Restriction estimates using polynomial partitioning II

By Larry Guth. 2016

148 citations recorded by the source corpus.

We improve the estimates in the restriction problem in dimension $n \ge 4$. To do so, we establish a weak version of a $k$-linear restriction estimate for any $k$. The exponents in this weak $k$-linear estimate are sharp for all $k$ and $n$.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3b210a3f9795b2b4febf092ee48725d6cb25c8a9

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In Applied Mathematics

1. Ruixiang Zhang
2. Xiumin Du
3. Hong Wang
4. Yumeng Ou
5. Xiaochun Li
6. Alex Iosevich
7. Larry Guth
8. Bobby Wilson

In Geometry and Topology

1. Xiumin Du
2. Ruixiang Zhang
3. Hong Wang
4. Yumeng Ou
5. Xiaochun Li
6. Alex Iosevich
7. Bobby Wilson
8. Larry Guth

Citations in the corpus

• POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES (2018, methodology): “In [9], via polynomial partitioning developed in [11, 12] and linear and bilinear refined Strichartz, some new weighted restriction estimates were established, and as applications improved results were obtained for the Falconer distance set problem and the spherical average decay rates of the Fourie”
• Weighted restriction estimates and application to Falconer distance set problem (2018, methodology): “ of solutions to wave equations, by applying [12, Proposition 1.5]. We omit the details. The key ingredients in our proofs are the method of polynomial partitioning developed by the second author [8] [9] and (linear and bilinear) refined Strichartz estimates obtained by Li and the first two authors in”
• On Falconer’s distance set problem in the plane (2018, background): “ a tube T ;v of radius R1{2 and length R. (6)If we think of as a cap in S, then the long axis of T ;v is normal to S. Also T ;v intersects the plane x d0 at the point pv;0q. See Section 3 of [14] for background on this wave packet decomposition, including proofs of these standard facts. Now we are ”
• Sharp L2 estimates of the Schrödinger maximal function in higher dimensions (2018, methodology): “Instead of doing decoupling in dimension n+ 1, we are going to do a broad-narrow analysis following Bourgain-Guth [6], Bourgain [3], Bourgain-Demeter [5] and Guth [19]: for each K(2)-cube, one of the following two has to happen: (i) It is broad in the sense that there are n+1 contributing caps that ”