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Article

Homeomorphic Changes of Variable and Fourier Multipliers

Lebedev V., Olevskii A.

 We consider the algebras M_p of
Fourier multipliers and show that for every bounded continuous
function f on R^d there exists a self-homeomorphism
h of R^d such that the superposition foh$ is
in M_p(R^d) for all p, 1<p<\infty. Moreover,
under certain assumptions on a family K of continuous
functions, one h will suffice for all f\in K. A similar
result holds for functions on the torus T^d. This may
be contrasted with the known solution of Luzin's problem
related to the Wiener algebra.