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
It is easy to see that the oscillatory factor e<sup>λP（x,y）</sup> makes it impossible to establish the weighted norm inequality of the oscillatory integral operators （1.1） by the method as in the case...It is easy to see that the oscillatory factor e<sup>λP（x,y）</sup> makes it impossible to establish the weighted norm inequality of the oscillatory integral operators （1.1） by the method as in the case of Calderon-Zygmund operators. The purpose of this note is to establish the weighted norm inequality of （1, 1） by the aid of interpolation of operators with change of measures （see [3] ）. Let us now state a general result.展开更多
In this paper,we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases.We first give a full characterization of the existence of trace spac...In this paper,we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases.We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces,and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces(on hyperplanes)which are defined by using integral averages over selected layers of dyadic cubes.展开更多
In this paper some new results concerning the C_p classes introduced by Muckenhoupt(1981)and later extended by Sawyer(1983),are provided.In particular,we extend the result to the full expected range p>0,to the weak...In this paper some new results concerning the C_p classes introduced by Muckenhoupt(1981)and later extended by Sawyer(1983),are provided.In particular,we extend the result to the full expected range p>0,to the weak norm,to other operators and to their vector-valued extensions.Some of those results rely upon sparse domination,which in the vector-valued case are provided as well.We will also provide sharp weighted estimates for vector-valued extensions relying on those sparse domination results.展开更多
I. INTRODUCTIONLet S<sup>2n+1</sup> be the （2n+1）- dimensional standard sphere in complex （n+1） space C<sup>n+1</sup>. Let T: S<sup>2+1</sup>→S<sup>2n+1</sup> be th...I. INTRODUCTIONLet S<sup>2n+1</sup> be the （2n+1）- dimensional standard sphere in complex （n+1） space C<sup>n+1</sup>. Let T: S<sup>2+1</sup>→S<sup>2n+1</sup> be the transformation defined by T（z<sub>0</sub>, z<sub>1</sub>, …, z<sub>n</sub>） = （e （2πi）/p Z<sub>0</sub>, e （2πi）/p Z<sub>1</sub>, …, e （2πi）/p z<sub>n</sub>）, where Z<sub>0</sub>, Z<sub>1</sub>, …, Z<sub>n</sub> are complex numbers with. T acts freely on S<sup>2n+1</sup> and generates a cyclic group Z<sub>p</sub> of order p, and the orbit space is a standard Lens space L<sup>n</sup>（p）.展开更多
In this report, we shall discuss the calculations of degree of equivariant mappings about Z_p actions and give a generalization of Borsuk-Ulam theorem about Z_p actions. At first, some notations are introduced. For no...In this report, we shall discuss the calculations of degree of equivariant mappings about Z_p actions and give a generalization of Borsuk-Ulam theorem about Z_p actions. At first, some notations are introduced. For nonnegative integers m,n,〈m,n〉 denotes the greatest common divisor of m and n. m|n means that m is a factor of n. We fix a positive integer p, and展开更多
基金The author is partially supported by the National Science Foundation of U.S.,Grant DMS96-22996
文摘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.
基金Project supported by the National Natural Science Foundation of China.
文摘It is easy to see that the oscillatory factor e<sup>λP（x,y）</sup> makes it impossible to establish the weighted norm inequality of the oscillatory integral operators （1.1） by the method as in the case of Calderon-Zygmund operators. The purpose of this note is to establish the weighted norm inequality of （1, 1） by the aid of interpolation of operators with change of measures （see [3] ）. Let us now state a general result.
基金partly supported by NNSF of China(Grant No.11822105)partly supported by NNSF of China(Grant Nos.12071121 and 11720101003)supported by NNSF of China(Grant No.12101226)。
文摘In this paper,we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases.We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces,and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces(on hyperplanes)which are defined by using integral averages over selected layers of dyadic cubes.
基金supported by the Basque Government through the Basque Excellence Research Centre 2018–2021 ProgramAgencia Estatal de Investigacion/European Regional Development Fund of UE(Grant No.MTM 2017-82160-C2-1-P),Acronym“Harmonic Analysis and Quantum Mechanics”+4 种基金Spanish Ministry of Economy and Competitiveness through Basque Center for Applied Mathematics Severo Ochoa Excellence Accreditation(Grant No.SEV-2013-0323)Universidad Nacional del Sur(Grant No.11/X752)Agencia Nacional de Promocion Cientifica y Tecnologica of Argentina(Grant No.PICT 2014-1771)Juan de la Cierva-Formacion2015(Grant No.FJCI-2015-24547)Consejo Nacional de Investigaciones Cientificas y Tecnicas/Proyectos de Investigacion Plurianuales of Argentina(Grant No.11220130100329CO)。
文摘In this paper some new results concerning the C_p classes introduced by Muckenhoupt(1981)and later extended by Sawyer(1983),are provided.In particular,we extend the result to the full expected range p>0,to the weak norm,to other operators and to their vector-valued extensions.Some of those results rely upon sparse domination,which in the vector-valued case are provided as well.We will also provide sharp weighted estimates for vector-valued extensions relying on those sparse domination results.
文摘I. INTRODUCTIONLet S<sup>2n+1</sup> be the （2n+1）- dimensional standard sphere in complex （n+1） space C<sup>n+1</sup>. Let T: S<sup>2+1</sup>→S<sup>2n+1</sup> be the transformation defined by T（z<sub>0</sub>, z<sub>1</sub>, …, z<sub>n</sub>） = （e （2πi）/p Z<sub>0</sub>, e （2πi）/p Z<sub>1</sub>, …, e （2πi）/p z<sub>n</sub>）, where Z<sub>0</sub>, Z<sub>1</sub>, …, Z<sub>n</sub> are complex numbers with. T acts freely on S<sup>2n+1</sup> and generates a cyclic group Z<sub>p</sub> of order p, and the orbit space is a standard Lens space L<sup>n</sup>（p）.
文摘In this report, we shall discuss the calculations of degree of equivariant mappings about Z_p actions and give a generalization of Borsuk-Ulam theorem about Z_p actions. At first, some notations are introduced. For nonnegative integers m,n,〈m,n〉 denotes the greatest common divisor of m and n. m|n means that m is a factor of n. We fix a positive integer p, and
