Weighted function spaces and Dunkl transform

TL;DR

This paper introduces Besov-Dunkl spaces, characterizes them, and establishes weighted Lp-estimates for the Dunkl transform, demonstrating that functions in these spaces have Dunkl transforms in L1, extending classical Fourier analysis results.

Contribution

It develops new weighted function spaces based on Dunkl convolution, provides their characterizations, and derives Lp-estimates for the Dunkl transform in these spaces.

Findings

Dunkl transform of functions in Besov-Dunkl spaces is in L1.

Weighted Lp-estimates of the Dunkl transform are established.

Characterizations of Besov-Dunkl spaces via function decomposition.

Abstract

We introduce first weighted function spaces on Rd using the Dunkl convolution that we call Besov-Dunkl spaces. We provide characterizations of these spaces by decomposition of functions. Next we obtain in the real line and in radial case on Rd weighted Lp-estimates of the Dunkl transform of a function in terms of an integral modulus of continuity which gives a quantitative form of the Riemann-Lebesgue lemma. Finally, we show in both cases that the Dunkl transform of a function is in L1 when this function belongs to a suitable Besov-Dunkl space.