Hausdorff operators on function spaces

Abstract

The authors mainly study the Hausdorff operators on Euclidean space ℝn. They establish boundedness of the Hausdorff operators in various function spaces, such as Lebesgue spaces, Hardy spaces, local Hardy spaces and Herz type spaces. The results reveal that the Hausdorff operators have better performance on the Herz type Hardy spaces \(H\dot K_q^{\alpha ,p} (\mathbb{R}^n )\) than their performance on the Hardy spaces Hp(ℝn) when 0 < p < 1. Also, the authors obtain some new results and reprove or generalize some known results for the high dimensional Hardy operator and adjoint Hardy operator.