In this paper, we establish stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincare inequalities. In particular, we establish the connection between parabolic Harnack inequalities and two-sided heat kernel estimates, as well as with the Holder regularity of parabolic functions for symmetric non-local Dirichlet forms.

中文翻译：

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对称非局部狄利克雷形式的抛物线 Harnack 不等式的稳定性

在本文中，我们建立了一般体积加倍条件下度量测度空间上对称非局部狄利克雷形式的抛物线 Harnack 不等式的稳定性。我们根据跳跃核、截止 Sobolev 不等式的变体和 Poincare 不等式获得了它们的稳定等效特征。特别是，我们建立了抛物线 Harnack 不等式和两侧热核估计之间的联系，以及与对称非局部狄利克雷形式的抛物线函数的 Holder 正则性之间的联系。