Non-autonomous right and left multiplicative perturbations and maximal regularity

We consider the problem of maximal regularity for non-autonomous Cauchy problems u'(t) B(t)A(t)u(t) + P(t)u(t) = f(t), u(0) = u(o), and u'(t) A(t)B(t)u(t) + P(t)u(t) = f(t), u(0) = u(o). In both cases, the time dependent operators A(t) are associated with a family of sesquilinear forms, and the multiplicative left or right perturbations B(t) as well as the additive perturbation P(t) are families of bounded operators on the Hilbert space considered. We prove maximal L-p-regularity results and other regularity properties for the solutions of the previous problems under minimal regularity assumptions on the forms and perturbations.