Large deviations for a stochastic Cahn-Hilliard equation in H\"older norm

We consider a stochastic Cahn-Hilliard partial differential equation driven by a space-time white noise. We prove the Large Deviations Principle (LDP) for the law of the solutions in the H"older norm. We use the weak convergence approach that reduces the proof to establishing basic qualitative properties for controlled analogues of the original stochastic system.