Mean value property and integrability for parabolic operators

在此期间，我们研究了三个主题：(1)温度的均值密度，(2)抛物线Bergman空间的结构，(3)热方程的多项式解。(1)关于均值性质，讨论了有界均值密度的存在性及其与Dirichlet正则性的联系。这些结果发表在Colloq.Math上。98(2003), 87-98。(2)在大阪数学中定义了α-抛物型Bergman空间，并收集了其基本性质。42(2005), 106-133。例如惠更斯性质、基本解的估计、对偶关系和再现核的显式公式。这些结果被推广到条带结构域。我们在松江国际势理论研讨会（2004）上宣布了这一结果。研究结果将于2006年发表在《ASPM》杂志上。去年，我们讨论了Gleason问题和抛物型Bergman空间上的Toeplitz算子。(3)讨论了热多项式边值问题可解的区域。这种定义域的存在性与埃尔米特多项式的零集密切相关。我们在2005年保加利亚普罗迪夫举行的第二届国际应用数学会议上宣布了这一结果。

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).