Best constants for some operators associated with the Fourier and Hilbert transforms
Volume 157 / 2003
Studia Mathematica 157 (2003), 237-278
MSC: Primary 42B10; Secondary 46E30.
DOI: 10.4064/sm157-3-2
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, and $I$ is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane.
We also obtain the $L^p$-norms of the operators $a I + b H$, where $H$ is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real $a, b$. Best constants in other related inequalities are found.
In a more general framework, we present an alternative proof of the important theorem of Cole relating best constant inequalities involving the Hilbert transform and the existence of subharmonic minorants, which extends to several variables and plurisubharmonic minorants.
