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.
