Unions of cubes in $\mathbb{R}^{n}$, combinatorics in $\mathbb{Z}^{n}$ and the John-Nirenberg and John-Strömberg inequalities

Michael Cwikel

Published 2017-02-02Version 1

Suppose that the $d$-dimensional unit cube $Q$ is the union of three disjoint "simple" sets $E$, $F$ and $G$ and that the volumes of $E$ and $F$ are both greater than half the volume of $G$. Does this imply that, for some cube $W$ contained in $Q$. the volumes of $E\cap W$ and $F\cap W$ both exceed $s$ times the volume of $W$ for some absolute positive constant $s$? Here, by "simple" we mean a set which is a union of finitely many dyadic cubes. We prove that an affirmative answer to this question would have deep consequences for the important space $BMO$ of functions of bounded mean oscillation introduced by John and Nirenberg. The notion of a John-Str\"omberg pair is closely related to the above question, and the above mentioned result is obtained as a consequence of a general result about these pairs. We also present a number of additional results about these pairs.