The Hausdorff operators on the real Hardy spaces HP(R)

We prove that the Hausdorff operator generated by a function phi is bounded on the real Hardy space H-p(R), 0 < p less than or equal to 1, if the Fourier transform phi of phi satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesaro operator of order alpha on H-p(R), 2/(2alpha + 1) < p less than or equal to 1. Our proof is based on the atomic decomposition and molecular characterization of H-p(R).