Potential theoretic approach to parabolic equations

本文对抛物型方程及其相关问题进行了研究，得到了如下结果：1.平均值性质。铃木用平均值性质刻画了热球。他的结果密度函数的平均值属性。2.拟共形映射(1)Sakan改进了海因茨关于调和映射和拟共形映射的结果（2）Masaoka证明了拟共形映射的海因茨型Rieman曲面的极小Martin边界的刚性性质. Masaoka给出了各类调和函数与海因茨型Rieman曲面的Martin边界的关系.热态射（1）西尾和下村给出了流形之间热态射的一个微分方程。(2)Shimomura对某个半黎曼流形上的热态射进行了分类.下村指出Appell变换是唯一存在多温的映射. Nishio，Shimomura和Suzuki在分数阶抛物型方程中引入了抛物型Bergman空间的概念，并在完备性，再生核，对偶空间等方面得到了基本结果.

We have studied on parabolic equations and related subjects to obtain the following results :1. Mean value property.Suzuki characterized the heat ball by using mean value properties. He had a result on density functions of mean value properties.2. Quasiconformal mappings.(1) Sakan improved the Heinz's results for harmonic and quasiconformal mappings(2) Masaoka showed that the rigidity property of the minimal Martin boundary of Rieman surfaces of Heinz type for quasiconformal mappings.3. Martin boundary.Masaoka gave the relation of various classes of harmonic functions with Martin boundary of Rieman surfaces of Heinz type.4. Caloric morphism(1) Nishio and Shimomura gave a differential equation which characterizes caloric morphisms between manifolds.(2) Shimomura classifed caloric morphisms on a certain semi-riemannian manifold.5. Polytemperature.Shimomura showed that the Appell transformation is the only map which presearves polytemperatures.6. Parabolic equation of fractional orderNishio, Shimomura and Suzuki introduced the notion of parabolic Bergman spaces and obtained the fundamental results on completeness, reproducing kernels, dual spaces, etc.