Studies of The Structures of Analytic Function Spaces and Their Operators

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(1) Ohno has investigatcd the 1)robleln of components of composition operators on H^* and obtained almost conIl)Iete answers with Professor IL Zhao. We had a talk at Poster Session in International Matlnnaticiazhs Goiigrcss 1998, Berlin, Garmany. After thc talk, wc summarized results to a paper with Profcssor beta.D). MacChicr and(l submitted it. Moreover Olino have studied weighted composition operators on some sup-norm function spaces, the disk algebra, H^* and the Bloch space. And R.Zha.o added results in the case of the case little Bloch space. We are ready to submit a manuscript.(2) Funabashmi studied time geometrical properties of the SP(1)-orbits which are realized by the special kinds of three actions to tIme nearly Kachler 6-sphere. We proved that each orbit is a manifold and somne orbit equips the contact CR-structure having the three distinct principal curvatures. Those results will be contributed in the imear future. We also studied about contact CR-submanifolds immersed in … More Sasakian space forms. Our maun results is that some totally contact Al-umbilical contact CR-submanifold is realized as the 2-dimensional torus immersed in the 3-dimensional sphere. For those results, we will contribute the paper entitled "On totally contact Al-umbilical contact CR-submanifolds" in collaboration with S.Funabashi, J.H.Kwon and J.S.Pak.(3) Let S^6 be the 6-dimensional unit sphere centered at the origin in a 7-dimensional Euclidean space. Hashimnoto identified 7-dimensional Euchidean space with purely imaginary octon ions ImO (or Cayley algebra). Taking account of algebraic properties of octonions we can define the homogeneous almost Itermitian structure on S^6, We denote by G_2 the Lie group of autornorphisms of O.Then we have S^6 = G_2/SU(3). This almost colnj)lex structure satisfy the nearly Kahler condition. ((*xJ)X = 0) where * is the Levi-Civita connection of S^6, and X is any vector field of S^6. We shall give some rigidity theorem of invariant submanifolds up to the action of G_2 amid deterirmine its geometrical invariants. Also, we shall give many examples of 3-dimensional CR-snbmanifolds of S^6 explicitely. We obtained some results related to 4-dimensional CR-submanifolds of S^6.(4) Ishmizaki has beemi studying the value distribution theory of meromorphic functions. Applications this theory to com np hex differential equations are of our interest. Algebraic differential equations admitting admissible solutiomis and complex oscillation theory have been comisidered. We are also concerned with functional equations in the complex plane. Results of existence and growth conditions on transcendental meromorphic solutions of Schmrdder's type functional equations, which are some generalizations due to Wittich, are obtained. Moreover, we investigated to lmypertranscendency of merornorphic solutiohs of a certain functional equation. Characterization of the set of meromorphic. functions has been studied from the unicity tlmeoretical poimits of view.(5) Eto investigated homnological properties of monoid rings, especially affine semigroup rings. To do it, lie comistructedI free resolntiomis of them in two cases conibinatorically. They are found in papers "a free resolutions of a binomial ideal" and "finite free resolutions of rnonoid rings". Less