The Hausdorff operator is bounded on the real Hardy space H-1(R)

We prove that the Hausdorff operator generated by a function phi is an elem ent of L-1 ( R) is bounded on the real Hardy space H-1 ( R). The proof is b ased on the closed graph theorem and on the fact that if a function f in L- 1(R) is such that its Fourier transform (f) over cap(t) equals 0 for t < 0 (or for t >0), then f is an element of H-1 (R).