$l^p$ decoupling for restricted $k$-broadness

TL;DR

This paper develops an $l^p$ decoupling technique for restricted $k$-broadness, advancing the understanding of Fourier restriction estimates through polynomial partitioning and broad norm analysis.

Contribution

It introduces an $l^p$ decoupling result for restricted $k$-broad parts, bridging the gap between broad estimates and regular $L^p$ restriction estimates.

Findings

Established an $l^p$ decoupling theorem for restricted $k$-broadness.

Applied the decoupling result to recover Guth's linear restriction estimates.

Extended the technique to analyze error terms in broad norm decompositions.

Abstract

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 ($k=2,\cdots,m$). 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.