The John-Nirenberg inequality and a Sobolev inequality in general domains
The classical John-Nirenberg lemma [\textit{F. John} and \textit{L. Nirenberg}, Commun. Pure Appl. Math. 14, 415-426 (1961; Zbl 0102.043)] states that if the mean oscillation of a function \(u\) in a cube \(D\) is uniformly bounded in each subcube, parallel to \(D\), then the oscillation of \(u\) is exponentially integrable in \(D\); more precisely, \[ \int_ D\exp(c| u(x)-a_ D|)dx\leq\text{const}.| D|,\tag{1} \] where \(a_ D\) is the integral average of \(u\) in \(D\). In the paper under review the author is looking for a class of domains \(D\) for which it holds that (1) is true as soon as the mean oscillation of \(u\) is uniformly bounded in each, say dyadic, cube \(Q\) such that, for a fixed constant \(\tau>1\), the enlarged cube \(\tau Q\) is still a subcube of \(D\). It appears that these domains are exactly the ones that satisfy a quasihyperbolic boundary condition. Hence e.g. the John domains belong to the class in question.
scientific article; zbMATH DE number 399044