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

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