On multiplicative perturbations of C0 -groups and C0-cosine operator functions

Certain convolution operators of the form (K f) (t) = A∈t0t L(t-s) f(s) ds , where A is the infinitesimal generator of either a C0 -group or a C0 -cosine family in a Banach space E , are considered. We obtain several lifting results guaranteeing that the continuity of K from Lp to Lq implies the continuity of K from Lp to L∈fty . These results are applied to the study of multiplicative perturbations of C0 -groups and C0 -cosine families in Banach spaces and to the study of the Maximal Regularly Property (MRP) in Lp , 1 ≤ p ≤ +∈fty , for second-order Cauchy problem. It is proved that the MRP is equivalent to the boundedness of the infinitesimal generator.