The Sobolev space $W_2^{1/2}$: Simultaneous improvement of functions by a homeomorphism of the circle

TL;DR

The paper investigates the limitations of using a single circle homeomorphism to improve all functions in the Lipschitz 1/2 class into the Sobolev space W_2^{1/2}, showing such a universal transformation does not exist.

Contribution

It establishes that no single homeomorphism of the circle can simultaneously transform all Lipschitz 1/2 functions into the Sobolev space W_2^{1/2}.

Findings

No universal homeomorphism exists for this transformation.

The result applies specifically to the class of Lipschitz 1/2 functions.

The study clarifies limitations in function space transformations via homeomorphisms.

Abstract

It is known that for every continuous real-valued function $f$ on the circle $\mathbb T=\mathbb R/2\pi\mathbb Z$ there exists a change of variable, i.e., a self-homeomorphism $h$ of $\mathbb T$, such that the superposition $f\circ h$ is in the Sobolev space $W_2^{1/2}(\mathbb T)$. We obtain new results on simultaneous improvement of functions by a single change of variable in relation to the space $W_2^{1/2}(\mathbb T)$. The main result is as follows: there does not exist a self-homeomorphism $h$ of $\mathbb T$ such that $f\circ h\in W_2^{1/2}(\mathbb T)$ for every $f\in \mathrm{Lip}_{1/2}(\mathbb T)$. Here $\mathrm{Lip}_{1/2}(\mathbb T)$ is the class of all functions on $\mathbb T$ satisfying the Lipschitz condition of order $1/2$.