There are deep works on the Bergman and the Szego kernels by Fefferman, Sato-Kashiwara-Komatsu-Hirachi. In this lecture we construct explicitly the singularity of Sego kernel following the line developed by Ho ramnder, Sostrand, Boutet-de- Monvel. Main tool is the Fourier integral operators, which will be explained in the lecture. Except the elementary of toplogical vector spaces, no prerequiste is required.
As an introductory question, let us consider the light clock which Einstein used in his definition of an ideal clock in special theory of relativity. A light clock consists of two mirrors stood parallel to each other with light running mirrored to each other continuously. The time is then measured as the number of counts that the light hits the mirrors. This clock is placed stationary to an inertial frame of reference, and the time of the frame is defined by the number of the light-hits of this clock. Insofar as the light is considered as a classical wave and the frame is an inertial one which has no acceleration, this clock can measure the time of the frame. One feature of this clock is that the time is defined by utilizing the distance between the two mirrors and the velocity of light in vacuum which is assumed as an absolute constant in special theory of relativity. Thus time is measured only after the distance between the mirrors and the velocity of light are given, and it is not that time measures the motion or velocity of light wave. Hitoshi Kitada Department of Mathematical Sciences University of Tokyo Komaba, Meguro, Tokyo 153-8914 Japan