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(L1): A bar right arrow T(chi(A)) for each Borel set A subset of (-1,1). This integral representation, together with various non-trivial properties of m(L1), 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.