### For REFEREES

 Well-posedness of degenerate differential equations with infinite delay in H\"older continuous function spaces Keywords:${C}^\alpha$-well-posedness, degenerate differential equations, Infinite delay, $\dot{C}^\alpha$-Fourier multiplier, H\"older continuous function spaces. Abstract: Using operator-valued $\dot{C}^\alpha$-Fourier multiplier results on vector-valued H\"older continuous function spaces, we give a characterization for the $C^\alpha$-well-posedness of the first order degenerate differential equations with infinite delay $(Mu)''(t) = Au(t) + \int_{-\infty}^t a(t-s)Au(s)ds + f(t)$ ($t\in\R$), where $A, M$ are closed operators on a Banach space $X$ such that $D(A)\cap D(M)\neq \{0\}$, $a\in L^1_{\rm{loc}}(\R_+)\cap L^1(\mathbb{R}_+; t^\alpha dt)$.