Restriction and spectral multiplier theorems on asymptotically conic manifolds

TL;DR

This paper extends restriction and spectral multiplier theorems to asymptotically conic manifolds with variable coefficients, providing sharp estimates and generalizing classical results from Euclidean spaces.

Contribution

It proves the first restriction theorem for the Laplacian plus potential on asymptotically conic manifolds, generalizing Euclidean results to variable coefficient settings.

Findings

Established restriction estimates for spectral measures on asymptotically conic manifolds.

Derived sharp spectral multiplier and Bochner-Riesz summability results.

Extended classical Euclidean theorems to a broader geometric setting.

Abstract

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(\lambda)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq 2(n+1)/(n+3)$, where $p'$ is the conjugate exponent to $p$, with operator norm scaling as $\lambda^{n(1/p - 1/p') - 1}$. We prove a geometric generalization in which the Laplacian on $\RR^n$ is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setting. It is closely related to, but stronger than, Sogge's discrete $L^2$ restriction theorem, which is an $O(\lambda^{n(1/p - 1/p') - 1})$ estimate on the $L^p \to L^{p'}$ operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including Bochner-Riesz summability results, which are sharp for $p$ in the range above.