Boundedness of the Dunkl–Hausdorff operator in Lebesgue spaces

Abstract

In this paper, the $L_v^p(ℝ)$-boundedness of the Dunkl–Hausdorff operator

$$H_{\alpha ,\varphi }f(x)={\displaystyle {\int }_{ℝ}\frac{\left|\varphi (t)\right|}{|t{|}^{2\alpha +2}}f\left(\frac{x}{t}\right)dt}$$

has been characterized and for a certain type of weight $v$, the operator norm $\parallel H_{\alpha ,\varphi }{\parallel }_{L_v^p(ℝ)\to L_v^p(ℝ)}$ has been obtained. This covers several of the existing results. Analogous results in two dimensions have also been proved.