Sufficient conditions, of both smoothness and algebraic type, for the boundedness of multidimensional Hausdorff operators in Hardy spaces
H
p
(
R
n
)
H^p (\mathbb {R}^n)
,
n
≥
2
n\ge 2
,
0
>
p
>
1
0>p>1
, are given. Positive and negative examples illustrate all aspects of the behavior of such operators and reveal the special nature of the case
p
>
1
p>1
.