$L^p$ weighted Fourier restriction estimates

We obtain some sharp $L^p$ weighted Fourier restriction estimates of the form $\mathrm{\parallel }Ef{\mathrm{\parallel }}_{L^p(B^{n+1}(0,R),Hdx)}⪅R^{\beta }\mathrm{\parallel }f{\mathrm{\parallel }}_2$, 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 $\sqrt{R}$.