Measure theoretic aspects of the finite Hilbert transform

The finite Hilbert transform , when acting in the classical Zygmund space (over ), was intensively studied in [8]. In this note, an integral representation of is established via the ‐valued measure for each Borel set . This integral representation, together with various non‐trivial properties of , allows the use of measure theoretic methods (not available in [8]) to establish new properties of . For instance, as an operator between Banach function spaces is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for plays a crucial role.