Estimates of fractional Hausdorff operators on weighted Lebesgue and Hardy spaces

In this article, we obtain some necessary and sufficient conditions for the boundedness of fractional Hausdorff operators $h_{\mathrm{\Phi },\beta }$ on weighted Lebesgue spaces $(0\le \beta <1)$, which are fractional variants of Bandaliev-Safarova [Hacet. J. Math. Stat., 2021, 50, 1334-1346]; it is found that a new constraint for $\beta$ should be added and it holds automatically for non-fractional variants in [HJMS, 2021] $(\beta =0)$. Then, we further obatin the boundedness of fractional Hausdorff operators $h_{\mathrm{\Phi },\beta }$ on power-weighted Hardy spaces which are fractional variants of Ruan-Fan [Math. Nachr., 2017, 290, 2388-2400]. Ruan-Fan obtained the relevant boundedness results by means of the radial maximal function characterization of Hardy spaces, while in this paper, two different relevant results are obtained respectively by using the radial maximal function characterization and the Riesz characterization of power-weighted Hardy spaces.