On the generalized Dunkl Dini–Lipschitz spaces

Abstract

In this paper we introduce function spaces denoted by $Li{p}_{k,n}(\text{β},\alpha ,p)$ ($0<\text{β}<1,\alpha \ge 0$) as subspaces of $L^p$ associated with $(k,n)$-generalized Fourier transform arising from the Dunkl theory, that we call generalized Dunkl Dini–Lipschitz spaces. Firstly, for $1<p\le 2$, we provide characterizations of these spaces by means modulus of continuity. Moreover, we obtain behaviour estimations related to $(k,n)$-generalized Fourier transform for $0<\text{β}<1$ and $\alpha \ge 0$, and for $\alpha =0$ we obtain an analogue of Titchmarsh's theorem in $L^p$. In the sequel, we introduce the modulus of smoothness related to the $(k,n)$-generalized Fourier transform for which we derive some of its properties on the Sobolev type space denoted by ${\mathcal{W}}_{p,n}^l$.

Keywords:

• $(k,n)$-generalized Fourier transform
• Dini–Lipschitz space
• modulus of continuity
• modulus of smoothness
• Sobolev space

Mathematics Subject Classifications:

• 42B10
• 26A16
• 26A15

Acknowledgments

The author is grateful to the reviewers for the useful comments and the valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.