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Heat kernels for reflected diffusions with jumps on inner uniform domains

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain $D$ in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When $D$ is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration are symmetric reflected diffusions with jumps on $D$, whose infinitesimal generators are non-local (pseudo-differential) operators $L$ on $D$ of the form $$ L u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} \left(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\right) + \lim_{\eps \downarrow 0} \int_{\{y\in D: \, ρ_D(y, x)>\eps\}} (u(y)-u(x)) J(x, y)\, dy $$ satisfying "Neumann boundary condition". Here, $ρ_D(x,y)$ is the length metric on $D$, $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on $D$ that is uniformly elliptic and bounded, and $$ J(x,y):= \frac{1}{Φ(ρ_D(x,y))} \int_{[α_1, α_2]} \frac{c(α, x,y)} {ρ_D(x,y)^{d+α}} \,ν(dα) , $$ where $ν$ is a finite measure on $[α_1, α_2] \subset (0, 2)$, $Φ$ is an increasing function on $[ 0, \infty )$ with $c_1e^{c_2r^β} \le Φ(r) \le c_3 e^{c_4r^β}$ for some $β\in [0,\infty]$, and $c(α, x, y)$ is a jointly measurable function that is bounded between two positive constants and is symmetric in $(x, y)$.