Upper Bounds of Heat Kernels on Doubling Spaces

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Energy inequalities for cutoff functions and some applications (2012, background): “> s) ≥ s(1− ε)e−λs, which yields (5.11). Theorem 5.5. Suppose X satisﬁes VD and UHK(Ψ). Then X satisﬁes FK(Ψ) and CSA(Ψ). Proof. The proof that UHK(Ψ) plus VD implies FK(Ψ) is as in Section 5.5 of [GH], where the case Ψ(r) = rβ is given. To prove CSA(Ψ) we will show that X satisﬁes the hypotheses”
On the equivalence of parabolic Harnack inequalities and heat kernel estimates (2012, background): “We do not pursue this approach here, but refer the reader to [30], [31], [32] for the theory of time dependent Dirichlet forms.”
Stability of heat kernel estimates for symmetric non-local Dirichlet forms (2016, methodology): “2 Proof. (1) =⇒ (2) =⇒ (3) can be proved similarly to [GH, Lemmas 5.4 and 5.5] by choosing a= CV(x,r)ν/φ(r) in the paper. (3) =⇒ (1) can be proved similarly to the approach of [GH, p. 553]. Note that [GH] discusses the case φ(r) = rβ, but the generalization to φis easy by using (1.13). Under VD and ”
Estimates of heat kernels for non-local regular Dirichlet forms (2014, background): “3) (UEloc) ⇔ (DUE) + (S) + (“locality”), where “locality” means that the form (E ,F) is local (see [16] and Remark 2.”
TWO-SIDED ESTIMATES OF HEAT KERNELS ON METRIC MEASURE SPACES (2012, methodology): “Note that (RVD) is essential for (EF≤) (see [31], Theorem 2.”