Weighted $L^2$ estimates with applications to $L^p$ problems

We establish some weighted $L^2$ estimates for the Fourier extension operator in ${\mathbb{R}}^2$ and discuss several applications to $L^p$ problems. These include estimates for the maximal Schrödinger operator and the maximal extension operator, decay of circular $L^p$-means of Fourier transform of fractal measures, and an $L^p$ analogue of the Mizohata-Takeuchi conjecture.