Abstract. For a wide family of multivariate Hausdorff operators, a new condition for the boundedness of an operator from this family on the real Hardy space $H^1({\msbm R}^n)$ is proved.

AMS Subject Classification (1991): 47B38, 42B10; 42B35

Keyword(s): Hausdorff operator, real Hardy space, atomic decomposition, eigenvalues

Received April 24, 2008. (Registered under 6049/2009.)