Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H (I center dot) on the real line a"e. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L (p) (a"e) and the Hardy space H (1)(a"e). The key idea is to reformulate H (I center dot) as a Caldern-Zygmund convolution operator, from which its boundedness is proved by verifying the Hormander condition of the convolution kernel. Secondly, to prove the boundedness on the Hardy space H (p) (a"e) 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) (a"e) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H (1)(a"e).