TWO-SIDED ESTIMATES OF HEAT KERNELS ON METRIC MEASURE SPACES

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We prove equivalent conditions for two-sided sub-Gaussian estimates of heat kernels on metric measure spaces.

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Energy inequalities for cutoff functions and some applications (2012, methodology): “n 1.2. We say that (X,d,m) satisﬁes volume doubling (VD) if there exists a constant CD such that for every x ∈ X, r > 0, V(x,2r) ≤ CDV(x,r). (1.6) We next introduce the Faber-Krahn inequality: see [GT], Section 3.3 for more details. For any open set D ⊂ X, FD is deﬁned to be the closure in F of t”
Two-sided estimates of heat kernels of jump type Dirichlet forms (2018, background): “3) were obtained in [26] and [19], but they contain an elliptic Harnack inequality that is difficult to verify.”