Global heat kernel estimates for symmetric Markov processes dominated by stable-like processes in exterior C¹,η open sets

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In this paper, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in exterior C¹,η open sets for all t> 0. The processes are symmetric pure jump Markov processes with jumping kernel intensity κ(x, y)ψ(|x-y|)⁻¹|x-y|⁻ᵈ⁻α where α∈(0,2), ψ is an increasing function on [0, ∞) with ψ(r)=1 on 0<r≤ 1 and c₁eᶜ²ʳβ≤ ψ(r)≤ c₃eᶜ⁴ʳβ on r>1 for β∈[0, ∞]. A symmetric function κ(x, y) is bounded by two positive constants and |κ(x, y)-κ(x,x)|≤ c₅ |x-y|ρ for |x-y|<1 and ρ>α/2. As a corollary of our main result, we estimates sharp two-sided Green function for this process in C¹,η exterior open sets.