The Bohr--P\'al Theorem and the Sobolev Space $W_2^{1/2}$

TL;DR

This paper explores the limits of the Bohr--Pál theorem, showing that for complex-valued functions with certain smoothness, it is impossible to always transform them into the Sobolev space $W_2^{1/2}$ via homeomorphisms.

Contribution

It demonstrates that the refined version of the Bohr--Pál theorem does not extend to complex-valued functions with Lipschitz regularity below 1/2.

Findings

The refined Bohr--Pál theorem applies only to real-valued functions.

For complex-valued functions with Lipschitz exponent less than 1/2, no homeomorphism can embed them into $W_2^{1/2}$.

The result delineates the boundary of the theorem's applicability.

Abstract

The well-known Bohr--P\'al 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--P\'al 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 $f\circ h\notin W_2^{1/2}(\mathbb T)$ for every homeomorphism $h$ of $\mathbb T$.