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

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