This paper focusses on the robust or absolute stability analysis of feedback interconnections consisting of a linear time-invariant system and an uncertain linear or nonlinear component. Building on the framework of integral quadratic constraints, we establish a novel local stability result using general dynamic multipliers that enables the verification of hard time-domain state and output constraints. This is achieved by merging frequency-domain descriptions of uncertainties with time-domain dissipation arguments as they typically arise in Lyapunov theory. The general applicability of the results provided in this paper is illustrated by means of several examples that demonstrate the effectiveness of the presented approach, even if compared to techniques that are tailored to particular uncertainty classes.