Weighted integrability of the Cherednik-Opdam transform in terms of the moduli of smoothness

In this paper, we give sufficient conditions for functions defined on the space \(L^{p}(\mathbb {R},d\mu )\) $L^p(\mathbb{R},d\mu )$ , \(1<p\le 2\) $1<p\le 2$ , providing the weighted integrability of their Cherednik-Opdam transforms. These results generalize a famous Titchmarsh’s theorem and Younis’ theorem for functions from Lipschitz classes.