Measure theoretic aspects of the finite Hilbert transform

The finite Hilbert transform $T$, when acting in the classical Zygmund space $LlogL$ (over $(-1,1)$), was intensively studied in [8]. In this note, an integral representation of $T$ is established via the $L^1(-1,1)$-valued measure $m_{L^1}:A\mapsto T\left({\chi }_A\right)$ for each Borel set $A\subseteq (-1,1)$. This integral representation, together with various non-trivial properties of $m_{L^1}$, allows the use of measure theoretic methods (not available in [8]) to establish new properties of $T$. For instance, as an operator between Banach function spaces $T$ is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for $T$ plays a crucial role.