Topics in Linear and Multilinear Harmonic Analysis

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

The author proposes to study a variety of problems in harmonic analysis related to linear and multilinear singular integral operators. More specifically, the principal investigator proposes to embark on a study of multipliers for translation-invariant multilinear operators, both broad enough to cover known examples, but also deep enough to include very singular operators such as the bilinear Hilbert transform. A key point of the author's research will be the characteristic function of the unit disc thought of as a bilinear multiplier and its relation to other important operators in Fourier analysis such as the ball multiplier and Carleson's operator. A related study of maximal multilinear multipliers will also be pursued. Deep relations between Carleson's operator in two dimensions and the maximal disc multiplier will be sought. In particular it will be investigated whether the analysis developed in the study of the maximal bilinear disc multiplier will shed light on the problem of almost everywhere convergence 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 integral multiples of a simple basic frequency and these can be used to disassemble arrangements of complicated sounds. In mathematics, harmonic analysis has a similar objective i.e. the study of complicated objects via their decomposition into simpler well-understood basic blocks. Irregularities of signals and images are better located once these are decomposed into small pieces and studied via Fourier analysis. For instance, noise and blurring are easily located with the application of the Fourier transform, but nowadays even more challenging feats can be achieved. This proposal is concerned with the study of certain linear and multilinear multiplier 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 or television transmission by a meteorological phenomenon are examples of such nonsmooth multiplier operators. The protection against the loss of information can be mathematically modeled in a quantitative way (integrability to a power) which is proposed to be studied here. This constitutes the first goal 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 they often reflect useful esoteric combinatorial or geometric information. Furthermore, they provide improved error estimates often needed in numerical implementation.