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
D
in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When
D
D
is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration is symmetric reflected diffusions with jumps on
D
D
, whose infinitesimal generators are non-local (pseudo-differential) operators
L
\mathcal {L}
on
D
D
of the form
\[
L
u
(
x
)
=
1
2
∑
i
,
j
=
1
d
∂
∂
x
i
(
a
i
j
(
x
)
∂
u
(
x
)
∂
x
j
)
+
lim
ε
↓
0
∫
{
y
∈
D
:
ρ
D
(
y
,
x
)
>
ε
}
(
u
(
y
)
−
u
(
x
)
)
J
(
x
,
y
)
d
y
\mathcal {L} u(x)\! =\!\frac 12 \!\sum _{i, j=1}^d\! \frac {\partial }{\partial x_i}\! \left (\!\!a_{ij}(x) \frac {\partial u(x)}{\partial x_j}\!\right ) \!+ \lim _{\varepsilon \downarrow 0}\! \int _{\{y\in D: \, \rho _D(y, x)>\varepsilon \}}\!\! (u(y)-u(x)) J(x, y)\, dy
\]
satisfying “Neumann boundary condition”. Here,
ρ
D
(
x
,
y
)
\rho _D(x,y)
is the length metric on
D
D
,
A
(
x
)
=
(
a
i
j
(
x
)
)
1
≤
i
,
j
≤
d
A(x)=(a_{ij}(x))_{1\leq i,j\leq d}
is a measurable
d
×
d
d\times d
matrix-valued function on
D
D
that is uniformly elliptic and bounded, and
\[
J
(
x
,
y
)
≔
1
Φ
(
ρ
D
(
x
,
y
)
)
∫
[
α
1
,
α
2
]
c
(
α
,
x
,
y
)
ρ
D
(
x
,
y
)
d
+
α
ν
(
d
α
)
,
J(x,y)≔\frac {1}{\Phi (\rho _D(x,y))} \int _{[\alpha _1, \alpha _2]} \frac {c(\alpha , x,y)} {\rho _D(x,y)^{d+\alpha }} \,\nu (d\alpha ),
\]
where
ν
\nu
is a finite measure on
[
α
1
,
α
2
]
⊂
(
0
,
2
)
[\alpha _1, \alpha _2] \subset (0, 2)
,
Φ
\Phi
is an increasing function on
[
0
,
∞
)
[ 0, \infty )
with
c
1
e
c
2
r
β
≤
Φ
(
r
)
≤
c
3
e
c
4
r
β
c_1e^{c_2r^{\beta }} \le \Phi (r) \le c_3 e^{c_4r^{\beta }}
for some
β
∈
[
0
,
∞
]
\beta \in [0,\infty ]
, and
c
(
α
,
x
,
y
)
c(\alpha , x, y)
is a jointly measurable function that is bounded between two positive constants and is symmetric in
(
x
,
y
)
(x, y)
.