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

TL;DR

This paper develops weighted $L^2$ estimates for the Fourier extension operator in two dimensions and applies them to various $L^p$ problems, including maximal operators, fractal measures, and conjectures.

Contribution

It introduces new weighted $L^2$ estimates for Fourier extension operators and demonstrates their applications to several open $L^p$ problems in harmonic analysis.

Findings

Weighted $L^2$ estimates for Fourier extension in $ r^2$

Applications to maximal Schrödinger and extension operators

Decay estimates for Fourier transforms of fractal measures

Abstract

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 Schr\"odinger 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.