A John–Nirenberg type inequality for

The John–Nirenberg inequality characterizes functions in the space BMO in terms of the decay of the distribution function of their oscillations over a cube. In this paper we prove separate necessary and sufficient John–Nirenberg type inequalities for functions in the space Q α ( R n ) , introduced by Essén, Janson, Peng and Xiao, who conjectured a version of this inequality. Our results are a modified version of their conjecture, and we give a counterexample to show the necessity for this modification. The counterexample also shows that these necessary and sufficient conditions cannot be reconciled.