$\mathcal{V}_{{{\mathrm b}},\operatorname{\mathrm {phg}}}$ denote the polyhomogeneous vector fields tangent to the boundary ${\partial}M$, means that $E$ lies in the algebra of differential operators generated $\mathcal{V}_{{{\mathrm b}},\operatorname{\mathrm {phg}}}$. Concretely, it satisfies $$\label{eq:errors-in-operator3}
  E = a x {\partial}_x + b^i {\partial}_{y^i} + c^{\alpha} {\partial}_{z^\alpha} + d,$$ for polyhomogeneous, bounded endomorphisms $a, b^i, c^\alpha, d$, and where repeated indices are summed over. In general, an element $Q \in
{\mathrm{Diff}}^m_{{{\mathrm b}},\operatorname{\mathrm {phg}}}(M; {{}^{\textrm{ice}}\Lambda^{*}})$ also satisfies , and is given locally by polyhomogeneous linear combinations of $x {\partial}_x, {\partial}_y,
{\partial}_z$, i.e.$$Q = \sum_{i + |\alpha| + |\beta| \le m} a_{i,\alpha, \beta} (x {\partial}_x)^i
{\partial}_y^\alpha {\partial}_z^\beta,$$ where $a_{i, \alpha, \beta}$ is a polyhomogeneous bounded endomorphism of ${{}^{\textrm{ice}}\Lambda^{*}}$.

The Hodge-Laplacian $\Delta = \eth^2 = d \delta + \delta d$ can now be decomposed along the same lines.

\[thm:Hodge-Laplacian\] Locally over the base, $\Delta$ can be decomposed as follows $$\label{eq:Hodge-Laplacian-decomp}
    \Delta = \Delta_0 + x^{-k}{\widetilde{P}} + x^{-1} {\widetilde{E}},$$ where $\Delta_0$ acts on forms decomposed as in by $$\begin{aligned}
\label{eq:Hodge-Laplacian-decomp1}
\begin{split}
\Delta_0 &= \operatorname{Id}_{2 \times 2} {\left(}- {\partial}_{x}^{2} - \frac{kf}{x}
{\partial}_{x} + \frac{1}{x^{2k}} \Delta^{Z,y} + \Delta^{Y} {\right)}\\ & \qquad
+ {\left(}\begin{array}{cc}
  k\operatorname{\mathbf{N}}(1 - k(f - \operatorname{\mathbf{N}})) x^{-2} & - 2k x^{-k - 1} d^{Z} \\
  - 2 k x^{-k 