论文标题
Dirichlet空间中热方程的粗糙性低纤维化性
Rough Hypoellipticity for the Heat Equation in Dirichlet Spaces
论文作者
论文摘要
本文旨在证明在一些相当弱的其他假设下，在迪里奇空间的背景下，热方程解决方案的局部界限和连续性。我们考虑对称的局部常规差异形式，这些形式满足了（a）截止函数的存在的轻度假设，（b）局部超包假设，以及（c）弱的外野外上限。在这种情况下，热方程的局部弱解及其时间衍生物被证明是局部界限的。如果Semigroup接受局部连续的密度函数，则它们是局部连续的。提供了结果的应用，包括讨论局部界限的热核的存在； $ l^\ infty $结构的结构结果是热方程式的古代解决方案。最后一节提出了一种特殊情况，其中$ l^\ infty $ off-Diagonal上限来自Semigroup的超包属性。本文是[7]的延续。
This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms which satisfy mild assumptions concerning (a) the existence of cut-off functions, (b) a local ultracontractivity hypothesis, and (c) a weak off-diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous density function. Applications of the results are provided including discussion on the existence of locally bounded heat kernel; $L^\infty$ structure results for ancient solutions of the heat equation. The last section presents a special case where the $L^\infty$ off-diagonal upper bound follows from the ultracontractivity property of the semigroup. This paper is a continuation of [7].
