Combinatorial Inequalities and Subspaces of L1

TL;DR

This paper explores combinatorial inequalities and demonstrates that certain product spaces of Orlicz spaces are uniformly isomorphic to subspaces of L_1 under specific conditions involving separation by a power function.

Contribution

It establishes new combinatorial inequalities and characterizes when product spaces of Orlicz spaces embed into L_1, advancing understanding of their geometric structure.

Findings

Product spaces l^n_M(l^n_N) are uniformly isomorphic to subspaces of L_1

Embedding occurs when M and N are separated by t^r, 1<r<2

Provides conditions for isomorphism based on Orlicz functions

Abstract

Let M and N be Orlicz functions. We establish some combinatorial inequalities and show that the product spaces l^n_M(l^n_N) are uniformly isomorphic to subspaces of L_1 if M and N are "separated" by a function t^r, 1<r<2.