Exponential dichotomy for a class of asymptotically autonomous delayed differential equations

In this work we give a criterion to have an exponential dichotomy over all $\mathbb{R}$ for delayed systems $x'(t)=L(t)x_t$, where $L_{\pm}=\lim_{t\to\pm\infty}L(t)$, and the systems $x'(t)=L_{\pm}x_t$ are autonomous and hyperbolic. The proof is based in a suitable choice of weighted Sobolev spaces $L^p(\mathbb{R},\mu)$ and $W^{1,p}(\mathbb{R},\mu)$, where the involves differential operators become Fredholm's operators and this property are strongly exploited.