The authors study the singular integral operatorT_~Ω,α f(x)=p.v.∫_~Rn b(｜y｜)Ω(y′)｜y｜^-n-α f(x-y)dy,defined on all test functions f,where b is a bounded function,α>0,Ω(y′) is an integrable function on t...The authors study the singular integral operatorT_~Ω,α f(x)=p.v.∫_~Rn b(｜y｜)Ω(y′)｜y｜^-n-α f(x-y)dy,defined on all test functions f,where b is a bounded function,α>0,Ω(y′) is an integrable function on the unit sphere S^n-1 satisfying certain cancellation conditions.It is proved that,for n/(n+α)<p<∞,T_~Ω,α is a bounded operator from the Hardy-Sobolev space Hp_α to the Hardy space Hp.The results and its applications improve some theorems in a previous paper of the author and they are extensions of the main theorems in Wheeden's paper(1969).The proof is based on a new atomic decomposition of the space Hp_α by Han,Paluszynski and Weiss(1995).By using the same proof,the singluar integral operators with variable kernels are also studied.展开更多
The authors define the directional hyper Hilbert transform and give ita mixed norm estimate. The similar conclusions for the directional fractional integral of one dimension are also obtained in this paper. As an appl...The authors define the directional hyper Hilbert transform and give ita mixed norm estimate. The similar conclusions for the directional fractional integral of one dimension are also obtained in this paper. As an application of the above results, the authors give the Lp-boundedness for a class of the hyper singular integrals and the fractional integrals with variable kernel. Moreover, as another application of the above results, the authors prove the dimension free estimate for the hyper Riesz transform. This is an extension of the related result obtained by Stein.展开更多
文摘The authors study the singular integral operatorT_~Ω,α f(x)=p.v.∫_~Rn b(｜y｜)Ω(y′)｜y｜^-n-α f(x-y)dy,defined on all test functions f,where b is a bounded function,α>0,Ω(y′) is an integrable function on the unit sphere S^n-1 satisfying certain cancellation conditions.It is proved that,for n/(n+α)<p<∞,T_~Ω,α is a bounded operator from the Hardy-Sobolev space Hp_α to the Hardy space Hp.The results and its applications improve some theorems in a previous paper of the author and they are extensions of the main theorems in Wheeden's paper(1969).The proof is based on a new atomic decomposition of the space Hp_α by Han,Paluszynski and Weiss(1995).By using the same proof,the singluar integral operators with variable kernels are also studied.
基金the 973 Project of China(No.G1999075105)the National Natural ScienceFoundation of China(No.19631080,No.10271016)the Zhejiang Provincial Natural ScienceFoundation of China(No.RC97017,No.197042).
文摘The authors define the directional hyper Hilbert transform and give ita mixed norm estimate. The similar conclusions for the directional fractional integral of one dimension are also obtained in this paper. As an application of the above results, the authors give the Lp-boundedness for a class of the hyper singular integrals and the fractional integrals with variable kernel. Moreover, as another application of the above results, the authors prove the dimension free estimate for the hyper Riesz transform. This is an extension of the related result obtained by Stein.
