Potential theoretic approach to parabolic equations

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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.