Strong and weak (Lp,Lq)-admissibility

We consider the notion of a strong exponential dichotomy with respect to a family of norms for a dynamics with continuous time and we characterize it in terms of the admissibility of the pair (Lp,Lq) for p,q∈[1,∞) with p⩾q. Moreover, we consider both strong and weak admissibility, in the sense that the solutions are, respectively, classical solutions of a nonautonomous linear equation or mild solutions of an integral equation obtained from perturbing an evolution family. We then use the characterization of the notion of a strong exponential dichotomy to establish its robustness in a very simple manner.