The Bohr-Pál theorem and the Sobolev space $W{₂}^{1/2}$

Vladimir Lebedev. "The Bohr-Pál theorem and the Sobolev space $W₂^{1/2}$." Studia Mathematica 231.1 (2015): 73-81. <http://eudml.org/doc/285805>.

@article{VladimirLebedev2015,
abstract = {The well-known Bohr-Pál theorem asserts that for every continuous real-valued function f on the circle there exists a change of variable, i.e., a homeomorphism h of onto itself, such that the Fourier series of the superposition f ∘ h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space $W₂^\{1/2\}()$. This refined version of the Bohr-Pál theorem does not extend to complex-valued functions. We show that if α < 1/2, then there exists a complex-valued f that satisfies the Lipschitz condition of order α and at the same time has the property that $f ∘ h ∉ W₂^\{1/2\}()$ for every homeomorphism h of .},
author = {Vladimir Lebedev},
journal = {Studia Mathematica},
keywords = {harmonic analysis; homeomorphisms of the circle; superposition operators; Sobolev spaces},
language = {eng},
number = {1},
pages = {73-81},
title = {The Bohr-Pál theorem and the Sobolev space $W₂^\{1/2\}$},
url = {http://eudml.org/doc/285805},
volume = {231},
year = {2015},
}

TY - JOUR
AU - Vladimir Lebedev
TI - The Bohr-Pál theorem and the Sobolev space $W₂^{1/2}$
JO - Studia Mathematica
PY - 2015
VL - 231
IS - 1
SP - 73
EP - 81
AB - The well-known Bohr-Pál theorem asserts that for every continuous real-valued function f on the circle there exists a change of variable, i.e., a homeomorphism h of onto itself, such that the Fourier series of the superposition f ∘ h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space $W₂^{1/2}()$. This refined version of the Bohr-Pál theorem does not extend to complex-valued functions. We show that if α < 1/2, then there exists a complex-valued f that satisfies the Lipschitz condition of order α and at the same time has the property that $f ∘ h ∉ W₂^{1/2}()$ for every homeomorphism h of .
LA - eng
KW - harmonic analysis; homeomorphisms of the circle; superposition operators; Sobolev spaces
UR - http://eudml.org/doc/285805
ER -