Harmonic Analysis Methods and Operator Decomposability in Modern Analysis

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FOR DMS-9705475 ABSTRACT OF PROJECT BERKSON The project will be concerned with the development of a unified abstract operator theory aimed at expanding the role of harmonic analysis in the structures of modern analysis. Harmonic analysis will be used to provide spectral decomposability of operators (and associated weakened forms of orthogonality) in a wide variety of contexts. The mechanism for combining the structural decomposability features of harmonic analysis and operator theory will be the transference theory initiated by A.P. Calder n and Coifman-Weiss. Studies aimed at expanding the scope and vistas of transference theory will be involved, as will applications of transference theory to harmonic analysis. The central theme of the project will be a unified overview providing new avenues for applying classical analysis to its own settings and to general mathematical analysis. The broad scope of the methods will be aimed at new vistas and discoveries applicable to a wide variety of mainstream mathematical subjects. From the standpoint of human resources, the NSF support awarded for the operations of the Wabash Modern Analysis Seminar, which functions as a Mid-Western regional seminar attracting national and international participation, will provide ready access to the frontiers of knowledge for many young researchers and graduate students, while furthering a wide array of research programs.