  We give relatively simple sufficient conditions on a Fourier multiplier, so
that it maps functions with the H$\ddot{o}$lder property with respect to a part
of the variables to functions with the H$o$lder property with respect to all
variables. With the using of these sufficient conditions we prove the
solvability in H$\ddot{o}$lder classes of the initial-boundary value problems
for the linearized Cahn-Hilliard equation with dynamic boundary conditions of
two types. For the solutions of these problems Schauder estimates are obtained.
  The final expanded version of this paper is available at AIMS Journals,
Evolution Equations and Control Theory (EECT) at
http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11870
