Fourier extension estimates on a strip in ${\mathbb{R}}^2$

Abstract

Given a smooth curve with nonzero curvature $\Sigma \subset {\mathbb{R}}^2$, let $E_{\Sigma }$ denote the associated Fourier extension operator. For both general compact curves and the parabola, we characterize the pairs $(p,q)\in [1,\infty {]}^2$ for which the estimates $\Vert E_{\Sigma }f{\Vert }_{L^q(\Omega )}\le C\Vert f{\Vert }_{L^p(\Sigma )}$ and $(\mathcal{R}(|E_{\Sigma }f{|}^q){)}^{\frac{1}{q}}\le C\Vert f{\Vert }_{L^p(\Sigma )}$ hold, where $\Omega$ is a strip in ${\mathbb{R}}^2$ and $\mathcal{R}$ denotes the Radon transform. This work continues the study of mass concentration of $x\mapsto E_{\Sigma }f(x)$ near lines in ${\mathbb{R}}^2$, initiated by Bennett and Nakamura and later extended by Bennett, Nakamura, and the second author, where expressions of the form $(\mathcal{R}(|E_{\Sigma }f{|}^2){)}^{\frac{1}{2}}$ were studied.

arXiv comment18 pages, 2 figures. v2: references added, typos corrected