Best constants for some operators associated with the Fourier and Hilbert transforms

Abstract

We determine the norm in $L^p ({{\mathbb R}}_+)$, $1< p< \infty $, of the operator $I - {\cal F}_{\rm s} {\cal F}_{\rm c}$, where ${\cal F}_{\rm c}$ and ${\cal F}_{\rm s}$ are respectively the cosine and sine Fourier transforms on the positive real axis,