Boundedness of Dunkl-Hausdorff operator in Lebesgue spaces

TL;DR

This paper characterizes the boundedness of the Dunkl-Hausdorff operator on weighted Lebesgue spaces, providing explicit norm calculations and extending results to two-dimensional cases.

Contribution

It offers a complete characterization of the operator's boundedness and computes the exact operator norm for certain weights, extending previous results to higher dimensions.

Findings

Dunkl-Hausdorff operator is bounded on specific weighted Lebesgue spaces.

Explicit formulas for the operator norm are derived.

Results are extended to two-dimensional settings.

Abstract

In this paper, the $L^p_v(\R)$-boundedness of the Dunkl-Hausdorff operator $\displaystyle H_{\al,\phi} f(x)=\ent\frac{ |\phi(t)|}{|t|^{2\al+2}}f\lf(\frac{x}{t}\rh) dt $ has been characterized and for a certain type of weight $v$, the precise value of the norm $\|H_{\al,\phi}\|_{L^p_v(\R)\to L^p_v(\R)}$ has been obtained. This covers several of the existing results. Analogous results in two dimensions have also been proved.