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 an inner uniform domain D D in length metric space. The is intrinsic strongly local Dirichlet form. When Euclidean space, prototype special case processes under consideration , whose infinitesimal generators are non-local (pseudo-differential) operators alttext="script upper L"> L encoding="application/x-tex">\mathcal {L} form u ( x stretchy="false">) = 1 2 ? i , j d mathvariant="normal">? ( a ) + movablelimits="true" form="prefix">lim ?<!-- ? stretchy="false">?<!-- ? <mml:mn>0 ?<!-- ? fence="false" stretchy="false">{ y ?<!-- ? <mml:mo>: width="thinmathspace" ?<!-- ? <mml:mo>&gt; stretchy="false">} ?<!-- ? <mml:mi>J {L} u(x)\! =\!\frac 12 \!\sum _{i, j=1}^d\! \frac {\partial }{\partial x_i}\! \left (\!\!a_{ij}(x) u(x)}{\partial x_j}\!\right ) \!+ \lim _{\varepsilon \downarrow 0}\! \int _{\{y\in D: \, \rho _D(y, x)&gt;\varepsilon \}}\!\! (u(y)-u(x)) J(x, y)\, dy \] satisfying “Neumann boundary condition”. Here, alttext="rho right-parenthesis"> encoding="application/x-tex">\rho _D(x,y) A less-than-or-equal-to d"> A ?<!-- ? encoding="application/x-tex">A(x)=(a_{ij}(x))_{1\leq i,j\leq d} measurable alttext="d times ×<!-- × encoding="application/x-tex">d\times d matrix-valued function that uniformly elliptic and bounded, colon-equal normal Phi left-bracket alpha 2 right-bracket c Superscript nu comma"> ? mathvariant="normal">?<!-- ? stretchy="false">[ ?<!-- ? stretchy="false">] c ?<!-- ? encoding="application/x-tex">J(x,y)?\frac {1}{\Phi (\rho _D(x,y))} _{[\alpha _1, \alpha _2]} {c(\alpha , x,y)} {\rho _D(x,y)^{d+\alpha }} \,\nu (d\alpha ), where alttext="nu"> encoding="application/x-tex">\nu finite measure alttext="left-bracket subset-of ?<!-- ? encoding="application/x-tex">[\alpha _2] \subset (0, 2) alttext="normal Phi"> encoding="application/x-tex">\Phi increasing infinity mathvariant="normal">?<!-- ? encoding="application/x-tex">[ 0, \infty ) alttext="c e r Super beta 3 4 beta"> e r ?<!-- ? <mml:mn>3 4 encoding="application/x-tex">c_1e^{c_2r^{\beta \le \Phi (r) c_3 e^{c_4r^{\beta }} some alttext="beta right-bracket"> encoding="application/x-tex">\beta \in [0,\infty ] encoding="application/x-tex">c(\alpha x, y) jointly bounded between two positive constants alttext="left-parenthesis encoding="application/x-tex">(x, .