# Decompositions of Nakano norms by ODE techniques

## TL;DR

This paper explores the structure of Nakano type variable exponent Lebesgue spaces using ODE techniques, providing new decompositions and embeddings into sums of projection bands, with implications for understanding their isomorphism properties.

## Contribution

It introduces a novel approach to decompose Nakano norms via ODE-based variable Lebesgue norms and establishes embeddings into tractable sums of projection bands.

## Key findings

- Embedding of variable Lebesgue spaces into sums of projection bands
- Isomorphism constants for these embeddings
- Effect of transformations on ODE-determined norms

## Abstract

We study decompositions of Nakano type varying exponent Lebesgue norms and spaces. These function spaces are represented here in a natural way as tractable varying $\ell^p$ sums of projection bands. The main results involve embedding the varying Lebesgue spaces to such sums, as well as the corresponding isomorphism constants. The main tool applied here is an equivalent variable Lebesgue norm which is defined by a suitable ordinary differential equation introduced recently by the author. We also analyze the effect of transformations changing the ordering of the unit interval on the values of the ODE-determined norm.