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 an elliptic operator, here $L^{p,w}_i$ is a weighted $L^{p}_i$ space.

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?