C1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers
- Cite this article as:
- Lebedev, V. & Olevskiî, A. Geometric and Functional Analysis (1994) 4: 213. doi:10.1007/BF01895838
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We prove that if aC1 smooth change of variable ϕ:ℝ→ℝ generates a bounded composition operatorf→f°ϕ in the spaceAp(ℝ)=Lp,p≠2, then φ is linear (affine).
We also prove that for a nonlinearC1 mapping φ, the norms of exponentialseiλϕ as Fourier multipliers inLp(ℝ) tend to infinity (λ∈ℝ,|λ|→∞). In both results the condition φ∈C1 is sharp, it cannot be replaced by the Lipschitz condition.