Mean value property and integrability for parabolic operators

点击翻译按钮获取中文摘要

In this period, we studied three subjects : (1) mean value density for temperatures, (2) structure of parabolic Bergman spaces, (3) polynomial solution for the heat equation.(1) As for the mean value property, we discussed the existence of bounded mean value densities and the connection with the Dirichlet regularity. These results were published in Colloq.Math. 98 (2003), 87-98.(2) We define the α-parabolic Bergman space and collected its basic properties in Osaka Math.J. 42 (2005), 106-133. For example, Huygens property, estimates of the fundamental solution, the duality relation and the explicit formula of the reproducing kernel. These results were extended to strip domains. We announced it at International Workshop of Potential Theory at Matsue (2004). The result will be published in ASPM in 2006. Last year, we dealt with the Gleason problem and the Toeplitz operators on parabolic Bergman spaces.(3) We discussed the domain where the heat polynomial boundary value problem is solvable. The existence of such a doman is closely related to the zero sets of Hermite polynomials. We announced this result at the Second International Conference of Applied Mathematics at Provdiv, Bulgaria (2005).