Abstract
We prove that if aC 1 smooth change of variable ϕ:ℝ→ℝ generates a bounded composition operatorf→f°ϕ in the spaceA p(ℝ)=L p,p≠2, then φ is linear (affine).
We also prove that for a nonlinearC 1 mapping φ, the norms of exponentialse iλϕ as Fourier multipliers inL p(ℝ) tend to infinity (λ∈ℝ,|λ|→∞). In both results the condition φ∈C 1 is sharp, it cannot be replaced by the Lipschitz condition.
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Lebedev, V., Olevskiî, A. C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers. Geometric and Functional Analysis 4, 213–235 (1994). https://doi.org/10.1007/BF01895838
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DOI: https://doi.org/10.1007/BF01895838