Let R+ be the set of all non-negative real numbers, I ∈ {R,R+} and U = {U(t,s) : t ≥ s ∈ I} be a strongly continuous and exponentially bounded evolution family of bounded linear operators acting on a Banach space X. Let E be a normed function space over I satisfying
some properties, see section 2. We prove that if �XI(•)||U(•,s)x|| defines an element of the space E for some s ∈ I and some x ∈ X, then there exists N(s, x) > 0 such that

*mathematical equation

Some related results for periodic evolution families are also proved.