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Regular version of the site

Article

The Bohr--Pal Theorem and the Sobolev Space W_2^{1/2}

Studia Mathematica. 2015. Vol. 231. No. 1. P. 73-81.

The well-known Bohr--Pal theorem
asserts that for every continuous real-valued function f on
the circle T there exists a change of variable, i.e.,
a homeomorphism h of T onto itself, such that the
Fourier series of the superposition foh converges
uniformly. Subsequent improvements of this result imply that
actually there exists a homeomorphism that brings f into the
Sobolev space W_2^{1/2}(T). This refined version of
the Bohr--Pal theorem does not extend to complex-valued
functions. We show that if \alpha<1/2, then there exists a
complex-valued f that satisfies the Lipschitz condition of
order \alpha and at the same time has the property that
foh is not in W_2^{1/2}(T) for every homeomorphism
h of T.