Comprehensive Study of Functional and Real Analysis

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This is a comprehensive research project involving all active mathematicians in the fields of Functional and Real Analysis. The aim is to have a proper perspective of recent progress in these fields in Japan as well as in the world so that coordinated research activities are possible for the future. For this purpose we made five groups according to the subjects.Each group conducted standing seminars and a nationwide meeting each year. The Investigators met three times every year in order to maintain close correspondence between the groups, and organized a joint symposium each year.In view of this character of the research project it is difficult to summarize our results in a few words but the following could be said. Until recently both Functional Analysis and Real Analysis are so much specialized that many results are profound but interest only few specialists of particular fields. However, we are having now a new trend in which those profound results are applied to an unexpected problem to make a breakthrough of a field which is stagnated for a long time. Our research project has promoted this desirable tendency of researches.For example, some of the function spaces appearing in Real Analysis have been proved to play a essential role in the theory of partial differential equations, although they were introduced only by theoretical necessity in Real Analysis. Toeplitz operators, which were introduced as examples of operators not equivalent to multiplications, are recognized as the same as pseudodifferential operators, and have found applications to the representation theory of Lie groups. Many approaches have been made for non-linear problems. A new lights was shed on the classical WKB method of perturbation.