Duality of ODE-determined norms

Recently a new approach to varying exponent $L^{p(\cdot)}$ space norms employing weak solutions to first order ordinary differential equations was initiated by the author. The duality of these ODE-determined $L^{p(\cdot)}$ spaces is analyzed here. The superreflexivity of these spaces is characterized under the anticipated conditions. A universal space construction is also given for these spaces.