高维Hausdorff算子在H^(p)(R^(n))上的有界性  

The Multidimensional Hausdorff Operators on H^(p)(R^(n))

作　　者：何少勇 朱相荣 Shao Yong HE;Xiang Rong ZHU(College of Mathematics and Computer Science,Zhejiang Normnal University,Jinhua 321004,P.R.China)

机构地区：[1]浙江师范大学数学与计算机科学学院,金华321004

出　　处：《数学学报（中文版）》2021年第2期289-300,共12页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11871436)。

摘　　要：本文主要研究以下形式的Hausdorff算子H_(Φ)f(x)=∫_(R^(n))Φ(u_(1)….,u_(n))f(u_(1)x_(1),…,u_(n)x_(n))du_(1)…du_(n),其中Φ是R^(n)上的缓增分布.当n≥2,0<p<1,若Φ是Schwartz函数,我们得到H_(Φ)在H^(p)(R^(n))上有界当且仅当Φ≡0.进一步,当n≥2,n/n+1<p<1,如果Φ仅仅是连续函数,并且H_(Φ)有合适定义,那么H_(Φ)在H^(p)(R^(n))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)(R^(n))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.We consider the following Hausdorff operator H_(Φ)f(x)=∫_(R^(n))Φ(u_(1),…,u_(n))·f(u_(1)x_(1),…,u_(n)x_(n))du_(1)…du_(n),whereΦcan be considered as a distribution on R^(n).When n≥2 andΦis a Schwartz fu_(n)ction,we show that H_(Φ)is bou_(n)ded on H^(p)(R^(n))for some p∈(0,1)if and only ifΦ≡0.Furthermore,when n≥2,ifΦis just a continuous fu_(n)ction and H_(Φ)can be defined suitable,then we can also prove that H_(Φ)is bou_(n)ded on H^(p)(R^(n))for some p∈(n/n+1,1)if and only ifΦequals to a constant.These facts mean that H_(Φ)is very complicated on H^(p)(R^(n))(n≥2).Moreover,we establish a result of the bou_(n)dedness of H_(Φ)on Lp(R^(n)),p>1.The key idea used here is to reformulate H_(Φ)as a convolution operator.

关 键 词：Hausdorff算子 LEBESGUE空间 HARDY空间 

分 类 号：O174.2[理学—数学]