\(l^p\) decoupling for restricted k-broadness
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To prove Fourier restriction estimate using polynomial partitioning, Guth introduced the concept of k-broad part of regular \(L^p\) norm and obtained sharp k-broad restriction estimates. To go from k-broad estimates to regular \(L^p\) estimates, Guth employed \(l^2\) decoupling result. In this article, similar to the technique introduced by Bourgain-Guth, we establish an analogue to go from regular \(L^p\) norm to its \((m+1)\)-broad part, as the error terms we have the restricted k-broad parts (\(k=2,\ldots ,m\)). To analyze the restricted k-broadness, we prove an \(l^p\) decoupling result, which can be applied to handle the error terms and recover Guth’s linear restriction estimates.
This material is based upon work supported by the National Science Foundation under Grant no. 1638352, as well as support from the Shiing-Shen Chern Foundation while the first author was in residence at the IAS. The authors wish to express their indebtedness to Larry Guth for his hospitality when visiting MIT and his inspirations in mathematics.
- 5.Guth, L.: Restriction estimates using polynomial partitioning II. arXiv:1606.07682