; secondary 43A15; 43A32

The multiplier of the interval [−1,1] for the Dunkl transform on the real line

“…Of course, the target is to extend the harmonic analysis of the Fourier transform to a more general context. For instance, let us cite [6,9,12], where many other references can be found. The behavior of the Bessel functions is very well known.…”

The Dunkl-Wigner transforms on the real line

“…The Proposition 2.2 allows us to establish the following properties for the Dunkl convolution on R (see [9,17]). …”

Linear and Bilinear Multiplier Operators for the Dunkl Transform

“…Obviously, away from the origin k is not integrable function. Therefore the boundedness of H α cannot be obtained by the L p -version given in [13].…”

“…From which he deduce that the distribution F −1 α (m) is an integrable function at infinity with respect to the measure dμ α (x) = (2 α+1 Γ(α + 1)) −1 |x| 2α+1 dx (1.1) where α > −1/2. Let us point out that the paper of F. Soltani [13] contained a false application (Example 3) as we can check in [2]. The aim of this work is to prove the Hörmander multiplier theorem for the Dunkl transform in its great generality by using the Hörmander's technique.…”

“…Later, another version of L p -multiplier theorem associated with the Dunkl transform is given in [13], using the J. Ph. Anker argument [1].…”