ODE for $L^p$ norms

Jarno Talponen¹

¹ Department of Physics and Mathematics University of Eastern Finland Box 111 FI-80101 Joensuu, Finland

Studia Mathematica, Tome 236 (2017) no. 1, pp. 63-83

Résumé

In this paper we relate the geometry of Banach spaces to the theory of differential equations, apparently in a new way. We will construct Banach function space norms arising as weak solutions to ordinary differential equations (ODE) of the first order. This provides as a special case a new way of defining varying exponent $L^p$ spaces, different from the Musielak–Orlicz type approach. We explain heuristically how the definition of the norm by means of a particular ODE is justified. The resulting class of spaces includes the classical $L^p$ spaces as a special case. A noteworthy detail regarding our $L^{p(\cdot )}$ norms is that they satisfy Hölder’s inequality (properly).

Détail

DOI : 10.4064/sm8561-8-2016

Keywords: paper relate geometry banach spaces theory differential equations apparently construct banach function space norms arising weak solutions ordinary differential equations ode first order provides special defining varying exponent spaces different musielak orlicz type approach explain heuristically definition norm means particular ode justified resulting class spaces includes classical spaces special noteworthy detail regarding cdot norms satisfy lder inequality properly