Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H_? on the real line ?. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L^p(?) and the Hardy space H¹(?). The key idea is to reformulate H_? as a Calderón-Zygmund convolution operator, from which its boundedness is proved by verifying the Hörmander condition of the convolution kernel. Secondly, to prove the boundedness on the Hardy space H^p(?) with 0 < p < 1; we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H^p(?) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H¹(?).