A Combinatorial Approach to Musielak-Orlicz Spaces

TL;DR

This paper introduces a combinatorial method to generate Musielak-Orlicz spaces using inequalities and matrix averages, providing new approximation results for these norms.

Contribution

It presents a novel combinatorial approach to construct Musielak-Orlicz spaces and offers approximation results for their norms, advancing the theoretical understanding of these spaces.

Findings

Established a relation between matrix averages and Musielak-Orlicz norms

Provided an approximation theorem for Musielak-Orlicz norms

Extended results to Orlicz spaces as a special case

Abstract

In this paper we show that, using combinatorial inequalities and Matrix-Averages, we can generate Musielak-Orlicz spaces, i.e., we prove that $1/\pi \sum_{\pi} \max\limits_{1 \leq i \leq n} \abs{x_i y_{i\pi(i)}} \sim \norm{x}_{\Sigma M_i}$, where the Orlicz functions $M_1,...,M_n$ depend on the matrix $(y_{ij})_{i,j=1}^n$. We also provide an approximation result for Musielak-Orlicz norms which already in the case of Orlicz spaces turned out to be very useful.