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 , … , 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.
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