Heat Kernel Estimate for Δ+Δα/² in C¹,¹ open sets

الأصل · EN

We consider a family of pseudo differential operators {Δ+ aαΔα/²; a∈ (0, 1]} on ᵈ for every d≥ 1 that evolves continuously from Δ to Δ+ Δα/², where α∈ (0, 2). It gives rise to a family of Lévy processes {Xᵃ, a∈ (0, 1]} in ᵈ, where Xᵃ is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. We establish sharp two-sided estimates for the heat kernel of Δ+ aα Δα/² with zero exterior condition in a family of open subsets, including bounded C¹, ¹ (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric α-stable process with weight a in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in a in the sense that the constants in the estimates are independent of a∈ (0, 1] so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking a→ 0. Integrating the heat kernel estimates in time t, we recover the two-sided sharp uniform Green function estimates of Xᵃ in bounded C¹,¹ open sets in ᵈ, which were recently established in CKSV2 by using a completely different approach.