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We consider the algebras $M_p$ of Fourier multipliers and show that everybounded continuous function $f$ on $\mathbb R^d$ can be transformed by anappropriate homeomorphic change of variable into a function that belongs to$M_p(\mathbb R^d)$ for all $p$, $1<p<\infty$. Moreover, under certainassumptions on a family $K$ of continuous functions, one change of variablewill suffice for all $f\in K$. A similar result holds for functions on thetorus $\mathbb T^d$. This may be contrasted with the known result on the Wieneralgebra, related to Luzin's rearrangement problem.

Homeomorphic Changes of Variable and Fourier Multipliers