We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.

中文翻译：

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光滑流形上粗略度量的热核和规律性

我们考虑平滑流形的粗略度量和由这些度量引起的相应拉普拉斯算子。我们证明了全局连续的热核存在并且在空间和时间上是局部连续的。这是通过局部抛物线 Harnack 估计来完成的，用于散度形式的算子的弱解，在加权 Sobolev 空间中具有有界可测系数。