A restriction estimate using polynomial partitioning

Abstract

If S S is a smooth compact surface in R 3 \mathbb {R}^3 with strictly positive second fundamental form, and E S E_S is the corresponding extension operator, then we prove that for all p > 3.25 p > 3.25 , ‖ E S f ‖ L p ( R 3 ) ≤ C ( p , S ) ‖ f ‖ L ∞ ( S ) \| E_S f\|_{L^p(\mathbb {R}^3)} \le C(p,S) \| f \|_{L^\infty (S)} . The proof uses polynomial partitioning arguments from incidence geometry.