Research on Martin and Kuramochi boundaries of covering surfaces

1.(1)对于Riemann曲面R和R的有限张无限覆盖曲面W，通过R和W的Martin边界，Masaoka和Segawa共同得到了R和W上有界调和函数空间相同的充要条件。(2)对于Riemann曲面R，R的有限张无限覆盖曲面W和R的极小Kuramochi边界点p，Masaoka和Segawa通过极小精细拓扑得到了W在P上的极小Kuramochi边界点个数的刻画。利用极小精细拓扑学，Masaoka和Segawa得到了R的有限张无限覆盖曲面W和R的极小Martin边界点p的极小Martin边界点个数的一个刻画。(4)对于R的Riemann曲面R和有限张无限覆盖曲面W，利用R和W的Kuramochi边界，Masaoka得到了R和W上具有有限Dirichlet积分的调和函数空间相同的充要条件。Ishida证明了，对于C中具有n个边界分支(n[大于或等于]3)的Denjoy域G和具有n个边界分支且使得G‘通过映射f共形映射到G的Denjoy子域G’，G/f(G‘)没有内点。Tsuji研究了非线性双曲型方程的柯西问题。Segawa研究了Riemann球面上无限薄片的单连通无限覆盖曲面的类型问题。(1)Nishio给出了α阶抛物方程解的均值性质。(2)Nishio证明了，在适当的条件下，对于Brelot意义下调和空间的Martin‘s紧化和Loeb紧化，它们的调和边界重合。

1. (1) For a Riemann surface R and a finitely sheeted unlimited covering surface W of R, by Martin boundaries of R and W, Masaoka obtained jointly with Segawa a necessary and sufficient condition for the spaces of bounded harmonic functions on R and W being same.(2) For a Riemann surface R, a finitely sheeted unlimited covering surface W of R and a minimal Kuramochi boundary point p of R, by minimal fine topology, Masaoka obtained jointly with Segawa a charcterization of the number of minimal Kuramochi boundary points of W over p.(3) For a Riemann surface R, a finitely sheeted unlimited covering surface W of R and a minimal Martin boundary point p of R, by minimal fine topology, Masaoka obtained jointly with Segawa a characterization of the number of minimal Martin boundary points of W over p.(4) For a Riemann surface R and a finitely sheeted unlimited covering surface W of R, by Kuramochi boundaries of R and W, Masaoka obtained a necessary and sufficient condition for the spaces of harmonic functions with finite Dirichlet integrals on R and W being same.2. Ishida showed that, for a Denjoy domain G in C with n boundary components (n 【greater than or equal】 3) and a Denjoy subdomain G' of G with n boundary components such that G' is mapped conformally into G by a map f, G / f(G') has no interior points.3. Tsuji studied the Cauchy problem for nonlinear hyperbolic equations.4. Segawa studied the type problem for a infinitely sheeted simply connected unlimited covering surface of the Riemann sphere.5. (1) Nishio gave a mean value property for solutions of parabolic equation of order α.(2) Nishio showed that, under an appropriate condition, for the Martin's and Loeb's compactifications of a harmonic space in the sense of Brelot, their harmonic boundaries coincide.