Researches in complex analysis and harmonic analysis

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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).