Tchebyshev's characteristic of rearrangement invariant space

TL;DR

This paper introduces Tchebyshev's characteristic for rearrangement invariant spaces, identifies a class called regular r.i. spaces, and demonstrates that many known spaces are regular while providing examples of non-regular spaces.

Contribution

It defines Tchebyshev's characteristic, introduces regular r.i. spaces, and classifies several well-known spaces as regular, expanding understanding of their structural properties.

Findings

Most known r.i. spaces are regular.

Examples of non-regular r.i. spaces are constructed.

Tchebyshev's characteristic helps classify r.i. spaces.

Abstract

We introduce and investigate in this short article a new characteristic of rearrangement invariant (r.i.) (symmetric) space, namely so-called Tchebychev's characteristic. We reveal an important class of the r.i. spaces - so called regular r. i. spaces and show that the majority of known r.i. spaces: Lebesgue-Riesz, Grand Lebesgue Spaces, Orlicz, Lorentz and Marcinkiewicz r.i. spaces are regular. But we construct after several examples of r.i. spaces without the regular property.