Topics in Linear and Multilinear Harmonic Analysis

线性和多线性谐波分析主题

• 批准号: 0099881
• 负责人: Loukas Grafakos
• 金额: $ 9.6万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0099881&HistoricalAwards=false
• 关键词: Topics; Linear; Multilinear; Harmonic; Analysis

The author proposes to study a variety of problems in harmonic analysisrelated to linear and multilinear singular integral operators.More specifically, the principal investigator proposes to embark on astudy of multipliers for translation-invariant multilinear operators, both broadenough to cover known examples, but also deep enough to include very singularoperators such as the bilinear Hilbert transform. A key point of the author'sresearch will be the characteristic function of the unitdisc thought of as a bilinear multiplier and its relation to otherimportant operators in Fourier analysis such as the ball multiplier andCarleson's operator. A related study of maximal multilinear multiplierswill also be pursued. Deep relations between Carleson's operator in two dimensions and the maximal disc multiplier will be sought. In particular it willbe investigated whether the analysis developed in the study of the maximal bilinear disc multiplier will shed light on the problem of almost everywhereconvergence of Fourier series in two dimensions. Problems in linear harmonic analysis that will be investigated include estimates for rough singular integrals and sharp inequalities for operators such as the discrete Hilbert transform and the Balayage operator associated with Carleson measures.In music, harmonics are simple tones whose oscillations are integralmultiples of a simple basic frequency and these can be used todisassemble arrangements of complicated sounds.In mathematics, harmonic analysis has a similar objective i.e.the study of complicated objects via their decomposition into simplerwell-understood basic blocks. Irregularities of signals and imagesare better located once these are decomposed into small pieces andstudied via Fourier analysis. For instance, noise and blurring are easily locatedwith the application of the Fourier transform, but nowadays evenmore challenging feats can be achieved. This proposal isconcerned with the study of certain linear and multilinearmultiplier operators using decomposition techniques.Multiplier operators are defined by altering the frequency of signals via multiplication with a fixed and often nonsmooth function.In practice, the abrupt interruption of radio communication ortelevision transmission by a meteorological phenomenonare examples of such nonsmooth multiplier operators.The protection against the loss of information can bemathematically modeled in a quantitative way (integrability to apower) which is proposed to be studied here. This constitutes the firstgoal of the proposed research. A secondary issue considered in this proposal is obtaining sharp estimates for some important and useful inequalities. Sharp estimates enrich our understanding of these inequalities as theyoften reflect useful esoteric combinatorial or geometric information.Furthermore, they provide improved error estimates often needed innumerical implementation.

作者提议研究与线性和多线性单数积分运算符有关的谐波分析中的各种问题。更具体地说，首席研究人员提议提出敏捷的乘数，用于翻译不变的多线性操作员，既涵盖已知的示例，又有足够的深度，但也足够深入，但也包括非常奇异的birlularoperoperators，例如bylarlorlularoperartors the byloperear the Brilert the Blilebert tronfroment the Brobert tronfroment。作者研究的关键点将是单位盘视为双线性乘数的特征功能及其与其他重要操作员的关系，例如傅立叶分析，例如球乘数和卡莱森的操作员。 还将追求一项最大多线性乘数的相关研究。将寻求卡莱森运营商之间的深厚关系与最大光盘乘数。特别是它将研究最大双线性圆盘倍增器研究中进行的分析是否会阐明傅立叶级数几乎在二维中的所有方面的问题。线性谐音分析中将进行研究的问题包括对诸如离散的希尔伯特变换和与Carleson措施相关的Balayage运算符等操作员的粗糙奇异积分和急剧不平等的估计值。在音乐中，谐波是简单的音调，其振荡是简单的基本频率的整体频率，并且可以使用这些简单的基本频率，并且可以使用复杂的构成Insimess。对象通过将其分解为简单的基本块。 一旦将它们分解为小块，并通过傅立叶分析将信号和图像的不规则性置于更好的位置。例如，噪声和模糊的位置很容易找到傅立叶变换的应用，但是如今，甚至可以实现更具挑战性的壮举。该提案与使用分解技术对某些线性和多电压运算符的研究相关联。多型运算符是通过通过固定且通常不平滑的功能来改变信号的频率来定义以定量方式（与apower的可集成性）进行bem仪建模，并建议在此处研究。 这构成了拟议的研究的第一个目标。该提案中考虑的次要问题是获得一些重要且有用的不平等现象的敏锐估计。敏锐的估计丰富了我们对这些不平等现象的理解，因为它们通常反映了有用的深奥组合或几何信息。furthermore，它们提供了改进的错误估计，通常需要实施无数的实施。