A sharp $k$-plane Strichartz inequality for the Schrödinger equation

Abstract

We prove that $$\Vert X(|u{|}^2){\Vert }_{L_{t,\ell }^3}\le C\Vert f{\Vert }_{L^2({\mathbb{R}}^2)}^2,$$ where $u(x,t)$ is the solution to the linear time-dependent Schrödinger equation on ${\mathbb{R}}^2$ with initial datum $f$, and $X$ is the (spatial) X-ray transform on ${\mathbb{R}}^2$. In particular, we identify the best constant $C$ and show that a datum $f$ is an extremiser if and only if it is a gaussian. We also establish bounds of this type in higher dimensions $d$, where the X-ray transform is replaced by the $k$-plane transform for any $1\le k\le d-1$. In the process we obtain sharp $L^2(\mu )$ bounds on Fourier extension operators associated with certain high-dimensional spheres, involving measures $\mu$ supported on natural "co-$k$-planarity" sets.

18 pages. Author accepted version. To appear in Trans. Amer. Math. Soc