Fourier restriction for the additive Brownian sheet

TL;DR

This paper investigates the Fourier restriction problem for the fractal surface generated by the additive Brownian sheet, providing new estimates and conditions that advance understanding of Fourier analysis on fractal sets.

Contribution

It introduces the first non-trivial Fourier restriction estimates for the additive Brownian sheet's graph, including a precise formula for its Fourier spectrum and improved bounds over classical results.

Findings

Established sufficient conditions for Fourier transform boundedness on the fractal surface.

Derived necessary conditions using Fourier spectrum and Knapp examples.

Provided a formula for the Fourier spectrum of the additive Brownian sheet's graph.

Abstract

The Fourier restriction problem asks when it is meaningful to restrict the Fourier transform of a function to a given set. Many of the key examples are smooth co-dimension 1 manifolds, although there is increasing interest in fractal sets. Here we propose a natural intermediary problem where one considers the fractal surface generated by the graph of the additive Brownian sheet in $\mathbb{R}^k$. We obtain the first non-trivial estimates in this direction, giving both a sufficient condition on the range of $q\in[1,2]$ for the Fourier transform to be $L^{q}(\mathbb{R}^{k+1})\to L^2(G(W))$ bounded and a necessary condition for it to be $L^{q}(\mathbb{R}^{k+1})\to L^p(G(W))$ bounded. The sufficient condition is obtained via the Fourier spectrum, which is a family of dimensions that interpolate between the Fourier and Hausdorff dimensions. Our main technical result, which is of interest in its own right, gives a precise formula for the Fourier spectrum of the natural measure on the graph of the additive Brownian sheet, and we apply this result to the Fourier restriction problem. Our restriction estimate is stronger than the estimate obtained from the well-known Stein--Tomas restriction theorem for all $k\geq3$. We obtain the necessary condition in two different ways, one via the Fourier spectrum and one via an appropriate Knapp example.