Restrictions of higher derivatives of the Fourier transform

Abstract

We consider several problems related to the restriction of (∇ k)f to a surface Σ ⊂ R d with nonvanishing Gauss curvature. While such restrictions clearly exist if f is a Schwartz function, there are few bounds available that enable one to take limits with respect to the L p (R d) norm of f. We establish three scenarios where it is possible to do so: • When the restriction is measured according to a Sobolev space H −s (Σ) of negative index, we determine the complete range of indices (k, s, p) for which such a bound exists. • Among functions wheref vanishes on Σ to order k − 1, the restriction of (∇ k)f defines a bounded operator from (this subspace of) L p (R d) to L 2 (Σ) provided 1 ≤ p ≤ 2d+2 d+3+4k. • When there is a priori control off | Σ in a space H (Σ), > 0, this implies improved regularity for the restrictions of (∇ k)f. If is large enough, then even ∇f L 2 (Σ) can be controlled in terms of f H (Σ) and f L p (R d) alone. The proofs are based on three main tools: the spectral synthesis work of Y. Domar, which provides a mechanism for L p approximation by "convolving along surfaces in spectrum", a new bilinear oscillatory integral estimate valid for ordinary L p functions, and a convexity-type property of the quantity (∇ k)f H −s (Σ) as a function of k and s that allows one to employ the control of f H (Σ). Contents 65 5. Robust estimates 72 6. Sharpness 85 7. Additional lemmas and supplementary material 91 Acknowledgment 95 References 95