Let $(M, g)$ be a Riemannian manifold with cylindrical ends, i.e., the ends of $(M, g)$ are modelled on $([0, \infty)_s\times Y, g=ds^2 +g_{Y}).$ I want to know whether the Hodge decomposition for $L^2_{\delta}$ forms is true or not. It would be appreciated if anyone can provide me with references on the question. The details of my question are as follows:
Let $E \to M$ be a vector bundle. To do analysis on $(M,g)$, it is natural to consider the weight Sobolev space $W^{k,p}_{\delta}(E)$. More precisely, let $\delta: M \to \mathbb{R}$ be a function with support on the ends of $M$ and $\delta=constant$ when $s \ge R_0$. Then the weight Sobolev $W^{k,p}_{\delta}$ is the completion of $C_c^{\infty}(M, E)$ with respect to norm $$|u|^p_{W^{k,p}_{\delta}}=\sum_{m=0}^k \int_M|e^{ \delta s} \nabla^mu|^p vol_g. $$
Now I focus on $E=\Lambda^qTM^*$. Let $d$ be the exterior derivative and $d^{*}_{\delta}$ be the adjoint with respect to the inner product $L^2_{\delta}$. Define a Laplacian operator by $$D=d+d^*_{\delta}.$$
My question: Is the Hodge decomposition true for Sobolev space $L^2_{\delta}(M, \Lambda^qTM^*)$(at least for very small $\delta$), i.e., $$L^2_{\delta}(\Lambda^qTM^*)=Im\{d: W^{1,2}_{\delta}(\Lambda^{q-1}TM^*) \to L^{2}_{\delta}(\Lambda^{q-1}TM^*) \} \oplus Ker D \oplus Ker\{d_{\delta}^*: W^{1,2}_{\delta}(\Lambda^{q+1}TM^*) \to L^{2}_{\delta}(\Lambda^{q}TM^*) \}.$$