The Daugavet property in the Musielak-Orlicz spaces

TL;DR

This paper characterizes Musielak-Orlicz spaces with the Daugavet property, showing only specific classical and combined spaces possess it, including $L_1$, $L_{ olinebreak ext{infty}}$, and their direct sums.

Contribution

It provides a complete characterization of the Daugavet property in Musielak-Orlicz spaces, including weighted, variable exponent, and Orlicz spaces, under standard norms.

Findings

Only $L_1$, $L_{ ext{infty}}$, and their direct sums have the Daugavet property.

Complete characterizations for weighted interpolation, Nakano, and Orlicz spaces.

The Daugavet property is exclusive to these specific Musielak-Orlicz spaces.

Abstract

We show that among all Musielak-Orlicz function spaces on a $\sigma$-finite non-atomic complete measure space equipped with either the Luxemburg norm or the Orlicz norm the only spaces with the Daugavet property are $L_1$, $L_{\infty}$, $L_1\oplus_1 L_{\infty}$ and $L_1\oplus_{\infty} L_{\infty}$. We obtain in particular complete characterizations of the Daugavet property in the weighted interpolation spaces, the variable exponent Lebesgue spaces (Nakano spaces) and the Orlicz spaces.