Let $X$ be a Riemannian $n$-manifold with tubular end $\mathbb R^+\times Y$, where $Y$ is a closed $n-1$-manifold. Suppose $L:L^{p,w}_2(X)\to L^{p,w}(X)$ is the Laplacian operator which is translate invariant on the cylindrical end, here $L^{p,w}_i$ is a weighted $L^{p}_i$ space, i.e. $w$ is a function on $X$ such that on the cylindrical end part $w(t,y)=\sigma t$ and $L^{p,w}_i=e^{-w}L^P_i$.

Q: WHY: If $L$ acting on $L^{p,w}_2(\mathbb R\times Y)\to L^{p,w}(\mathbb R\times Y)$ is Fredholm, then $L:L^{p,w}_2(X)\to L^{p,w}(X)$ is Fredholm?

I saw such a statement(Lemma 2.4) on Froyshov's paper: K A Frøyshov, Monopoles over 4–manifolds containing long necks. I, Geom. Topol. 9 (2005) 1–93 MR2115668