A Combinatorial Approach to Musielak-Orlicz Spaces

In this paper we show that, using combinatorial inequalities and Matrix-Averages, we can generate Musielak-Orlicz spaces, i.e., we prove that $1/\pi \sum_{\pi} \max\limits_{1 \leq i \leq n} \abs{x_i y_{i\pi(i)}} \sim \norm{x}_{\Sigma M_i}$,wheretheOrliczfunctions$M_1,…,M_n$ 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.