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The linear restriction conjecture for curved hypersurfaces

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Let d≥2d\geq 2, let UU be a compact neighbourhood of the origin in Rd−1\mathbb{R}^{d-1}, and let Σ:U→Rd\Sigma:U\to\mathbb{R}^d be a smooth parametrisation of a (d−1)(d-1)-dimensional submanifold SS with everywhere non-vanishing Gaussian curvature. Define the extension operator by

Eg(ξ)=∫Ug(x)eiξ⋅Σ(x) dx.\mathcal{E}g(\xi)=\int_U g(x)e^{i\xi\cdot\Sigma(x)}\,dx.

For 1≤p,q≤∞1\leq p,q\leq\infty, write p′p' for the Hölder conjugate of pp. Linear restriction conjecture. If q>2dd−1q>\frac{2d}{d-1} and p′≤d−1d+1qp'\leq\frac{d-1}{d+1}q, then there exists a constant 0<C<∞0<C<\infty, depending only on dd and Σ\Sigma, such that

∥Eg∥Lq(Rd)≤C∥g∥Lp(U)\|\mathcal{E}g\|_{L^q(\mathbb{R}^d)}\leq C\|g\|_{L^p(U)}

for all g∈Lp(U)g\in L^p(U). This is the classical restriction problem for hypersurfaces with non-vanishing Gaussian curvature; the conjectured exponent range remains unresolved in general.

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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The linear restriction conjecture for curved hypersurfaces

    Let SS be a hypersurface in Rd\mathbb{R}^d with induced Lebesgue measure dσd\sigma, and let p′p' be the Hölder conjugate of pp. Linear restriction conjecture. If SS has everywhere non-vanishing Gaussian curvature, 1q<d−12d\frac{1}{q}<\frac{d-1}{2d} and 1q≤d−1d+11p′\frac{1}{q}\leq\frac{d-1}{d+1}\frac{1}{p'}, then

    ∥fdσ^∥Lq(Rd)≲∥f∥Lp(dσ).\|\widehat{fd\sigma}\|_{L^q(\mathbb{R}^d)}\lesssim\|f\|_{L^p(d\sigma)}.

    This is the classical Fourier restriction problem for curved hypersurfaces; the stated range is conjectural in general.

    source: Jonathan Bennett, “Aspects of Multilinear Harmonic Analysis Related to Transversality”, arXiv:1405.5369 (2014).

References

Primary source

Jonathan Bennett, Anthony Carbery and Terence Tao, “On the Multilinear Restriction and Kakeya conjectures”, arXiv:math/0509262 (2005).

fourier-analysisfourier-restrictionharmonic-analysisrestriction-theory
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