A restriction estimate in ${\mathbb{R}}^3$ using brooms

Abstract

If $f$ is a function supported on the truncated paraboloid in ${\mathbb{R}}^3$ and $E$ is the corresponding extension operator, then we prove that for all $p>3+3/13$, $\Vert Ef{\Vert }_{L^p({\mathbb{R}}^3)}\le C\Vert f{\Vert }_{L^{\infty }}$. The proof combines Wolff's two ends argument with polynomial partitioning techniques. We also observe some geometric structures in wave packets.

arXiv comment53 pages, revised version incorporating referees' suggestions

References in corpus (1)

• Some Problems in Harmonic Analysis

Cited by in corpus (9)

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• Extremizability of Fourier restriction to the paraboloid
• A note on Fourier restriction and nested Polynomial Wolff axioms
• On the Bochner-Riesz operator in ${\mathbb{R}}^3$
• Fourier restriction in low fractal dimensions
• New bounds for Stein's square function in ${\mathbb{R}}^3$
• The almost optimal multilinear restriction estimate for hypersurfaces with curvature: the case of $n-1$ hypersurfaces in R^n
• New Kakeya estimates using the polynomial Wolff axioms
• Factorisation in Restriction theory and near extremisers