Researches in complex analysis and harmonic analysis

该研究项目由1994年和1995年的“Tyowa Kaiseki Semina”（=谐波分析研讨会）的参与者完成。该研讨会已举办了10多年，每年12月底举行，正式成员约10人，每年参加者约20人。(1)谐波分析：具有临界指标的Bochner-Riesz算子的估计（S.Sato）;傅立叶乘法器类的性质（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）;某些奇异振荡积分和某些Littlewood-Paley型函数的估计（S.Sato，K.Yabuta）;几个经典正交函数系的性质（Y.Kanjin，K.Ohashi）。(2)真实的分析：Kakeya极大函数的估计（S.Igari，H.Tanaka）;内插和外插理论的推广（T.Sobukawa，T.Miyamoto）。(3)各种函数空间的性质，这些函数空间中各种算子的有界性及其应用（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)函数的几个复杂的变量：特征的布洛赫功能，调和分析退化二阶椭圆偏微分算子强pseudoclave域，估计切柯西-黎曼方程和柯西-塞戈投影（H.荒井）。(6)用调和分析和真实的分析方法研究各种偏微分方程（H.Arai，K.Kurata）。(7)分形研究（坂、盐田、川村）。

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).