- 1} \delta^{Z} & k(f - \operatorname{\mathbf{N}})(1 - k \operatorname{\mathbf{N}}) x^{-2}
\end{array} {\right)},
\end{split}\end{aligned}$$ where $${\widetilde{P}} = \delta_Z Q_1  + d_Z Q_2  + Q_3 \delta_Z + Q_4 d_Z$$ and where the $Q_i\in {\mathrm{Diff}}^1_{{{{\mathrm b}},\operatorname{\mathrm {phg}}}}(M)$ and ${\widetilde{E}} \in {\mathrm{Diff}}^2_{{{{\mathrm b}},\operatorname{\mathrm {phg}}}}(M)$, as described below Proposition \[thm:normaloperatordeRham\].

Proof of Propositions \[thm:normaloperatordeRham\] and \[thm:Hodge-Laplacian\] {#sec:proof-of-operator-decomps}
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To prove Proposition \[thm:normaloperatordeRham\], we will compare the Hodge–de Rham operator for $g$ to that of the locally defined warped product metric $$\label{eq:local-product-metric}
    g_p =  
{\left(}\begin{array}{ccc}
  dx & dy^i & x^{k} dz^\alpha
\end{array}
{\right)}{\left(}\begin{array}{ccc}
1 & 0 & 0 \\
0 & (h_{ij}) & 0 \\
0 & 0 & ({\textbf{k}}_{\alpha \beta})
\end{array}
{\right)}{\left(}\begin{array}{c}
  dx \\ dy^i \\ x^{k} dz^\alpha
\end{array}
{\right)}.$$ We compute the Hodge-de Rham operator for $g_p$ now. First of all, the Hodge star operator of $g_p$, $\star_p$, acts on forms decomposed as in as $$\label{eq:star-warped-produect}
  \star_p = 
{\left(}\begin{array}{cc}
  0 & (-1)^{\operatorname{\mathbf{N}}(b - q_Y)} \star_Y \star_Z \\
 (-1)^{q_Y + \operatorname{\mathbf{N}}(b - q_Y + 1)} \star_Y \star_Z & 0
\end{array}
{\right)},$$ where $q_Y$ denotes multiplication by the degree in $Y$ and $\operatorname{\mathbf{N}}$ denotes multiplication by the degree in $Z$, and $b = \dim Y$. Using , we see that the Hodge-de Rham operator for 