The Bohr–Pál theorem and the Sobolev space $W_2^{1/2}$
Volume 231 / 2015
Abstract
The well-known Bohr–Pál theorem asserts that for every continuous real-valued function $f$ on the circle $\mathbb T$ there exists a change of variable, i.e., a homeomorphism $h$ of $\mathbb T$ onto itself, such that the Fourier series of the superposition $f\circ h$ converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings $f$ into the Sobolev space $W_2^{1/2}(\mathbb T)$. This refined version of the Bohr–Pál theorem does not extend to complex-valued functions. We show that if $\alpha \lt 1/2$, then there exists a complex-valued $f$ that satisfies the Lipschitz condition of order $\alpha $ and at the same time has the property that $f\circ h\notin W_2^{1/2}(\mathbb T)$ for every homeomorphism $h$ of $\mathbb T$.
