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We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay where A is a closed operator in a Banach space X, α ∈ C, and a, b ∈ L 1 (R + ). We use Fourier multipliers to characterize maximal regularity for (P ). Using known results on Fourier multipliers, we find suitable conditions on the kernels a and b under which necessary and sufficient conditions are given for the problem (P ) to have maximal regularity on L p (T, X),doi:10.4064/sm184-2-1 fatcat:5svljbfgrrgvph3eal4wk6cavu