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 [2] and later extended by Bennett, Nakamura, and the second author in [3], where expressions of the form \((\mathcal {R}(|E_{\Sigma }f|^{2}))^{\frac{1}{2}}\) were studied.
