Mean value property and integrability for parabolic operators

抛物线算子的均值性质和可积性

• 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
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540163/
• 关键词: Bergman space; Bergman projection; parabolic operator; heat equation; mean value property; Bergman空間; Gleason問題; Toeplitz作用素; Carleson測度; 平均値の定理; ベルグマン核; 調和関数の可積分性; 調和関数; 再生核

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).

在此期间，我们研究了三个主题：（1）温度的平均值密度，（2）抛物线伯格曼空间的结构，（3）热方程式的多项式解。（1）对于平均值特性，我们讨论了界面的平均值密度和与Dirichlet的连接的存在。这些结果发表在Colloq.math中。 98（2003），87-98。（2）我们定义了α-助型伯格曼空间，并在Osaka Math.J.中收集了其基本特性。 42（2005），106-133。例如，Huygens属性，基本解决方案的估计，二元性关系和繁殖内核的显式公式。这些结果扩展到了带状域。我们在Matsue（2004）的国际潜在理论研讨会上宣布了它。结果将于2006年在ASPM上发布。去年，我们在抛物线伯格曼空间上处理了格里森问题和Toeplitz运营商。（3）我们讨论了可以解决热点多项式边界价值问题的域。这种doman的存在与Hermite多项式的零集密切相关。我们在保加利亚Provdiv的第二次国际应用数学会议（2005年）上宣布了这一结果。

项目成果

a-pazabahic Berguan spaces

阿扎巴巴贝尔瓜空间

• 发表时间: 2005
• 期刊: Osaka J. Math. 42
• 作者: M.Nishio;K.Shimomura;N.Suzuki
• 通讯作者: N.Suzuki

L^P boundedness of Bergman projection for α-parabolic operator

α-抛物线算子的 Bergman 投影的 L^P 有界

• 期刊: Proceeding of IWPT in Matsue (発表予定)
• 作者: M.Nishio;N.Suzuki;K.Shimomura
• 通讯作者: K.Shimomura

M.Nishio: "A characterization of caloric morphisms between manifolds"Ann.Acad.Sci.Fenn.Math.. 28. 111-122 (2003)

M.Nishio：“流形之间热量态射的表征”Ann.Acad.Sci.Fenn.Math.. 28. 111-122 (2003)

L^p-boundedness of Bergman projection for α-parabolic operators

α-抛物线算子的伯格曼投影的 L^p-有界性

• 发表时间: 2006
• 期刊: Proceedings of the International Workshop on Potential Theory, ASPM (in press)
• 作者: M.Nishio;K.Shimomura;N.Suzuki
• 通讯作者: N.Suzuki

α-parabolic Bergman spaces and their reproducing kernels

α-抛物线 Bergman 空间及其再现核

• 发表时间: 2004
• 期刊: Suriken-Kokyuroku 1352
• 作者: K.Shimomura;N.Suzuki;M.Nishio
• 通讯作者: M.Nishio