Non-Homogeneous Harmonic Analysis, two weight estimates, and spectral problems

非齐次谐波分析、两次权重估计和谱问题

基本信息

  • 批准号:
    0501067
  • 负责人:
  • 金额:
    --
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2005
  • 资助国家:
    美国
  • 起止时间:
    2005-06-01 至 2009-05-31
  • 项目状态:
    已结题

项目摘要

ABSTRACT.PI's propose to concentrate their efforts on several classical problems inAnalysis and Spectral Theory that remained unsolved for the last 20--50years, due to the lack of appropriate technical tools. Among the problemsare:* bilipschitz equivalence for higher dimensional analogues of analyticcapacity; * two weight estimates for the Hilbert Transform;* well-posedness of the inverse scattering problem for the discreteSchrodinger operator,i.e., uniqueness of the inverse nonlinear Fourier transform;* selected problems of noncommutative harmonic analysisAlthough the problems span several different areas of analysis andmathematical physics, our recent research revealed striking connectionsbetween the proposed problems. To put it briefly, they all are unified bythe fact that in all of them the same type of singular kernels (usually theCauchy kernel) appears. Also, the problems share the same difficulty, thekernel got "spoiled'' by multiplication by virtually arbitrary functions(weights). Recent developments in the non-homogeneous harmonic analysis,which treats exactly this type of situations, made successful solution ofthe proposed problems plausible.Harmonic analysis investigates complex processes by representing them as asum of elementary ones (sinusoidal waves, wavelets) with well understoodbehavior. A central part of modern harmonic analysis deals with "singularintegral operators" of one type or another. Such operators are pervasive inthe scientific landscape: they turn up in mathematical physics, probability,engineering, image processing, etc. While the theory of singular integraloperators is now well developed (starting with works of Calderon and Zygmundand continued by numerous researchers after them), it deals with theoperators defined on a nice "smooth" set, like the usual Euclidean space.However, in many problems one needs to investigate such operators on a "bad"set, like surfaces with singularities and even on more pathological sets.The non-homogeneous harmonic analysis was introduced by the PI's to dealexactly with such situations: recent solution by X. Tolsa of the famoussubbaditivity problem for the analytic capacity is one of the mostimpressive applications of this PI's theory of nonhomogeneous analysis. PI'spropose to attack several classical problems, where the framework of thenon-homogeneous harmonic analysis appear naturally.
摘要。PI建议将他们的精力集中在分析和光谱理论中的几个经典问题上,这些问题由于缺乏适当的技术工具,在过去的20- 50年里一直没有得到解决。这些问题包括:*解析能力的高维类似物的bilipschitz等价;* Hilbert变换的两个权值估计;*离散chrodinger算子逆散射问题的适定性,即。,非线性傅里叶逆变换的唯一性;虽然这些问题跨越了分析和数学物理的几个不同领域,但我们最近的研究揭示了所提出问题之间惊人的联系。简而言之,它们都是统一的,因为它们都有相同类型的奇异核(通常是柯希核)出现。而且,这些问题有相同的困难,内核被几乎任意函数(权重)的乘法“破坏”了。非齐次谐波分析的最新发展,正是处理这类情况,使所提出的问题的成功解决成为可能。谐波分析通过将复杂过程表示为具有良好理解行为的基本过程(正弦波,小波)的总和来研究复杂过程。现代谐波分析的一个核心部分是处理一类或另一类的“奇异积分算子”。这样的运算符在科学领域无处不在:它们出现在数学物理、概率、工程、图像处理等领域。虽然奇异积分算子的理论现在发展得很好(从Calderon和zygmund的工作开始,并由许多研究者在他们之后继续),但它处理的是在一个漂亮的“光滑”集合上定义的算子,就像通常的欧几里得空间一样。然而,在许多问题中,人们需要在“坏”集合上研究这样的算子,比如有奇点的曲面,甚至在更多的病态集合上。PI引入了非齐次谐波分析来精确地处理这种情况:X. Tolsa最近解决了著名的解析能力次性问题,这是PI的非齐次分析理论最令人印象深刻的应用之一。PI提出了几个经典问题,其中非齐次谐波分析的框架自然出现。

项目成果

期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

Alexander Volberg其他文献

NONCOMMUTATIVE BOHNENBLUST–HILLE INEQUALITY IN THE HEISENBERG–WEYL AND GELL-MANN BASES WITH APPLICATIONS TO FAST LEARNING
海森堡-韦尔和盖尔曼基中的非交换 Bohnenblust-Hille 不等式及其在快速学习中的应用
  • DOI:
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Joseph Slote;Alexander Volberg;Haonan Zhang
  • 通讯作者:
    Haonan Zhang
Dimension-free discretizations of the uniform norm by small product sets
  • DOI:
    10.1007/s00222-024-01306-9
  • 发表时间:
    2024-12-19
  • 期刊:
  • 影响因子:
    3.600
  • 作者:
    Lars Becker;Ohad Klein;Joseph Slote;Alexander Volberg;Haonan Zhang
  • 通讯作者:
    Haonan Zhang
Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension-one Hausdorff measure ✩
如果谐波测度相对于同维一豪斯多夫测度绝对连续,则它是可校正的 ✩
  • DOI:
  • 发表时间:
  • 期刊:
  • 影响因子:
    0
  • 作者:
    C. Acad;Sci;Ser. I Paris;Jonas Azzam;Steve Hofmann;J. M. Martell;S. Mayboroda;Mihalis Mourgoglou;X. Tolsa;Alexander Volberg
  • 通讯作者:
    Alexander Volberg
On the sign distributions of Hilbert space frames
  • DOI:
    10.1007/s13324-019-00304-y
  • 发表时间:
    2019-05-06
  • 期刊:
  • 影响因子:
    1.600
  • 作者:
    Nikolai Nikolski;Alexander Volberg
  • 通讯作者:
    Alexander Volberg

Alexander Volberg的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

{{ truncateString('Alexander Volberg', 18)}}的其他基金

Collaborative Research: Non-homogeneous Harmonic Analysis, Spectral Theory, and Weighted Norm Estimates
合作研究:非齐次谐波分析、谱理论和加权范数估计
  • 批准号:
    2154402
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Collaborative research: Weighted Estimates with Matrix Weights and Non-Homogeneous Harmonic Analysis
合作研究:矩阵权重加权估计和非齐次谐波分析
  • 批准号:
    1900268
  • 财政年份:
    2019
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Collaborative Research: Calderon-Zygmund Operators in Highly Irregular Environments, and Applications
合作研究:高度不规则环境中的 Calderon-Zygmund 算子及其应用
  • 批准号:
    1600065
  • 财政年份:
    2016
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Collaborative Research: Universality Phenomena and Some Hard Problems of Non-homogeneous Harmonic Analysis
合作研究:非齐次谐波分析的普遍性现象和一些难题
  • 批准号:
    1265549
  • 财政年份:
    2013
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Collaborative Research: Bellman function, Harmonic Analysis and Operator Theory
合作研究:贝尔曼函数、调和分析和算子理论
  • 批准号:
    0758552
  • 财政年份:
    2008
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Multidimensional and Non-Homogeneous Harmonic Analysis: Bellman Functions, Pertubations of Normal Operators and Two Weight Estimates of Singular Integrals
多维非齐次调和分析:贝尔曼函数、正规算子的摄动和奇异积分的两个权重估计
  • 批准号:
    0200713
  • 财政年份:
    2002
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Three Measures on Fractals
数学科学:分形的三种测度
  • 批准号:
    9302728
  • 财政年份:
    1993
  • 资助金额:
    --
  • 项目类别:
    Standard Grant

相似国自然基金

代数的 Leading homogeneous (monomial) 代数及其应用研究
  • 批准号:
    10971044
  • 批准年份:
    2009
  • 资助金额:
    26.0 万元
  • 项目类别:
    面上项目

相似海外基金

Collaborative Research: Non-homogeneous Harmonic Analysis, Spectral Theory, and Weighted Norm Estimates
合作研究:非齐次谐波分析、谱理论和加权范数估计
  • 批准号:
    2154335
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Collaborative Research: Non-homogeneous Harmonic Analysis, Spectral Theory, and Weighted Norm Estimates
合作研究:非齐次谐波分析、谱理论和加权范数估计
  • 批准号:
    2154321
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Collaborative Research: Non-homogeneous Harmonic Analysis, Spectral Theory, and Weighted Norm Estimates
合作研究:非齐次谐波分析、谱理论和加权范数估计
  • 批准号:
    2154402
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Collaborative research: Weighted Estimates with Matrix Weights and Non-Homogeneous Harmonic Analysis
合作研究:矩阵权重加权估计和非齐次谐波分析
  • 批准号:
    1856719
  • 财政年份:
    2019
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Collaborative research: Weighted Estimates with Matrix Weights and Non-Homogeneous Harmonic Analysis
合作研究:矩阵权重加权估计和非齐次谐波分析
  • 批准号:
    1900268
  • 财政年份:
    2019
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Collaborative research: Weighted Estimates with Matrix Weights and Non-Homogeneous Harmonic Analysis
合作研究:矩阵权重加权估计和非齐次谐波分析
  • 批准号:
    1900008
  • 财政年份:
    2019
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Visible actions on spherical homogeneous spaces and applications to non-commutative harmonic analysis
球面齐次空间上的可见行为及其在非交换调和分析中的应用
  • 批准号:
    17K14155
  • 财政年份:
    2017
  • 资助金额:
    --
  • 项目类别:
    Grant-in-Aid for Young Scientists (B)
Collaborative research: Universality phenomena and several hard problems of non-homogeneous Harmonic Analysis
合作研究:非齐次调和分析的普遍性现象及若干难题
  • 批准号:
    1265623
  • 财政年份:
    2013
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Collaborative research: Universality phenomena and some hard problems of non-homogeneous Harmonic Analysis
合作研究:非齐次调和分析的普遍性现象和一些难题
  • 批准号:
    1301579
  • 财政年份:
    2013
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Collaborative Research: Universality Phenomena and Some Hard Problems of Non-homogeneous Harmonic Analysis
合作研究:非齐次谐波分析的普遍性现象和一些难题
  • 批准号:
    1265549
  • 财政年份:
    2013
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了