Fourier restriction for the additive Brownian sheet

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.

Figures (4)

• Figure 1: Two realisations of the Brownian surface for $k=2$. Left: in the additive case. Right: in the non-additive case.
• Figure 2: Plots of the Fourier spectrum of the surface measure $\mu$ on the additive Brownian surface $G(W)$ for $k=1,\ldots,6$; see Theorem \ref{['thm:FSBM']}.
• Figure 3: Bounds for the range of $q$ for the Fourier extension estimate to hold and not hold for the Brownian sheet; see Theorem \ref{['thm:restrictionBM']} and Corollary \ref{['usingus']}. By "Hambrook--Łaba" we refer to the observation made in HL13 that no extension estimate will hold for $2\leqslant q < \frac{2d}{\dim_{\mathrm{S}} \mu} = \frac{2(k+1)}{k + 1/2}$. These plots should be understood as only applying to integer points in the domain, but we included the full curves for aesthetic reasons. In particular, using the Fourier spectrum bounds the sharp threshold between the solid curves and appealing to previous estimates bounds the sharp threshold between the dashed curves.
• Figure 4: Knapp example for an $\alpha$-Hölder function $f$, where $g$ is the characteristic function of the rectangle $R$ centred at $(s,f(s)) \in G(f)$ of side-lengths $\approx \delta\times\cdots\times\delta\times\delta^{\alpha}$. To the right is $R^*$ the dual rectangle of $R$.

Theorems & Definitions (10)

• Theorem 2.1
• Corollary 2.2
• Proposition 2.3: Xiao
• proof
• Theorem 2.4
• proof
• Corollary 2.5
• proof
• Theorem 2.6
• Corollary 2.7