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

B. Hollenbeck, N. J. Kalton, and I. E. Verbitsky. "Best constants for some operators associated with the Fourier and Hilbert transforms." Studia Mathematica 157.3 (2003): 237-278. <http://eudml.org/doc/286251>.

@article{B2003,
abstract = {We determine the norm in $L^\{p\}(ℝ₊)$, 1 < p < ∞, of the operator $I - ℱ_\{s\}ℱ_\{c\}$, where $ℱ_\{c\}$ and $ℱ_\{s\}$ 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 aI + bH, 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.},
author = {B. Hollenbeck, N. J. Kalton, I. E. Verbitsky},
journal = {Studia Mathematica},
keywords = {operator norm; re-expansion operator; cosine and sine Fourier transform; identity operator; Hilbert transform; subharmonic minorants; plurisubharmonic minorants},
language = {eng},
number = {3},
pages = {237-278},
title = {Best constants for some operators associated with the Fourier and Hilbert transforms},
url = {http://eudml.org/doc/286251},
volume = {157},
year = {2003},
}

TY - JOUR
AU - B. Hollenbeck
AU - N. J. Kalton
AU - I. E. Verbitsky
TI - Best constants for some operators associated with the Fourier and Hilbert transforms
JO - Studia Mathematica
PY - 2003
VL - 157
IS - 3
SP - 237
EP - 278
AB - We determine the norm in $L^{p}(ℝ₊)$, 1 < p < ∞, of the operator $I - ℱ_{s}ℱ_{c}$, where $ℱ_{c}$ and $ℱ_{s}$ 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 aI + bH, 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.
LA - eng
KW - operator norm; re-expansion operator; cosine and sine Fourier transform; identity operator; Hilbert transform; subharmonic minorants; plurisubharmonic minorants
UR - http://eudml.org/doc/286251
ER -