  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
  $$ L u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i}
\left(a_{ij}(x)
  \frac{\partial u(x)}{\partial x_j}\right) + \lim_{\eps \downarrow 0}
\int_{\{y\in D: \,
  \rho_D(y, x)>\eps\}} (u(y)-u(x)) J(x, y)\, dy $$ satisfying "Neumann boundary
condition". Here, $\rho_D(x,y)$ 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 $$ J(x,y):=
\frac{1}{\Phi(\rho_D(x,y))} \int_{[\alpha_1, \alpha_2]} \frac{c(\alpha, x,y)}
{\rho_D(x,y)^{d+\alpha}} \,\nu(d\alpha) , $$ where $\nu$ is a finite measure on
$[\alpha_1, \alpha_2] \subset (0, 2)$, $\Phi$ is an increasing function on $[
0, \infty )$ with $c_1e^{c_2r^{\beta}} \le \Phi(r) \le c_3 e^{c_4r^{\beta}}$
for some $\beta \in [0,\infty]$, and $c(\alpha , x, y)$ is a jointly measurable
function that is bounded between two positive constants and is symmetric in
$(x, y)$.
