Boundedness of multidimensional Hausdorff operators on

In this paper, we consider the H 1 -boundedness of multidimensional Hausdorff operators defined by H Φ , A f ( x ) = ∫ R n Φ ( u ) f ( A ( u ) x ) d u , where Φ ∈ L L o c 1 ( R n ) , A ( u ) = ( a i j ( u ) ) i , j = 1 n is an n × n matrix, and each a i j ( u ) is a measurable function of u . Let ‖ B ‖ = ∑ i , j = 1 n | b i j | for the matrix B = ( b i j ( u ) ) i , j = 1 n . We prove that H Φ , A is bounded from the Hardy space H 1 to itself if ∫ R n | Φ ( u ) det A − 1 ( u ) | ln ( 1 + ‖ A − 1 ( u ) ‖ n | det A − 1 ( u ) | ) d u < ∞ . Our result improves known results. In addition, we show that the above condition is optimal in the size condition.