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

## TL;DR

This paper characterizes the Fourier extension estimates for smooth curves with nonzero curvature on a strip in ^2, extending previous work on mass concentration near lines and providing precise conditions for these estimates.

## Contribution

It provides a complete characterization of the pairs (p,q) for which Fourier extension estimates hold on a strip in ^2 for general curves and the parabola, advancing understanding of mass concentration phenomena.

## Key findings

- Identifies (p,q) pairs satisfying extension estimates on a strip in ^2.
- Extends previous work on mass concentration near lines for Fourier extension operators.
- Provides new bounds and conditions for both general curves and the parabola.

## 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.