included into the latter is obvious), and we show that elements in the $L^2$ kernel of $\Delta^{g_{ice}}$ have expansions at ${\partial}M$ analogous to Taylor expansions but with non-integer powers, a statement which can be be interpreted as a sort of elliptic regularity at the boundary of $M.$

One application of these results, and to putative further work we describe below, is to the analysis on the Riemann moduli spaces $\mathcal M_{\gamma,\ell}$ of Riemann surfaces of genus $\gamma \geq 0$ with $\ell\geq0$ marked points. These spaces carry a natural $L^2$ metric, the Weil-Petersson metric $g_{WP}$, which near the interior of a divisor is an incomplete cusp edge metric with $k=3$. In general divisors may intersect with normal crossings, but in at least two cases only one divisor is present.

\[thm:moduli-space\] Let $\mathcal{M}_{1, 1}$ (also known as the moduli space of elliptic curves) and $\mathcal{M}_{0, 4}$ be the spaces of, respectively, once punctured Riemann surfaces of genus $1$ and $4$ times punctured Riemann surfaces of genus zero, modulo conformal diffeomorphism. Then the Hodge-Laplacian $\Delta^{g_{WP}}$ on differential forms is essentially self-adjoint on $L^2$ with core domain $C^{\infty}_{c, orb}$ (see Section \[thm:heatkernel\]) with discrete spectrum and Weyl asymptotics, and if ${\overline{\mathcal{M}}}_{1,1}$ and ${\overline{\mathcal{M}}}_{0,4}$ denote 