  A unified approach to the concept of a Hausdorff operator is proposed in such
a way that a number of classical and new operators feet into the given
definition. Conditions are given for the boundedness of the operators under
consideration in $L^p$ and in the atomic Hardy space $H^1$, and their
regularity property is investigated.
  Examples are considered. The author hopes that this approach will allow one
to unify the study of a lot of extensions and analogs of the classical
Hausdorff operator.
