$L^p$ weighted Fourier restriction estimates

Abstract: We obtain some sharp $L^p$ weighted Fourier restriction estimates of the form $|Ef|_{L^p(B^{n+1}(0,R),Hdx)} \lessapprox R^{\beta}|f|_2$, where $E$ is the Fourier extension operator over the truncated paraboloid, and $H$ is a weight function on $\mathbb R^{n+1}$ which is $n$-dimensional up to scale $\sqrt R$.