On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

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On Harnack Inequality and Hölder Regularity for Isotropic Unimodal Lévy Processes (2013, methodology): “Such an estimate is sufficient in the proof of the Harnack inequality only if a density of Lévy measure satisfies similar conditions as in [6].”
Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms (2016, background): “nd then passing n→ ∞ if needed, without loss of generality, we may and do assume that t0 = 0, and that the function uis bounded on Q, see Step 3 in the proof of [CKK1, Theorem 5.2] (e.g. page 1085 in [CKK1]). Furthermore, by looking at au+ bfor suitable constants aand b, we may and do assume that in”
Dirichlet heat kernel for unimodal Lévy processes (2014, background): “ν(x) = ν0(|x|) ≈ f(1/|x|)/|x|d, where f is nonincreasing and satisﬁes WLSC(α,θ,c) and WUSC (α,θ,C). Then by [9, Proposition 28], ψ(x) ≈ f(x) for |x| > θ, hence ψ satisﬁes (local) WLSC and WUSC. By [13] for θ>0 and [18] for θ= 0 we get estimates for the heat kernel. This implies that x→ pt(x) i”
Dirichlet heat kernel estimates for rotationally symmetric Lévy processes (2013, unknown): “Thus, we have the following two-sided estimates for p(t, x, y) from [9, 10].”