A combinatorial approach to Musielak--Orlicz spaces

Abstract

In this paper we show that, using combinatorial inequalities and Matrix-Averages, we can generate Musielak-Orlicz spaces, i.e., we prove that $\underset{\pi}{\mbox{Ave}} \max\limits_{1 \leq i \leq n} |x_i y_{i\pi(i)}| \sim \|x\|_{\Sigma M_i}$, where the Orlicz functions $M_1,\ldots,M_n$ 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.