  We prove the existence of infinitely many classical periodic solutions for a
class of semilinear wave equations with periodic boundary conditions. Our
argument relies on some new estimates for the linear problem with periodic
boundary conditions, by combining Littlewood-Paley techniques, the
Hausdorff-Young theorem, and a variational formulation due to Rabinowitz. We
also develop a new approach to the regularity of the distributional solutions,
by employing Gagliardo-Nirenberg estimates.
