On the restriction of the Fourier transform to curves: endpoint results and the degenerate case
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- by Michael Christ PDF
- Trans. Amer. Math. Soc. 287 (1985), 223-238 Request permission
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.References
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Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 287 (1985), 223-238
- MSC: Primary 42B10; Secondary 26A33
- DOI: https://doi.org/10.1090/S0002-9947-1985-0766216-6
- MathSciNet review: 766216