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 $\|E_{\Sigma}f\|_{L^q(\Omega)}\leq C\|f\|_{L^p(\Sigma)}$ and $(\mathcal{R}(|E_{\Sigma}f|^{q}))^{\frac{1}{q}}\leq C\|f\|_{L^p(\Sigma)}$ 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.