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On the restriction of the Fourier transform to curves: endpoint results and the degenerate case


Author: Michael Christ
Journal: Trans. Amer. Math. Soc. 287 (1985), 223-238
MSC: Primary 42B10; Secondary 26A33
MathSciNet review: 766216
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Abstract: For smooth curves $ \Gamma $ in $ {{\mathbf{R}}^n}$ with certain curvature properties it is shown that the composition of the Fourier transform in $ {{\mathbf{R}}^n}$ followed by restriction to $ \Gamma $ defines a bounded operator from $ {L^p}({{\mathbf{R}}^n})$ to $ {L^q}(\Gamma )$ for certain $ p,q$. The curvature hypotheses are the weakest under which this could hold, and $ p$ is optimal for a range of $ q$. In the proofs the problem is reduced to the estimation of certain multilinear operators generalizing fractional integrals, and they are treated by means of rearrangement inequalities and interpolation between simple endpoint estimates.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0766216-6
Keywords: Fourier transform, curvature, multilinear operator, interpolation
Article copyright: © Copyright 1985 American Mathematical Society