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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Taryn C. Flock (University of Birmingham)

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

We explore a natural interplay between the solution to the time-dependent free Schrödinger equation on $\mathbb{R}^d$ and the (spatial) $k$-plane transform for $1\leq k\leq d-1$. A first result is that $$\|X(|u|^2)\|_{L^3_{t,\ell}}\leq C\|f\|_{L^2(\mathbb{R}^2)}^2,$$ 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 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 an isotropic centered 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.