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COMPENSATED COMPACTNESS AND THE STRATIFIED LIE GROUP
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作者 Yu Liu 《Analysis in Theory and Applications》 2009年第2期101-108,共8页
In this paper we prove that the Jacobian J(F) of a map F(f1,…,f1 from Ginto Rt maps the product of Lebesgue space Lp1×…× Lp1 into local Hardy space hY(G),whereQ/Q+1〈r〈1,and Q is the homogeneous dim... In this paper we prove that the Jacobian J(F) of a map F(f1,…,f1 from Ginto Rt maps the product of Lebesgue space Lp1×…× Lp1 into local Hardy space hY(G),whereQ/Q+1〈r〈1,and Q is the homogeneous dimension of the stratified Lie group G. 展开更多
关键词 compensated compactness stratified Lie group Hardy space maximal function
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Hardy Type Estimates for Riesz Transforms Associated with Schrdinger Operators on the Heisenberg Group
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作者 Yu Liu Guobin Tang 《Analysis in Theory and Applications》 CSCD 2016年第1期78-89,共12页
Abstract. Let H^n be the Heisenberg group and Q = 2n+2 be its homogeneous dimen- sion. In this paper, we consider the Schr6dinger operator -△H^n +V, where △H^n is the sub-Laplacian and V is the nonnegative potenti... Abstract. Let H^n be the Heisenberg group and Q = 2n+2 be its homogeneous dimen- sion. In this paper, we consider the Schr6dinger operator -△H^n +V, where △H^n is the sub-Laplacian and V is the nonnegative potential belonging to the reverse H61der class Bql for ql _〉 Q/2. We show that the operators T1 = V(-△H^n-In +V)-1 and T2 = V1/2(-△H^n-V)-1/2 are both bounded from 1 n HL^1(H^n ) into L1(H^n). Our results are also valid on the stratified Lie group. 展开更多
关键词 Heisenberg group stratified Lie group reverse H61der class Riesz transform Schr6dinger operator.
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Weighted High Order Poincaré Inequalities on Stratified Lie Groups
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作者 HAN Yecong HUANG Tiren +1 位作者 XU Chengdan ZHU Tingjing 《数学进展》 CSCD 北大核心 2024年第5期1071-1083,共13页
In this paper,we focus on studying weighted Poincare inequalities on stratified Lie groups.We derive various Poincaréinequalities in the case 1<p=q<∞ in the high order Sobolev space Wm,p.We derive several ... In this paper,we focus on studying weighted Poincare inequalities on stratified Lie groups.We derive various Poincaréinequalities in the case 1<p=q<∞ in the high order Sobolev space Wm,p.We derive several Poincare inequalities that complement existing results,which have only been proved for the case 1<p<q<∞. 展开更多
关键词 A_(p)weight Poincaréinequality entire space stratified Lie group high order Sobolev space
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Polynomials,Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups 被引量:10
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作者 Guozhen Lu Department of Mathematics,Wayne State University,Detroit,MI 48202,USA 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2000年第3期405-444,共40页
This paper consists of three main parts.One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups.Despite ... This paper consists of three main parts.One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups.Despite the extensive research after Jerison’s work[3]on Poincaré-type inequalities for Hrmander’s vector fields over the years,our results given here even in the nonweighted case appear to be new.Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE’s involving vector fields.The main tools to prove such inequalities are approximating the Sobolev func- tions by polynomials associated with the left invariant vector fields on G.Some very useful properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights.Finding the existence of such polynomials is the second main part of this paper.Main results of these two parts have been announced in the author’s paper in Mathematical Research Letters[38]. The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on(εδ)domains. Some results of weighted Sobolev spaces are also given here.We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously.In particular,we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions.Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups. 展开更多
关键词 Poincaré inequalities Extension theorems Interpolation inequalities Anisotropic Sobolev spaces A_p weights δ)domains Vector fields Polynomials on stratified groups
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Some Estimates of Bilinear Hausdorff Operators on Stratified Groups
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作者 Qi Zhang Jiman Zhao 《Analysis in Theory and Applications》 CSCD 2020年第2期200-216,共17页
In this paper,we give four kinds of sharp estimates of two variants of bilinear Hausdorff operators on stratified groups,involving weighted Lebesgue spaces,classical Morrey spaces and central Morrey spaces.Meanwhile,s... In this paper,we give four kinds of sharp estimates of two variants of bilinear Hausdorff operators on stratified groups,involving weighted Lebesgue spaces,classical Morrey spaces and central Morrey spaces.Meanwhile,some necessary and sufficient conditions of boundness are obtained. 展开更多
关键词 stratified group Hausdorff operator BILINEAR sharp estimate weighted Lebesgue space Morrey space
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Boundedness of some multipliers on stratified Lie groups
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作者 Mingju LIU Deming LI 《Frontiers of Mathematics in China》 SCIE CSCD 2015年第1期137-146,共10页
Let £ be the sub-Laplacian on a stratified Lie group G, and let m be a function defined on [0, +∞). We give the boundedness of the multiplier operators m(£) on Herz-type Hardy spaces on G.
关键词 stratified group Herz-type Hardy space MULTIPLIER
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Gaussian BV Functions and Gaussian BV Capacity on Stratified Groups
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作者 Jizheng Huang Pengtao Li Yu Liu 《Analysis in Theory and Applications》 CSCD 2021年第3期311-329,共19页
Let G be a stratified Lie group and let{X_(1)……,X_(n1)}be a basis of the first layer of the Lie algebra of G.The sub-Laplacian△G is defined by△G=-n_(1)∑j=1 X^(2)j,The operator defined by△G-n_(1)∑j=1 Xjp/pXj is ... Let G be a stratified Lie group and let{X_(1)……,X_(n1)}be a basis of the first layer of the Lie algebra of G.The sub-Laplacian△G is defined by△G=-n_(1)∑j=1 X^(2)j,The operator defined by△G-n_(1)∑j=1 Xjp/pXj is called the Ornstein-Uhlenbeck operator on G,where p is a heat kernel at time 1 on G.In this paper,we investigate Gaussian BV functions and Gaussian BV capacities associated with the Ornstein-Uhlenbeck operator on the stratified Lie group. 展开更多
关键词 Gaussian p bounded variation capacity PERIMETER stratified Lie group
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Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrdinger operators 被引量:6
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作者 LIU Yu HUANG JiZheng DONG JianFeng 《Science China Mathematics》 SCIE 2013年第9期1895-1913,共19页
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
关键词 COMMUTATOR spaces of homogeneous type stratified Lie groups admissible function Hardy space reverse Ho¨lder inequality Riesz transform Schr¨odinger operators
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Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces 被引量:2
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作者 Xia LI Guo Zhen LU Han Li TANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2015年第7期1067-1085,共19页
In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebe... In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander's condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander's condition, but they also hold for Grushin vector fields as well with obvious modifications. 展开更多
关键词 Poincare inequalities the representation formula fractional integrals on homogeneousspaces vector fields satisfying Hormander's condition stratified groups high order non-isotropic Sobolev spaces with variable exponents Sobolev inequalities with variable exponents
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Potential Analysis on Carnot Groups,Part Ⅱ:Relationship between Hausdorff Measures and Capacities 被引量:1
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作者 CuoZhenLU 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2004年第1期25-46,共22页
In this paper,we establish the relationship between Hausdorff measures and Bessel capac- ities on any nilpotent stratified Lie group G (i.e.,Carnot group).In particular,as a special corollary of our much more general ... In this paper,we establish the relationship between Hausdorff measures and Bessel capac- ities on any nilpotent stratified Lie group G (i.e.,Carnot group).In particular,as a special corollary of our much more general results,we have the following theorem (see Theorems A and E in Section 1): Let Q be the homogeneous dimension of G.Given any set E(?)G,B_(α,p)(E)=0 implies (?)^(Q-αp+(?))(E)=0 for all (?)>0.On the other hand,(?)^(Q-αp)(E)<∞ implies B_(α,p)(E)=0.Conse- quently,given any set E(?)G of Hausdorff dimension Q-d,where 0<d<Q,B_(α,p)(E)=0 holds if and only if αp(?)d. A version of O.Frostman's theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadic decomposition on such a space (see Theorem 4.4 in Section 4). 展开更多
关键词 Sobolev spaces stratified groups Bessel capacities Hausdorff measures Radon measures
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