Abstract. Recently, sufficent conditions for the $H^p$ boundedness of the one-dimensional Hausdorff operator were given by Liflyand and Miyachi. In this paper, we obtain new sufficent conditions for the $H^p$ boundedness of the one-dimensional Hausdorff operator. The results of Liflyand and Miyachi and the results of this paper are mutually independent. More importantly, our method in the proof allows us to study the high dimensional Hausdorff operator and fractional Hausdorff operator. We then obtain $H^p({\msbm R}^n)\rightarrow L^q({\msbm R}^n)$ and $L^p(| x| ^{\gamma }dx)\rightarrow L^q(| x| ^{\gamma }dx)$ boundedness for the high dimensional (fractional) Hausdorff operator.

AMS Subject Classification (1991): 47B38, 47D05

Keyword(s): Hausdorff operator, Hardy spaces, Marcinkiewicz interpolation, Lipschitz spaces

Received December 19, 2011, and in revised form May 2, 2012. (Registered under 67/2011.)