Comprehensive Study of Functional and Real Analysis

这是一个全面的研究项目，涉及所有活跃的数学家在功能和真实的分析领域。其目的是对日本和世界在这些领域的最新进展有一个正确的看法，以便将来有可能开展协调一致的研究活动。为此，我们按主题分成五个小组，每个小组每年举行一次常设研讨会和一次全国性会议。调查员每年举行三次会议，以保持各小组之间的密切联系，并每年组织一次联合研讨会。鉴于研究项目的这一特点，很难用几句话概括我们的结果，但可以说如下。直到最近，泛函分析和真实的分析都是如此专业化，以至于许多结果是深刻的，但只有少数特定领域的专家感兴趣。然而，我们现在有一个新的趋势，即这些深刻的结果被应用到一个意想不到的问题，使一个长期停滞不前的领域取得突破。我们的研究工作推动了这一研究的发展趋势，例如，在真实的分析中出现的一些函数空间，尽管只是在理论上的需要才被引入到真实的分析中，但它们已经被证明在偏微分方程理论中起着重要的作用。托普利茨算子，作为不等价于乘法的算子的例子被引入，被认为是与伪微分算子相同的，并且已经发现了李群表示论的应用。对于非线性问题已经有许多方法。本文对经典的WKB摄动法作了新的解释。

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.