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

Abstract

We study the boundedness of the multiplier of the interval $[-1,1]$ for the Dunkl transform of order $\alpha >-1$ on weighted $L^p$ spaces, with $1<p<\mathrm{\infty }$. In particular, we get that it is bounded from $L^p(\mathbb{R},|x{|}^{2\alpha +1}\mathrm{d}x)$ into itself if and only if $4(\alpha +1)/(2\alpha +3)<p<4(\alpha +1)/(2\alpha +1)$ when $\alpha >-\frac{1}{2}$ or if and only if $1<p<\mathrm{\infty }$ when $-1<\alpha \le -\frac{1}{2}$.

Keywords:

• Dunkl operator
• Dunkl transform
• Bessel functions
• multipliers

AMS Subject Classifications:

• 42A45
• 42A38

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Juan L. Varona http://orcid.org/0000-0002-2023-9946

Additional information

Funding

The research of the authors is supported by the DGI (Ministerio de Economía y Competitividad, Spain) under Grant [MTM2012-36732-C03-02].