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Abstract 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.
Curbera et al. (Mon,) studied this question.