Weighted-BMO and the Hilbert transform

In 1967, E. M. Stein proved that the Hilbert transform is bounded from ${\displaystyle L^{\infty }}$ to BMO. In 1976, Muckenhoupt and Wheeden gave an analogue of Stein's result. They gave a necessary and sufficient condition for the boundedness of the Hilbert transform from ${\displaystyle L^{\infty }\_w}$. We improve the results of Muckenhoupt and Wheeden's and give a necessary and sufficient condition for the boundedness of the Hilbert transform from ${\displaystyle BM{O}_w}$ to ${\displaystyle BM{O}_w}$.