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 Lp(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 where f vanishes on Σ to order k − 1, the restriction of (∇ k ) f defines a bounded operator from (this subspace of)• When there is a priori control of f | 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 Lp(R d ) alone. The techniques underlying these results are inspired by the spectral synthesis work of Y. Domar, which provides a mechanism for Lp approximation by "convolving along surfaces", and the Stein-Tomas restriction theorem. Our main inequality is a bilinear form bound with similar structure to the Stein-Tomas T * T operator, generalized to accommodate smoothing along Σ and derivatives transverse to it. It is used both to establish basic H −s (Σ) bounds for derivatives of f and to bootstrap from surface regularity of f to regularity of its higher derivatives. p 2.1. Description of the annihilator and Domar's theory 2.2. Coincedence of Σ L k p and the spaces defined as kernels of restriction operators 2.3. Proofs of "if" part in Theorem 1.4 and Theorem 1.6 2.4. Proof of Corollary 1.14 3. Proof of Theorem 1.16 3.1. Pointwise estimates of the kernel 3.2. Interpolation 3.3. Strichartz estimates 4. Robust estimates 4.1. Introduction to "numerology" 4.2. Two simple examples