In Terry Tao's notes on the Hilbert transform it is stated that the Hilbert transform is not bounded on the spaces $L^1$ and $L^\infty$. I'm currently working on a problem where it would be convenient to find a bound for $$||H(f)||_{L^\infty}$$ in terms of e.g. a finite sum of $L^p$-norms of $f$, bounds in terms of derivatives of $f$ (if we assume $f$ to be differentiable) etc.
Does someone know such bounds? It would be also interesting to know if we can obtain bounds when restricting our function $f$ to some special class.