Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈...Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.展开更多
Continuous, SL(n) and translation invariant real-valued valuations on Sobolev spaces are classified.The centro-affine Hadwiger's theorem is applied. In the homogeneous case, these valuations turn out to be L^p-nor...Continuous, SL(n) and translation invariant real-valued valuations on Sobolev spaces are classified.The centro-affine Hadwiger's theorem is applied. In the homogeneous case, these valuations turn out to be L^p-norms raised to p-th power(up to suitable multipication scales).展开更多
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
For a commutative ring R with a unit, an R-homology rose is a topological space whose homology groups with R-coefficients agree with those of a bouquet of circles. In this paper, we study some special properties of th...For a commutative ring R with a unit, an R-homology rose is a topological space whose homology groups with R-coefficients agree with those of a bouquet of circles. In this paper, we study some special properties of the fundamental groups of R-homology roses and their covering spaces, from which we obtain some results supporting the Carlsson conjecture on free(Zp)ractions. In addition, we discuss how to search candidates of the counterexamples of Wall's D(2)-problem among R-homology roses and R-acyclic spaces and propose some candidates.展开更多
基金National Natural Science Foundation of China (Grant Nos. 10901018 and 11001002)the Shanghai Leading Academic Discipline Project (Grant No. J50101)the Fundamental Research Funds for the Central Universities
文摘Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.
基金supported by Austrian Science Fund Project(Grant No.P23639-N18)National Natural Science Foundation of China(Grant No.11371239)
文摘Continuous, SL(n) and translation invariant real-valued valuations on Sobolev spaces are classified.The centro-affine Hadwiger's theorem is applied. In the homogeneous case, these valuations turn out to be L^p-norms raised to p-th power(up to suitable multipication scales).
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 No.11371188)the PAPD(Priority Academic Program Development)of Jiangsu Higher Education Institutions
文摘For a commutative ring R with a unit, an R-homology rose is a topological space whose homology groups with R-coefficients agree with those of a bouquet of circles. In this paper, we study some special properties of the fundamental groups of R-homology roses and their covering spaces, from which we obtain some results supporting the Carlsson conjecture on free(Zp)ractions. In addition, we discuss how to search candidates of the counterexamples of Wall's D(2)-problem among R-homology roses and R-acyclic spaces and propose some candidates.
