If $(X,g)$ is a $\delta$-aymptotically Calabi space for $n=2$, one can define a weight function

(8.4)

$\displaystyle\rho^{(k+\alpha)}_{\delta,\nu}(\bm{x})=e^{\delta z(\bm{x})}\cdot(z(\bm{x}))^{\frac{\nu+k+\alpha}{2}},$

and defined the weighted space $C^{k,\alpha}_{\delta,\nu}(X)$ exactly as in Definition 8.1. On such a space the Laplace operator is a bounded linear mapping

(8.5)

$\displaystyle\Delta:C^{k,\alpha}_{\delta,\nu}(X)\rightarrow C^{k-2,\alpha}_{\delta,\nu+2}(X).$

It is expected that this operator is Fredholm if $\delta$ is sufficiently small and non-zero, and for arbitrary $\nu$. The analysis in Section 4 can likely be extended to prove this stronger result, but for the purposes of this paper we do not need this. Likewise, we expect that

(8.6)

$\displaystyle\mathscr{D}:C^{k,\alpha}_{\delta,\nu}(\Omega^{1})\rightarrow C^{k-1,\alpha}_{\delta,\nu+1}(\Omega^{0}\oplus\Omega^{2}_{+})$

is Fredholm for $\delta$ sufficiently small, but this would require a much more elaborate separation of variables argument.