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Mean value property and integrability for parabolic operators

抛物线算子的均值性质和可积性

基本信息

批准号：
15540163
负责人：
SUZUKI Noriaki
金额：
$ 1.66万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

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).

在这期间，我们研究了三个问题：（1）温度的平均值密度，（2）抛物Bergman空间的结构，（3）热方程的多项式解。(1)关于均值性质，我们讨论了有界均值密度的存在性及其与Dirichlet正则性的关系。这些结果发表在Colloq. Math.98（2003），87-98。(2)我们在Osaka Math.J. 42（2005），106-133中定义了α-抛物Bergman空间并收集了它的基本性质。例如，惠更斯性质，基本解的估计，对偶关系和再生核的显式公式。这些结果被扩展到带域。我们在2004年松江国际势理论研讨会上宣布了这一点。调查结果将于2006年在ASPM上发表。去年，我们处理了格里森问题和抛物Bergman空间上的Toeplitz算子。(3)讨论了热多项式边值问题的可解区域。这种区域的存在性与Hermite多项式的零点集密切相关。我们在2005年保加利亚Provdiv举行的第二届国际应用数学会议上宣布了这一结果。