On multiplicative perturbations of C 0 -groups and C 0 -cosine operator functions

Certain convolution operators of the form (K f) (t) = A∈t ₀ ^t L(t-s) f(s) ds , where A is the infinitesimal generator of either a C ₀ -group or a C ₀ -cosine family in a Banach space E , are considered. We obtain several lifting results guaranteeing that the continuity of K from L ^p to L ^q implies the continuity of K from L ^p to L ^∈ fty . These results are applied to the study of multiplicative perturbations of C ₀ -groups and C ₀ -cosine families in Banach spaces and to the study of the Maximal Regularly Property (MRP) in L ^p , 1 ≤ p ≤ +∈fty , for second-order Cauchy problem. It is proved that the MRP is equivalent to the boundedness of the infinitesimal generator. April 30, 1999