We study the maximal super-singular integral operator T＊Ω,α,β（f）（x,y）=sup ∈1〉0,∈2〉0｜∫｜u｜〉ε1,｜v｜〉ε2 b1（｜u｜）b2（｜u｜）Ω（u＇,u＇）/｜u｜^n＋α｜u｜^m＋β-f（x-u,y-u）dudu｜defined on all f ...We study the maximal super-singular integral operator T＊Ω,α,β（f）（x,y）=sup ∈1〉0,∈2〉0｜∫｜u｜〉ε1,｜v｜〉ε2 b1（｜u｜）b2（｜u｜）Ω（u＇,u＇）/｜u｜^n＋α｜u｜^m＋β-f（x-u,y-u）dudu｜defined on all f ∈ S（R^n ×R^m）, where 0 ≤ α,β〈∞, b1 b2 ∈ L∞（R＋1 ）,Ω satisfies certain cancellation conditions and Ω∈L1（S^n-1×S^m-1）in the case α,β〉0;Ω∈L（log＋L）（S^n-1×S^m-1）in the case αβ=0 and α＋β 〉0. It is proved that, for 1〈p〈∞.T＊Ω,α,βis a bounded operator from the homogeneous Sobolev space Lα,β^p（R^n×R^m）to the Lebesgue space L^p（R^n×R^m）.展开更多
In this paper,we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space Lpγ(Rn) to the Lebesgue space Lp(Rn),which is a substantial improvement and an exte...In this paper,we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space Lpγ(Rn) to the Lebesgue space Lp(Rn),which is a substantial improvement and an extension of some known results.展开更多
The purpose of this paper is to study the mapping properties of the singular Radon transforms with rough kernels. Such singular integral operators are proved to be bounded on Lebesgue spaces.
In this paper,the relationship between the extended family and several mixing properties in measuretheoretical dynamical systems is investigated.The extended family eF related to a given family F can be regarded as th...In this paper,the relationship between the extended family and several mixing properties in measuretheoretical dynamical systems is investigated.The extended family eF related to a given family F can be regarded as the collection of all sets obtained as"piecewise shifted"members of F.For a measure preserving transformation T on a Lebesgue space(X,B,μ),the sets of"accurate intersections of order k"defined below are studied,Nε(A0,A1,...,Ak)=n∈Z+:μk i=0T inAiμ(A0)μ(A1)μ(Ak)<ε,for k∈N,A0,A1,...,Ak∈B and ε>0.It is shown that if T is weakly mixing(mildly mixing)then for any k∈N,all the sets Nε(A0,A1,...,Ak)have Banach density 1(are in(eFip),i.e.,the dual of the extended family related to IP-sets).展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 10871173, 10931001)
文摘We study the maximal super-singular integral operator T＊Ω,α,β（f）（x,y）=sup ∈1〉0,∈2〉0｜∫｜u｜〉ε1,｜v｜〉ε2 b1（｜u｜）b2（｜u｜）Ω（u＇,u＇）/｜u｜^n＋α｜u｜^m＋β-f（x-u,y-u）dudu｜defined on all f ∈ S（R^n ×R^m）, where 0 ≤ α,β〈∞, b1 b2 ∈ L∞（R＋1 ）,Ω satisfies certain cancellation conditions and Ω∈L1（S^n-1×S^m-1）in the case α,β〉0;Ω∈L（log＋L）（S^n-1×S^m-1）in the case αβ=0 and α＋β 〉0. It is proved that, for 1〈p〈∞.T＊Ω,α,βis a bounded operator from the homogeneous Sobolev space Lα,β^p（R^n×R^m）to the Lebesgue space L^p（R^n×R^m）.
基金supported by National Natural Science Foundation of China (Grant Nos.10901017 and 10931001)Program for New Century Excellent Talents in University (Grant No. NCET-11-0574)Doctoral Fund of Ministry of Education of China (Grant No. 20090003110018)
文摘In this paper,we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space Lpγ(Rn) to the Lebesgue space Lp(Rn),which is a substantial improvement and an extension of some known results.
基金the National Natural Science Fundation of China(Nos.10371087,10671041).
文摘The purpose of this paper is to study the mapping properties of the singular Radon transforms with rough kernels. Such singular integral operators are proved to be bounded on Lebesgue spaces.
基金supported by National Natural Science Foundation of China(Grant Nos.10926038 and 11201157)Fundamental Research Funds for the Central Universities(Grant No.2012ZZ0073)
文摘In this paper,the relationship between the extended family and several mixing properties in measuretheoretical dynamical systems is investigated.The extended family eF related to a given family F can be regarded as the collection of all sets obtained as"piecewise shifted"members of F.For a measure preserving transformation T on a Lebesgue space(X,B,μ),the sets of"accurate intersections of order k"defined below are studied,Nε(A0,A1,...,Ak)=n∈Z+:μk i=0T inAiμ(A0)μ(A1)μ(Ak)<ε,for k∈N,A0,A1,...,Ak∈B and ε>0.It is shown that if T is weakly mixing(mildly mixing)then for any k∈N,all the sets Nε(A0,A1,...,Ak)have Banach density 1(are in(eFip),i.e.,the dual of the extended family related to IP-sets).
