Topics in Linear and Multilinear Harmonic Analysis

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线性不变多线性算子的乘子，既要足够广泛地覆盖已知的例子，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。作者研究的重点将是单位圆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。 最大多线性乘法器的相关研究也将继续进行。本文将探讨二维Carleson算子与最大圆盘乘子之间的深层关系。特别是它将调查是否在最大的双线性圆盘乘子的研究中开发的分析将揭示在二维傅立叶级数几乎处处收敛的问题.在线性谐波分析的问题，将被调查包括估计粗糙奇异积分和尖锐不等式的运营商，如离散希尔伯特变换和Balayage运营商与Carleson措施。在音乐中，谐波是简单的音调，其振荡是一个简单的基频的整数倍，这些可以用来拆解安排复杂的声音。在数学中，谐波分析具有类似的目的，即通过将复杂对象分解为更简单、更易于理解的基本块来研究复杂对象。 一旦将信号和图像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干扰。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。本文主要研究某些线性和多线性乘子算子 使用分解技术。乘子算子是通过与固定且通常不平滑的函数相乘来改变信号的频率来定义的。在实践中，无线电通信或电视传输因气象现象而突然中断，就是这种非平滑乘子算子的例子。对信息丢失的保护可以用定量的方法进行数学建模（可积性到幂），这是建议在这里进行研究。 这构成了拟议研究的第一个目标。在这个建议中考虑的一个次要问题是获得一些重要和有用的不等式的尖锐估计。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.