Homeomorphic Changes of Variable and Fourier Multipliers

TL;DR

The paper demonstrates that through suitable homeomorphic changes of variables, any bounded continuous function can be transformed into a Fourier multiplier function across all $p$, extending the understanding of Fourier multiplier algebras.

Contribution

It introduces a method to transform bounded continuous functions into Fourier multipliers via homeomorphic changes of variables, applicable uniformly to entire families of functions.

Findings

Any bounded continuous function can be transformed into a Fourier multiplier by a homeomorphism.

A single change of variable can suffice for entire families of functions under certain conditions.

Results extend to functions on the torus, contrasting with classical Wiener algebra results.

Abstract

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 $p$, $1<p<\infty$. 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.