Duality of ODE-determined norms - Talponen, Jarno (Mathematica Scandinavica, 2019)
Recently the author initiated a novel approach to varying exponent Lebesgue space L^p(·) norms. In this approach the norm is defined by means of weak solutions to suitable first order ordinary differential equations (ODE). The resulting norm is equivalent with constant 2 to a corresponding Nakano norm but the norms do not coincide in general and thus their isometric properties are different. In this paper the duality of these ODE-determined L^p(·) spaces is investigated. It turns out that the duality of the classical L^p spaces generalizes nicely to this class of spaces. Here the duality pairing and Hölder’s inequality work in the isometric sense which is a notable feature of these spaces. The uniform convexity and smoothness of these spaces are characterized under the anticipated conditions. A kind of universal space construction is also given for these spaces.