that it can be approximated in the graph norm by compactly supported smooth forms. This we do using the heat kernel.

Recall that the heat kernel $H$ is a section of the form bundle $\Pi \colon \operatorname{End}(\Lambda) {\longrightarrow}M^\circ \times
M^\circ \times [0, \infty)$, where $M^\circ$ is the interior of $M$ and $\operatorname{End}(\Lambda)$ is the vector bundle whose fiber over $(p, q,
t)$ is $\operatorname{End}(\Lambda^*_q(M) ; \Lambda^*_p(M))$, smooth on the interior $M^\circ \times M^\circ \times [0, \infty)_t$, which solves $$\label{eq:fundamental-solution}
  ({\partial}_t + \Delta^{g_{ice}}) H = 0 \mbox{ and } H_t {\longrightarrow}\operatorname{Id}, \mbox{strongly as } t
  \downarrow 0.$$ For a compactly supported smooth differential form $\alpha$, the differential form $$\beta(\omega, t) := \int_M
H(w, {\widetilde{w}}, t)\alpha({\widetilde{w}}) \operatorname{dVol}_{g_{ice}}({\widetilde{w}})$$ solves the heat equation $({\partial}_t + \Delta^{g_{ice}}) \beta = 0$ with initial data $\beta
\rvert_{t = 0} = \alpha$. One consequence of our precise description of $H$ in Theorem \[thm:heatkernel\] below will be the following.

\[thm:heatkernelmap\] On a Witt incomplete cusp edge space $(M, g_{ice})$ with metric satisfying the assumptions in below together with , there exists a fundamental solution to the heat equation $H_{t} = H(w, {\widetilde{w}}, t)$ in the sense of such that for $t>0$ $$\label{eq:heat-mapsto-Dmin}
  H_{t} \colon L^{2}(M; \Omega^{*}(M)) {\longrightarrow}\mathcal{D}_{\min},$$ and such 