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

Abstract:

We prove that \begin{equation*} \|X(|u|^2)\|_{L^3_{t,\ell }}\leq C\|f\|_{L^2(\mathbb {R}^2)}^2, \end{equation*} 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 $1\leq k\leq d-1$. 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.