Measure theoretic aspects of the finite Hilbert transform

The finite Hilbert transform T, when acting in the classical Zygmund space (over $(-1,1)$), was intensively studied in {curbera-okada-ricker-log}. In this note an integral representation of T is established via the L¹(-1,1)-valued measure A↦ T(χA) for each Borel set A⊆(-1,1). This integral representation, together with various non-trivial properties of , allow the use of measure theoretic methods (not available in {curbera-okada-ricker-log}) 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.