Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

Abstract

In this paper, we study the boundedness of the Hausdorff operator Hϕ on the real line ℝ. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space Lp(ℝ) and the Hardy space H1(ℝ). The key idea is to reformulate Hϕ as a Calderon-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 Hp(ℝ) 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 Hp(ℝ) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H1(ℝ).