A restriction estimate using polynomial partitioning

By Larry Guth. 2014

204 citations recorded by the source corpus.

If $S$ is a smooth compact surface in $\mathbb{R}^3$ with strictly positive second fundamental form, and $E_S$ is the corresponding extension operator, then we prove that for all $p > 3.25$, $\| E_S f\|_{L^p(\mathbb{R}^3)} \le C(p,S) \| f \|_{L^\infty(S)}$. The proof uses polynomial partitioning arguments from incidence geometry.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b9d5162762f6193790f9938fa20200db59e55a5d

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Restriction estimates using polynomial partitioning II (2016, methodology): “ towards this conjecture in dimension n≥ 4. The best previous estimate was proven by Bourgain and the author in [BG]. The polynomial partitioning approach to the restriction problem was introduced in [G], which gave new bounds in dimension n= 3. The present paper combines this idea with the multilin”
Sharp estimates for oscillatory integral operators via polynomial partitioning (2017, background): “The first is the following theorem of Wongkew [37] (see also [13], [39]), which controls the size of a neighbourhood of a variety.”
A sharp Schrödinger maximal estimate in R2 (2017, methodology): “In [8] and [9], the second author applied this technique to restriction estimates in Fourier analysis.”
Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (2016, background): “In particular one can mention milestone papers by Bourgain [3], Wolff [27], Tao [23], Bennett, Carbery, and Tao [2], Bourgain and Guth [4], and Guth [11] which also contain further important references.”
Improved Fourier restriction estimates in higher dimensions (2018, methodology): “• In Section 9, Guth’s polynomial partitioning argument from [11, 12] is reformulated as a recursive algorithm.”