Let I be an interval contained in R and let φ:I→ R be a given function. The paper contains the proof of the sharp estimate ||φ-(1/|I|)∫_I φ||_W(I)≦ 2||φ||_BMO(I), where W(I) is the weak-L^∞ space introduced by Bennett, DeVore and Sharpley. The proof exploits Bellman function method: the above inequality is deduced from the existence of a special function possessing certain majorization and concavity properties.

The research was partially supported by the NCN grant DEC-2012/05/B/ST1/00412.