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

Journal of Mathematical Analysis and Applications. 2026. Vol. 563. No. 2. Article 130787.

Lebedev V.

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 $1/2$. 

Keywords: гармонический анализ, гомеоморфизмы окружности, операторы суперпозиции, homeomorphisms of the circle, superposition operators, Sobolev spaces, пространства Соболева, harmonic analysis