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.

作者建议研究调和分析中与线性和多线性奇异积分算子相关的各种问题。更具体地说，主要研究者建议着手研究平移不变的多线性算子的乘法器，既要广泛到涵盖已知的例子，又要足够深入，包括非常奇异的算子，例如双线性希尔伯特变换。作者研究的一个重点是作为双线性乘子的单位圆盘的特征函数及其与傅里叶分析中其他重要算子（例如球乘子和卡尔森算子）的关系。 还将进行最大多重线性乘数的相关研究。我们将寻求二维卡尔森算子与最大圆盘乘数之间的深层关系。特别是，我们将研究最大双线性圆盘乘法器研究中开发的分析是否将揭示二维傅里叶级数几乎处处收敛的问题。将研究的线性和声分析中的问题包括估计粗糙奇异积分和算子的尖锐不等式，例如离散希尔伯特变换和与卡尔森测量相关的Balayage算子。在音乐中，和声是简单的音调，其振荡是简单基本频率的整数倍，这些可用于分解复杂声音的排列。在数学中，和声分析具有类似的目标，即研究复杂的声音 对象通过分解为更简单、易于理解的基本块来实现。 一旦信号和图像被分解成小块并通过傅立叶分析进行研究，就可以更好地定位信号和图像的不规则性。例如，通过应用傅立叶变换可以轻松定位噪声和模糊，但现在可以实现更具挑战性的壮举。该提案涉及使用分解技术研究某些线性和多线性乘法算子。乘法算子是通过与固定且通常不光滑的函数相乘来改变信号频率来定义的。在实践中，气象现象导致的无线电通信或电视传输的突然中断就是此类非光滑乘法算子的例子。防止信息丢失的保护可以 以定量方式进行数学建模（apower 的可积性），建议在此进行研究。 这是本研究的第一个目标。该提案中考虑的第二个问题是对一些重要且有用的不平等现象进行准确的估计。精确的估计丰富了我们对这些不等式的理解，因为它们通常反映了有用的深奥组合或几何信息。此外，它们提供了数值实现中经常需要的改进的误差估计。