Weighted restriction estimates using polynomial partitioning

Abstract

We use the polynomial partitioning method of Guth [J. Amer. Math. Soc. 29 (2016) 371-413] to prove weighted Fourier restriction estimates in r3 with exponents p that range between 3 and 3.25, depending on the weight. As a corollary to our main theorem, we obtain new (non-weighted) local and global restriction estimates for compact C∞ surfaces S∪R3 with strictly positive second fundamental form. For example, we establish the global restriction estimate ∥Ef∥Lp(r3) ≲∥f∥Lq (S) in the full conjectured range of exponents for p>3.25 (up to the sharp line), and the global restriction estimate ∥Ef∥Lp (ω)≲ ∥f∥L2 (S) for p>3 and certain sets Ω∪R3 of infinite Lebesgue measure. As a corollary to our main theorem, we also obtain new results on the decay of spherical means of Fourier transforms of positive compactly supported measures on R3 with finite α-dimensional energies. © 2017 London Mathematical Society.