On the equivalence of parabolic Harnack inequalities and heat kernel estimates

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Stability of heat kernel estimates for symmetric non-local Dirichlet forms (2016, background): “inf Q+u, (1.22) where Q− := (t 0 + φ(C 1R),t 0 + φ(C 2R)) × B(x 0,R) and Q + := (t 0 + φ(C 3R),t 0 + φ(C 4R))× B(x 0,R). We note that the above PHI(φ) is called a weak parabolic Harnack inequality in [BGK2], inthe sense that(1.22) holdsforsome C 1,··· ,C 5. Itis called aparabolicHarnack inequality i”
TWO-SIDED ESTIMATES OF HEAT KERNELS ON METRIC MEASURE SPACES (2012, background): “For comparison, let us observe that, under the same standing assumptions, it was proved in [10] that (UE) + (NLE )⇔ (PHI F ), where (PHIF ) stands for the parabolic Harnack inequality for caloric functions.”
Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms (2016, background): “with C 6 = C 6(C 1,...,C 5) <∞. Since our underlying metric measure space may not be geodesic, one can not expect to deduce parabolic Harnack inequality from weak parabolic Harnack inequality. See [BGK] for related discussion on diﬀusions. (ii) We will show in Proposition 4.4 that under VD, RVD a”