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

B. Hollenbeck¹ ; N. J. Kalton² ; I. E. Verbitsky²

¹ Department of Mathematics Emporia State University Emporia, KS 66801, U.S.A.
² Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A.

Studia Mathematica, Tome 157 (2003) no. 3, pp. 237-278

Résumé

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.

Détail

DOI : 10.4064/sm157-3-2

Keywords: determine norm mathbb infty operator cal cal where cal cal respectively cosine sine fourier transforms positive real axis identity operator solves problem posed nbsp nbsp birman bir which originated scattering theory unbounded obstacles plane obtain p norms operators where hilbert transform conjugate function operator circle real line arbitrary real best constants other related inequalities found general framework present alternative proof important theorem cole relating best constant inequalities involving hilbert transform existence subharmonic minorants which extends several variables plurisubharmonic minorants