Researches in complex analysis and harmonic analysis

这项研究项目是由1994年和1995年的“Tyowa Kaiseki研讨会”(=谐波分析研讨会)的与会者完成的。这次研讨会已经举办了10多年，在每年的12月底，正式成员约10人，每年的参与者约20人。以下是研究项目取得的主要成果。(1)调和分析：具有临界指数的Bochner-Riesz算子的估计(S.Sato)；傅立叶乘子的性质(S.Igari，E.Sato，Y.Kanjin)；具有非负傅立叶变换的函数的性质(K.Tachizawa，T.Kawazoe，Y.Onoe)；一些奇异振荡积分和一些Littlewood-Paley型函数的估计(S.Sato，K.Yabuta)；几个经典的正交函数系的性质(Y.Kanjin，K.Ohashi)。(2)实分析：Kakeya极大函数的估计(S.Igari，H.Tanaka)；内插和外推理论的推广(T.Sobukawa，T.Miyamoto)。(3)各种函数空间的性质，各种算子在这些函数空间中的有界性及其应用(A.Miyachi，T.Mizuhara，E.Nakai，J.Tateoka，T.Kitada，M.Satake)。(4)小波理论：微局部小波理论及其应用(S.Moritoh)；与半单李群表示有关的小波理论(T.Kawazoe)。(5)多复变函数：Bloch函数的刻划，退化二阶椭圆偏微分算子强伪凸域的调和分析，切向Cauchy-Riemann方程和Cauchy-Sego投影(H.Arai)的估计。(6)用调和分析和实数分析的方法研究各种偏微分方程(H.Arai，K.Kurata)。(7)关于分形的研究(K.Saka，Y.Shiota，K.Kawamura)。

This research project was done by the participants of the "Tyowa Kaiseki Semina" (=Harmonic Analysis Seminar) of 1994 and of 1995. This seminar has been held over 10 years, in the end of December in each year, the regular members are about 10, and the participants of each year are about 20.The following are the main results obtained in the research project. (1) Harmonic Analysis : Estimates for the Bochner-Riesz operator with the critical index (S.Sato) ; Properties of the class of Fourier multipliers (S.Igari, E.Sato, Y.Kanjin) ; Properties of the functions with nonnegative Fourier transforms (K.Tachizawa, T.Kawazoe, Y.Onoe) ; Estimates for some singular oscillating integrals and for some Littlewood-Paley type functions (S.Sato, K.Yabuta) ; Properties of several classical orthogonal systems of functions (Y.Kanjin, K.Ohashi). (2) Real Analysis : Estimates for the Kakeya maximal function (S.Igari, H.Tanaka) ; Generalization of the theory of interpolation and extrapolation (T.Sobukawa, T.Miyamoto). (3) Properties of various function spaces, boundedness of various operators in those function spaces, and their applications (A.Miyachi, T.Mizuhara, E.Nakai, J.Tateoka, T.Kitada, M.Satake). (4) Wavelet Theory : Microlocal wavelet theory and its applications (S.Moritoh) ; Wavelet theory related to the representations of semisimple Lie groups (T.Kawazoe). (5) Functions of Several Complex Variables : Characterizations of the Bloch functions, Harmonic analysis of the degenerate second order elliptic partial differential operators strongly pseudoconvex domains, Estimates for the tangential Cauchy-Riemann equation and for the Cauchy-Sego projection (H.Arai). (6) Studies of various partial differential equations by the use of the methods of harmonic analysis and real analysis (H.Arai, K.Kurata). (7) Studies on fractals (K.Saka, Y.Shiota, K.Kawamura).