Damian Kubiak | Searcharxiv

arXiv2015

The Daugavet property in the Musielak-Orlicz spaces

We show that among all Musielak-Orlicz function spaces on a $σ$-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.

math.FA↗

arXiv2011

On the dual of Cesàro function space

The goal of this paper is to present an isometric representation of the dual space to Cesàro function space $C_{p,w}$, $1<p<\infty$, induced by arbitrary positive weight function $w$ on interval $(0,l)$ where $0<l\leqslant\infty$. For this purpose given a strictly decreasing nonnegative function $Ψ$ on $(0,l)$, the notion of essential $Ψ$-concave majorant $\hat f$ of a measurable function $f$ is introduced and investigated. As applications it is shown that every slice of the unit ball of the Cesàro function space has diameter 2. Consequently Cesàro function spaces do not have the Radon-Nikodym property, are not locally uniformly convex and they are not dual spaces.

math.FA↗