Product properties of Hilbert transforms

AbstractLet fϵLp(R), gϵLq(R) with 1<p<∞, 1<q<∞ and let Hf, Hg be their respective Hilbert transforms. We give a simple proof of the identity Hf · Hg − f · g = H(f · Hg + g · Hf) a.e. and of its inverse in the case (1p) + (1q) ⩽ 1 which includes the cases already considered by Cossar and Tricomi. We next derive applications, especially to boundary values of analytic functions