Some estimates on the Hausdorff operator

Abstract

Recently, sufficent conditions for the Hp boundedness of the one-dimensional Hausdorff operator were given by Liflyand and Miyachi. In this paper, we obtain new sufficent conditions for the Hp boundedness of the one-dimensional Hausdorff operator. The results of Liflyand and Miyachi and the results of this paper are mutually independent. More importantly, our method in the proof allows us to study the high dimensional Hausdorff operator and fractional Hausdorff operator. We then obtain Hp(ℝn) → Lq(ℝn) and Lp(|x|γdx) → Lq(|x|γdx) boundedness for the high dimensional (fractional) Hausdorff operator.