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A restriction estimate in $\mathbb{R}^3$ using brooms
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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 $p> 3+ 3/13$, $\|Ef\|_{L^p(\mathbb{R}^3)}\leq C \|f\|_{L^{\infty}}$. The proof combines Wolff's two ends argument with polynomial partitioning techniques. We also observe some geometric structures in wave packets.
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Cited by 1 Pith paper
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Improved bounds for the Kakeya maximal conjecture in higher dimensions
New multiscale polynomial Wolff axioms lead to Kakeya maximal estimates for p ≥ 1 + O(1/n), improving prior bounds in dimensions n=5 and n≥7.
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