Comprehensive Study of Functional and Real Analysis

这是一个全面的研究项目，涉及泛函和实分析领域的所有活跃的数学家。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，以便今后能够开展协调的研究活动。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会议。为了保持小组之间的密切联系，研究人员每年召开三次会议，并每年组织一次联合研讨会。鉴于研究项目的这一特点，我们的成果很难用几句话来概括，但可以这样说。直到最近，泛函分析和实际分析都是如此专门化，以至于许多结果都是深刻的，但只有少数特定领域的专家感兴趣。然而，我们现在正有一种新的趋势，将这些深刻的成果应用于一个意想不到的问题，以突破一个长期停滞不前的领域。我们的研究项目促进了这一可取的研究趋势。例如，实分析中出现的一些函数空间已经被证明在偏微分方程组理论中发挥着重要作用，尽管它们是在实分析中仅因理论上的必要性而引入的。Toeplitz算子作为不等价于乘法的算子的例子被引入，被认为与伪微分算子相同，并在李群的表示理论中得到了应用。对于非线性问题，人们已经提出了许多方法。经典的WKB微扰方法被赋予了新的光芒。

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.