The paper Restriction for Schrödinger equations, by Fabio Nicola, is now available on arXiv (arXiv:2505.12527).
We prove a transference principle, asserting that every restriction estimate satisfied by the Fourier transform in \(\mathbb{R}^d\) is likewise valid for the propagator of certain Schrödinger equations. We consider smooth Hamiltonians with at most quadratic growth, and also a class of nonsmooth Hamiltonians, encompassing potentials that are Fourier transforms of complex (finite) measures.
Roughly speaking, if the initial datum belongs to \(L^p(\mathbb{R}^d)\), for \(p\) in a suitable range of exponents, the solution \(u(t,\cdot)\) (for each fixed \(t\), with the exception of certain particular values) can be meaningfully restricted to compact curved submanifolds of \(\mathbb{R}^d\).
The underlying property responsible for this phenomenon is the boundedness
\(L^p \to (\mathcal{F}L^p)_{\text{loc}}\), with \(1 \leq p \leq 2\), of the propagator, which is derived from almost diagonalization and dispersive estimates in certain function spaces defined in terms of wave packet decompositions in phase space.