Strong and weak \((L^p, L^q)\)-admissibility (Q401436)

The purpose of this paper is to characterize the notion of a strong exponential dichotomy with respect to a family of norms in terms of the admissibility of the pair \((L^p, L^q)\) for \(p, q \in [1, \infty)\) with \(p \geq q\). Given a nonautonomous linear equation NEWLINE\[NEWLINEx' = A(t)x,NEWLINE\]NEWLINE the admissibility property corresponds to assume that for each \(y \in L^q\), there exists a unique \(x \in L^p\) that is absolutely continuous on each compact interval and satisfies NEWLINE\[NEWLINEx' = A(t)x + y(t)NEWLINE\]NEWLINE almost everywhere with respect to the Lebesgue measure. This property is called strong admissibility (the authors also consider the notion of weak admissibility that is expressed in terms of the existence of mild solutions for a perturbed integral equation). The authors' approach consists in characterizing the notion of a strong exponential dichotomy with respect to a sequence of norms in terms of the invertibility of certain operators. The characterization of the notion of a strong exponential dichotomy is used to establish its robustness in a very simple manner.