A new approach to the Fourier extension problem for the paraboloid

Abstract

We propose a new approach to the Restriction Conjectures. It is based on a discretization of the Extension Operators in terms of quadratically modulated wave packets. Using this new point of view, and by combining natural scalar and mixed norm stopping times performed simultaneously, we prove that all the $k$-linear Extension Conjectures are true for every $1 \leq k \leq d+1$ if one of the functions involved has a tensor structure. Assuming this structure for one of the functions, we show that all multilinear conjectures also hold under a weak transversality hypothesis (as opposed to classical transversality), and that one can improve the conjectured threshold in some cases. Finally, we prove that our results are sharp under weak transversality.