We give relatively simple sufficient conditions for a Fourier multiplier in order that it maps functions with the Hölder property in a part of the variables into functions with the Hölder property in all variables. With the use of these sufficient conditions, we prove the solvability in Hölder classes of the initial-boundary-value problems for a linearized Cahn–Hilliard equation with dynamic boundary conditions of two types. For the solutions of these problems, the Schauder estimates are obtained.