Abstract. The Hausdorff operators defined by suitable signed measures on $R=(-\infty,\infty )$ are shown to be bounded on $L^p(R^n)$, on the real Hardy space $H^1(R^n)$, and on the space of bounded mean oscillation $BMO(R^n)$. An example is given that negatively resolves a related conjecture of Móricz.

AMS Subject Classification (1991): 47B38, 46A30

Keyword(s): Fourier transform, Riesz transforms, Hausdorff operator, Ces?ro operator, Lebesgue spaces, Hardy Spaces, Bounded Mean Oscillation

Received December 14, 2001, and in revised form June 6, 2002. (Registered under 2904/2009.)