Fredholm operators on little Lipschitz algebras

Let $X$ and $Y$ be compact metric spaces and $\alpha,\beta\in (0,1)$. We consider the weighted composition operators between the little Lipschitz algebras $\mathrm{lip}_\alpha(X)$ and $\mathrm{lip}_\beta(Y)$ and give a set of conditions which is necessary and sufficient for these operators to be Fredholm. We completely characterize disjointness preserving Fredholm operators between these algebras.