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## Studia Mathematica

2015 | 231 | 1 | 73-81
Tytuł artykułu

### The Bohr-Pál theorem and the Sobolev space $W₂^{1/2}$

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The well-known Bohr-Pál 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 f ∘ h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space $W₂^{1/2}(𝕋)$. 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 $f ∘ h ∉ W₂^{1/2}(𝕋)$ for every homeomorphism h of 𝕋.
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Tom
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73-81
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2015
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• National Research University Higher School of Economics, 34 Tallinskaya St., Moscow, 123458, Russia
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