Mathematical Sciences: Abstract Operator Theory and Transference Methods in Modern Analysis

伯克森教授将继续他近年来的工作，从转移方法的观点出发，发展统一的抽象算子理论，旨在处理一般调和分析中出现的分解的多样性。具体目标包括：关于极大算子的弱型和强型界的迁移理论的扩展；针对特定空间类的迁移估计的发展；傅立叶乘子扩张定理；利用无界表示研究迁移及其与加权范数不等式的关系；以及UMD空间在局部紧交换群作用下的分解。这个项目的数学框架是谐波分析，它可以被认为是理论信号处理。傅立叶为周期信号发明的经典策略是将其分解成纯音，即某个单位频率的倍数。这给出了一个数字列表，然后可以对其进行操作，并将其转换回修改后的信号。有时，经典的策略并不是最合适的，最好使用其他在数学上不那么直接的分解。伯克森教授的工作旨在更好地理解执行谐波分析的各种策略。

Professor Berkson will continue his work of recent years on the development, from the viewpoint of transference methods, of a unified abstract operator theory designed to treat the diversity of decompositions arising in general harmonic analysis. Specific objectives include: expansion of transference theory in regard to weak type and strong type bounds for maximal operators; the development of transference estimates tailored to specific classes of spaces; Fourier multiplier extension theorems; investigation of transference by unbounded representations and its relationship to weighted norm inequalities; and the decomposition of UMD spaces under the action of a locally compact abelian group. The mathematical framework for this project is harmonic analysis, which may be thought of as theoretical signal- processing. The classic strategy, invented by Fourier for periodic signals, is to decompose into pure tones, multiples of some unit frequency. This gives a list of numbers which can then be manipulated and turned back into a modified signal. Sometimes the classic strategy is not the most apt, and it becomes preferable to use some other decomposition that is less straightforward mathematically. Professor Berkson's work is aimed at better understanding of a wide variety of strategies for performing harmonic analysis.