Boundedness of multidimensional Hausdorff operators on H1(Rn)H1(Rn)

In this paper, we consider the H1H1-boundedness of multidimensional Hausdorff operators defined by HΦ,Af(x)=∫RnΦ(u)f(A(u)x)du,HΦ,Af(x)=∫RnΦ(u)f(A(u)x)du, where Φ∈LLoc1(Rn), A(u)=(aij(u))i,j=1n is an n×nn×n matrix, and each aij(u)aij(u) is a measurable function of uu. Let ‖B‖=∑i,j=1n|bij| for the matrix B=(bij(u))i,j=1n. We prove that HΦ,AHΦ,A is bounded from the Hardy space H1H1 to itself if ∫Rn|Φ(u)detA−1(u)|ln(1+‖A−1(u)‖n|detA−1(u)|)du<∞. Our result improves known results. In addition, we show that the above condition is optimal in the size condition.