Topics in Linear and Multilinear Harmonic Analysis

点击翻译按钮获取中文摘要

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.