Topics in Linear and Multilinear Harmonic Analysis

作者提出研究与线性和多线性奇异积分算子有关的调和分析中的各种问题。更具体地说，首席研究员建议着手研究平移不变多线性算子的乘数，既要足够广泛地涵盖已知的例子，又要足够深入地包括非常奇异的算子，如双线性希尔伯特变换。本文研究的重点是单位圆盘作为双线性乘子的特征函数及其与傅里叶分析中其他重要算子（如球乘子和carleson算子）的关系。最大多线性乘数的相关研究也将进行。寻求二维Carleson算子与最大圆盘乘子之间的深层关系。特别要研究的是，在最大双线性圆盘乘法器的研究中发展起来的分析是否会对傅里叶级数在二维中几乎处处收敛的问题有所启示。将研究的线性调和分析中的问题包括粗糙奇异积分的估计和算子（如离散希尔伯特变换和与Carleson测度相关的Balayage算子）的尖锐不等式。在音乐中，谐波是简单的音调，其振荡是简单基本频率的整数倍，可以用来分解复杂声音的排列。在数学中，谐波分析也有类似的目的，即通过将复杂的物体分解成更简单、更容易理解的基本块来研究它们。一旦将信号和图像分解成小块并通过傅里叶分析进行研究，就可以更好地定位信号和图像的不规则性。例如，噪声和模糊很容易定位与傅里叶变换的应用，但现在甚至更具有挑战性的壮举可以实现。这一建议是关于研究某些线性和多线性乘子算子使用分解技术。乘法器算子的定义是通过与固定且通常是非平滑的函数相乘来改变信号的频率。在实际应用中，由气象现象引起的无线电通信或电视传输的突然中断就是这种非光滑乘数算子的例子。防止信息丢失的保护可以用定量的方法（幂可积性）进行数学建模，本文将对其进行研究。这是本研究的第一个目标。本建议考虑的第二个问题是对一些重要和有用的不平等进行精确估计。尖锐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的组合或几何信息。此外，它们提供了数值实现中经常需要的改进的误差估计。

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.