The finite Hilbert transform acting on $L^{\mathrm{\infty }}$

The action of the finite Hilbert transform defined on $L^{\mathrm{\infty }}(-1,1)$ and taking its values in the Zygmund space $L_{\text{exp}}(-1,1)$ is studied in detail. This is a reciprocal situation to the investigation recently undertaken in [11] of the finite Hilbert transform defined on the Zygumd space $L\text{log}L(-1,1)$ and taking its values in $L^1(-1,1)$. The fact that both $L^{\mathrm{\infty }}(-1,1)$ and $L_{\text{exp}}(-1,1)$ fail to be separable generates new features not present in[11].