Duality of ODE-determined norms

  • Jarno Talponen

Abstract

Recently the author initiated a novel approach to varying exponent Lebesgue space $L^{p(\cdot)}$ 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 $L^{p(\cdot)}$ 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.

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Published
2019-01-13
How to Cite
Talponen, J. (2019). Duality of ODE-determined norms. MATHEMATICA SCANDINAVICA, 124(1), 61-80. https://doi.org/10.7146/math.scand.a-109390
Section
Articles