I am considering the boundary values of a bounded holomorphic functions. Suppose $w$ is a bounded holomorphic function in upper half plane, with continuous and bounded boundary value $f$ on real axis. $f$ is smooth, for example, in $C^2$, and bounded. We know that for bounded $f$, hilbert transform $Hf\in BMO$, and in general not in $L^{\infty}$.

My question is: With so many properties of $w$, would it be the case that $Hf\in L^{\infty}$? Is there a sufficient and necessary for $Hf\in L^{\infty}$ for $f\in L^{\infty}$?