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

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 $1/2$.

Minor improvements are made for more clarity