EN
We prove that the Hausdorff operator generated by a function ϕ is bounded on the real Hardy space $H^{p}(ℝ)$, 0 < p ≤ 1, if the Fourier transform ϕ̂ of ϕ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on $H^{p}(ℝ)$, 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of $H^{p}(ℝ)$.