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调和分析及其应用
  • 项目名称：调和分析及其应用
  • 项目类别：重点项目
  • 批准号：10931001
  • 申请代码：A010504
  • 项目来源：国家自然科学基金
  • 研究期限：2010-01-01-2013-12-31
  • 项目负责人：丁勇
  • 负责人职称：教授
  • 依托单位：北京师范大学
  • 批准年度：2009
中文摘要：

调和分析是现代数学的重要组成部分，调和分析的思想与方法已成为偏微分方程等数学领域中重要的工具。本项目组将在已有的调和分析理论及其在偏微分方程中的应用的基础上，拟深入研究积分核的低正则性下抛物型和变量核的奇异积分算子有界性；建立在多参数结构下积分核低正则性的奇异积分算子及带旗核的奇异积分算子的实变理论；给出在低正则性下积分算子交换子有界性和紧性的特征刻画；发展一套高维Hardy算子及一类振荡积分算子的实变理论；建立超奇异双Hilbert变换及正规李群上奇异积分算子的有界性；研究多线性算子的各类有界性问题。应用调和分析的理论和方法研究非线性色散方程的适定性、散射理论及低正则性问题与流体动力学方程的数学理论，建立Quaternionic Heisenberg群上相关于波方程的广义Strichartz不等式，并研究调和映照热流弱解的正则性。

中文主题词： 奇异积分及交换子；Hardy和Carleson算子；色散与流体动力学方程；四元数Heisenberg群；紧流形上热流方程
英文摘要：

Singular Integral and commutat；Hardy and Carleson operators；dispersive and fluid equation；Quaternionic Heisenberg group；heat flow on compact manifold

英文主题词： Singular Integral and commutat；Hardy and Carleson operators；dispersive and fluid equation；Quaternionic Heisenberg group；heat flow on compact manifold
结论摘要：

项目组深入研究了各类粗糙核积分算子（沿子流形的粗糙奇异积分算子、Littlewood-Paley算子、振荡积分、Carleson 型极大算子等）的有界性及加权有界性问题，其中一些结果改进了E.M.Stein的工作；系统研究了低正则性积分核的奇异积分算子（抛物型奇异积分、变量核奇异积分、超奇异积分等）交换子在几类函数空间上的有界性和紧性特征，本质改进并推广了Calderon的结果；建立了高维Hardy算子（及Hausdorff算子）的实变理论并将Christ-Grafakos关于Hardy算子不等式的最佳常数推进到加权情形；得到了高维超奇异双Hilbert变换的有界性；给出了多线性积分算子及其交换子的有界性。作为调和分析理论和方法的应用，首次建立了相关于一般可测复系数的高阶齐次椭圆算子的Riesz变换和Hardy空间理论；研究了几类非线性色散方程和带高频振荡初值可压 Navier-Stokes方程的适定性，改进了Danchin的结果；系统研究了四元数Heisenberg群上的调和分析问题并建立了关于波方程的广义Strichartz不等式；给出了紧Riemannian流形上热流方程强解的衰减估计。

成果综合统计
成果类型
数量
  • 期刊论文
  • 会议论文
  • 专利
  • 获奖
  • 著作
  • 120
  • 0
  • 0
  • 0
  • 2
期刊论文
Sharp Weighted Estimates for a Class of n-dimensional Hardy-Steklov Operators
A remark to the L(2) boundedness of parametric Marcinkiewicz integral
L-P BOUNDEDNESS OF CARLESON TYPE MAXIMAL OPERATORS WITH NONSMOOTH KERNELS
Boundedness for Commutators of Rough Hypersingular Integrals with Variable Kernels
Weighted boundedness for rough Marcinkiewicz integrals with different weights
Estimate for generalized unimodular multipliers on modulation spaces
一类振荡积分算子交换子的加权有界性
WEIGHTED L-p BOUNDEDNESS OF CARLESON TYPE MAXIMAL OPERATORS
Characterizations of Hardy spaces associated to higher order elliptic operators
Uniform weighted estimates for oscillatory singular integrals
Estimates on fractional power dissipative equations in function spaces
A CHARATERIZATION OF COMMUTATORS FOR PARABOLIC SINGULAR INTEGRALS
L-2 boundedness for maximal commutators with rough variable kernels
Commutators of Marcinkiewicz integral with rough kernels on Sobolev spaces
COMPACTNESS FOR COMMUTATORS OF MARCINKIEWICZ INTEGRALS IN MORREY SPACES
On Singular Integral Operators with Rough Kernel Along Surface
Rough singular integrals supported on submanifolds
BOUNDEDNESS OF THE MARCINKIEWICZ INTEGRALS WITH ROUGH KERNEL ASSOCIATED TO SURFACES
Commutators of Littlewood-Paley Operators on the Generalized Morrey Space
Commutator of hypersingular integral with rough kernels on Sobolev spaces
Singular Integrals on Product Homogeneous Groups
Compactness of the commutators of parabolic singular integrals
L^p boundedness for Littlewood-Paley operators with rough variable kernels
The boundedness for commutators of maximal hypersingular integrals with rough kernels
Weyl transform and generalized spectrogram associated with quaternion Heisenberg group
Abstract framework for John-Nirenberg inequalities and applications to Hardy spaces
Clifford Algebra-Valued Wavelet Transform on Multivector Fields
Weyl transforms on the upper half plane
A Sharp Bilinear Estimate for the Bourgain-Type Space with Application to the Benjamin Equation
Sharp well-posedness for the Benjamin equation
Bourgain space X^{s,b} with applications to the KdV type equations
Dispersive estimates with geometry of finite type
单边Triebel-Lizorkin空间及其应用
Weighted estimates for the multilinear singular integral operators with non-smooth kernels
Commutators of Bochner-Riesz operators for a class of radial functions
Commutators of n-dimensional rough Hardy operators
Sharp bounds for m-linear Hardy and Hilbert operators
The multilinear Marcinkiewicz interpolation theorem revisited: The behavior of the constant
Weighted estimates for iterated commutators of multilinear fractional operators
Endpoint estimates for n-dimensional Hardy operators and their commutators
A Weighted Variant of Riemann—Liouville Fractional Integrals on R^n
Explicit constants for Hardy's inequality with power weight on n-dimensional product spaces
A characterization of λ-central BMO space
Sharp estimates of p-adic Hardy and Hardy-Littlewood-Polya operators
振荡积分算子及其交换子在加权Morrey空间上的有界性
Sharp bounds for Hardy type operators on higher-dimensional product spaces
Mp weights for bilinear Hardy operators on Rn
Some characterizations of Campanato spaces via commutators on Morrey spaces
Conjectures and problems on Bochner-Riesz means
Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependen
The?self-dual?Chern-Simons?Higgs?equation?on?a?compact?Riemann?surface?with?boundary
Liouville theorems for the ancient solution of heat flows
拟调和球的不存在性
A mean field equation with symmetry on torus
高维Hardy算子交换子的加权估计
Multilinear Calderon Zygmund operators on weighted Hardy spaces
Continuity property for the commutators of multilinear Calderón–Zygmund operators
Boundedness of multidimensional Hausdorff operators on H1(Rn)
Maximal operator for multilinear Calder\'{o}n-Zygmund singular integral operators on weighted Ha
Hausdorff operators on Euclidean spaces
Multilinear version of reversed H\"{o}lder inequality and its applications to multilinear Calder\'{o
Weighted Trudinger inequality associated with rough multilinear fractional type operators
Smoothness of Extremizers of a Convolution inequality
On regularizing-rate estimates for solutions to the heat flow with initial value problem
The Equation △u + △R· △u = 8πc(1 - heu ) on a Riemann Surface
Hausdorff算子在Triebel-Lizorkin型空间上的有界性
Estimates for unimodular multipliers on modulation Hardy spaces
Hausdorff operators on function spaces
\lambda-Central BMO Estimates for Commutators ofMultilinear Maximal Operators
Weighted Estimates for the iterated Commutators of Multilinear Maximal and Fractional Type Operators
Weighted Estimates for Vector-Valued Commutators of Multilinear Operators
Estimates for wave and Klein-Gordon equations on modulation spaces
Weighted Estimates for Commutators of Vector-Valued Maximal Multilinear Operators
L2-boundedness of Hilbert transforms along variable curves
Boundedness of Hausdorff operators on some product Hardy type spaces
On the Boundedness of multilinear Littlewood-Paley g_{\lambda}^{*} function
The maximal super-singular integral operators on product spaces
具有非倍测度的Littlewood-Paley g_\lambda*函数的多线性交换子的端点估计
A bilinear fractional integral on compact Lie groups
Modulation space estimates for the fractional integral operators
Central oscillating multipliers on compact Lie groups
Transference on some non-convolution operators from Euclidean spaces to torus
Oscillatory integrals on unit square along surfaces
Hardy space estimates for the wave equation on compact Lie groups
The Cauchy integral operator on weighted Hardy space
Rough bilinear fractional integrals with variable kernels
Boundedness of singular integrals along surfaces on Lebesgue spaces
The Cauchy integral operator on the predual of a Morrey space
Boundedness of Marcinkiewicz integral on Triebel-Lizorkin spaces
一类广义加权Hardy算子交换子的有界性
Some endpoint estimates for the multiplier operator
The Boundedness for Commutator of Fractional Integral Operator With Rough Variable Kernel
Well-posedness and ill-posedness for a fifth-order shallow water wave equation
The asymptotic behavior of Chern-Simons Higgs model on a compact Riemann surface with boundary
强奇异 Calderón-Zygmund算子及其交换子在Hardy型空间上的有界性
Strichartz estimates on the quaternion Heisenberg group
Integrability of the general product Hardy operators on the product Hardy spaces
Global well-posedness for the 3D rotating Navier-Stokes equations with highly oscillating initial da
Riesz potentials, Riesz transforms on Lipschitz spaces in compact Lie groups
The boundedness for commutators of a class of maximal hypersingular integrals with variable kernels
Marcinkiewicz积分在加权Hardy空间和加权Herz型Hardy空间上的弱型估计
极大函数交换子在广义Morrey空间的估计
关于一个Sobolev不等式的一点注记
加权Hardy算子及其交换子在广义Morrey空间上的有界性
模空间上的多线性乘子估计
Multiple weighted estimates for maximal vector-valued commutator of multilinear Calderon-Zygmund singular integrals
具有非倍测度的Littlewood-Paley gλ^＊函数的多线性交换子的端点估计
著作
Bochner-Riesz Means on Euclidean Spaces
Bochner-Riesz Means on Euclidean Spaces
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