Let $f\in \operatorname{BMO}(\partial \Delta)$, then there exists a Carleson measure $\mu$ in $\Delta$ such that $$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\partial\Delta,$$ where $\Delta$ is the unit disc and $P_{z}(\zeta)$ is the Poisson kernel.
The author did not give the proof but said that this proposition is an immediate consequence of the $\operatorname{BMO}$, $H^1$ duality. Could anyone give me some hints? Thanks very much!
