Homeomorphic Changes of Variable and Fourier Multipliers

We consider the algebras $M_p$ of Fourier multipliers and show that every bounded continuous function $f$ on ${\mathbb{R}}^d$ can be transformed by an appropriate homeomorphic change of variable into a function that belongs to $M_p({\mathbb{R}}^d)$ for all $p$, $1<p<\mathrm{\infty }$. Moreover, under certain assumptions on a family $K$ of continuous functions, one change of variable will suffice for all $f\in K$. A similar result holds for functions on the torus ${\mathbb{T}}^d$. This may be contrasted with the known result on the Wiener algebra, related to Luzin’s rearrangement problem.