Good-\(\lambda\) inequalities, rearrangements, and the John-Nirenberg theorem (Q2367625)

The goal of this work is to prove an integral inequality of the form \[ {1 \over | Q_ 0 |} \int_{Q_ 0} \left | f(x)-{1 \over | Q_ 0 |} \int_{Q_ 0} f\right |^ pdx \leq C{1 \over | Q_ 0 |} \int_{Q_ 0} \left | f(x)-{1 \over | Q_ 0 |} \int_{Q_ 0} f \right | dx, \] for \(Q_ 0\) a cube in \(\mathbb{R}^ n\), with a constant \(C\) that does not depend on the dimension \(n\). The condition on \(f\) is that the nonincreasing rearrangement of \(f\chi_{Q_ 0}\) has bounded mean oscillation on \(\mathbb{R}\). The proof is based on a good-\(\lambda\) inequality satisfied by \(f\).