ODE for $L^p$ norms
Tom 236 / 2017
Studia Mathematica 236 (2017), 63-83
MSC: Primary 46E30, 34A12; Secondary 46B10, 31B10.
DOI: 10.4064/sm8561-8-2016
Opublikowany online: 25 November 2016
Streszczenie
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).
