Boundedness of multidimensional Hausdorff operators in 𝐻^{𝑝} spaces, 0<𝑝<1

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 .