高维Hausdorff算子在Hp(Rn)上的有界性

本文主要研究以下形式的Hausdorff算子H_Φf(x)=∫_RⁿΦ(u₁….,u_n)f(u₁x₁,…,u_nx_n)du₁…du_n,其中Φ是Rⁿ上的缓增分布.当n≥2,0Φ在H^p(Rⁿ)上有界当且仅当Φ≡0.进一步,当n≥2,n/n+1Φ有合适定义,那么H_Φ在H^p(Rⁿ)上有界当且仅当Φ是常数.这些结果都表明Hausdorff算子H_Φ在H^p(Rⁿ)上的有界性很复杂.此外,我们将H_Φ转化成卷积型算子,得到H_Φ在Lebesgue空间上有界的一些新的结果.

The Multidimensional Hausdorff Operators on Hp(Rn)

We consider the following Hausdorff operator H_Φf(x)=∫_RⁿΦ(u₁,…,u_n)·f(u₁x₁,…,u_nx_n)du₁…du_n,whereΦcan be considered as a distribution on Rⁿ.When n≥2 andΦis a Schwartz fu_nction,we show that H_Φis bou_nded on H^p(Rⁿ)for some p∈(0,1)if and only ifΦ≡0.Furthermore,when n≥2,ifΦis just a continuous fu_nction and H_Φcan be defined suitable,then we can also prove that H_Φis bou_nded on H^p(Rⁿ)for some p∈(n/n+1,1)if and only ifΦequals to a constant.These facts mean that H_Φis very complicated on H^p(Rⁿ)(n≥2).Moreover,we establish a result of the bou_ndedness of H_Φon Lp(Rⁿ),p>1.The key idea used here is to reformulate H_Φas a convolution operator.