No self-homeomorphism h of the circle exists such that f ∘ h lies in W_2^{1/2}(T) for every f in Lip_{1/2}(T).
Change of variable and the rapidity of decrease of Fourier coefficients
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abstract
We consider the class $C(T)$ of continuous real-valued functions on the circle. For certain classes of functions naturally characterised by the rapidity of decrease of Fourier coefficients we investigate whether it is possible to bring families of functions in $C(T)$ into these classes by a change of variable. This paper was originally published in Matematicheski\v{\i} Sbornik, 181:8 (1990), 1099--1113 (Russian). The English translation, published in Mathematics of the USSR, Sbornik, 70:2 (1991), 541--555, is to a large extent inconsistent with the original text. Herein the author provides a corrected translation.
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The Sobolev space $W_2^{1/2}$: Simultaneous improvement of functions by a homeomorphism of the circle
No self-homeomorphism h of the circle exists such that f ∘ h lies in W_2^{1/2}(T) for every f in Lip_{1/2}(T).