H\"older continuity of Lyapunov exponent for a family of smooth Schr\"odinger cocycles

We prove the H"older continuity of the Lyapunov exponent for quasi-periodic Schr"odinger cocycles with a $C^2$ cos-type potential and any fixed Liouvillean frequency, provided the coupling constant is sufficiently large. Moreover, the H"older exponent is independent of the frequency and the coupling constant.