We give necessary and sufficient conditions of L p -maximal regularity (resp. B s p,q -maximal regularity or F s p,q -maximal regularity) for the second order delay equations:where A is a closed operator in a Banach space X, F and G are delay operators on L p ([−2π, 0]; X) (resp. B s p,q ([−2π, 0]; X) or F s p,q ([−2π, 0]; X)).