The authors consider convolution operators composed with infinitesimal generators of strongly continuous groups or cosine families. Continuity of these operators is studied within the scale of \(L^p\)-spaces, namely the lifting problem to guarantee that the continuity retains to hold with respect to finer topologies on the range space. The results are applied to multiplicative perturbations of the families considered and to a maximal regularity property for second-order Cauchy problem. The latter is shown to be equivalent to the boundedness of the infinitesimal generator extending the known case \(p=\infty\).