On the topological stability of continuous functions in certain spaces related to Fourier series

We show that the following properties of a continuous function $f$ on the circle $\mathbb T$ are equivalent: the sequence $\widehat{f\circ h}$ of the Fourier coefficients of the superposition $f\circ h$ belongs to the weak $l^1$ for every homeomorphism $h$ of the circle onto itself; $f$ is a function of bounded quadratic variation. We obtain similar results for spaces of functions whose sequence of Fourier coefficients belongs to the weak $l^p$, $1<p<2$, for spaces $A_p$ of functions $f$ with $\widehat{f}\in l^p$, for the Sobolev spaces $W_2^\lambda$, and for other spaces of functions on $\mathbb T$. Under rather general assumptions on a space $\mathbb X$ of functions on the circle, we give a necessary condition for a given continuous function $f$ to stay in $\mathbb X$ for every change of variable. We also consider the multidimensional case, which is essentially different from the one-dimensional case. In particular, we show that if $p<2$ and $f$ is a continuous function on the torus $\mathbb T^d$, $d\geqslant2$, such that $f\circ h\in A_p(\mathbb T^d)$ for every homeomorphism $h\colon \mathbb T^d\to\mathbb T^d$, then $f$ is constant.