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On the restriction of the Fourier transform to curves: Endpoint results and the degenerate case. (English) Zbl 0563.42010
The author states the results of his paper in the abstract of the paper. For smooth curves ${\Gamma }$ in ${ℝ}^{n}$ with certain curvature properties he shows that the composition of the Fourier transform in ${ℝ}^{n}$ followed by restriction to ${\Gamma }$ defines a bounded operator from ${L}^{p}\left({ℝ}^{n}\right)$ to ${L}^{q}\left({\Gamma }\right)$ for certain p and q. The curvature hypotheses are the weakest under which this can 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.
Reviewer: R.D.Carmichael
##### MSC:
 42B10 Fourier type transforms, several variables 26D10 Inequalities involving derivatives, differential and integral operators
##### Keywords:
curves; Fourier transform; fractional integrals