Note on order-isomorphic isometric embeddings of some recent function spaces

TL;DR

This paper studies recent variable exponent $L^p$ spaces defined via ODEs, showing they can be embedded into a universal $ ext{ell}^p$ space, revealing their structure through order-isomorphic isometric embeddings.

Contribution

It demonstrates that these variable exponent $L^p$ spaces are finitely representable in a universal $ ext{ell}^p$ space and can be embedded into an ultrapower of this space in a natural way.

Findings

Spaces are finitely representable in a universal $ ext{ell}^p$ space.

Existence of order-isomorphic isometric embeddings into ultrapowers.

Unified approach to embedding these function spaces.

Abstract

We investigate certain recently introduced ODE-determined varying exponent $L^p$ spaces. It turns out that these spaces are finitely representable in a concrete universal varying exponent $\ell^p$ space. Moreover, this can be accomplished in a natural unified fashion. This leads to order-isomorphic isometric embeddings of all of the above $L^p$ spaces to an ultrapower of the above varying exponent $\ell^p$ space.