Random Constructions for Sharp Estimates of Mizohata-Takeuchi Type

TL;DR

This paper uses high-dimensional probability to construct a broad class of weights that satisfy sharp Mizohata-Takeuchi type Fourier restriction estimates, showing that most weights are near optimal with high probability.

Contribution

It introduces a probabilistic construction of weights that achieve sharp Mizohata-Takeuchi estimates, expanding the understanding of weighted Fourier restriction phenomena.

Findings

Most weights satisfy sharp Mizohata-Takeuchi estimates with high probability.

The constructed weights are large and diverse, covering a broad class.

The results demonstrate the typicality of sharp estimates in a probabilistic sense.

Abstract

A Mizohata-Takeuchi type estimate is a type of weighted Fourier restriction estimate. Using tools from high dimensional probability, we construct a large class of weights that satisfy sharp estimates of Mizohata-Takeuchi type. One can interpret our result as saying that with high probability, a generic weight satisfies a sharp inequality of Mizohata-Takeuchi type (up to an epsilon-loss).