The finite Hilbert transform acting in the Zygmund space LlogL

Abstract

. The finite Hilbert transform T is a singular integral operator which maps the Zygmund space L log L := L log L ( − 1 , 1) continuously into L 1 := L 1 ( − 1 , 1). By extending the Parseval and Poincar´e-Bertrand formulae to this setting, it is possible to establish an inversion result needed for solving the airfoil equation T ( f ) = g whenever the data function g lies in the range of T within L 1 (shown to contain L (log L ) 2 ). Until now this was only known for g belonging to the union of all L p spaces with p > 1. It is established (due to a result of Stein) that T cannot be extended to any domain space beyond L log L whilst still taking its values in L 1 , i.e., T : L log L → L 1 is optimally defined.