The sharp Fourier restriction range of the additive Brownian surface

The graph of the additive Brownian sheet is a canonical random fractal hypersurface in $\mathbb{R}^{k+1}$, and Fraser and de Orellana (arXiv:2601.05802, 2026) initiated its Fourier restriction theory: they computed the Fourier spectrum of the natural surface measure and used it to derive the first non-trivial restriction estimates, beating Stein–Tomas for $k\geq 3$. However, only partial sufficient and necessary regions for the restriction estimate $\|\widehat{f\mu}\|_{L^{q}(\mathbb{R}^{k+1})}\lesssim\|f\|_{L^{p}(\mu)}$ are known, and the gap between them is open for every $k\geq 1$. The problem is to determine, almost surely, the exact set of exponents $(p,q)$ for which the estimate holds for the surface measure on the additive Brownian surface.

Background

The Fourier restriction problem asks for which exponent pairs the Fourier transform of an $L^{q'}$ function can be meaningfully restricted to a given set: given a finite Borel measure $\mu$ supported on a Lebesgue-null set of $\mathbb{R}^{d}$, one asks when $\|\widehat{f}\|_{L^{p'}(\mu)}\lesssim\|f\|_{L^{q'}(\mathbb{R}^{d})}$ holds, which by duality is equivalent to the extension estimate $\|\widehat{f\mu}\|_{L^{q}(\mathbb{R}^{d})}\lesssim\|f\|_{L^{p}(\mu)}$. The classical theory concerns smooth hypersurfaces such as the sphere and paraboloid, where the Stein–Tomas theorem gives sharp $L^{2}$-based estimates. Mockenhaupt (2000), Mitsis (2002), and Bak–Seeger (2011) extended Stein–Tomas to general fractal measures satisfying a Fourier decay condition and a Frostman condition (a uniform bound $\mu(B(x,r))\lesssim r^{s}$ on the mass of small balls), and Hambrook and Łaba (2013) showed these hypotheses cannot in general be weakened, in particular ruling out extension estimates for $2\leq q<2d/\dim_{\mathrm{S}}\mu$, where $\dim_{\mathrm{S}}\mu$ is the Sobolev dimension of the measure, defined via average Fourier decay as the supremum of $s$ such that $\int|\widehat{\mu}(\xi)|^{2}|\xi|^{s-d}\,d\xi<\infty$. A key open frontier is restriction for sets that are fractal but still carry topological manifold structure. The additive Brownian sheet is the random field $W(t)=\sum_{i=1}^{k}W^{i}(t_{i})$ on $[0,1]^{k}$, where the $W^{i}$ are independent one-parameter Brownian motions; it is the standard tractable proxy for the full $(k,1)$-Brownian sheet, the multi-parameter generalisation of Brownian motion. Its graph $G(W)=\{(t,W(t)):t\in[0,1]^{k}\}\subset\mathbb{R}^{k+1}$ — the additive Brownian surface — is a topological co-dimension 1 manifold that is almost surely nowhere differentiable, has Hausdorff dimension $k+1/2$ (Yoder; Kahane-type results), and is not a Salem set (a set whose Fourier dimension, the supremum of $s$ such that some measure on it has Fourier decay $|\xi|^{-s/2}$, equals its Hausdorff dimension). It carries a natural surface measure $\mu$, the lift of Lebesgue measure on $[0,1]^{k}$ to the graph. Fraser and Sahlsten (2018) had earlier determined the Fourier-analytic structure of the $k=1$ Brownian graph, and Fraser, Orponen, and Sahlsten (2014) proved that the graph of any function on $[0,1]^{k}$ has Fourier dimension at most $k$. Fraser and de Orellana (arXiv:2601.05802) introduced the Fourier spectrum of $\mu$ — the family of dimensions $\dim^{\theta}_{\mathrm{F}}\mu$ interpolating between Fourier dimension ($\theta=0$) and Hausdorff-type dimension ($\theta=1$) via energies of convolutions $\mu*\mu$ averaged at scale $|\xi|^{-1}$ — and proved the exact formula $\dim^{\theta}_{\mathrm{F}}\mu=\min\{k+\theta/2,\,2+k\theta\}$. Combining this with the restriction theorem of Carnovale, Fraser, and de Orellana (arXiv:2412.14896), they obtained that, almost surely, the $L^{2}$-based extension estimate $\|\widehat{f\mu}\|_{L^{q}(\mathbb{R}^{k+1})}\lesssim\|f\|_{L^{2}(\mu)}$ holds for $q>4$ when $k=1$ and for $q>(8k+2)/(3k)$ when $k\geq 2$ — coinciding with the Stein–Tomas threshold $q>4$ for $k=1$ and $q>3$ for $k\geq 2$, and improving on it for every $k\geq 3$. In the converse direction they proved, via the Fourier spectrum and via an explicit Knapp example (a counterexample built by concentrating $f$ on the preimage of a ball where the Hölder-continuous sheet stays nearly flat, of width $r^{1/2}$), that the estimate fails almost surely for $q<2+1/k$, and more generally that $\|\widehat{f\mu}\|_{L^{q}}\lesssim\|f\|_{L^{p}(\mu)}$ fails whenever $p<2kq/(2k(q-1)-1)$. These necessary and sufficient regions do not meet: for instance at $p=2$ and $k=2$ the estimate holds for $q>3$ and fails for $q<5/2$ and for $q<12/5$ by the Hambrook–Łaba Sobolev-dimension obstruction, leaving a genuine gap for every $k\geq 1$. The authors note that the exponent $2+1/k$ coming from the Fourier-spectrum obstruction equals the conjectured sharp restriction endpoint for many smooth manifolds, suggesting but not proving that the necessary side is sharp here.

Problem Statement

Determine, almost surely with respect to the randomness of the additive Brownian sheet $W(t)=\sum_{i=1}^{k}W^{i}(t_{i})$ on $[0,1]^{k}$, the exact set of exponents $(p,q)\in[1,\infty]^{2}$ for which the Fourier extension estimate $\|\widehat{f\mu}\|_{L^{q}(\mathbb{R}^{k+1})}\lesssim\|f\|_{L^{p}(\mu)}$ holds for all $f\in L^{p}(\mu)$, where $\mu$ is the surface measure (the lift of $k$-dimensional Lebesgue measure) on the graph $G(W)=\{(t,W(t)):t\in[0,1]^{k}\}$. This is Question 1.1 of Fraser–de Orellana (arXiv:2601.05802). Currently the estimate is known to hold for $q>4$ when $k=1$ and $q>(8k+2)/(3k)$ when $k\geq 2$ at $p=2$, and to fail for $q<2+1/k$ at $p=2$ and for $p<2kq/(2k(q-1)-1)$ in general; the gap between these regions is open for every $k\geq 1$.

References

1. Jonathan M. Fraser and Ana E. de Orellana, "Fourier restriction for the additive Brownian sheet", arXiv:2601.05802 (2026), DOI 10.48550/arXiv.2601.05802, https://arxiv.org/abs/2601.05802
2. Marc Carnovale, Jonathan M. Fraser, and Ana E. de Orellana, "Fourier restriction estimates based on $L^q$-dimensions: beyond Stein–Tomas", arXiv:2606.07143 (2026), https://arxiv.org/abs/2606.07143
3. Itamar Oliveira and Ana E. de Orellana, "Knapp-type obstructions in multilinear fractal Fourier extension", arXiv:2602.08568 (2026), https://arxiv.org/abs/2602.08568
4. Marc Carnovale, Jonathan M. Fraser, and Ana E. de Orellana, "$L^2$ restriction estimates from the Fourier spectrum", arXiv:2412.14896 (2024), https://arxiv.org/abs/2412.14896
5. Jonathan M. Fraser and Tuomas Sahlsten, "On the Fourier analytic structure of the Brownian graph", Analysis & PDE 11(1) (2018), 115–132, DOI 10.2140/apde.2018.11.115, https://doi.org/10.2140/apde.2018.11.115
6. Jonathan M. Fraser, Tuomas Orponen, and Tuomas Sahlsten, "On Fourier Analytic Properties of Graphs", International Mathematics Research Notices 2014(10) (2014), 2730–2745, DOI 10.1093/imrn/rnt015, https://doi.org/10.1093/imrn/rnt015
7. Kyle Hambrook and Izabella Łaba, "On the Sharpness of Mockenhaupt's Restriction Theorem", Geometric and Functional Analysis 23(4) (2013), 1262–1277, DOI 10.1007/s00039-013-0240-9, https://doi.org/10.1007/s00039-013-0240-9
8. Götz Mockenhaupt, "Salem sets and restriction properties of Fourier transforms", Geometric and Functional Analysis 10 (2000), DOI 10.1007/pl00001662, https://doi.org/10.1007/pl00001662
9. Jong-Guk Bak and Andreas Seeger, "Extensions of the Stein–Tomas Theorem", Mathematical Research Letters 18(4) (2011), DOI 10.4310/mrl.2011.v18.n4.a14, https://doi.org/10.4310/mrl.2011.v18.n4.a14
10. Themis Mitsis, "A Stein–Tomas restriction theorem for general measures", Publicationes Mathematicae Debrecen 60(1–2) (2002), 89–99, DOI 10.5486/pmd.2002.2543, https://doi.org/10.5486/pmd.2002.2543
11. Chun-Kit Lai and Cheuk Yin Lee, "Fourier dimension of the graph of fractional Brownian motion with $H\geq 1/2$", arXiv:2502.09032 (2025), https://arxiv.org/abs/2502.09032
12. Cheuk Yin Lee and Samy Tindel, "On the Fourier dimension of fractional Brownian graphs", arXiv:2510.09818 (2025), https://arxiv.org/abs/2510.09818